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+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A
+BANACH SPACE
+MIGUEL MART´IN, YO¨EL PERREAU, AND ABRAHAM RUEDA ZOCA
+Abstract. We introduce extensions of ∆-points and Daugavet points in which slices are replaced by
+relative weakly open subsets (super ∆-points and super Daugavet points) or by convex combinations of
+slices (ccs ∆-points and ccs Daugavet points). These notions represent the extreme opposite to denting
+points, points of continuity, and strongly regular points. We first give a general overview on these
+new concepts and provide some isometric consequences on the spaces. As examples: if a Banach space
+contains a super ∆-point, then it does not admit an unconditional FDD (in particular, unconditional
+basis) with suppression constant smaller than two; if a real Banach space contains a ccs ∆-point, then
+it does not admit a one-unconditional basis; if a Banach space contains a ccs Daugavet point, then
+every convex combination of slices of its unit ball has diameter two. We next characterize the notions in
+some classes of Banach spaces showing, for instance, that all the notions coincide in L1-predual spaces
+and that all the notions but ccs Daugavet points coincide in L1-spaces. We next remark on some
+examples which have previously appeared in the literature and provide some new intriguing examples:
+examples of super ∆-points which are as closed as desired to strongly exposed points (hence failing
+to be Daugavet points in an extreme way); an example of a super ∆-point which is strongly regular
+(hence failing to be a ccs ∆-point in the strongest way); a super Daugavet point which fails to be a ccs
+∆-point. The extensions of the diametral notions to point in the open unit ball and the consequences
+on the spaces are also studied. Last, we investigate the Kuratowski measure of relative weakly open
+subsets and of convex combinations of slices in the presence of super ∆-points or ccs ∆-points, as well
+as for spaces enjoying diameter 2 properties. We conclude the paper with a section on open problems.
+Contents
+1.
+Introduction
+2
+2.
+Notation and preliminary results
+5
+3.
+Characterisations of diametral-notions and implications on the geometry of the ambient
+space
+9
+3.1.
+Spaces with a one-unconditional basis and beyond
+12
+3.2.
+Absolute sums
+16
+4.
+Examples and counterexamples of diametral elements
+19
+4.1.
+Characterization in C(K)-spaces, L1-preduals, and M¨untz spaces
+19
+4.2.
+Characterization in L1-spaces
+22
+4.3.
+Remarks on some examples from the literature
+24
+4.4.
+A super ∆-point which fails to be a Daugavet point in an extreme way
+26
+4.5.
+A super ∆-point which is a strongly regular point
+27
+4.6.
+A super Daugavet point which is not ccs ∆-point
+28
+Date: January 11th, 2023.
+The first and third named authors were supported by grant PID2021-122126NB-C31 funded by MCIN/AEI/
+10.13039/501100011033 and “ERDF A way of making Europe”, by Junta de Andaluc´ıa I+D+i grants P20 00255
+and FQM-185,
+and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by
+MCIN/AEI/10.13039/501100011033. The second named author was supported by the Estonian Research Council grant
+SJD58.
+1
+arXiv:2301.04433v1  [math.FA]  11 Jan 2023
+
+2
+MART´IN, PERREAU, AND RUEDA ZOCA
+4.7.
+A summary of relations between the properties
+35
+5.
+Diametral-properties for elements of the open unit ball
+35
+6.
+Kuratowski measure and large diameters
+38
+6.1.
+Kuratowski measure and diameter two properties.
+39
+6.2.
+Kuratowski measure and ∆-notions
+41
+7.
+Commented open questions
+42
+Acknowledgments
+45
+References
+45
+1. Introduction
+It is fair to say that one of the most studied properties of Banach spaces is the Radon-Nikod´ym
+property (RNP) because it has shown to be very useful; due to the large amount of its geometric,
+analytic, and measure theoretic characterisations; in several fields of Banach space theory such as
+representation of bounded linear operators, representation of dual spaces or representation of certain
+tensor product spaces (see [18, 21]).
+A famous geometric characterization of the Radon-Nikod´ym property is related to the size of slices.
+Recall that a slice of a bounded non-empty subset C of a Banach space X is simply the (nonempty)
+intersection of C with a half-space. A Banach space X has the RNP if and only if every non-empty
+closed and bounded subset of X admits slices of arbitrarily small diameter (see e.g. [18]).
+A closely related and equally important geometric property of Banach spaces is the point of con-
+tinuity property. Recall that a Banach space X has the point of continuity property (PCP) if every
+non-empty closed and bounded subset of X admits non-empty relatively weakly open subsets of ar-
+bitrarily small diameter. Let us emphasize here as an example the striking equivalence between the
+Radon-Nikod´ym property and the weak∗ version of the point of continuity property for dual spaces,
+and the related characterization of Asplund spaces as preduals of RNP spaces (see e.g. [19]). In his
+proof of the determination of the Radon-Nikod´ym property by subspaces with a finite dimensional
+decomposition (FDD) in [17], Bourgain also introduced an important weakening of the point of conti-
+nuity property, that he called property “(∗)”, and that is nowadays referred to as the convex point of
+continuity property. Recall that a Banach space X has the convex point of continuity property (CPCP)
+if every non-empty closed, convex and bounded subset of X admits non-empty relatively weakly open
+subsets of arbitrarily small diameter.
+In fact, Bourgain implicitly used in his work the notion of strong regularity which, as he showed,
+is implied by the CPCP. Recall that a Banach space X is strongly regular (SR) if every non-empty
+closed, convex and bounded subset of X contains convex combinations of slices of arbitrarily small
+diameter. Observe that the convexity of the subset is required in this definition in order to guarantee
+that it contains all the convex combinations of its slices. It later turned out that strong regularity
+had important applications to the famous (still open) question of the equivalence between the Radon-
+Nikod´ym property and the Krein-Milman property. Recall that a Banach space X has the Krein-
+Milman property (KMP) if every non-empty closed, convex and bounded subset C of X admits an
+extreme point. The RNP implies the KMP (see e.g. [18, Theorem 3.3.6]), and it follows from [48] that
+every strongly regular space with the KMP has the RNP. Also recall that it was proved in [29] the
+RNP and the KMP are equivalent in dual spaces.
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+3
+From the definitions it follows that RNP⇒PCP⇒CPCP and it is also known that CPCP⇒SR.
+None of the above implications reverse (see e.g [49] and references therein). In order to show that
+strong regularity is implied by the CPCP, Bourgain made an important geometric observation, namely
+that in every non-empty bounded and convex subset of a Banach space X, every non-empty relatively
+weakly open subset contains a convex combination of slices. We will discuss this “Bougain Lemma”
+and its applications to the subject of the present paper in more details in Section 2.
+Another classical refinement of the above characterization of the Radon-Nikod´ym property is related
+to the notion of denting points. Recall that a point x0 of a bounded subset C of X is a denting point
+of C if there are slices of C containing x0 of arbitrarily small diameter. A Banach space X has the
+RNP if and only if every closed, convex and bounded subset contains a denting point. Actually, every
+nonempty closed, convex and bounded subset C of a Banach space X with the RNP is equal to the
+closure of the convex hull of the set of its denting points (see e.g. [18, Corollary 3.5.7]).
+For the PCP and the CPCP, a similar role is played by points of weak-to-norm continuity. Given
+a bounded subset C of X, we say that a point x0 ∈ C is a point of weak-to-norm continuity (point
+of continuity in short) if the identity mapping i: (C, w) −→ (C, τ) is continuous at the point x0 or,
+equivalently, if x0 belongs to relatively weakly open subsets of C of arbitrarily small diameter. Note
+that a classical result by Lin-Lin-Troyanski [39] establishes that a point x0 ∈ C is a denting point if,
+and only if, x0 is simultaneously a point of continuity and a extreme point of C. In a space with the
+PCP every non-empty closed and bounded subset contains a point of continuity; and the set of all
+points of continuity of a given closed, convex and bounded subset C of a Banach space X with the
+CPCP is weakly dense in C (see e.g. [22, Theorem 1.13]).
+In relation to strong regularity, a point x0 of a bounded, convex subset C of X is a point of strong
+regularity if there are convex combinations of slices of C containing x0 of arbitrarily small diameter.
+Then the set of all points of strong regularity of a given closed, convex and bounded subset C of a
+strongly regular Banach space X is norm dense in C (see [25, Theorem 3.6]). Let us observe that
+points of strong regularity may be in the interior of a set, while denting points (and points of continuity
+in the infinite-dimensional case) belong always to the border of the set.
+In [3] the extreme opposite notion to denting point of the unit ball was introduced in the following
+sense: an element x in the unit sphere of a Banach space X is a ∆-point if we can find in every slice
+of BX containing x points which are at distance from x as close as we wish to the maximal possible
+distance in the ball (distance 2). A similar yet stronger notion appeared simultaneously in relation to
+another quite famous property of Banach spaces, the Daugavet property. Recall that a Banach space
+X has the Daugavet property (DPr) if the Daugavet equation
+(DE)
+∥Id + T∥ = 1 + ∥T∥
+holds for every rank-one operator T : X −→ X, where Id denotes the identity operator.
+In this
+case, all weakly compact operators also satisfy (DE). We refer the reader to the seminal paper [35]
+for background. Recent results can be found in [42] and references therein. The Daugavet property
+admits a beautiful geometric characterization involving slices related to the notion of Daugavet points:
+an element x on the unit sphere of a Banach space X is a Daugavet point if in every slice of BX (not
+necessarily containing the point x) there are points which are at distance from x as close as we wish to
+2. With this definition in mind, [35, Lemma 2.1] states that X has the DPr if and only if all elements
+in SX are Daugavet points. Let us comment that the Daugavet property imposes severe restriction
+on the Banach space: if X is a Banach space with the DPr, then it fails the RNP and it has no
+unconditional basis (actually, it cannot be embedded into a Banach space with unconditional basis).
+
+4
+MART´IN, PERREAU, AND RUEDA ZOCA
+On the other hand, ∆- and Daugavet points have proved to be far more flexible than the global
+properties that they define. For example, there exists a Banach space with the RNP and a Daugavet
+point [51] (see paragraph 4.3.1), there exists a Banach space with a one-unconditional basis and a large
+subset of Daugavet points [6] (see paragraph 4.3.3), and there is an MLUR Banach space for which all
+elements in its unit sphere are ∆-points, which contains convex combinations of slices of arbitrarily
+small diameter, but satisfying that every convex combination of slices intersecting its unit sphere has
+diameter two [2] (see paragraph 4.3.2). Nonetheless, it has been recently proved that ∆-points have
+some influence on the isometric structure of the space. For example, it is shown in [5] that uniformly
+non-square spaces do not contain ∆-points; actually, it has been very recently proved in [37] that
+a ∆-point cannot be a locally uniformly non-square point. Also, combining the results from [5] and
+[52], asymptotic uniformly smooth spaces and their duals do not contain ∆-points. However, it is still
+an important open problem to understand whether ∆- or Daugavet point have any influence on the
+isomorphic structure of the space.
+In this paper, our main aim is to study natural strengthening of the notions of Daugavet- and
+∆-points obtained by replacing slices by non-empty relatively weakly open subsets (“super points”)
+or convex combination of slices (“ccs points”) in order to provide new diametral notions which are
+extreme opposites to points of continuity and to strongly regular points, respectively. See Definitions
+2.5 and 2.4 for details. Our main goal will be to understand the influence, for a given Banach space,
+of the existence of such points on its geometry, and to study the different diametral notions in several
+families of Banach spaces. A particular emphasis will be put on trying to distinguish between all the
+various formally different notions.
+Let us end this section by giving a brief description about the organization of the paper and the
+main results obtained. Section 2 contains the necessary notation (which is standard, anyway), needed
+definitions, and some preliminary results. We include in Section 3 some characterizations of the newer
+diametral point notions and some necessary conditions on the existence of such points. In particular,
+we study the existence of super ∆-points and ccs ∆-points in spaces with a one-unconditional basis.
+We first give an analogue for ccs ∆-points to a result from [6] which implicitly states that such spaces
+contain no super ∆-points. Second, we provide sharper and improved versions of this super ∆ result
+in the context of unconditional FDDs with a small unconditional constant, and more generally in the
+context of spaces in which special families of operators are available. The section finishes with the
+study of the behaviour of super ∆-points and super Daugavet points with respect to absolute sums
+somehow analogous to the known one for ∆-points and Daugavet points; however, not all the results
+extend to ccs ∆-points and ccs Daugavet points, but we also give some partial results. Section 4 is
+devoted to examples and counterexamples. We first characterize the diametral notions in some families
+of classical Banach spaces: we show that all notions are equivalent in L1-preduals and M¨untz spaces
+(Subsection 4.1); all notions but ccs Daugavet points also coincide in L1-spaces (Subsection 4.2).
+We next give in Subsection 4.3 some remarks on examples which have previously appeared in the
+journal literature, discussing the new diametral notions on them, and showing that they may help to
+distinguish between the diametral notions. The most complicated and tricky examples are produced
+in the last three subsection of this section: super ∆-points which are as closed as desired to strongly
+exposed points (hence failing to be Daugavet points in an extreme way) in Subsection 4.4; a super ∆-
+point which is strongly regular (hence failing to be ccs ∆-point in an extreme way) in Subsection 4.5;
+super Daugavet points which belong to convex combinations of slices of diameter as small as desired
+(hence failing to be ccs ∆-points in an extreme way). We finish this subsection with a summary of
+relations between all the diametral notions. The idea in Section 5 is to generalize the diametral notions
+to elements of the open unit ball, and use these notions to characterize some geometric properties. In
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+5
+particular, we properly localize the result by Kadets that the DSD2P is equivalent to the Daugavet
+property. Section 6 deals with Kuratowki index of non-compactness of slices, relative weakly open
+subsets, and convex combinations of slices. We get that every relative weakly open subset (respectively,
+every convex combination of slices) in a space with the diameter 2 property (respectively, with the
+strong diameter 2 property) has Kuratowski measure 2; these results extends the analogous result for
+slices and the the local diameter 2 property proved in [20, Proposition 3.1]. Also, we show that every
+relative weakly open subset that contains a super ∆-point has Kuratowski measure 2, and a similar
+result is obtained with convex combinations of relative weakly open subsets containing a ccw ∆-point;
+these results extend [52, Corollary 2.2]. Finally, Section 7 is devoted to collect some interesting open
+questions and some remarks on them.
+2. Notation and preliminary results
+We will use standard notation as in the books [8], [23], and [24], for instance. Given a Banach
+space X, BX (respectively, SX) stands for the closed unit ball (respectively, the unit sphere) of
+X.
+We denote by X∗ the topological dual of X and we write JX : X −→ X∗∗ for the canonical
+injection.
+We denote by dent (BX) and ext (BX) the sets of all denting points of BX and of all
+extreme points of BX, respectively. The set of preserved extreme points of BX (i.e. those x ∈ BX such
+that JX(x) ∈ ext (BX∗∗)) is denoted by pre-ext (BX). For Banach spaces X and Y , L(X, Y ), F(X, Y ),
+K(X, Y ) denote, respectively, the set of all (bounded linear) operator, the finite-rank operators, and
+the compact operators. The properties in which we are interested only deal with the real structure of
+the involved Banach spaces, but we do not restrict the study to real spaces in order to consider real or
+complex examples. We will use the notation K to denote either R or C, Re(z) to denote the real part
+of z (which is just the identity when dealing with a real space), and T to represent the set of scalars
+of modulus one.
+Given a non-empty subset C of X, we will denote by co(C) the convex hull of C and by span(C)
+the linear hull of C. Also we denote by co(C) (respectively, span(C)) the norm closure of the convex
+hull (respectively, of the linear hull) of C. By a slice of C we will mean any subset of C of the form
+S(x∗, δ; C) := {x ∈ C : Re x∗(x) > M − δ}
+where x∗ ∈ X∗ is a continuous linear functional on X, δ > 0 is a positive real number, and M :=
+supx∈C Re x∗(x).
+For slices of the unit ball we will simply write S(x∗, δ) := S(x∗, δ; BX).
+By a
+relatively weakly open subset of C we mean as usual any subset of C obtained as the (non-empty)
+intersection of C with an open set of X in the weak topology.
+If C is assumed to be convex we will mean by a convex combination of slices of C (ccs of C in
+short) any subset of C of the form
+�n
+i=1 λiSi,
+where λ1, . . . , λn ∈ (0, 1] are such that �n
+i=1 λi = 1 and Si is a slice of C for every i ∈ {1, . . . , n}.
+Observe that convex combinations of slices are convex sets. We define in the same way convex com-
+binations of relatively weakly open subsets of C (ccw of C in short).
+The following lemma from [30] is a very useful tool when working with ∆-points.
+Lemma 2.1 ([30, Lemma 2.1]). Let X be a Banach space, and let x∗ ∈ SX∗ and α > 0. For every
+x ∈ S(x∗, α) and every 0 < β < α there exists y∗ ∈ SX∗ such that
+x ∈ S(y∗, β) ⊆ S(x∗, α).
+
+6
+MART´IN, PERREAU, AND RUEDA ZOCA
+We also often rely on the following result, due to Bourgain, and that we already mentioned in the
+introduction. We provide a proof below, following the one from [25, Lemma II.1], for the sake of
+completeness and for further discussions.
+Lemma 2.2 (Bourgain). Let X be a Banach space and let C be a bounded convex closed subset of X.
+Then, every non-empty relatively weakly open subset W of C contains a convex combination of slices
+of C.
+Proof. Assume with no loss of generality that W :=
+m�
+i=1
+S(fi, αi, C), write �C = JX(C)
+w∗
+⊂ X∗∗, and
+W ∗∗ :=
+m
+�
+i=1
+S
+�
+JX∗(fi), αi; �C
+�
+,
+which is a non-empty relatively weak∗ open subset of �C. By the Krein-Milman theorem (see e.g. [24,
+Theorem 3.37]), it follows that �C = co(ext (BX∗∗))
+w∗
+, so co(ext (BX∗∗)) ∩ W ∗∗ ̸= ∅. Pick a convex
+combination of extreme points �n
+i=1 λie∗∗
+i
+contained in W ∗∗. By the continuity of the sum we can find,
+for every 1 ⩽ i ⩽ n, a weak-star open subset W ∗∗
+i
+with e∗∗
+i
+∈ W ∗∗
+i
+and such that �n
+i=1 λiW ∗∗
+i
+⊂ W ∗∗.
+Now, since each e∗∗
+i
+is an extreme point of �C, we have by Choquet’s lemma (see [24, Lemma 3.40],
+for instance) that there are weak-star slices S
+�
+JX∗(gi), βi; �C
+�
+with e∗∗
+i
+∈ S
+�
+JX∗(gi), βi; �C
+�
+⊆ W ∗∗
+i
+for
+every i ∈ {1, . . . , n}. Henceforth, �n
+i=1 λiS
+�
+JX∗(gi), βi; �C
+�
+⊆ �n
+i=1 λiW ∗∗
+i
+⊆ W ∗∗. Now, if we take
+U :=
+n
+�
+i=1
+λiS(gi, βi, C)
+it is not difficult to prove that U ⊆ W, as desired.
+□
+Remark 2.3. Observe that, in general, it is unclear from the above proof whether or not, if we fix
+x ∈ W, we can guarantee that there exists a convex combination of slices U of C such that x ∈ U ⊆ W.
+On the other hand, the result holds true if x ∈ W ∩ co(pre-ext (C)) in view of the above proof.
+Indeed, if such situation, if we write x = �n
+i=1 λixi ∈ W satisfying that x1, . . . , xn ∈ pre-ext (C)
+and λ1, . . . , λn ∈ (0, 1] with �n
+i=1 λi = 1, by the weak continuity of the sum, we can find, for every
+1 ⩽ i ⩽ n, a non-empty relatively weakly open subset Vi with xi ∈ Vi for every i and such that
+x = �n
+i=1 λixi ∈ �n
+i=1 λiVi ⊆ W. Now, observe that, since each xi is a preserved extreme point of C,
+slices of C containing xi are a neighbourhood basis for xi in the weak topology. Hence, we can find,
+for 1 ⩽ i ⩽ n, a slice Si of C with xi ∈ Si ⊆ Vi, and so x = �n
+i=1 λixi ∈ �n
+i=1 λiSi ⊆ �n
+i=1 λiVi ⊆ W,
+so U := �n
+i=1 λiSi is the desired convex combination of slices.
+Throughout the text, we will often be discussing various “diameter 2 properties”.
+We use the
+notation introduced in [7]. A Banach space X has the local or slice diameter 2 property (LD2P) if
+every slice of BX has diameter 2; X has the diameter two property (D2P) if every non-empty relatively
+weakly open subset of BX has diameter 2; finally, X has the strong diameter 2 property (SD2P)
+whenever every ccs of BX has diameter 2 (and then, every ccw has diameter 2 due to Lemma 2.2).
+For definitions and for examples concerning those properties, we refer to [2, 14, 15, 45]. In particular,
+let us comment that the three properties are different, a result which was not easy to show, see [14].
+Our paper is closely related to the diametral versions of those properties which have been implicitly
+studied for a long time in the literature, but whose formal definitions and names where fixed in [13].
+A Banach space X has the diametral local diameter 2 property (DLD2P) if for every slice S of BX
+and every x ∈ S ∩ SX, supy∈S ∥x − y∥ = 2; if slices are replaced by non-empty relatively weakly
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+7
+open subsets of BX, we obtain the diametral diameter 2 property (DD2P). It is immediate that these
+properties are not satisfied by any finite-dimensional space. Clearly, DLD2P implies LD2P, DD2P
+implies D2P (and none of these implications reverses, e.g. X = c0), and DD2P implies DLD2P. It is
+unknown whether the DLD2P and the DD2P are equivalent; in fact it is even unknown whether the
+DLD2P implies the D2P. For the analogous definition using ccs, we have to discuss a little bit. Even
+for an infinite-dimensional space X, it is not true that every ccs of BX intersects SX; actually, this
+happens if and only if X has a property stronger than the SD2P (see [41, Theorem 3.4]). Thus, the
+definition of the diametral strong diameter 2 property (DSD2P) given in [13] deals with all points in
+BX as follows: for every ccs C and every x ∈ C, supy∈C ∥x − y∥ = ∥x∥ + 1. This definition allows to
+show that DSD2P implies the SD2P. But, actually, it has been recently shown by V. Kadets [34] that
+the DSD2P is equivalent to the Daugavet property. We will discuss this in detail in Section 5. On the
+other hand, we will use the following property which is weaker than the DSD2P: a Banach space X
+has the restricted DSD2P if for every ccs C and every x ∈ C ∩ SX, supy∈C ∥x − y∥ = 2. This property
+is strictly weaker than the DSD2P, see Paragraph 4.3.2.
+Let us now introduce all the notions of diametral points that we will consider in the text. Let us
+start with the more closely related ones to the definitions above.
+Definition 2.4. Let X be a Banach space and let x ∈ SX. We say that
+(1) [3] x is a ∆-point if supy∈S ∥x − y∥ = 2 for every slice S of BX containing x,
+(2) x is a super ∆-point if supy∈V ∥x − y∥ = 2 for every non-empty relatively weakly open subset
+V of BX containing x,
+(3) x is a ccs ∆-point if supy∈C ∥x − y∥ = 2 for every slice ccs C of BX containing x.
+∆-points were introduced in [3] as a natural localization of the DLD2P (i.e. X has the DLD2P if
+and only if every element of SX is a ∆-point). The other two definitions are new. Clearly, super
+∆-points are the natural localization of the DD2P: X has the DD2P if and only if every element of
+SX is a super ∆-point. Besides, ccs ∆-points are the localization of the restricted DSD2P: X has the
+restricted DSD2P if and only if every element of SX is a ccs ∆-point.
+In relation with the Daugavet property, we have the following notions for points.
+Definition 2.5. Let X be a Banach space and let x ∈ SX. We say that
+(1) [3] x is a Daugavet point if supy∈S ∥x − y∥ = 2 for every slice S of BX,
+(2) x is a super Daugavet point if supy∈V ∥x − y∥ = 2 for every non-empty relatively weakly open
+subset V of BX,
+(3) x is a ccs Daugavet point if supy∈C ∥x − y∥ = 2 for every ccs C of BX.
+Let us recall that Daugavet points were introduced in [3] as a natural localization of the Daugavet
+property in the sense that a Banach space X has the Daugavet property if and only if every point in SX
+is a Daugavet point ([35, Lemma 2.1]). From the geometric characterization given in [50, Lemma 3]
+and the implicit result contained in its proof, it follows that super Daugavet points as well as ccs
+Daugavet points are also natural localizations of the Daugavet property.
+Since every slice of BX is relatively weakly open, and since by Bourgain’s lemma (see Lemma 2.2)
+every non-empty relatively weakly open subset of BX contains a ccs of BX, we clearly have the diagram
+of Figure 1.
+We will show throughout the text that none of the above implications reverses, see Subsection 4.7
+for a description of all the relations and the counterexamples. However, let us point out right away
+
+8
+MART´IN, PERREAU, AND RUEDA ZOCA
+ccs Daugavet
+ccs ∆
+super Daugavet
+super ∆
+∆
+Daugavet
+Figure 1. Relations between the diametral notions
+that we do not know whether there exists ccs ∆-points which are not super ∆. In view of Remark 2.3
+such examples may exist since Bourgain’s lemma is not localizable. Also let us point out that it follows
+again from Bourgain’s lemma that a ccs Daugavet point x ∈ SX also satisfies supy∈D ∥x − y∥ = 2 for
+every ccw D of BX. Again this is not clear for ccs ∆-points and we could thus naturally distinguish
+between ccs ∆-points and “ccw ∆-points”. Since we do not have concrete examples at hand, we will
+focus on convex combination of slices and specifically point out any available ccw behavior throughout
+the text.
+Let us also comment that it is clear that if every ccs of the unit ball of a given Banach space is weakly
+open (respectively, has non-empty relative weak interior), then every super ∆-point (respectively, every
+super Daugavet point) in this space is a ccs ∆-point (respectively, a ccs Daugavet point). Several
+properties of this kind where introduced and studied in [1], [4], and [41]. We refer to those papers for
+some background and for examples.
+Remark 2.6. There are natural weak∗ versions in dual spaces of all the notions of diametral-points
+introduced in the present section where slices and relatively weakly open subsets are respectively
+replaced with weak∗ slices (i.e. slices defined by elements of the predual) and relatively weak∗ open
+subsets. With obvious terminology, it then follows from [35, Lemma 2.1] and from [50, Lemma 3] that
+a Banach space X has the Daugavet property if and only if every element in SX∗ is a weak∗ Daugavet
+point if and only if every element in SX∗ is a weak∗ ccs Daugavet point. It also follows from [2,
+Theorem 3.6] that X has the DLD2P if and only if every point in SX∗ is a weak∗ ∆-point. However,
+the relationship between the DD2P in X and weak∗ super ∆-points in SX∗ is currently unknown.
+Observe that a direct consequence of those results is that weak∗ diametral points and their weak
+counterparts might differ in a very strong way since, for instance, the unit ball of the space C[0, 1]∗
+admits denting points. Yet clearly all the results from the following sections concerning the different
+notions of diametral-points admit obvious analogues for their weak∗ counterparts. We leave the details
+to the reader to avoid unnecessary repetitions, but let us still point out that it follows from Goldstine’s
+theorem and from the lower weak∗ semicontinuity of the norm in dual spaces that there is a natural
+correspondence between diametral-properties of points in SX and weak∗ properties of their image in
+the bidual under the canonical embedding JX. Namely:
+(1) x ∈ SX is a Daugavet point (respectively, a ccs Daugavet point) if and only if JX(x) is a weak∗
+Daugavet point (respectively, a weak∗ ccs Daugavet point).
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+9
+(2) x ∈ SX is a super Daugavet point if and only if JX(x) is a weak∗ super Daugavet point if
+and only if for every y ∈ BX there exists a net (y∗∗
+s ) in BX∗∗ which converges to JX(y) in the
+weak∗ topology and such that ∥πX(x) − y∗∗
+s ∥ −→ 2 (see Section 3).
+(3) x ∈ SX is a ∆-point (respectively, a super ∆-point) if and only if JX(x) is a ∆-point (respec-
+tively, a super ∆-point).
+(4) x ∈ SX is a ccs ∆-point if and only if JX(x) is a weak∗ ccs ∆-point.
+Let us point out that (3) essentially follows from the obvious fact that ∆-points and super ∆-points
+naturally pass to superspaces, that is if Y is a subspace of X and if x ∈ SY if a ∆-point (respectively,
+a super ∆-point) in Y , then x is a ∆-point (respectively, a super ∆-point) in X . This property is
+unclear for ccs ∆-points, so the assertion (4) is not analogous to assertion (3).
+3. Characterisations of diametral-notions and implications on the geometry of the
+ambient space
+In view of the definitions of diametral-points, it is natural to expect that the presence of any kind
+of Daugavet- or ∆-element in a given Banach space will affect, by the severe restrictions it inflicts
+on the nature of the considered point, its global isometric geometry or even its topological structure.
+However, previous studies in the context have shown that the situation is much more complicated
+than one could expect at first sight.
+For example, let us comment that a Banach space X with
+the RNP and admitting a Daugavet point, and a Banach space with a one-unconditional basis and
+admitting a weakly dense subset of Daugavet points, were respectively constructed in [51] and in [6].
+In this section, we provide useful characterizations of the new diametral notions, and investigate the
+immediate effect of the presence of such points on the geometry of the considered space.
+We start by an intuitive but not completely trivial observation.
+Observation 3.1. By definition, it is clear that super ∆-points do not exist in finite dimensional
+spaces because the weak and norm topology coincide in this context.
+Also, it was proved in [5,
+Theorem 4.4] that finite dimensional spaces do also fail to contain ∆-points (hence ccs ∆-points).
+In fact they fail to contain them in a stronger way, see [5, Corollary 6.10]. Consequently, the study
+of diametral-notions only makes sense in infinite dimension, and from now on we will assume unless
+otherwise stated that all the Banach spaces we consider are infinite dimensional.
+Let us next prove a bunch of characterisations for super Daugavet- and super ∆-points.
+Let X be a Banach space. For every x ∈ SX and for every ε > 0, let us define
+∆ε(x) := {y ∈ BX : ∥x − y∥ > 2 − ε}.
+We recall the following characterization of Daugavet- and ∆-points from [3].
+Lemma 3.2 ([3, Lemma 2.1 and 2.2]). Let X be a Banach space.
+(1) An element x ∈ SX is a Daugavet point if and only if BX = co ∆ε(x) for every ε > 0.
+(2) An element x ∈ SX is a ∆-point if and only if x ∈ co ∆ε(x) for every ε > 0.
+We have similar characterisations for super points.
+Lemma 3.3. Let X be a Banach space.
+(1) An element x ∈ SX is a super Daugavet point if and only if BX = ∆ε(x)
+w for every ε > 0.
+(2) An element x ∈ SX is a super ∆-point if and only if x ∈ ∆ε(x)
+w for every ε > 0.
+
+10
+MART´IN, PERREAU, AND RUEDA ZOCA
+Proof. Observe that for given x ∈ SX, y ∈ BX, and ε > 0, we have that y belongs to the weak
+closure of the set ∆ε(x) if and only if ∆ε(x) has non-empty intersection with any neighborhood of
+y in the relative weak topology of BX. Thus y belongs to ∆ε(x)
+w for every ε > 0 if and only if
+supz∈V ∥x − z∥ = 2 for every relatively weakly open subset V of BX containing y. The conclusion
+easily follows.
+□
+For any given x ∈ BX, we denote by V(x) the set of all neighborhoods of x for the relative weak
+topology of BX. We can provide characterizations of super points using nets which is just a localization
+of [13, Proposition 2.5].
+Proposition 3.4. Let X be an infinite-dimensional Banach space.
+(1) An element x ∈ SX is a super Daugavet point if and only if for every y ∈ BX there exists a
+net (ys) in BX which converges weakly to y and such that ∥x − ys∥ −→ 2.
+(2) An element x ∈ SX is a super ∆-point if and only if there exists a net (xs) in BX which
+converges weakly to x and such that ∥x − xs∥ −→ 2.
+In both cases we can moreover force the nets to be in SX.
+Proof. Let us fix x ∈ SX. Given any y ∈ BX, it is clear that if there exists a net (ys) in BX which
+converges weakly to y and such that ∥x − ys∥ −→ 2, then y belongs to the weak closure of ∆ε(x) for
+every ε > 0. Conversely let us pick y ∈ BX satisfying this property. We turn S := V(y) × (0, ∞) into
+a directed set by (V, ε) ⩽ (V ′, ε′) if and only if V ′ ⊂ V and ε′ ⩽ ε. By the assumptions we have that
+V ∩ ∆ε(x) is a non-empty subset of BX for every couple s := (V, ε) in S. Picking any ys in this set
+will then provide the desired net.
+Finally observe that for x ∈ BX and ε > 0, we have that BX\∆ε(x) = {y ∈ BX : ∥x − y∥ ⩽ 2 − ε}
+is weakly closed by the lower semi-continuity of the norm, so that ∆ε(x) is a relatively weakly open
+subset of BX. Thus we have that V ∩ ∆ε(x) is a non-empty relatively weakly open subset of BX for
+every couple s := (V, ε) in S. Since X is infinite dimensional, this set has to intersect SX, and we can
+actually pick ys in V ∩ ∆ε(x) ∩ SX.
+□
+Remark 3.5. In [36] an example of a Banach space satisfying simultaneously the Daugavet property
+and the Schur property was provided. Such example shows that there is no hope to get a version of
+the above result involving sequences.
+Observe that the following result, similar to [32, Lemma 2.1 and 2.2], is included in the preceding
+proof.
+Proposition 3.6. Let X be a Banach space and let x ∈ SX.
+(1) If x is a super Daugavet point, then for every ε > 0 and every non-empty relatively weakly
+open subset V of BX we can find a non-empty relatively weakly open subset U of BX which is
+contained in V and such that ∥x − y∥ > 2 − ε for every y ∈ U.
+(2) If x is a super ∆-point, then for every ε > 0 and every non-empty relatively weakly open subset
+V of BX containing x we can find a non-empty relatively weakly open subset U of BX which
+is contained in V and such that ∥x − y∥ > 2 − ε for every y ∈ U.
+Proof. Fix any x ∈ SX and any y ∈ BX which belongs to the weak closure of ∆ε(x) for every ε > 0.
+Then, for every V ∈ V(y) and every ε > 0, we have that U := V ∩ ∆ε(x) is a non-empty relatively
+weakly open subset of BX.
+□
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+11
+It is clear from the definition that denting points of BX cannot be ∆-points. Also it was first
+observed in [32, Proposition 3.1] that every Daugavet point in a Banach space X has to be at distance
+2 from every denting point of the unit ball of X. This elementary observation turned out to play
+an important role in the study of Daugavet points in Lipschitz-free spaces in [32] and [51]. We have
+similar observations for super points.
+Lemma 3.7. Let X be a Banach space and let x ∈ SX. If x is a super ∆-point, then x cannot be
+a point of continuity. If, moreover, x is a super Daugavet point, then x has to be at distance 2 from
+every point of continuity of BX.
+Proof. If and element y of BX is a point of continuity, then it is contained in relatively weakly open
+subsets of BX of arbitrarily small diameter. Clearly no super ∆-point can have this property, and any
+super Daugavet point has to be at distance 2 from any such points.
+□
+This lemma provides quite a few examples of Banach spaces which fail to contain super points.
+Following [26] let us recall that X has the Kadets property if the norm topology and the weak topology
+coincide on SX, and that X has the Kadets-Klee property if weakly convergent sequences in SX are
+norm convergent. Let us also recall that any LUR space has the Kadets-Klee property, and that any
+space with the Kadets-Klee property which fails to contain ℓ1 has the Kadets property. By proposition
+3.4 we clearly have the following result.
+Proposition 3.8. If X has the Kadets property, then X fails to contain super ∆-points.
+As a corollary we obtain the following. Recall that a Banach space is asymptotic uniformly convex
+(AUC in short) [31] if its modulus of asymptotic uniform convexity
+δX(t) := inf
+x∈SX
+sup
+dim X/Y <∞
+inf
+y∈SY ∥x + ty∥ − 1
+is strictly positive for every t > 0.
+Corollary 3.9. Let X be AUC. Then, X fails to contain super ∆-points.
+Proof. In an AUC space, every element of the unit sphere is a point of continuity of BX, see [31,
+Proposition 2.6].
+□
+Remark 3.10. It was proved in [5, Theorem 3.4] that any reflexive AUC space fails to contain ∆-
+points. Also, combining the observations from [5, End of Section 4] about weak∗ quasi-denting points
+in the unit ball of AUC∗ duals and [52, Corollary 2.4] about the maximality of the Kuratowski index
+of weak∗ slices containing weak∗ ∆-points, we have that every AUC∗ dual space fails to contain weak∗
+∆-points. However, note that it is currently unknown whether non-reflexive AUC spaces (and, in
+particular, whether the dual of the James tree spaces JT∗) may contain Daugavet- or ∆-points.
+It turns out that Daugavet points are characterized by this distance to denting points in RNP
+spaces (because the unit ball of an RNP space X can be written as the closed convex hull of the set
+of its denting points) as well as in Lipschitz-free spaces ([32, Theorem 3.2] for compact metric spaces
+and [51, Theorem 2.1] for a general statement). In the same way we can characterize super Daugavet
+points in terms of this distance to points of continuity of BX is spaces with the CPCP.
+Proposition 3.11. If a Banach space X has the CPCP, then a point x ∈ SX is a super Daugavet
+point if and only if it is at distance 2 from any point of continuity of BX.
+
+12
+MART´IN, PERREAU, AND RUEDA ZOCA
+Proof. If X has the CPCP, then the set of all points of continuity of BX is weakly dense in BX (see for
+example [22, Proposition 3.9]), that is, every non-empty relatively weakly open subset of BX contains
+a point of continuity. The conclusion follows easily.
+□
+For ccs points, the situation is quite different. Indeed, although ccs ∆-points can clearly not be
+points of strong regularity, we have by [41, Theorem 3.1] that X has the SD2P if and only if every
+convex combination of slices of BX contains elements of norm arbitrarily close to 1. It readily follows
+that any space X which contains a ccs Daugavet point satisfies the SD2P, so it is very far from being
+strongly regular. We will provide more details on this topic in Section 5, but for later reference let us
+state the following.
+Proposition 3.12. Let X be a Banach space. If X contains a ccs Daugavet point, then it has the
+SD2P (it fails to be strongly regular).
+Next, we show that extreme points have a nice behaviour with respect to diametral notions.
+Proposition 3.13. Let X be a Banach space and let x ∈ SX.
+(1) If x ∈ pre-ext (BX) and it is a ∆-point, then x is a super ∆-point.
+(2) If x ∈ ext (BX) and it is a super ∆-point, then x is a ccs ∆-point.
+(3) In particular, if x ∈ pre-ext (BX) is a ∆-point, then x is a super ∆-point as well as a ccs
+∆-point.
+Proof. It follows from Choquet’s lemma (see for example [23, Lemma 3.69]) that slices form neighbor-
+hood bases in the relative weak topology of the unit ball of a Banach space for its preserved extreme
+points, so (1) immediately follows. For (2), if x is extreme and belongs to a ccs C := �n
+i=1 λiSi of BX
+then x ∈ �n
+i=1 Si, which is a relatively weakly open subset of BX.
+□
+Remark 3.14. Observe that, in fact, any extreme super ∆-point is “ccw ∆-point” as we discussed in
+Section 2. Also, Choquet’s lemma implies that every extreme weak∗ ∆-point in a dual space is weak∗
+ccw ∆-point.
+3.1. Spaces with a one-unconditional basis and beyond. In [6], it was proved that no real
+Banach space with a subsymmetric basis contains a ∆-point. On the other hand, an example of a
+Banach space with a one-unconditional basis that contains a ∆-point was provided, and a more involved
+example of a Banach space with a one-unconditional basis that contains many Daugavet-points was
+constructed. We will discuss this second example in detail in Subsection 4.3.3.
+In the process, it was also implicitly shown that real Banach spaces with a one-unconditional basis
+cannot contain super ∆-points. In the present subsection, we prove that the same goes for ccs ∆-
+points. Also, we provide sharper and more general versions of [6, Proposition 2.12]. In the first part
+of this section, we follow [6] and restrict ourselves to real Banach spaces.
+Let X be a real Banach space with a Schauder basis (ei)i⩾1. We denote by (e∗
+i )i⩾1 the corresponding
+sequence of biorthogonal functionals.
+Recall that (ei)i⩾1 is said to be unconditional if the series
+�
+i⩾1 e∗
+i (x)ei converges unconditionally for every x ∈ X.
+Also, recall that an unconditional basis
+(ei)i⩾1 is said to be one-unconditional if
+�����
+�
+i⩾1
+θie∗
+i (x)ei
+����� =
+�����
+�
+i⩾1
+e∗
+i (x)ei
+�����
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+13
+for every (θi)i⩾1 ∈ {−1, 1}N and for every x ∈ X. Moreover, if
+�����
+�
+i⩾1
+θie∗
+i (x)eni
+����� =
+�����
+�
+i⩾1
+e∗
+i (x)ei
+�����
+for every (θi)i⩾1 ∈ {−1, 1}N, for every x ∈ X, and for every strictly increasing sequence (ni)i⩾1 in N,
+then the basis is called subsymmetric.
+Observe that for spaces with a one-unconditional basis, it is enough, in order to study the various
+Daugavet- and ∆-notions, to work in the positive sphere
+S+
+X := {x ∈ SX : e∗
+i (x) ⩾ 0 ∀i}
+of the space X. Also, the following result is well known.
+Lemma 3.15. Let X be a real Banach space with a one-unconditional basis (ei)i⩾1, and let (ai)i⩾1
+and (bi)i⩾1 be sequences of real numbers. If the series �
+i⩾1 biei converges, and if |ai| ⩽ |bi| for every
+i, then �
+i⩾1 aiei converges as well, and we have
+�����
+�
+i⩾1
+aiei
+����� ⩽
+�����
+�
+i⩾1
+biei
+����� .
+Let us now recall a few notation and preliminary results from [6]. Let X be a real Banach space
+with a normalized one-unconditional basis (ei)i⩾1. For every subset A of N, we denote by PA the
+projection on span{ei, i ∈ A}. Then for every x ∈ X, we define
+M(x) := {A ⊂ N: ∥PA(x)∥ = ∥x∥ , and
+��PA(x) − e∗
+j(x)ej
+�� < ∥x∥ ∀j ∈ A}.
+The set M(x) can be seen as the set of all minimal norm-giving subsets of the support of x. We denote
+respectively by MF(x) and M∞(x) the subsets of all finite and infinite elements of M(x). It follows
+from [6, Lemma 2.7] that the set M(x) is never empty, and from [6, Proposition 2.15] that no element
+x ∈ SX satisfying M∞(x) = ∅ can be a ∆-point.
+For every non-empty ordered subset A := {a1 < a2 < . . . } of N, and for every n ∈ N smaller than
+or equal to |A|, we denote by A(n) := {a1, . . . , an} the subset consisting of the n first elements of A.
+We will implicitly assume in the following that the elements of M(x) are ordered subsets of N. The
+next two results were proved in [6, Lemma 2.8 and Lemma 2.11].
+Lemma 3.16. Let X be a real Banach space with a normalized one-unconditional basis (ei)i⩾1 and
+let x ∈ SX. For every n ∈ N, the sets
+�
+A ∈ M(x): |A| ⩽ n
+�
+and
+�
+A(n): A ∈ M(x), and |A| > n
+�
+are both finite.
+Lemma 3.17. Let X be a real Banach space with a normalized one-unconditional basis and let x ∈ SX.
+For every subset E of N such that E ∩ A ̸= ∅ for every A ∈ M(x), we have ∥x − PE(x)∥ < 1.
+With those tools at hand, we can now prove an analogue to [6, Proposition 2.13] for convex combi-
+nation of slices.
+Proposition 3.18. Let X be a Banach space with a normalized one-unconditional basis and x ∈ S+
+X.
+Then, there exists δ > 0 and a ccs C of BX containing x such that supy∈C ∥x − y∥ ⩽ 2 − δ.
+
+14
+MART´IN, PERREAU, AND RUEDA ZOCA
+Proof. Let x ∈ S+
+X, and define E = �
+A∈M(x) A(1). From Lemma 3.16 and Lemma 3.17, we have
+that E is a finite subset of N and that ∥x − PE(x)∥ < 1. In particular, there exists γ > 0 such that
+∥x − PE(x)∥ ⩽ 1 − γ. For every i ∈ E, we define
+Si := S
+�
+e∗
+i , 1 − e∗
+i (x)
+2
+�
+.
+Then we consider the ccs
+C :=
+1
+|E|
+�
+i∈E
+Si.
+Since x ∈ S+
+X, we clearly have that x ∈ �
+i∈E Si and, in particular, that x ∈ C.
+So let us pick y :=
+1
+|E|
+�
+i∈E yi in C.
+Then we have e∗
+i (yi) >
+e∗
+i (x)
+2
+for every i.
+In particular,
+e∗
+i (yi) ⩾ 0, and
+��e∗
+i (yi) − e∗
+i (x)
+�� ⩽ e∗
+i (yi). Indeed, for any given non-negative real numbers α and β
+with β ⩾ α
+2 , we have
+|β − α| = β − α ⩽ β
+if β ⩾ α, and
+|β − α| = α − β ⩽ α − α
+2 = α
+2 ⩽ β
+if β ⩽ α. So in either case, |β − α| ⩽ β as desired.
+It then follows from Lemma 3.15 that
+��yi − e∗
+i (x)ei
+�� ⩽
+��yi�� ⩽ 1 and, finally,
+∥x − y∥ ⩽
+����x − x
+|E|
+���� +
+����
+x
+|E| − PE(x)
+|E|
+���� +
+����
+PE(x)
+|E|
+− y
+����
+⩽ 1 − 1
+|E| + 1 − γ
+|E|
++ 1
+|E|
+�
+i∈E
+��e∗
+i (x)ei − yi�� ⩽ 2 − γ
+|E|.
+The conclusion follows with δ :=
+γ
+|E|. In particular, note that since x belongs to the relative weakly
+open set �
+i∈E Si ⊂ C, we also get that x is not super ∆, recovering the result from [6].
+□
+So combining [6, Proposition 2.13] and Proposition 3.18, we immediately get that spaces with a
+normalized one-unconditional basis fail to contain super ∆-points and ccs ∆-points. So let us state
+the following here for future reference.
+Theorem 3.19. Let X be a real Banach space with a normalized one-unconditional basis. Then X
+does not contain super ∆-points, and X does not contain ccs ∆-points.
+In the rest of the subsection, we aim at providing sharper and improved versions of [6, Proposi-
+tion 2.13]. In particular we will go back to working with either real or complex Banach spaces. The
+main result of this study is the following proposition.
+Proposition 3.20. Let X be a Banach space, and let us assume that there exists a subset A ⊆ F(X, X)
+satisfying that sup
+�
+∥Id − T∥ : T ∈ A
+�
+< 2 and that for every ε > 0 and every x ∈ X, there exists
+T ∈ A such that ∥x − Tx∥ < ε. Then, X contains no super ∆-point.
+Let us provide a lemma which is a localization of the above result from which its proof is immediate.
+Lemma 3.21. Let X be a Banach space, and let x ∈ SX. If there exists a finite-rank operator T on
+X such that ∥x − Tx∥ + ∥Id − T∥ < 2, then x is not a super ∆-point.
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+15
+Proof. Consider ε > 0 such that K := ∥x − Tx∥ + ∥Id − T∥ + ε < 2. Since T has finite rank, we can
+find N ⩾ 1, w1, . . . , wN ∈ SX and f1, . . . , fN ∈ X∗ such that T(z) = �N
+n=1 fn(z)wn for every z ∈ X.
+Let us consider
+W :=
+�
+y ∈ BX : |fn(x − y)| <
+ε
+2n+1 ∀n ∈ {1, . . . , N}
+�
+.
+W is a neighborhood of x in the relative weak topology of BX, and for every y ∈ W, we have
+∥x − y∥ ⩽ ∥x − Tx∥ + ∥Tx − Ty∥ + ∥y − Ty∥
+⩽ ∥x − Tx∥ + ∥Id − T∥ +
+N
+�
+n=1
+|fn(x − y)| ∥wn∥
+⩽ ∥x − Tx∥ + ∥Id − T∥ + ε
+N
+�
+n=1
+1
+2n+1 ⩽ K < 2.
+□
+N.B. It is unclear whether an analogue to Lemma 3.21 can be given for ccs ∆-points. So we do not
+know whether Proposition 3.20 extends to this notion.
+As particular cases of Proposition 3.20, we have the following ones. Recall that a sequence (En)n⩾1
+of finite dimensional subspaces of a given Banach space X is called a finite dimensional decomposition
+(FDD) for X if every element x ∈ X can be represented in a unique way as a series x := �
+n⩾1 xn
+with xn ∈ En for every n ⩾ 1. Such an FDD is said to be unconditional if the above series converges
+unconditionally for every x ∈ X. In this case, it is well known that the family (PA)A⊂N, where PA is the
+projection given by PA(x) := �
+n∈A xn, is uniformly bounded, and the constant KS := supA⊂N ∥PA∥ is
+called the suppression-unconditional constant of the FDD. We refer to [40, Section 1.g] for the details
+and to [8, Section 3.1] for the particular case of unconditional bases.
+Corollary 3.22. A Banach space X fails to have super ∆-points provided one of the following condi-
+tions is satisfied.
+(1) There exists a family A ⊆ F(X, X) satisfying that sup
+�
+∥Id − T∥ : T ∈ A
+�
+< 2 and that the
+identity mapping belongs to its strong operator topology (SOT) closure.
+(2) There exists a family {Pλ}λ∈Λ of finite rank projections on X such that X = �
+λ∈Λ Pλ(X),
+and such that supλ∈Λ ∥Id − Pλ∥ < 2.
+(3) The space X admits a FDD with suppression-unconditional constant less than 2. In particular,
+if X admits an unconditional basis with suppression-unconditional constant less than 2.
+Let us observe that the value 2 in the above results is sharp in several ways.
+Remark 3.23.
+(1) The space C[0, 1] admits a monotone Schauder basis, so there exists a sequence
+{Pn}n⩾1 of norm one finite rank projections on this space which converges to Id in SOT
+topology. As C[0, 1] has the Daugavet property, all elements in SX are super Daugavet points.
+Observe that ∥Id − Pn∥ = 2 for every n ⩾ 1 by the DPr.
+(2) Let X be an arbitrary Banach space. For every x ∈ SX choose fx ∈ SX∗ such that fx(x) = 1,
+and define Px(z) = fx(z)x for every z ∈ X. Then {Px : x ∈ SX} is a family of norm one
+rank-one projections on X, X = �
+x∈SX Px(X), and ∥Id − Px∥ ⩽ 2 for every x ∈ SX.
+(3) The space c admits ccs Daugavet points (hence super Daugavet points), see Theorem 4.2, but
+it is easy to check that its usual basis is 3-unconditional and 2-suppression unconditional.
+(4) It is shown in [30] that a Banach space has the DLD2P if and only if ∥Id − P∥ ⩾ 2 for every
+rank-one projection P. It follows that the suppression constant of an unconditional basis on
+a Banach space with the DLD2P has to be greater than or equal to 2. Let us mention here
+
+16
+MART´IN, PERREAU, AND RUEDA ZOCA
+that there is no local version of this result, as there are Banach spaces with one-unconditional
+basis and containing many Daugavet points [6] (see Paragraph 4.3.3).
+3.2. Absolute sums. In this subsection we look at the transfer of the diametral points through
+absolute sums of Banach spaces. Let us first recall the following definition.
+Definition 3.24. A norm N on R2 is absolute if N(a, b) = N(|a| , |b|) for every (a, b) ∈ R2 and
+normalized if N(0, 1) = N(1, 0) = 1.
+If X and Y are Banach spaces, and if N is an absolute normalized norm on R2, we denote by
+X ⊕N Y the product space X × Y endowed with the norm ∥(x, y)∥ = N(∥x∥ , ∥y∥). It is easy to check
+that X ⊕N Y is a Banach space, and that its dual can be expressed as (X ⊕N Y )∗ ≡ X∗ ⊕N∗ Y ∗ where
+N∗ is the absolute norm given by the formula N∗(c, d) = maxN(a,b)=1 |ac|+|bd|. Classical examples of
+absolute normalized norms on R2 are the ℓp norms for p ∈ [1, ∞]. Information on absolute norms can
+be found in [16, §21] and [43] and references therein, for instance. Let us recall that for every absolute
+normalized sum N, given non-negative a, b, c, d in R with a ⩽ b and c ⩽ d we have N(a, b) ⩽ N(c, d).
+In particular, ∥·∥∞ ⩽ N ⩽ ∥·∥1.
+Similar to the DD2P (see [13, Theorem 2.11]) and to ∆-points [28], super ∆-points transfer very
+well through absolute sums.
+Proposition 3.25. Let X and Y be Banach spaces, and let N be an absolute normalized norm.
+(1) If x ∈ SX and y ∈ SY are super ∆-points, then (ax, by) is a super ∆-point in X ⊕N Y for
+every (a, b) ∈ R2 with N(a, b) = 1.
+(2) If x ∈ SX is a super ∆-point, then (x, 0) is a super ∆-point in X ⊕N Y . If y ∈ SY is a super
+∆-point, then (0, y) is a super ∆-point in X ⊕N Y .
+Proof. (1). We can find two nets (xs)s∈S and (yt)t∈T respectively in SX and SY such that xs
+w
+−→ x,
+yt
+w
+−→ y, and ∥x − xs∥ , ∥y − yt∥ −→ 2. Now, if we take (a, b) ∈ R2 with N(a, b) = 1 we clearly have
+(axs, byt)
+w
+−→
+(s,t)∈S×T (ax, by) and ∥(ax, by) − (axs, byt)∥ = N (a ∥x − xs∥ , b ∥y − yt∥) −→ 2N(a, b) = 2,
+so (ax, by) is a super ∆-point in X ⊕N Y . For (2), we just repeat the previous proof with a = 1 and
+b = 0 or with a = 0 and b = 1 and so we only need one of the points to be super ∆-point.
+□
+For super Daugavet points the situation is more complicated and we need to distinguish between
+different kinds of absolute norms. The following definitions can be found, for instance, in [28].
+Definition 3.26. Let N be an absolute normalized norm on R2.
+(1) N has property (α) if for every a, b ∈ R+ with N(a, b) = 1 we can find a neighborhood W of
+(a, b) in R2 with sup(c,d)∈W c < 1 or sup(c,d)∈W d < 1 and such that any couple (c, d) ∈ R2
++
+satisfying N(c, d) = 1 and N ((a, b) + (c, d)) = 2 belongs to W.
+(2) N is A-octahedral if there exists a, b ∈ R+ such that N(a, b) = 1 and N ((a, b) + (c, d)) = 2 for
+c = max{e ∈ R+ : N(e, 1) = 1}
+and
+d = max{f ∈ R+ : N(1, f) = 1}.
+(3) N is positively octahedral if there exists a, b ∈ R+ such that N(a, b) = 1 and
+N ((a, b) + (0, 1)) = N ((a, b) + (1, 0)) = 2.
+Positively octahedral norms where introduced in [27] in order to characterize the absolute norms for
+which the corresponding absolute sum is octahedral. It is clear that property (α) and A-octhaedrality
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+17
+exclude each other and that every positively octahedral absolute normalized norm is A-octahedral
+(while there clearly exists absolute A-octahedral norms which are not positively octahedral). Moreover
+it was proved in [28, Proposition 2.5] that every absolute normalized norm on R2 must either satisfy
+property (α) or be A-octahedral.
+For ℓp-norms, we have that ∥·∥1 and ∥·∥∞ are both positively
+octahedral, and that ∥·∥p satisfies property (α) for every p ∈ (1, ∞).
+Observe that if an absolute normalized norm N on R2 is positively octahedral, and if (a, b) is as in
+the above definition, then the intersection of the unit sphere of N with the positive quadrant of R2 is
+equal to the union of the segments [(1, 0), (a, b)] and [(0, 1), (a, b)] (see [45, Section 3.3.1] for pictures).
+In particular, it follows that N((a, b)+(c, d)) = 2 for every non-negative c, d with N(c, d) = 1. Similar
+to the results from [3, Section 4] concerning Daugavet points, we have the following.
+Proposition 3.27. Let X and Y be Banach spaces, and let N be an absolute normalized norm.
+(1) [3, Proposition 4.6] If N has property (α), then X ⊕N Y has no Daugavet point (hence, in
+particular, no super Daugavet points).
+(2) If N is positively octahedral and if x ∈ SX and y ∈ SY are super Daugavet points, then (ax, by)
+is a super Daugavet point in X ⊕N Y for every (a, b) ∈ R2
++ as in the above definition.
+Proof. (2). Assume that N is positively octahedral, take (a, b) ∈ R2
++ as in the definition, and let
+x ∈ SX and y ∈ SY be super Daugavet points. For any given (u, v) ∈ X ⊕N Y of norm ∥(u, v)∥ = 1 we
+can find two nets (us)s∈S and (vt)t∈T respectively in SX and SY such that ∥u∥us
+w
+−→ u, ∥v∥vt
+w
+−→ v,
+and ∥x − us∥ , ∥y − vt∥ −→ 2. Then (∥u∥ us, ∥v∥ vt)
+w
+−→
+(s,t)∈S×T (u, v). Since
+∥ax − ∥u∥ us∥ = ∥(x − us) − [(1 − a)x − (1 − ∥u∥)us]∥
+⩾ ∥x − us∥ − (1 − a + 1 − ∥u∥),
+= a + ∥u∥ − (2 − ∥x − us∥),
+and, in the same way,
+∥by − ∥v∥ vt∥ ⩾ b + ∥v∥ − (2 − ∥y − vt∥),
+we have
+∥(ax − ∥u∥ us, by − ∥v∥ vt)∥ = N (∥ax − ∥u∥ us∥ , ∥by − ∥v∥ vt∥)
+⩾ N (a + ∥u∥ − (2 − ∥x − us∥), b + ∥v∥ − (2 − ∥y − vt∥))
+−→ N ((a + ∥u∥ , b + ∥v∥) = 2.
+This shows that (ax, by) is a super Daugavet point in X ⊕N Y .
+□
+Remark 3.28. Note that if (a, b) = (1, 0) (respectively, (a, b) = (0, 1)) in the previous statement (for
+example, when N = ∥·∥1), then we only need to assume that x (respectively, y) is super Daugavet
+in order to get that (x, 0) (respectively, (0, y)) is super Daugavet in X ⊕N Y . Also, if N = ∥·∥∞,
+then we only need to assume that x (respectively, y) is super Daugavet in order to obtain that (x, βy)
+(respectively, (αx, y)) is super Daugavet in X ⊕N Y for every β ∈ [0, 1] (respectively, α ∈ [0, 1]).
+In [28, Theorem 2.2] it is proved that regular Daugavet points do also transfer through A-octahedral
+sums. We do not know if a similar result can be obtained for super Daugavet points. Indeed, observe
+that if N is an A-octahedral norm, and if c, d, and (a, b) are as in the above definition, then the
+intersection of the unit sphere of N with the positive quadrant of R2 is equal to the union of the
+segments [(1, 0), (1, d)], [(1, d), (a, b)], [(0, 1), (c, 1)] and [(c, 1), (a, b)]. In particular, N((a, b)+(e, f)) =
+2 for every couple (e, f) on the segments [(1, d), (a, b)] and [(c, 1), (a, b)], but this is no longer true
+
+18
+MART´IN, PERREAU, AND RUEDA ZOCA
+on the segments [(1, 0), (1, d)] and [(0, 1), (c, 1)] and the argument in the above proof does not work
+anymore.
+The situation for ccs ∆-points and ccs Daugavet point is not clear and the proofs of the above results
+do not seem to admit easy extensions. For instance, it follows from the next result that Remark 3.28
+is not valid for ccs Daugavet points.
+Proposition 3.29. Let X be an arbitrary Banach space, let Y be a Banach space containing an
+strongly exposed point y0 ∈ SY , and let E := X ⊕1 Y . Then, there are convex combinations of slices of
+BE around 0 of arbitrarily small diameter. In particular, E fails to contain ccs Daugavet points and
+also fails to have the SD2P.
+Proof. Let y∗
+0 ∈ SY ∗ strongly exposes y0. Given ε > 0, there is 0 < δ < ε such that ∥y − y0∥ < ε
+whenever y ∈ BY satisfies Re y∗
+0(y) > 1 − δ. Consider f = (0, y∗
+0) ∈ SE∗ and write
+C := 1
+2 (S(f, δ; BE) + S(−f, δ, BE))
+Take u := 1
+2(u1 + u2) ∈ C with u1 ∈ S(f, δ; BE) and u2 ∈ S(−f, δ; BE). So if write u1 := (x1, y1) and
+u2 := (x2, y2), we have
+Re y∗
+0(y1) = Re f(x1, y1) > 1 − δ and
+Re y∗
+0(y2) = Re f(x2, y2) < −1 + δ.
+On the one hand, it follows that ∥y1−y0∥ < ε and ∥y2+y0∥ < ε. On the other hand, ∥y1∥, ∥y2∥ > 1−δ,
+hence ∥x1∥ < δ < ε and ∥x2∥ < δ < ε. Summarizing, we have
+∥u∥ = 1
+2
+�
+∥x1 + x2∥ + ∥y1 + y2∥
+�
+⩽ 1
+2(2ε + 2ε) = 2ε.
+□
+Remark 3.30. It is straightforward to adapt the previous proof to ℓp-sums for 1 < p < ∞.
+However, note that the situation is very different for ℓ∞-sums.
+Theorem 3.31. Let X and Y be Banach spaces, and let E := X ⊕∞ Y . If x ∈ SX is a ccs Daugavet
+point, then (x, y) ∈ SE is a ccs Daugavet point for every y ∈ BY .
+Proof. Let C := �n
+i=1 λiSi be a ccs of BE. For every i ∈ {1, . . . , n}, we can write Si := S(fi, δi) with
+fi := (x∗
+i , y∗
+i ) ∈ SE∗ satisfying 1 = ∥fi∥ = ∥x∗
+i ∥ + ∥y∗
+i ∥. Consider on the one side
+˜Si :=
+�
+s ∈ BX : Re x∗
+i (s) > ∥x∗
+i ∥ − δi
+2
+�
+,
+and pick on the other side any ti ∈ BY such that Re y∗
+i (ti) > ∥y∗
+i ∥ − δi
+2 . Since ˜C := �n
+i=1 λi ˜Si is a ccs
+of BX, we can find for every ε > 0 an element s := �n
+i=1 λisi in ˜C such that ∥x − s∥ > 2 − ε. Then,
+if we let t := �n
+i=1 λiti, we get (si, ti) ∈ BE and
+Re fi(si, ti) = Re x∗
+i (si) + y∗
+i (ti) > ∥x∗
+i ∥ + ∥y∗
+i ∥ − δi = 1 − δi
+for every i, so that (si, ti) ∈ Si, and (s, t) = �n
+i=1 λi(si, ti) ∈ C. Finally,
+∥(x, y) − (s, t)∥ ⩾ ∥x − s∥ > 2 − ε,
+so (x, y) is a ccs Daugavet point as stated.
+□
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+19
+4. Examples and counterexamples of diametral elements
+In this section we aim to include a number of examples and counterexamples of diametral elements
+on the unit sphere of Banach spaces. We first characterize the notion in some spaces which have
+natural relations with the Daugavet property, such as L1-preduals spaces, M¨untz spaces, and L1-
+spaces.
+Next, we will remark on some examples which have previously appear in the literature,
+including some improvements in some cases (as for Lipschitz free spaces). Finally, we will include
+some complicated examples which will be needed to see that no implication in Figure 1 in page 8
+reverses and also to negate some other possible implications between the notions. A summary of all
+the relations between properties will be included in Subsection 4.7.
+4.1. Characterization in C(K)-spaces, L1-preduals, and M¨untz spaces. It was shown in
+[3, Theorems 3.4 and 3.7] that the notions of ∆-point and Daugavet point coincide for L1-preduals.
+The authors first characterize the ∆-points in C(K) spaces and then get the result for L1-preduals by
+using the principle of local reflexivity. Later on, a characterization of ∆-points (equivalently, Daugavet
+points) of L1-preduals was provided in [42, Theorem 3.2] which implicitly prove that actually ∆-points,
+and super Daugavet points coincides in this setting. Let us state this result here for further reference.
+Let us observe that the authors of [3] works with real Banach spaces, but it is immediate that the
+proof of [3, Theorems 3.4] works in the complex case as well; the paper [42] works in both the real
+and the complex case.
+Proposition 4.1 ([3, Theorems 3.4 and 3.7], [42, Theorem 3.2]). Let X be an L1-predual and let x ∈
+SX. The following assertions are equivalent.
+(1) x is a Daugavet point.
+(2) x is a ∆-point
+(3) For every δ > 0, the weak∗ slice S(JX(x), δ; BX∗) contain infinitely many pairwise linearly
+independent extreme points of BX∗.
+(4) For every element y ∈ BX, there exists a sequence (x∗∗
+n ) in BX∗∗ such that ∥x − x∗∗
+n ∥ −→ 2
+and
+�����
+�
+n⩾1
+an(y − x∗∗
+n )
+����� ⩽ 2 ∥a∥∞
+for every a := (an) ∈ c00.
+(5) For every element y ∈ BX, there exists a sequence (x∗∗
+n ) in BX∗∗ which converges weak∗ to y
+and such that ∥x − x∗∗
+n ∥ −→ 2.
+In the case that X = C(K) for a Hausdorff topological space K, the above is also equivalent to:
+(6) x attains its norm at an accumulation point of K.
+We will show that, in fact, ∆-points also coincide with the ccs versions for L1 preduals.
+Our
+approach will be analogous to the one used in [3] for ∆-points and Daugavet points: we first prove the
+result for C(K) spaces and then deduce it for all L1-preduals using that the bidual of an L1-predual
+is a C(K)-space. In the case of C(K) spaces, we first prove a sufficient condition for ccs Daugavet
+points which, for the same price, can be proved for vector-valued spaces. Recall that given a compact
+Hausdorff topological space K and a Banach space X, C(K, X) denotes the Banach space of those
+continuous functions from K to X endowed with the supremum norm.
+
+20
+MART´IN, PERREAU, AND RUEDA ZOCA
+Theorem 4.2. Let K be a compact Hausdorff topological space, X a Banach space, and let t0 be an
+accumulation point of K. If a function f ∈ SC(K,X) satisfies ∥f(t0)∥ = 1, then f is a ccs Daugavet
+point.
+Proof. Pick x∗ ∈ SX∗ such that Re x∗(f(t0)) = ∥f(t0)∥ = 1.
+Let C := �L
+i=1 λiSi be a convex
+combination of slices of BC(K). For every i ∈ {1, . . . , L}, pick a function gi ∈ Si. Since K is compact
+and t0 is an accumulation point of K we have the following.
+Claim. There exists a sequence (Un)n⩾0 of open neighborhoods of t0 such that:
+(1) U0 = K,
+(2) Un+1 is a proper subset of Un for every n ⩾ 0,
+(3) Re(x∗ ◦ f)|Un ⩾ 1 − 1
+n and
+���gi|Un − gi(t0)
+��� ⩽ 1
+n for every i ∈ {1, . . . , L} and every n ⩾ 1.
+Indeed, we construct the sequence inductively. Let U0 := K and assume that U0, . . . , Un are con-
+structed for some n ⩾ 0.
+Since K is normal, we can find an open subset U of Un such that
+t0 ∈ U ⊂ U ⊂ Un.
+Also since t0 is an accumulation point of K and since K is Hausdorff, we
+can find an open subset V of U such that V is a proper subset of U (pick any point in U distinct from
+t0 and separate the two points with open sets). By continuity of f and of the finitely many g′
+is, we
+can then find an open subset W of V such that
+Re(x∗ ◦ f)|W > 1 −
+1
+n + 1
+and
+���gi|W − gi(t0)
+��� <
+1
+n + 1
+for every i ∈ {1, . . . , L}. The set Un+1 := W does the job.
+Now, let us pick (Un)n⩾0 as in the claim and let us define Fn := Un\Un+1 for every n ⩾ 0. By
+construction, the F ′
+ns are closed non-empty subsets of K and cover K\
+��
+n⩾0 Un
+�
+, and each Fn may
+only intersects its neighbors Fn−1 and Fn+1. By Urysohn’s lemma, for every n ⩾ 1 we can find a
+function pn ∈ C(K) satisfying:
+(1) 0 ⩽ pn ⩽ 1,
+(2) pn|Fn+1 = 1,
+(3) pn|F0∪···∪Fn−1∪Un+3 = 0.
+The sequence (pn) is normalized and converges pointwise to 0, so it converges weakly to 0. Moreover,
+observe that
+∥gi − (1 + gi(t0))pn∥∞ ⩽ 1 + 1
+n
+for every i ∈ {1, . . . , L} since
+���gi|Un − gi(t0)
+��� ⩽
+1
+n and pn|(K\Un) = 0 by construction. So all the
+functions
+gi,n :=
+n
+n + 1 (gi − (1 + gi(t0))pn)
+belong to BC(K) and the sequences (gi,n)n∈N converges weakly to gi for every i ∈ {1, . . . , L}. Since
+the finitely many S′
+is are all weakly open, we may thus find some N ⩾ 1 such that gi,n ∈ Si for every
+i and every n ⩾ N. In particular, the function
+gn :=
+L
+�
+i=1
+λigi,n
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+21
+belongs to C for every n ⩾ N. To conclude, fix t ∈ Fn+1 ⊂ Un+1 ⊂ Un and observe that
+Re x∗f(t) ⩾ 1 − 1
+n
+and that
+Re x∗gi,n(t) =
+n
+n + 1 Re x∗ (gi(t) − (1 + gi(t0))
+⩽
+n
+n + 1 Re x∗
+�
+gi(t0) + 1
+n − (1 + gi(t0))
+�
+= −1 +
+2
+n + 1
+for every i ∈ {1, . . . L}. Hence,
+∥f − gn∥∞ ⩾ Re x∗
+�
+f(t) −
+L
+�
+i=1
+λi Re(gi,n(t))
+�
+⩾ 2 − 1
+n −
+2
+n + 1.
+□
+Combining the previous result with Proposition 4.1, we get the promised characterization of diame-
+tral points in C(K) spaces.
+Corollary 4.3. Let K be a Hausdorff topological compact space. Then the six concepts of diametral
+points are equivalent in C(K).
+For vector-valued spaces, the situation is not that easy, but we may provide with some results.
+Observe that, clearly, if t0 is an isolated point of a compact Hausdorff topological space K and X is
+a Banach space, then C(K, X) = C(K \ {t0}, X) ⊕∞ X.
+Remark 4.4. Let K be a Hausdorff topological compact space, let X be a Banach space. and let
+f ∈ C(K, X) be a function with ∥f∥ = 1.
+(1) If f ∈ C(K, X) with ∥f∥ = 1 attains its norm at an accumulation point of K, then f is a ccs
+Daugavet point (by Theorem 4.2) and hence, f satisfies the six diametral notions.
+(2) If f ∈ C(K, X) with ∥f∥ = 1 attains its norm at an isolated point t0 and f(t0) is a Dau-
+gavet (respectively, super Daugavet, ccs Daugavet) point, then f is a Daugavet (respectively,
+super Daugavet, ccs Daugavet) point (by [3, Section 4], Remark 3.28, and Theorem 3.31,
+respectively).
+(3) Suppose that K contains an isolated point t0, let x0 ∈ SX, and let f ∈ C(K, X) be given by
+f(t0) = x0 and f(t) = 0 for every t ∈ K \ {t0}. Then:
+(3.1) If x0 is a ∆- (respectively, super ∆-) point of X, then f is a ∆- (respectively, super ∆-)
+point of C(K, X) (by [3, Section 4] and Proposition 3.25, respectively).
+(3.2) If x0 is a Daugavet (respectively, super Daugavet, ccs Daugavet) point of X, then f
+is a Daugavet (respectively, super Daugavet, ccs Daugavet) point of C(K, X) (by [45,
+Proposition 3.3.11], Remark 3.28 Theorem 3.31, respectively).
+(3.3) If f is a ∆- (respectively, Daugavet) point of C(K, X), then x0 is a ∆- (respectively,
+Daugavet) point of X (by [45, Theorem 3.4.4], [45, Theorem 3.3.13], respectively).
+(4) It is now easy to show that the six diametral notions do not coincide in C(K, X) spaces.
+Indeed, let K be a compact Hausdorff topological space containing an isolated point t0, let X
+a Banach space containing a ∆-point x0 which is not a Daugavet point (e.g. any x0 in the unit
+sphere of X = C[0, 1] ⊕2 C[0, 1]), see Propositions 3.25 and 3.27), and consider the function
+f ∈ C(K, X) given by f(t0) = x0 and f(t) = 0 for every t ∈ K \ {t0}. Then, f is a ∆-point by
+(3.1) but it is not a Daugavet point by (3.3).
+We are now ready to extend Corollary 4.3 to general L1-predual spaces.
+
+22
+MART´IN, PERREAU, AND RUEDA ZOCA
+Corollary 4.5. Let X be an L1-predual and let x ∈ SX be a ∆-point. Then, x is a ccs Daugavet
+point. Hence the six diametral notions are equivalent for L1-preduals.
+Proof. If x is a ∆-point in X, then as mentioned in item (3) of Remark 2.6, we have that JX(x) is
+a ∆-point in X∗∗. Now, X∗∗ is isometric to a C(K) space so Theorem 4.2 gives that JX(x) is a ccs
+Daugavet point in X∗∗. Then, using now item (4) of Remark 2.6 (or using a straightforward argument
+based on the principle of local reflexivity as in [3, Theorem 3.7]), we get that x is a ccs Daugavet point
+in X.
+□
+Let us observe that the proof of Theorem 4.2 also works for M¨untz spaces (by using [3, Lemma 3.10]
+to provide suitable replacements for the functions pn). We recall that given an an increasing sequence
+Λ = (λn)∞
+n=0 of non-negative real numbers with λ0 = 0 such that �∞
+i=1
+1
+λi < ∞, then the real Banach
+space
+M(Λ) := span{tλn : n ⩾ 0} ⊆ C[0, 1]
+is called the M¨untz space associated with Λ. Excluding the constant functions form M(Λ), we have
+the subspace M0(Λ) := span{tλn : n ⩾ 1} of M(Λ).
+So, adapting the proof of Theorem 4.2 to M¨untz spaces (for real scalar-valued functions attaining
+its norm at 1 ∈ [0, 1]) and also using [3, Proposition 3.12], we get the following result analogous to
+Corollaries 4.3 and 4.5.
+Corollary 4.6. Let X = M(Λ) or X = M0(Λ) for an increasing sequence Λ of non-negative real
+numbers with λ0 = 0 such that �∞
+i=1
+1
+λi < ∞. Then, every ∆-point of X is a ccs Daugavet point (and
+hence the six diametral notions are equivalent).
+4.2. Characterization in L1-spaces. In [3, Theorem 3.1] the equivalence between the notions of
+Daugavet point and ∆-point was obtained for elements of σ-finite L1-spaces in the real case. Actually,
+it is not complicated to extend the results to arbitrary measures and also to the complex case.
+Proposition 4.7 ([3, Theorem 3.1] for the σ-finite real case). Let (Ω, Σ, µ) be a measure space, and
+let f be a norm one element in L1(µ). Then, the following assertions are equivalent.
+(1) f is a Daugavet point.
+(2) f is a ∆-point.
+(3) The support of the function f contains no atom.
+Observe that (1) implies (2) is immediate. For (2) implies (3), suppose that f is a ∆-point and let
+A be an atom of finite measure (the only ones that can be contained in the support of an integrable
+function). Then, we clearly have that L1(µ) = L1(µ|Ω\A) ⊕1 K (as integrable functions are constant
+on atoms), and we may write f = (f1, c) for suitable f1 ∈ L1(µ|Ω\A) and c = f(A) ∈ K. If c ̸= 0, then
+∥f1∥ ̸= 1 and it follows from [45, Theorem 3.4.4] that 1 ∈ K is a ∆-point, a contradiction. This shows
+that the support of f does not contain any atom.
+To get that (3) implies (1), we actually prove the following more general result. Recall that given a
+measured space (Ω, Σ, µ) and a Banach space X, L1(µ, X) denotes the Banach space of all B¨ochner-
+integrable functions from Ω to X.
+Theorem 4.8. Let (Ω, Σ, µ) be a measured space, let X be a Banach space, and let f be a norm one
+element in L1(µ, X). If the support of the function f contains no atom, then f is a super Daugavet
+point.
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+23
+Proof. Let us write S := supp f which contains no atom by hypothesis. Let us first prove that f is
+a super Daugavet point. Since S contains no atoms, we have that L1(µ|S, X) satisfies the Daugavet
+property (see e.g. [54, Example in p. 81]). In particular, f is a super Daugavet point in this space.
+Since L1(µ, X) = L1(µ|S, X) ⊕1 L1(µ|Σ\S, X), we get that f is a super Daugavet point in L1(µ, X) by
+the transfer results from Subsection 3.2 (see Remark 3.28).
+□
+Our next goal is to discuss the relationship with the ccs diametral notions. For real L1(µ)-spaces,
+and using a result from [1], we may actually get that real-valued integrable functions with atomless
+support are ccs ∆-points.
+Proposition 4.9. Let (Ω, Σ, µ) be a measured space and let f be a norm one element in the real space
+L1(µ). If the support of the function f contains no atom, then f is a ccs ∆-point.
+Proof. Take ε > 0 and D := �n
+i=1 λiSi a ccs of BL1(µ) containing f. Write f := �n
+i=1 λigi with gi ∈ Si
+for every i. Consider the measurable subset ˜S := supp f ∪ �n
+i=1 supp gi of Ω and let ˜µ be the σ-finite
+measure ˜µ := µ| ˜S on ( ˜S, Σ| ˜S). Then D induces a ccs ˜D of BL1(˜µ) by restriction of the support which
+contains the function ˜f which is just f viewed as an element of L1(˜µ) and hence, the support of ˜f
+does not contains atoms. Since ˜f belongs to the unit sphere of the real space L1(˜µ), we have by [1,
+Theorem 5.5] that ˜f is an interior point of ˜D for the relative weak topology of BL1(˜µ). As we have
+already shown that ˜f is a super Daugavet point in Theorem 4.8 (and hence a super ∆-point), we can
+find ˜g ∈ ˜D such that
+�� ˜f − ˜g
+�� > 2 − ε. By just considering the extension g of ˜g to the whole Ω by 0,
+we get that g ∈ D and that ∥f − g∥ =
+�� ˜f − ˜g
+�� > 2 − ε.
+□
+Let us comment that it is not clear whether ccs ∆-points transfer through absolute sums, but we
+have used specific geometric properties of L1-spaces in the previous proof.
+Remark 4.10. Observe that since [1, Theorem 5.5] is also valid for convex combination of relative
+weakly open subsets of BL1(µ), we in fact have that every ∆-point in a real L1(µ) space is actually a
+ccw ∆-point.
+Putting together Proposition 4.7, Theorem 4.8, and Proposition 4.9, we get the following corollary.
+Corollary 4.11. Let (Ω, Σ, µ) be a measured space and let f be a norm one element in L1(µ). Then,
+the following notions are equivalent for f: ∆-points, Daugavet point, super ∆-point, and super Dau-
+gavet point. Moreover, in the real case, the previous four notions are also equivalent to being ccs
+∆-point.
+We now deal with ccs Daugavet points in L1(µ)-spaces. Observe that if Ω admits an atom A of
+finite measure, then we have L1(µ) ≡ L1(µ|Ω\A) ⊕1 K. In particular, in this case L1(µ) fails to have
+ccs Daugavet points by Proposition 3.29. We then have the following characterization of the presence
+of a ccs Daugavet point in an L1-space.
+Proposition 4.12. Let (Ω, Σ, µ) be a measure space. Then, the following assertions are equivalent.
+(1) L1(µ) has the Daugavet property.
+(2) L1(µ) contains a ccs Daugavet point.
+(3) L1(µ) has the SD2P.
+(4) µ admits no atom of finite measure.
+
+24
+MART´IN, PERREAU, AND RUEDA ZOCA
+Proof. (1)⇔(4) is well known (see [54, Section 2, Example (b)]); (1)⇒(2) is also known; (2)⇒(3) is
+contained in Proposition 3.12. Finally, (3)⇒(4) follows from Proposition 3.29 and the comment before
+the statement of this proposition.
+□
+4.3. Remarks on some examples from the literature.
+4.3.1. Two examples in Lipschitz-free spaces. In [51], Veeorg constructed a surprising example of a
+space satisfying the Radon-Nikod´ym property and containing a Daugavet point. We slightly improve
+this result by showing that this point is also a ccs ∆-point by proving a general fact about extreme
+∆-molecules in Lipschitz-free spaces. For the necessary definitions we refer to the cited paper [51] and
+to [9, 10, 32]; for further background on Lipschitz-free spaces, we refer to the book [53].
+For this purpose, we start by recalling the following characterization of molecules which are ∆-points
+on Lipschitz-free spaces from [32].
+Proposition 4.13 ([32, Theorem 4.7]). Let M be a pointed metric space and let x ̸= y ∈ M. The
+molecule mx,y is a ∆-point if and only if every slice S of BF(M) containing mx,y also contains for
+every ε > 0 a molecule mu,v with u ̸= v ∈ M satisfying d(u, v) < ε.
+In the case in which the molecule is an extreme point, we have the following improved result.
+Theorem 4.14. Let M be a pointed metric space, and let x ̸= y ∈ M. If the molecule mx,y is an
+extreme point and a ∆-point, then mx,y is a ccs ∆-point.
+Observe that this result cannot be obtained from Proposition 3.13: molecules of Lipschitz-free
+spaces which are preserved extreme points are denting points, hence very far from being ∆-points.
+To give the proof of the theorem, we need a result which is just an equivalent reformulation of a
+result in [32].
+Lemma 4.15 ([32, Theorem 2.6]). Let M be a pointed metric space, and let µ ∈ SF(M). For every
+ε > 0, there exists δ > 0 such that given u ̸= v ∈ M with d(u, v) < δ we have ∥µ ± mu,v∥ > 2 − ε.
+Using this result and a homogeneity argument similar to the one from [15, Lemma 2.3], we can
+provide the pending proof.
+Proof of Theorem 4.14. Let C := �n
+i=1 λiSi be a ccs of BF(M) containing mx,y and let ε > 0. Since
+mx,y is extreme, we have that mx,y ∈ �n
+i=1 Si, and by Proposition 4.13 every Si contains molecules
+of F(M) supported at arbitrarily close points. Using Lemma 4.15, we construct inductively for every
+η > 0 a finite sequence (mui,vi)n
+i=1 of molecules in F(M) such that
+(1) mui,vi ∈ Si for every i.
+(2)
+���mx,y − �k
+i=1 λimui,vi
+��� > 1 + �k
+i=1 λi − kε
+n for every k ⩽ n.
+Indeed, since S1 contains molecules of F(M) supported at arbitrarily close points, we can find by
+Lemma 4.15 u1 ̸= v1 ∈ M such that mu1,v1 ∈ S1 and ∥mx,y − mu1,v1∥ > 2 − ε
+n.
+It follows that
+∥mx,y − λ1mu1,v1∥ ⩾ ∥mx,y − mu1,v1∥−(1−λ1) > 1+λ1− ε
+n. Let us assume that mu1,v1, . . . , muk,vk are
+constructed as desired for a given k ∈ {1, . . . , n−1}. Since Sk+1 contains molecules of F(M) supported
+at arbitrarily close points, we can find by Lemma 4.15 uk+1 ̸= vk+1 ∈ M such that muk+1,vk+1 ∈ Sk+1
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+25
+and
+������
+mx,y − �k
+i=1 λimui,vi
+���mx,y − �k
+i=1 λimui,vi
+���
+− muk+1,vk+1
+������
+> 2 −
+ε
+n
+���mx,y − �k
+i=1 λimui,vi
+���
+.
+Then,
+������
+mx,y − �k+1
+i=1 λimui,vi
+���mx,y − �k
+i=1 λimui,vi
+���
+������
+⩾
+������
+mx,y − �k
+i=1 λimui,vi
+���mx,y − �k
+i=1 λimui,vi
+���
+− muk+1,vk+1
+������
+−
+�
+�1 −
+λk+1
+���mx,y − �k
+i=1 λimui,vi
+���
+�
+�
+> 1 +
+λk+1
+���mx,y − �k
+i=1 λimui,vi
+���
+−
+ε
+n
+���mx,y − �k
+i=1 λimui,vi
+���
+.
+By the assumption,
+�����mx,y −
+k+1
+�
+i=1
+λimui,vi
+����� >
+�����mx,y −
+k
+�
+i=1
+λimui,vi
+����� + λk+1 − ε
+n > 1 +
+k+1
+�
+i=1
+λi − (k + 1)ε
+n
+.
+As a consequence, µ := �n
+i=1 λimui,vi belongs to C and satisfies ∥mx,y − µ∥ > 2 − ε.
+□
+In particular, we have, as announced, that the molecule mx,y in the example from [51] is a ccs
+∆-point. Note that it cannot be a ccs Daugavet point by Proposition 3.12 since the space has the
+RNP, but we do not know whether it is a super ∆-point or even a super Daugavet point. Let us state
+the result for further reference.
+Example 4.16. Let M be the metric space constructed in [51, Example 3.1] and let x, y be the points
+described there. Then, F(M) has the RNP, the molecule mx,y is an extreme point of the unit ball of
+F(M) which is a Daugavet point. Hence, by our Theorem 4.14, mx,y is a ccs-∆-point.
+Another interesting example in the Lipschitz-free space setting is the following one which uses a
+metric space constructed by Aliaga, Noˆus, Petitjean, and Proch´azka [10].
+Example 4.17. Let M be the metric space from [10, Examples 4.2]. Then one can check that the
+molecule m0,q is an extreme point of BF(M) and, since the points 0 and q are discretely connectable, it
+follows from an easy adjustment of [32, Proposition 4.2] that this molecule is a ∆-point. In particular,
+it follows from Theorem 4.14 that this molecule is also a ccs ∆-point. However, it is not difficult to
+show that there exists denting points in BF(M) that are at distance strictly less than 2 to m0,q (take
+any among the molecules mxn
+i ,xn
+i+1), so this molecule is not a Daugavet point. Also observe that this
+space has the RNP since the metric space M is countable and complete [9, Theorem 4.6].
+Let us finally remark that the spaces F(M) of Examples 4.16 and 4.17 have the RNP, so they
+are strongly regular and hence strongly regular points are norm dense, but both examples have ccs
+∆-points. They cannot contain ccs Daugavet points by Proposition 3.12.
+Let us also comment that the use of Theorem 4.14 above cannot be omitted, as the molecule m0,q
+is not a preserved extreme point, hence Proposition 3.13 is again not applicable.
+
+26
+MART´IN, PERREAU, AND RUEDA ZOCA
+4.3.2. An example of a Banach space with the DD2P, the restricted DSD2P, but containing ccs of
+arbitrarily small diameter. In [2, Theorem 2.12], Abrahamsen, H´ajek, Nygaard, Talponen, and Troy-
+anski constructed a space X which has the DLD2P, which is midpoint locally uniformly rotund (in
+particular, satisfying that pre-ext (BX) = SX), and such that BX contains convex combinations of
+slices of arbitrarily small diameter. It then follows from Proposition 3.13 that every element of SX is
+actually a super ∆-point and a ccs ∆-point (that is, X has the DD2P and the restricted DSD2P). But
+containing ccs of arbitrarily small diameter, X fails the SD2P. The obvious explanation for the failure
+of the SD2P and the fact that every element in the unit sphere is a ccs ∆-point is that none of the
+convex combinations of slices of diameter strictly smaller than 2 intersects the unit sphere. On the
+other hand, the space X is constructed as the ℓ2-sum of spaces, and so X does not contain Daugavet
+points by [3, Proposition 4.6] (see Proposition 3.27).
+Observe further that X has the restricted DSD2P and the DD2P, but fails the DSD2P (which is
+equivalent to the Daugavet property by [34]).
+4.3.3. An example in a space with one-unconditional basis. Abrahamsen, Lima, Martiny, and Troy-
+anski constructed in [6, Section 4] a Banach space XM with one-unconditional basis which contains a
+subset DB ⊆ SXM satisfying:
+• Every element in DB is both a Daugavet point and a point of continuity;
+• BXM = co(DB);
+• DB is weakly dense in the unit ball.
+Observe that no element of DB is a super ∆-point (it is exactly the opposite!). By Theorem 3.19, no
+element of DB is a ccs ∆-point.
+4.4. A super ∆-point which fails to be a Daugavet point in an extreme way. In order to
+put into a context the following result, let us recall that Daugavet points are at distance 2 from any
+denting point (see [32, Proposition 3.1]). With this in mind, the following result can be interpreted as
+the existence of super ∆-points which fail to be Daugavet points in an extreme way.
+Theorem 4.18. Let X be a Banach space with the Daugavet property. Then, for every ε > 0, there
+exists an equivalent norm | · | and two points x, y ∈ B(X,|·|) such that
+(1) y is a super ∆-point.
+(2) x is strongly exposed.
+(3) |x − y| < ε.
+Proof. Take a subspace Y ⊆ X with dim(X/Y ) = 1. Observe that Y has the Daugavet property (see
+e.g. [50, Theorem 6 (a)]). Take x ∈ SX with 0 < d(x, Y ) < ε (this can be settled taking a non-zero
+element v ∈ X/Y with quotient norm smaller than ε). Now, we can find an element y ∈ SY such that
+∥x − y∥ < ε. By the Hahn-Banach theorem, we can take f ∈ SX∗ with Re f(x) > 0 and f = 0 on Y .
+This means that x belongs to the slice T := {z ∈ BX : Re f(z) > α} for some α > 0. Take δ > 0 such
+that ∥x−y∥
+1−δ
+< ε. By Lemma 2.1 we can find x∗ ∈ SX∗ such that x ∈ S(x∗, δ; BX) ⊆ T. By the above
+inclusion we conclude that S(x∗, δ; BX) ∩ BY = ∅ or, in other words, that Re x∗(z) ⩽ 1 − δ for every
+z ∈ BY . Set
+B := co(BY ∪ (1 − δ)BX ∪ {±x}).
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+27
+B is the unit ball of an equivalent norm |·| which satisfies, in view of the inclusions (1−δ)BX ⊆ B ⊆ BX,
+that
+∥x∥ ⩽ |x| ⩽
+1
+1 − δ∥x∥
+for every x ∈ X. Let us prove that | · |, x and y satisfies our requirements. First, observe that
+|x − y| ⩽ ∥x − y∥
+1 − δ
+< ε.
+Next, we claim that y is a super ∆ point. Indeed, since Y has the Daugavet property we can find a net
+{ys} ⊆ BY with {ys} −→ y weakly and ∥y − ys∥ −→ 2. Notice that the weak convergence {ys} −→ y
+is still guaranteed on X because i: (Y, ∥ · ∥) −→ (X, | · |) is weak to weak continuous as ∥ · ∥ and | · |
+are equivalent. Moreover, notice that ys ∈ BY ⊆ B for every s, so |ys| ⩽ 1 for every s. Finally,
+|ys − y| ⩾ ∥ys − y∥ −→ 2,
+and since y ∈ BY ⊆ B, we conclude |ys − y| −→ 2. From there, y is clearly a super ∆-point for the
+norm | · |.
+It remains to prove that x is strongly exposed. Indeed, we will prove that Re x∗ strongly exposes
+B at x, for which it is enough to prove that Re x∗ strongly exposes co(BY ∪ (1 − δ)BX ∪ {±x}) at
+x. Take z := αu + β(1 − δ)v + (γ − ω)x ∈ co(BY ∪ (1 − δ)BX ∪ {±x}) with α + β + γ + ω = 1.
+Observe that 1 − δ < Re x∗(x) ⩽ |x∗| ⩽ ∥x∗∥ due to the inclusion B ⊆ BX. Taking into account that
+Re x∗(u) ⩽ 1 − δ since u ∈ BY as BY ∩ S(x∗, δ; BX) = ∅, we conclude
+Re x∗(z) ⩽ (1 − δ)(α + β) + (γ − ω) Re x∗(x).
+Since Re x∗(x) > 1 − δ, we get that sup
+�
+Re x∗(z): z ∈ co(BY ∪ (1 − δ)BX ∪ {±x})
+�
+= Re x∗(x). If we
+take a sequence
+zn := αnun + βn(1 − δ)vn + (γn − ωn)x ∈ co(BY ∪ (1 − δ)BX ∪ {±x})
+with αn + βn + γn + ωn = 1 such that Re x∗(zn) −→ Re x∗(x), it follows from the previous argument
+that αn → 0, βn → 0, ωn → 0 and γn → 1, which means zn → x in norm.
+□
+Remark 4.19. Using the previous theorem and Proposition 3.25 it is easy to construct (considering
+ℓ2-sums, for instance) a Banach space X containing a sequence of super ∆-points (yn) such that the
+distance from yn to the set of strongly exposed points is going to zero.
+4.5. A super ∆-point which is a strongly regular point. In the present subsection, as well
+as in the next, we aim to distinguish the super and ccs notions of ∆- and Daugavet points. The
+following result shows that there are plenty of examples of spaces containing super ∆-points which are
+strongly regular points (hence far from being ccs ∆-points). We do the construction in for real spaces
+for simplicity.
+Theorem 4.20. Every real Banach space with the Daugavet property can be equivalently renormed
+so that the new unit ball has a point which is simultaneously super-∆ and a point of strong regularity
+(hence, far away of being ccs ∆-point).
+We will use the following immediate result which follows from the fact that a convex combination
+of ccs is again a ccs.
+Lemma 4.21. Let X be a Banach space and let C be a closed, convex, bounded subset of X. Then
+the set of strongly regular points of C is a convex set.
+
+28
+MART´IN, PERREAU, AND RUEDA ZOCA
+Proof of Theorem 4.20. Let X be a Banach space with the Daugavet property. Take a 1-codimensional
+subspace Y of X. Since Y is complemented in X then X = Y ⊕ R, so we will see X in such way. Take
+r > 0, y0 ∈ SY and f ∈ SX∗ such that f(y0) = 1, and consider on X = Y ⊕ R the equivalent norm
+| · | whose unit ball is B := co
+�
+BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}
+�
+. It readily follows that | · | agrees
+with the original norm ∥ · ∥ on the elements of the form (y, 0).
+We claim that (y0, 0) satisfies our requirements. First of all, let us prove that (y0, 0) is a super-∆
+point. Since Y is one-codimensional, it has the Daugavet property (see e.g. [50, Theorem 6 (a)]).
+Consequently, there exists a net (ys) −→ y0 weakly in BY such that ∥y0 −ys∥ −→ 2. Then, (ys, 0) −→
+(y0, 0) weakly in (X, | · |). Moreover, it is clear that (ys, 0) ∈ B for every s. Finally,
+|(ys, 0) − (y0, 0)| = |(ys − y0, 0)| = ∥ys − y0∥ −→ 2.
+Let us now prove that (y0, 0) is a point of strong regularity.
+To do so, it is enough, in view of
+Lemma 4.21, to show that (y0, ±r) is a strongly exposed point (we will prove that for (y0, r), being
+the other case completely analogous). Let us prove that Re(f, 1) strongly exposes (y0, r) in the set
+BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}. On the one hand, we have
+Re(f, 1)(y0, r) = Re f(y0) + r = 1 + r.
+On the other hand, given (y, 0) ∈ BY × 0 we have Re(f, 1)(y, 0) = f(y) ⩽ 1 < 1 + r. Moreover,
+Re(f, 1)(y0, −r) = 1 − r and Re(f, 1)(−y0, ±r) = −1 ± r < 1 + r. Consequently,
+sup{Re(f, 1)(a, b): (a, b) ∈ BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}, (a, b) ̸= (y0, r)}
+⩽ 1 < 1 + r = Re(f, 1)(y0, r).
+This is enough to guarantee that Re(f, 1) strongly exposes (y0, r) in B, so we are done.
+□
+4.6. A super Daugavet point which is not ccs ∆-point. The previous example shows that we
+can distinguish the notion of super ∆-point and the one of ccs ∆-point. It seems natural then that
+we should be able to distinguish the notions of super Daugavet point and the one of ccs ∆-point. In
+order to do so, we need to consider an involved construction but, as a consequence, we will prove
+that there are super Daugavet points which are contained in convex combinations of slices of small
+diameter. The construction will be very similar to that of [14, Theorem 2.4], with a slight variation
+which makes the resulting norm with a stronger Daugavet flavour. As in the previous subsection, we
+will only work with real spaces here.
+In order to do so, let us recall a construction from Argyros, Odell, and Rosenthal [11]. Pick a
+nonincreasing null sequence {εn} in R+. We construct an increasing sequence of closed, bounded and
+convex subsets {Kn} in the real space c0 and a sequence {gn} in c0 as follows: First define K1 = {e1},
+g1 = e1 and K2 = co(e1, e1 + e2). Choose l2 > 1 and g2, . . . , gl2 ∈ K2 an ε2-net in K2. Assume that
+n ⩾ 2 and that mn, ln, Kn, and {g1, . . . , gln} have been constructed, with Kn ⊆ Bspan{e1,...,emn} and
+gi ∈ Kn for every 1 ⩽ i ⩽ ln. Define Kn+1 as
+Kn+1 = co(Kn ∪ {gi + emn+i : 1 ⩽ i ⩽ ln}).
+Consider mn+1 = mn + ln and choose {gln+1, . . . , gln+1} ∈ Kn+1 so that {g1, . . . , gln+1} is an εn+1-net
+in Kn+1. Finally, we define K0 = ∪nKn. Then it follows that K0 is a non-empty closed, bounded
+and convex subset of c0 such that x(n) ⩾ 0 for every n ∈ N and ∥x∥∞ ⩽ 1 for every x ∈ K0 and so
+diam (K0) ⩽ 1.
+Now, for a fixed i, we have from the construction that {gi + emn+i}n is a sequence in K0 (for n
+large enough) which is weakly convergent to gi, and ∥(gi − emn+i) − gi∥ = ∥emn+i∥ = 1 holds for every
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+29
+n. Then diam (K0) = 1. We will freely use the set K0 and the above construction throughout the
+subsection. Observe that, from the above construction, it follows that
+K0 = {gi : i ∈ N}
+w = {gi : i ∈ N}.
+Observe finally that, by the inductive construction, gi has finite support for every i ∈ N.
+By [11, Theorem 1.2] we have that K0 contains convex combinations of slices of arbitrarily small
+diameter. However, all the points in K0 are “super Daugavet points” in the following sense.
+Proposition 4.22. For every x0 ∈ K0, every ε > 0, and every non-empty weakly open subset W of
+K0, there exists y ∈ W satisfying that ∥x0 − y∥ > 1 − ε = diam (K0) − ε.
+Proof. Take ε > 0 and a non-empty relatively weakly open subset of K0. By a density argument, we
+can find i ∈ N satisfying that ∥x0 − gi∥ < ε. Again by a density argument there exists gk ∈ W for
+certain k ∈ N.
+As we explained above, by the definition of K0 we have that the sequence gk +emn+k ∈ K0 for every
+n ∈ N. Since
+�
+gk + emn+k
+�
+n∈N −→ gk weakly, we can find n ∈ N large enough so that gk + emn+k ∈ W
+and mn + k /∈ supp(gi) ∪ supp(gk) (this is possible because the previous set is finite).
+So taking
+y = gk + emn+k, we get y(mm + k) = 1 and so
+∥gi − y∥ ⩾ y(mn + k) − gi(mn + k) = 1 − 0 = 1.
+As a consequence, ∥x0 − y∥ ⩾ ∥gi − y∥ − ∥gi − x0∥ > 1 − ε, and the proof is finished.
+□
+It is time to construct the announced renorming of C[0, 1]. Take a sequence of non-empty pairwise
+disjoint open subsets Vn of [0, 1] satisfying that 0 /∈ �
+n∈N
+Vn. By Urysohn lemma, we can find, for
+every n ∈ N, a function hn ∈ SC[0,1] with 0 ⩽ hn ⩽ 1 and such that supp(hn) ⊆ Vn. If we consider
+Z := span{hn : n ∈ N}, we get that Z is lattice isometrically isomorphic to c0 (indeed, the mapping
+en �−→ hn is an isometric Banach lattice isomorphism). Consequently, we can consider the set K0
+constructed in Z, obtaining that K0 ⊆ BC[0,1] is a set of positive functions (because the latter linear
+isometry preserves the lattice structure) which contains convex combination of slices of arbitrarily small
+diameter but enjoying the property exhibited in Proposition 4.22. Moreover, by the construction of
+the functions hn, f(0) = 0 for every f ∈ Z so, in particular, f(0) = 0 for every f ∈ K0.
+Now, take 0 < ε < 1 and write
+Bε := co
+�
+2
+�
+K0 − 1
+2
+�
+∪ 2
+�
+−K0 + 1
+2
+�
+∪ ((1 − ε)BC[0,1] + εBker(δ0))
+�
+,
+where 1 stands for the constant function 1 in C[0, 1].
+Consider ∥·∥ε the norm on (the real version of) C[0, 1] whose unit ball is Bε. As we have indicated,
+the renorming technique follows the scheme of the renorming given in [14, Theorem 2.4] with the
+difference that we use Bker(δ0) instead of Bc0 in the last term because ker(δ0) is a Banach space with
+the Daugavet property.
+We have the following result.
+Theorem 4.23. The space (X, ∥ · ∥ε) satisfies that:
+(1) Every element of 2(K0 − 1
+2 ) is a super Daugavet point.
+(2) For every η > 0 there exists a convex combination of slices D of Bε with D ∩ 2(K0 − 1
+2 ) ̸= ∅
+and such that diam (D) < η.
+
+30
+MART´IN, PERREAU, AND RUEDA ZOCA
+In particular, there are super Daugavet points which are not ccs−∆ points.
+Proof. (1). Take a ∈ K0, and let us prove that 2a − 1 is a super Daugavet point. In order to do so,
+pick a non-empty relatively weakly open subset W of Bε. Write
+A := 2(K0 − 1
+2 ) and B := (1 − ε)BC[0,1] + εBker(δ0).
+Since Bε = co(A ∪ −A ∪ B) we have that W has non-empty intersection with co(A ∪ −A ∪ B). Now
+observe that A−A
+2
+= K0 −K0 ⊆ Bker(δ0) ⊆ B so that co(A∪−A∪B) = co(A∪B)∪co(−A∪B) by [14,
+Lemma 2.4]. Consequently, either W ∩ co(A ∪ B) or W ∩ co(−A ∪ B) is non-empty. Let us distinguish
+by cases.
+Assume first that W ∩ co(A ∪ B) is non-empty, so find a′ ∈ K0, f ∈ BC[0,1], g ∈ Bker(δ0), and
+α, β ∈ [0, 1] with α + β = 1 satisfying that
+α(2a′ − 1) + β((1 − ε)f + εg) ∈ W.
+Take η > 0. By Proposition 4.22, there exists a net (as) −→ a′ weakly with as ∈ K0 for every s and
+satisfying that ∥a − as∥ −→ 1. Since (2as − 1) −→ 2a′ − 1 weakly, we can find s large enough so that
+α(2as − 1) + β((1 − ε)f + εg) ∈ W
+and
+∥(2a − 1) − (2as − 1)∥ = 2∥a − as∥ > 2 − η.
+Observe that 2a − 1 and 2as − 1 are functions in BC[0,1] since a, as are positive functions of norm at
+most one. Since ∥(2a − 1) − (2as − 1)∥ > 2 − η, there exists t0 ∈ [0, 1] and θ ∈ {−1, 1} such that
+θ(2a − 1)(t0) > 1 − η and θ(2as − 1)(t0) < −1 + η (observe that t0 ̸= 0 since a(t0) = as(t0) = 0 by
+construction). Consequently, the set
+U := {t ∈ [0, 1]: θ(2a − 1)(t) > 1 − η and θ(2as − 1)(t) < −1 + η}
+is a non-empty open subset of [0, 1], and we can construct a sequence of non-empty pairwise disjoint
+open sets Wn ⊆ U. Observe that 0 /∈ �
+n∈N Wn since 0 /∈ U. Take pn ∈ Wn for every n ∈ N. We
+can construct, for every n ∈ N, two functions fn and gn in the unit ball of C[0, 1] satisfying fn = f
+and gn = g in [0, 1] \ Wn and fn(pn) = gn(pn) = −θ. Observe that the sequence of functions (f − fn)
+have pairwise disjoint supports, so (f − fn) −→ 0 weakly or, in other words, (fn) −→ f weakly. A
+similar argument shows that (gn) −→ g weakly. Notice also that, given n ∈ N, since 0 /∈ Wn then
+gn(0) = g(0) = 0, so (gn) ⊆ ker(δ0). Henceforth α(2as − 1) + β((1 − ε)fn + εgn) is a sequence in Bε
+which converges in n weakly to α(2as − 1) + β((1 − ε)f + εg) ∈ W. Consequently, we can find n large
+enough such that α(2as −1)+β((1−ε)fn +εgn) ∈ W. Finally, observe that the inclusion Bε ⊆ BC[0,1]
+implies that ∥z∥ ⩽ ∥z∥ε, so
+��(2a − 1) − α(2as − 1) − β((1 − ε)fn + εgn)
+��
+ε ⩾
+��(2a − 1) − α(2as − 1) − β((1 − ε)fn + εgn)
+��
+⩾ θ((2a − 1) − α(2as − 1) − β((1 − ε)fn)(pn)
+= θ(2a − 1)(pn) − θα(2as − 1)(pn)
+− θβ((1 − ε)fn(pn) + θεgn(pn))
+> 1 − η − α(−1 + η) − β(−1)
+= 1 + α + β − (1 + α)η = 2 − 2η.
+Since η > 0 was arbitrary this finishes the case W ∩ co(A ∪ B) ̸= ∅.
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+31
+For the case W ∩ co(−A ∪ B) ̸= ∅, find a′ ∈ K0, f ∈ BC[0,1], g ∈ Bker(δ0), and α, β ∈ [0, 1] with
+α + β = 1 satisfying that
+α(−2a′ + 1) + β((1 − ε)f + εg) ∈ W.
+This case is simpler because ∥(2a − 1) − (−2a′ + 1)∥ ⩾ (2a − 1) − (−2a′ + 1)(0) = 2. Now, an
+approximation argument for fn and gn similar to that of the above case (working on a non-empty
+open subset of (0, 1) in order to get gn(0) = 0) finishes this case and, consequently, the proof of (1).
+(2). The first part of the proof will be a repetition of the argument of [14, Theorem 2.4]. Fix γ > 0.
+From [11, Theorem 1.2] there exist slices S1, · · · , Sn of K0 such that
+diam
+�
+1
+n
+n
+�
+i=1
+Si
+�
+< 1
+4(1 − ε)γ.
+We can assume that Si = {x ∈ K0 : x∗
+i (x) > 1 − �δ} where 0 < �δ < 1, x∗
+i ∈ C[0, 1]∗ and sup x∗
+i (K0) = 1
+holds for every i = 1, . . . , n. It is clear that
+sup x∗
+i
+�
+2(K0 − 1
+2 )
+�
+= 2(1 − x∗
+i (1
+2 )),
+for all i = 1, · · · , n. We put ρ, δ > 0 such that 1
+2ρ∥x∗
+i ∥ + δ < �δ, 2ρ < ε, ρ∥x∗
+i ∥ < 4δ, and (7−2ε)ρ
+(1−ε)
+< γ,
+for all i = 1, . . . , n. We consider the relatively weakly open set of Bε given by
+Ui :=
+�
+x ∈ Bε : x∗
+i (x) > 2
+�
+1 − δ − x∗
+i
+�1
+2
+��
++ 1
+2ρ∥x∗
+i ∥, x(0) = δ0(x) < −1 + ρ2
+�
+for every i = 1, . . . , n. It is clear that ∥x∗
+i ∥ε ⩽ ∥x∗
+i ∥ for every i = 1, . . . , n and ∥δ0∥ε = ∥δ0∥ = 1.
+Since ρ∥x∗
+i ∥ < 4δ, we have that 2(1 − x∗
+i ( 1
+2 )) > 2(1 − δ − x∗
+i ( 1
+2)) + 1
+2ρ∥x∗
+i ∥. Now, we have that
+sup x∗
+i (2(K0 − 1
+2 )) = 2(1 − x∗
+i ( 1
+2)), then there exists x ∈ K0 such that
+x∗
+i (2(x − 1
+2 )) > 2(1 − δ − x∗
+i (1
+2)) + 1
+2ρ∥x∗
+i ∥ and δ0(2(x − 1
+2)) = −1 < −1 + ρ2.
+This implies that Ui ̸= ∅ for every i = 1, . . . , n. In order to estimate the diameter of 1
+n
+�n
+i=1 Ui, it is
+enough to compute the diameter of
+1
+n
+n
+�
+i=1
+Ui ∩ co
+�
+2
+�
+K0 − 1
+2
+�
+∪ −2
+�
+K0 − 1
+2
+�
+∪ [(1 − ε)BX + εBker(δ0)]
+�
+.
+Since 2(K0 − 1
+2 ) and (1 − ε)BC[0,1] + εBker(δ0) are convex subsets of Bε, given x ∈ Bε, we can assume
+that x = λ12(a − 1
+2 ) + λ22(−b + 1
+2 ) + λ3[(1 − ε)x0 + εy0], where λi ∈ [0, 1] with �3
+i=1 λi = 1 and
+a, b ∈ K0, x0 ∈ BC[0,1], and y0 ∈ Bker(δ0).
+So given x, y ∈ 1
+n
+�n
+i=1 Ui, for i = 1, · · · , n, there exist ai, a′
+i, bi, b′
+i ∈ K0, λ(i,j), λ′
+(i,j) ∈ [0, 1] with
+j = 1, 2, 3 and, xi, x′
+i ∈ BC[0,1], and yi, y′
+i ∈ BKer(δ0), such that
+ui := 2λ(i,1)
+�
+ai − 1
+2
+�
++ 2λ(i,2)
+�
+−bi + 1
+2
+�
++ λ(i,3)[(1 − ε)xi + εyi]
+u′
+i := 2λ′
+(i,1)
+�
+a′
+i − 1
+2
+�
++ 2λ′
+(i,2)
+�
+−b′
+i + 1
+2
+�
++ λ′
+(i,3)[(1 − ε)x′
+i + εy′
+i]
+belong to Ui for every i ∈ {1, . . . , n}, and such that
+x = 1
+n
+n
+�
+i=1
+ui and y = 1
+n
+n
+�
+i=1
+u′
+i.
+
+32
+MART´IN, PERREAU, AND RUEDA ZOCA
+For i ∈ {1, . . . , n} we have that ui ∈ Ui so
+δ0(ui) = δ0
+�
+2λ(i,1)
+�
+ai − 1
+2
+�
++ 2λ(i,2)
+�
+−bi + 1
+2
+�
++ λ(i,3)[(1 − ε)xi + εyi]
+�
+< −1 + ρ2.
+Observe that, by construction,
+δ0
+�
+ai − 1
+2
+�
+= −1
+2, δ0
+�
+−bi + 1
+2
+�
+= 1
+2 and δ0((1 − ε)xi + εyi) = δ0((1 − ε)xi) ⩾ −(1 − ε).
+This implies that
+2λ(i,2) + λ(i,3)ε − 1 = −λ(i,1) + λ(i,2) − λ(i,3)(1 − ε) < −1 + ρ2.
+Since 2ρ < ε, we deduce that λ(i,2) + λ(i,3) < 1
+2ρ. As a consequence we get that
+(4.1)
+λ(i,1) > 1 − 1
+2ρ,
+and, similarly, we get that
+(4.2)
+λ′
+(i,1) > 1 − 1
+2ρ,
+for every i = 1, . . . , n. Now, the previous inequalities imply that
+∥x − y∥ε ⩽ 1
+n
+�����
+n
+�
+i=1
+2λ(i,1)
+�
+ai − 1
+2
+�
+− 2λ′
+(i,1)
+�
+a′
+i − 1
+2
+������
+ε
++ 1
+n
+n
+�
+i=1
+����2λ(i,2)
+�
+−bi + 1
+2
+�����
+ε
++ 1
+n
+n
+�
+i=1
+����2λ′
+(i,2)
+�
+−b′
+i + 1
+2
+�����
+ε
++ 1
+n
+n
+�
+i=1
+∥λ(i,3)[(1 − ε)xi + εyi]∥ε + 1
+n
+n
+�
+i=1
+∥λ′
+(i,3)[(1 − ε)x′
+i + εy′
+i]∥ε
+⩽ 1
+n
+�����
+n
+�
+i=1
+2λ(i,1)
+�
+ai − 1
+2
+�
+− 2λ′
+(i,1)
+�
+a′
+i − 1
+2
+������
+ε
++ 1
+n
+n
+�
+i=1
+�
+λ(i,2) + λ(i,3)
+�
++ 1
+n
+n
+�
+i=1
+�
+λ′
+(i,2) + λ′
+(i,3)
+�
+and, by using (4.1),(4.2),
+⩽ 1
+n
+�����
+n
+�
+i=1
+2λ(i,1)
+�
+ai − 1
+2
+�
+− 2λ′
+(i,1)
+�
+a′
+i − 1
+2
+������
+ε
++ ρ
+⩽ 2
+n
+�����
+n
+�
+i=1
+λ(i,1)ai − λ′
+(i,1)a′
+i
+�����
+ε
++ 1
+n
+n
+�
+i=1
+|λ(i,1) − λ′
+(i,1)|∥1∥ε + ρ
+⩽ 2
+n
+�����
+n
+�
+i=1
+λ(i,1)ai − λ′
+(i,1)a′
+i
+�����
+ε
++ (3 − 2ε)
+2(1 − ε)ρ.
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+33
+Now,
+�����
+n
+�
+i=1
+λ(i,1)ai − λ′
+(i,1)a′
+i
+�����
+ε
+⩽
+�����
+n
+�
+i=1
+(λ(i,1) − 1)ai
+�����
+ε
++
+�����
+n
+�
+i=1
+ai − a′
+i
+�����
+ε
++
+�����
+n
+�
+i=1
+(λ′
+(i,1) − 1)a′
+i
+�����
+ε
+⩽
+1
+1 − ε
+�����
+n
+�
+i=1
+ai − a′
+i
+����� +
+n
+�
+i=1
+1
+1 − ε|λ(i,1) − 1|∥ai∥ +
+n
+�
+i=1
+1
+1 − ε|λ′
+(i,1) − 1|∥a′
+i∥
+⩽
+1
+1 − ε
+�����
+n
+�
+i=1
+ai − a′
+i
+����� +
+1
+1 − εnρ.
+(In the previous estimate observe that ∥ai∥ ⩽ 1 and ∥a′
+i∥ ⩽ 1 since ai, a′
+i ∈ K0 ⊆ Bker(δ0) ⊆ Bε).
+Hence,
+(4.3)
+∥x − y∥ε ⩽
+2
+1 − ε
+�����
+1
+n
+n
+�
+i=1
+ai − a′
+i
+����� + (7 − 2ε)
+2(1 − ε)ρ.
+Now, in order to prove that the previous norm is small we will prove that both elements 1
+n
+�n
+i=1 ai, 1
+n
+�n
+i=1 a′
+i
+are elements of 1
+n
+�n
+i=1 Si, which has small diameter. To this end, note that
+x∗
+i
+�
+2λ(i,1)
+�
+ai − 1
+2
+�
++ 2λ(i,2)
+�
+−bi + 1
+2
+�
++ λ(i,3)[(1 − ε)xi + εyi]
+�
+> 2
+�
+1 − δ − x∗
+i
+�1
+2
+��
++ ρ
+2∥x∗
+i ∥,
+for every i ∈ {1, . . . , n}. Then,
+x∗
+i
+�
+2λ(i,1)
+�
+ai − 1
+2
+��
++ 1
+2ρ∥x∗
+i ∥ ⩾ x∗
+i
+�
+2λ(i,1)
+�
+ai − 1
+2
+��
++ λ(i,2)∥x∗
+i ∥ + λ(i,3)∥x∗
+i ∥
+⩾ x∗
+i
+�
+2λ(i,1)
+�
+ai − 1
+2
+��
++ λ(i,2)∥x∗
+i ∥ε + λ(i,3)∥x∗
+i ∥ε
+⩾ x∗
+i
+�
+2λ(i,1)
+�
+ai − 1
+2
+�
++ 2λ(i,2)
+�
+−bi + 1
+2
+�
++ λ(i,3)[(1 − ε)xi + εyi]
+�
+.
+We have that
+x∗
+i
+�
+2λ(i,1)
+�
+ai − 1
+2
+��
+> 2
+�
+1 − δ − x∗
+i
+�1
+2
+��
+,
+and hence
+x∗
+i (λ(i,1)ai) > 1 − δ − (1 − λ(i,1))x∗
+i
+�1
+2
+�
+⩾ 1 − δ − 1
+2ρ∥x∗
+i ∥.
+We recall that δ+ 1
+2ρ∥x∗
+i ∥ < �δ, so x∗
+i (λ(i,1)ai) > 1−�δ. It follows that x∗
+i (ai) > 1−�δ. Then, ai ∈ K0∩Si
+and, similarly, we get that a′
+i ∈ K0 ∩ Si, for every i = 1, . . . , n. Therefore,
+1
+n
+n
+�
+i=1
+ai, 1
+n
+n
+�
+i=1
+a′
+i ∈ 1
+n
+n
+�
+i=1
+Si.
+
+34
+MART´IN, PERREAU, AND RUEDA ZOCA
+Since the diameter of 1
+n
+�n
+i=1 Si is less than 1
+4(1 − ε)γ, we deduce that 1
+n∥ �n
+i=1 ai − a′
+i∥ < 1
+4(1 − ε)γ.
+Finally, we conclude from (4.3) and the above estimate that ∥x − y∥ε ⩽ γ. Hence, the set C :=
+1
+n
+�n
+i=1 Ui has diameter at most γ for the norm ∥ · ∥ε.
+Now, Bourgain’s lemma (see Lemma 2.2) ensures the existence of a convex combination of slices
+�pi
+j=1 αijTij ⊆ Ui for every 1 ⩽ i ⩽ n. Using this fact, we will find a convex combination of slices of
+B of diameter smaller than γ +
+4ρ2
+(1− ρ2
+ε )ε and such that every slice contains points of 2(K0 − 1
+2 ). Since
+ρ and γ can be taken as small as we wish, we will be done. In order to do so, fix 1 ⩽ i ⩽ n and define
+Ai :=
+�
+j ∈ {1, . . . , pi}: Tij ∩
+�
+2K0 − 1
+2
+�
+= ∅
+�
+;
+Bi := {1, . . . , pi} \ Ai.
+Given xij ∈ Tij we have that, for j ∈ Ai, that δ0(xij) ⩾ −1 + ε by the definition of the unit ball Bε.
+Since �pi
+j=1 αijxij ∈ �pi
+j=1 αijTij ⊆ Ui we derive −1 + ρ2 > δ0
+��pi
+j=1 αijxij
+�
+. Hence
+−1 + ρ2 >
+�
+j∈Ai
+αijδ0(xij) +
+�
+i∈Bi
+αijδ0(xij) ⩾ (−1 + ε)
+�
+j∈Ai
+αij −
+�
+j∈Bi
+αij = −1 + ε
+�
+j∈Ai
+αij.
+From the above inequality we infer that �
+j∈Ai αij < ρ2
+ε holds for every 1 ⩽ i ⩽ n. Now, we set
+Λi := �
+j∈Bi λij, which belongs to the interval [1 − ρ2
+ε , 1] for 1 ⩽ i ⩽ n and set
+D := 1
+n
+n
+�
+i=1
+�
+j∈Bi
+αij
+ΛI
+Tij.
+Observe that D is a convex combination of slices of Bε since every Tij is a slice of Bε and since
+1
+n
+n
+�
+i=1
+�
+j∈BI
+αij
+Λi
+αij = 1.
+We claim that D ⊆ C +
+2
+1− ρ2
+ε
+ρ2
+ε Bε. This is enough to finish the proof because the above condition
+implies that
+diam (D) ⩽ diam (C) +
+4
+1 − ρ2
+ε
+ρ2
+ε ⩽ γ +
+4
+1 − ρ2
+ε
+ρ2
+ε .
+So let us prove the above inclusion. Take z := 1
+n
+�n
+i=1
+�
+j∈Bi
+αij
+Λi xij ∈ D for certain xij ∈ Tij. Write
+z′ := 1
+n
+�n
+i=1
+�
+j∈Bi αijxij. Then
+|z − z′| ⩽ 1
+n
+n
+�
+i=1
+�
+j∈Bij
+����1 − 1
+Λi
+���� αij|xij| <
+1
+1 − ρ2
+ε
+ρ2
+ε .
+On the other hand, for 1 ⩽ i ⩽ n and j ∈ Ai take xij ∈ Tij. Define
+z′′ := 1
+n
+n
+�
+i=1
+pi
+�
+j=1
+αijxij ∈ 1
+n
+n
+�
+i=1
+pi
+�
+j=1
+αijTij ⊆ 1
+n
+n
+�
+i=1
+Ui = C.
+Moreover, we have
+|z′ − z′′| ⩽ 1
+n
+n
+�
+i=1
+�
+j∈Ai
+αij < ρ2
+ε .
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+35
+Consequently z = z′′ + (z − z′′) ∈ C +
+2
+1− ρ2
+ε
+ρ2
+ε Bε since
+|z − z′′| ⩽ |z − z′| + |z′ − z′′| <
+1
+1 − ρ2
+ε
+ρ2
+ε + ρ2
+ε <
+2
+1 − ρ2
+ε
+ρ2
+ε .
+□
+4.7. A summary of relations between the properties. Figure 2 below is an scheme which
+complements Figure 1 with the counterexamples following from known results and from the results in
+this section.
+ccs Daugavet
+ccs ∆
+super Daugavet
+super ∆
+∆
+Daugavet
+?
+/
+(d)
+/
+(a)
+/
+(b)
+/
+(e)
+/
+(c)
+/
+(f)
+Figure 2. Scheme of all relations between the diametral notions
+Let us list the corresponding counterexamples.
+(a) The example in Subsection 4.6.
+(b) The example in Subsection 4.5 negates this implication in the strongest possible way.
+(c) Any of the elements in DB in Paragraph 4.3.3. They also show directly that ∆-points are not
+necessarily ccs ∆-points.
+(d) Every element of the unit sphere of the space X given in Paragraph 4.3.2 is ccs ∆-point but
+not Daugavet point. Another example is the molecule m0,q of Example 4.17.
+(e) In X = C[0, 1]⊕2 C[0, 1], every element in the unit sphere is super ∆-point (Proposition 3.25);
+but X contains no Daugavet point (Proposition 3.27). Also, every element of the unit sphere
+of the space X given in Paragraph 4.3.2 is super ∆-point but not Daugavet point.
+(f) Any of the elements in DB in Paragraph 4.3.3.
+5. Diametral-properties for elements of the open unit ball
+As mentioned in Section 2, the DSD2P is equivalent to the Daugavet property by [34], but the
+ccs ∆-points on the unit sphere of a Banach space do not characterize the DSD2P, but the restricted
+DSD2P, which is not equivalent to the Daugavet property (see Paragraph 4.3.2). Actually, the elements
+in the open unit ball play a decisive role in the proof in [34] of the equivalence between the DSD2P
+and the Daugavet property. Our objective here is to introduce and study the diametral notions for
+interior points, providing interesting applications, and to investigate the behavior of Daugavet- and
+∆-elements on rays in the unit ball of a given Banach space.
+The definition of the Daugavet notions for elements in the open unit ball is the natural extension
+of the definitions for elements of norm one given in Definition 2.5.
+
+36
+MART´IN, PERREAU, AND RUEDA ZOCA
+Definition 5.1. Let X be a Banach space and let x ∈ BX. We say that
+(1) x is a Daugavet point if supy∈S ∥x − y∥ = ∥x∥ + 1 for every slice S of BX,
+(2) x is a super Daugavet point if supy∈V ∥x − y∥ = ∥x∥ + 1 for every non-empty relatively weakly
+open subset V of BX,
+(3) x is a ccs Daugavet point if supy∈C ∥x − y∥ = ∥x∥ + 1 for every ccs C of BX.
+It turns out that the existence of a non-zero Daugavet kind element actually forces the whole ray
+to which it belongs to be composed of similar elements.
+Proposition 5.2. Let X be a Banach space, and let x ∈ SX. The following assertions are equivalent.
+(1) x is a Daugavet- (resp. super Daugavet-, resp. ccs Daugavet-) point.
+(2) rx is a Daugavet- (resp. super Daugavet-, resp. ccs Daugavet-) point for every r ∈ [0, 1].
+(3) rx is a Daugavet- (resp. super Daugavet- , resp. ccs Daugavet-) point for some r ∈ (0, 1).
+Let us recall the following elementary but very useful result from [34] due to Kadets.
+Lemma 5.3 ([34, Lemma 2.2]). Let X be a normed space. If x, y ∈ X and ε > 0 satisfies that
+∥x + y∥ > ∥x∥ + ∥y∥ − ε,
+then for every a, b > 0, it is satisfied that
+∥ax + by∥ > a ∥x∥ + b ∥y∥ − max{a, b}ε.
+Proof of Proposition 5.2. We will only do the proof for Daugavet points, being the other cases com-
+pletely analogous. So let us first assume that x is a Daugavet point. Take r ∈ [0, 1], ε > 0, and S
+a slice of BX. Then, there exists y ∈ S such that ∥x − y∥ > 2 − ε. In particular, ∥y∥ > 1 − ε. As
+∥y∥ ⩽ 1,
+∥x − y∥ > 2 − ε ⩾ ∥x∥ + ∥y∥ − ε.
+It follows from Lemma 5.3 that
+∥rx − y∥ > ∥rx∥ + ∥y∥ − ε > ∥rx∥ + 1 − 2ε.
+Hence, rx is also a Daugavet point.
+Now, let us assume that rx is a Daugavet point for some r ∈ (0, 1). Again take ε > 0 and S slice
+of BX, and pick y ∈ S such that ∥rx − y∥ > ∥rx∥ + 1 − rε. In particular, ∥y∥ > 1 − rε. As ∥y∥ ⩽ 1,
+we have
+∥rx − y∥ > ∥rx∥ + ∥y∥ − rε.
+Hence, by Lemma 5.3, we get that
+∥x − y∥ > ∥x∥ + ∥y∥ − ε > 2 − (1 + r)ε
+and so x is a Daugavet point.
+□
+As mentioned in the discussion preceding Proposition 3.12, the presence of a ccs Daugavet point in
+a given Banach space forces the space to satisfy the SD2P. This can now be viewed as a consequence
+of the previous proposition and the following immediate reformulation of [41, Theorem 3.1].
+Proposition 5.4 ([41, Theorem 3.1]). Let X be a Banach space. Then, X has the SD2P if and only
+if 0 is a ccs Daugavet point.
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+37
+Note that c0 has the SD2P but has no non-zero Daugavet points (use Proposition 4.1, for instance).
+Compare Proposition 5.4 with the following obvious remark.
+Remark 5.5. A Banach space X is infinite-dimensional if and only if 0 is a super Daugavet point.
+We also mention that 0 is always a Daugavet point (in finite or infinite dimension), as every slice
+of the unit ball has to intersect the unit sphere.
+Let us also point out that [41, Theorem 3.1] admits the following scaled version.
+Proposition 5.6. Let X be a Banach space, and let r ∈ (0, 1]. Then, the following assertions are
+equivalent.
+(1) Every ccs of BX has diameter greater than or equal to 2r.
+(2) sup{∥x∥: x ∈ C} ⩾ r for every ccs C of BX.
+(3) sup{∥x∥: x ∈ D} ⩾ r for every symmetric ccs D of BX (so containing 0).
+Proof. Suppose (1) holds. Then, for any given ccs C of BX, and for any fixed ε > 0, there exists
+x, y ∈ C such that ∥x − y∥ > 2r − 2ε. In particular, it follows that ∥x∥ > r − ε or ∥y∥ > r − ε, giving
+(2). (2)⇒(3) is immediate. Suppose that (1) fails, that is, that there exists a ccs C of BX and ε > 0
+such that diam (C) ⩽ 2r − 2ε. We consider the ccs D of BX given by D := 1
+2(C − C). Then D is
+symmetric, and for every u := x−y
+2
+and u′ := x′−y′
+2
+in D we have ∥u − u′∥ =
+��� x+y′
+2
+− x′+y
+2
+���. Now,
+x, x′, y, y′ belong to C, and C is convex, so x+y′
+2
+and x′+y
+2
+do also belong to C, so ∥u − u′∥ ⩽ 2r − 2ε.
+Being D symmetric, it implies that D ⊂ (r − ε)BX, hence (3) fails.
+□
+The following is a nice consequence of the proposition above outside the diametral notions.
+Corollary 5.7. Let X be a Banach space. Then, BX contains ccs of arbitrarily small diameter if and
+only if 0 is a strongly regular point of BX.
+Now let us consider the ∆ notions for points of the open unit ball which are just the adaptation of
+the notions given in Definition 2.4.
+Definition 5.8. Let X be a Banach space and let x ∈ BX. We say that
+(1) x is a ∆-point if supy∈S ∥x − y∥ = ∥x∥ + 1 for every slice S of BX containing x,
+(2) x is a super ∆-point if supy∈V ∥x − y∥ = ∥x∥ + 1 for every non-empty relatively weakly open
+subset V of BX containing x,
+(3) x is a ccs ∆-point if supy∈C ∥x − y∥ = ∥x∥ + 1 for every slice ccs C of BX containing x.
+With this definitions in hands, we may get an improvement of Proposition 5.4 from Proposition 5.6.
+Corollary 5.9. Let X be a Banach space. Then, X has the SD2P if and only if 0 is ccs ∆-point.
+Compare the previous corollary with the following obvious remark which is analogous to Remark 5.5.
+Remark 5.10. A Banach space X is infinite-dimensional if and only if 0 is a super ∆-point.
+Observe that the definition of ccs ∆-points for elements in BX gives a localization of the DSD2P,
+that is, X has the DSD2P (and hence the Daugavet property [34]) if and only if all the elements of
+BX are ccs ∆-points. Recall that the DSD2P is not equivalent to the restricted DSD2P (meaning that
+all points in SX are ccs ∆-points), see Paragraph 4.3.2.
+
+38
+MART´IN, PERREAU, AND RUEDA ZOCA
+The following result is a localization of Kadets’ theorem [34] on the equivalence of the DSD2P and
+the DPr.
+Theorem 5.11. Let X be a Banach space and let x ∈ SX. If rx is a ccs ∆-point for every r ∈ (0, 1),
+then x is a ccs Daugavet point. Moreover, it is enough that inf{r ∈ (0, 1): rx is a ccs ∆-point} = 0.
+Proof. Fix a ccs C of BX and ε > 0. Since ˜C := 1
+2(C − C) is also a ccs of BX and since 0 ∈ ˜C is a
+norm interior point of ˜C by [41, Proposition 2.1], we have that rx belongs to ˜C for every r ∈ (0, δ)
+for some δ > 0. By hypothesis, there is r > 0 such that rx is a ccs ∆-point and rx ∈ ˜C. So there
+exists y ∈ ˜C such that ∥rx − y∥ > r + 1 − rε. Then if we write y := y1 − y2 with y1, y2 ∈ C, we have
+∥rx − y1∥ > r +1−rε or ∥rx − y2∥ > r +1−rε by the triangle inequality; in particular, ∥y1∥ > 1−rε
+and ∥y2∥ > 1 − rε. In both cases, we have that there is y ∈ C such that ∥rx − y∥ > r∥x∥ + ∥y∥ − rε
+and it follows from Lemma 5.3 that
+∥x − y∥ > ∥x∥ + ∥y∥ − ε > 2 − (1 + r)ε > 2 − 2ε.
+□
+At this point, it is natural to ask whether an equivalent formulation of Proposition 5.2 is valid for
+some of the various ∆-notions. For ccs ∆-points, the answer is negative as follows from Theorem 5.11
+and, for instance, the example in Paragraph 4.3.2. Another, maybe simpler, example showing that is
+the following one.
+Example 5.12. Let us assume that a positive measure µ admits an atom of finite measure and also
+has a non-empty non-atomic part. Then, the real space L1(µ) contains no ccs Daugavet point by
+Proposition 4.12. However, it contains elements in the unit sphere which are ccs ∆-points and super
+Daugavet points by Theorem 4.8 and Proposition 4.9. In particular, as a consequence of Theorem 5.11
+and of Proposition 5.2, there must exist f in the unit sphere which is a ccs ∆-point and t ∈ (0, 1) such
+that tf is not a ccs ∆-point but it is a super ∆-point.
+For ∆-points, we have the following result.
+Proposition 5.13. Let X be a Banach space, and let x ∈ SX. If x is a ∆-point, then rx is a ∆-point
+for every r ∈ (0, 1).
+Proof. Let us assume that x is a ∆-point and let us fix r ∈ (0, 1).
+Take ε > 0 and a slice S of
+BX containing rx. Now, either x belongs to S or −x belongs to S. In the first case, we can find
+y ∈ S such that ∥x − y∥ ⩾ 2 − ε, and using Lemma 5.3, we get ∥rx − y∥ ⩾ ∥rx∥ + 1 − 2ε. Else,
+∥rx − (−x)∥ = r + 1 = ∥rx∥ + 1 and we are done.
+□
+For super ∆-points, it is currently quite obscure whether they behave like ∆-points up on rays.
+6. Kuratowski measure and large diameters
+Let M be a metric space.
+The Kuratowski measure of non-compactness α(A) of a non-empty
+bounded subset A of M is defined as the infimum of all real numbers ε > 0 such that A can be covered
+by a finite number of subsets of M of diameter smaller than or equal to ε.
+From the definition, we clearly have α(A) = 0 if and only if A is totally bounded (a.k.a. precompact).
+It follows that every complete subset A of M with α-measure 0 is compact, and in particular, if M is
+a complete metric space, that α(A) = 0 if and only if A is compact, where A stands for the closure of
+the set A. The α-measure can be thus seen as a way to measure how far a given (non-empty) bounded
+and closed subset of M is from being a compact space. It was introduced by C. Kuratowski in [38] in
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+39
+order to provide a generalization of the famous intersection theorem from Cantor. A general theory
+on measures of non-compactness was later developed, and it turned out to provide important results
+in metric fixed point theory, and in particular to have applications in functional equations or optimal
+control. We refer e.g. to [12] for an introduction to the topic and for more precise applications.
+Observe that A ⊆ B implies α(A) ⩽ α(B), and that α(A) = α(A). Also note that α(A ∪ B) =
+max{α(A), α(B)} for every non-empty bounded subsets A, B of M. Furthermore, if M = X is a
+normed space, then α is known to enjoy additional useful properties: it is symmetric, translation
+invariant, positively homogeneous, sub-additive, and satisfies α(co A) = α(A). The α-measure has
+proved to be a powerful tool for the study of the geometry of Banach spaces and we refer e.g. to the
+works [46], [47] and [44] in connection with property (α), with drop property, and with an isomorphic
+characterization of reflexive Banach spaces.
+From the definition it is clear that the Kuratowski measure of A is smaller than or equal to its
+diameter. Obviously, equality does not always hold, but a fruitful relationship between the notion
+of ∆-points and the Kuratowski measure of slices was discovered in [5] and completed in [52]. In
+particular, the following result was obtained (see [52, Corollary 2.2]).
+Theorem 6.1. Let X be a Banach space and let x ∈ SX. If x is a ∆-point, then α(S) = 2 for every
+slice S of BX containing x. Besides, α(S(x, δ; BX∗)) = 2 in BX∗ for every δ > 0.
+Observe that the converse does not hold in general, as the following example shows.
+Example 6.2. Consider X := L1([0, 1]) ⊕∞ ℓ1. It follows that both X and X∗ enjoy the SD2P [15,
+Remark 2.6], so Theorem 6.3 below implies that given any slice S = S(x∗, δ; BX) we have α(S) = 2
+and the same holds for the slices in the dual. However, there are points which are not ∆-points because
+X fails the DLD2P [30, Theorem 3.2] since ℓ1 fails it, so it remains to take any point x ∈ SX which
+is not a ∆-point to get the desired counterexample.
+Observe that the connection between having big slices in diameter and having big slices in Kura-
+towski index goes beyond Theorem 6.1. The following result was first pointed out in [20].
+Theorem 6.3 ([20, Proposition 3.1]). Let X be a Banach space and let β ∈ (0, 2]. The following
+assertions are equivalent.
+(1) Every slice of BX has diameter greater than or equal to β.
+(2) Every slice of BX has Kuratowski measure greater than or equal to β.
+In this section, we aim to prove analogues to this result for relative weakly open subsets, as well
+as for convex combinations of slices or of weakly open sets, and to extend Veeorg’s result to super
+∆-points and ccw ∆-points.
+6.1. Kuratowski measure and diameter two properties. The analogue to Theorem 6.3 for
+non-empty weakly open subsets is the following.
+Theorem 6.4. Let X be a Banach space and let β ∈ (0, 2]. The following assertions are equivalent.
+(1) Every non-empty relatively weakly open subset of BX has diameter greater than or equal to β.
+(2) Every non-empty relatively weakly open subset of BX has Kuratowski measure greater than or
+equal to β.
+Proof. (2)⇒(1) is immediate, so let us prove (1)⇒(2). To this end, fix β ∈ (0, 2] and assume that
+every non-empty relatively weakly open subset of BX has diameter greater than or equal to β. Then
+
+40
+MART´IN, PERREAU, AND RUEDA ZOCA
+pick ε > 0, and let us prove by induction on n that for every non-empty relatively weakly open subset
+W of BX and for every finite collection C1, . . . , Cn of subsets of X with diam (Ci) ⩽ β − ε for every
+i, we have that W ̸⊂
+n�
+i=1
+Ci.
+For n = 1, it is clear since by assumption diam (W) ⩾ β > β − ε for every non-empty relatively
+weakly open subset W of BX.
+So assume that the result is true for every non-empty relatively weakly open subset W of BX
+and for every collection of n sets, and let us prove the result for collections of n + 1 sets. To this
+end, consider C1, . . . , Cn, Cn+1 be subsets of X with diam (Ci) ⩽ β − ε for every i. Observe that
+diam (Ci) = diam (Ci
+w) ⩽ β − ε by w-lower semicontinuity of the norm of X, so that we may and do
+assume that Ci is weakly closed for every i.
+Observe that by the case n = 1 we have that W ̸⊂ Cn+1, which means that W \ Cn+1 is non-
+empty. Moreover, it is a weakly open subset of BX since Cn+1 is assumed to be weakly closed, and by
+induction hypothesis we conclude that W\Cn+1 ̸⊂
+n�
+i=1
+Ci. In particular W ̸⊂
+n+1
+�
+i=1
+Ci and the theorem
+is proved.
+□
+Next, let us establish the analogue Theorem 6.3 for convex combinations of slices. To this end,
+observe that by Bourgain lemma (see Lemma 2.2) every convex combination of non-empty relatively
+weakly open subsets of BX contains a convex combination of slices of BX. This assertion makes valid
+the following lemma which allows us to focus our attention in convex combination of weakly open
+subsets.
+Lemma 6.5. Let X be a Banach space and r > 0.
+(1) The following are equivalent:
+(a) Every convex combination of slices of BX has diameter greater than or equal to r.
+(b) Every convex combination of non-empty relatively weakly open subsets of BX has diameter
+greater than or equal to r.
+(2) The following are equivalent:
+(a) α(C) ⩾ r holds for every convex combination C of slices of BX.
+(b) α(D) ⩾ r holds for every convex combination D of non-empty relatively weakly open
+subsets of BX.
+Now we are able to give the following result.
+Theorem 6.6. Let X be a Banach space and let β ∈ (0, 2]. The following assertions are equivalent.
+(1) Every convex combination of slices of BX has diameter greater than or equal to β.
+(2) Every convex combination of slices of BX has Kuratowski measure greater than or equal to β.
+Proof. (2)⇒(1) is immediate, so let us prove (1)⇒(2). To this end, fix β ∈ (0, 2] and assume that every
+convex combination of non-empty relatively weakly open subsets of BX has diameter greater than or
+equal to β. Then pick ε > 0, and let us prove by induction on n that for every D convex combination
+of non-empty relatively weakly open subsets of BX and for every finite collection C1, . . . , Cn of subsets
+of X with diam (Ci) ⩽ β − ε for every i, we have that D ̸⊂
+n�
+i=1
+Ci.
+For n = 1 it is clear since by assumption diam (D) ⩾ β > β − ε for every convex combination of
+non-empty relatively weakly open subsets of D of BX.
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+41
+Assume by inductive step that the result stands for n.
+Now pick D convex combination of non-empty relatively weakly open subsets of BX and a finite
+collection C1, . . . , Cn+1 of subsets of X with diam (Ci) ⩽ β − ε for every i. We can assume as in the
+proof of Theorem 6.4 that every Ci is weakly closed. Write D = �k
+i=1 λiWi. Observe that by the case
+n = 1 we have that D ̸⊆ Cn+1, so there exists z ∈ D \ Cn+1. Since z ∈ D we can write z = �k
+i=1 λixi
+where xi ∈ Wi holds for every 1 ⩽ i ⩽ k. Moreover, since z = �k
+i=1 λixi /∈ Cn+1, this means that
+z = �k
+i=1 λixi ∈ X \ Cn+1, and the latter is a weakly open set. By a weak-continuity argument of
+the sum we can find weakly open subsets Vi of BX, with xi ∈ Vi for every 1 ⩽ i ⩽ k, satisfying that
+z = �k
+i=1 λixi ∈ �n
+i=1 λiVi ⊆ X \ Cn+1. Up to taking smaller Vi, we can assume Vi ⊆ Wi for every
+i. Now call ˜D := �k
+i=1 λiVi, which is a convex combination of weakly open subsets of BX. By the
+inductive step we get that ˜D ̸⊆
+n�
+j=1
+Cj, so there exists y ∈ ˜D with y /∈ Cj for 1 ⩽ j ⩽ n. Observe that
+the condition Vi ⊆ Wi implies ˜D ⊆ D, so y ∈ D indeed. Moreover, ˜D ⊆ X \Cn+1 implies in particular
+y /∈ Cn+1. This implies that y ∈ D \
+n+1
+�
+i=1
+Ci, which is precisely what we wanted to prove.
+□
+6.2. Kuratowski measure and ∆-notions. We now prove an analogue to Theorem 6.1 for super
+∆-points.
+Theorem 6.7. Let X be a Banach space and let x ∈ SX be a super ∆-point. Then every non-empty
+relatively weakly open subset W of BX containing x satisfies that α(W) = 2.
+The proof will be an obvious consequence of the following result.
+Proposition 6.8. Let X be a Banach space, x ∈ SX be a super ∆ point, and W be a weakly open
+subset of BX such that x ∈ W. Then, for every ε > 0, there exists a sequence {xn} ⊆ W such that
+∥xi − xj∥ > 2 − ε holds for every i ̸= j.
+Proof. Set ε > 0. Let us construct by induction a sequence {xn} satisfying that ∥x − xi∥ > 2 − ε
+2 and
+such that ∥xi − xj∥ > 2 − ε for i ̸= j.
+Using that x is a super ∆ point select, by the definition of super ∆, a point x1 ∈ W with ∥x−x1∥ >
+2 − ε
+2.
+Now, assume that x1, . . . , xn have been constructed and let us construct xn+1. By the properties
+defining the sequence observe that, given 1 ⩽ i ⩽ n, we have ∥x−xi∥ > 2− ε
+2, so we can find gi ∈ SX∗
+with Re gi(x − xi) > 2 − ε
+2, which implies Re gi(x) > 1 − ε
+2 and Re gi(xi) < −1 + ε
+2. Consequently
+x ∈ V := W ∩
+n�
+i=1
+S
+�
+gi, ε
+2
+�
+,
+which is a weakly open set. Since x is super ∆ we can find xn+1 ∈ V such that ∥x−xn+1∥ > 2− ε
+2. In
+order to finish the construction we only must to prove that ∥xi−xn+1∥ > 2−ε holds for every 1 ⩽ i ⩽ n.
+But this is clear because, given 1 ⩽ i ⩽ n, the condition xn+1 ∈ V implies that Re gi(xn+1) > 1 − ε
+2,
+so
+∥xn+1 − xi∥ ⩾ Re gi(xn+1 − xi) > 1 − ε
+2 + 1 − ε
+2 = 2 − ε,
+and the proof is finished.
+□
+Note that a similar statement than Theorem 6.7 can be established for ccw ∆ points.
+
+42
+MART´IN, PERREAU, AND RUEDA ZOCA
+Theorem 6.9. Let X be a Banach space and let x ∈ SX be a ccw ∆-point. Then every non-empty
+convex combination D of relatively weakly open subsets of BX containing x satisfies that α(D) = 2.
+As in the previous case, the proof follows directly from the next result.
+Proposition 6.10. Let X be a Banach space, x ∈ SX be a ccw ∆ point, and D a ccw of BX such
+that x ∈ D. Then, for every ε > 0, there exists a sequence {xn} ⊆ D such that ∥xi − xj∥ > 2 − ε holds
+for every i ̸= j.
+Proof. Set ε > 0. Write D := �k
+i=1 λiWi with λi ̸= 0 for every i. Set δ :=
+ε
+2 min1⩽i⩽k λi . Let us construct
+by induction a sequence {xn} ⊆ D satisfying that ∥x − xi∥ > 2 − δ and such that ∥xi − xj∥ > 2 − ε
+for i ̸= j. Using that x is a ccw ∆ point select, by the definition of ccw ∆, a point x1 ∈ D with
+∥x − x1∥ > 2 − δ.
+Now assume that x1, . . . , xn have been constructed and let us construct xn+1. We can write x =
+�k
+j=1 λjxj and xi := �k
+j=1 λjxi
+j as being elements of D.
+By the properties defining the sequence, observe that, given 1 ⩽ i ⩽ n we have ∥x − xi∥ > 2 − δ, so
+we can find gi ∈ SX∗ with
+Re gi(x − xi) =
+k
+�
+j=1
+λj Re gi(xj − xi
+j) > 2 − δ = 2 −
+ε
+2 min1⩽j⩽n λj
+.
+A convexity argument implies that Re gi(xj − xi
+j) > 2 − ε
+2 holds for every 1 ⩽ j ⩽ k, which implies
+that
+Re gi(xj) > 1 − ε
+2 and
+Re gi(xi
+j) < −1 + ε
+2.
+Observe that
+xj ∈ Vi := Wi ∩
+n�
+i=1
+S
+�
+gi, ε
+2
+�
+,
+which is a weakly open set.
+Since x is a ccw ∆-point and x ∈ �k
+j=1 λjVj, we can find a point
+xn+1 = �k
+j=1 λjzj ∈ �k
+j=1 λjVj ⊆ D such that ∥x − xn+1∥ > 2 − δ. In order to finish the construction
+we only must to prove that ∥xi − xn+1∥ > 2 − ε holds for every 1 ⩽ i ⩽ n. Given 1 ⩽ j ⩽ k, the
+condition zj ∈ Vj implies Re gi(zj) > 1 − ε
+2. On the other hand, Re gi(xi
+j) < −1 + ε
+2, so
+∥xn+1 − xi∥ ⩾ Re gi(x − xi) =
+k
+�
+j=1
+λj Re gi(zj − xi
+j) > (2 − ε)
+k
+�
+j=1
+λj = 2 − ε,
+and the proof is finished.
+□
+7. Commented open questions
+The only implications between properties which is not known to hold or not is the following one
+(see Figure 2 in page 35).
+Question 7.1. Let X be a Banach space and let x ∈ SX be a ccs ∆-point. Is x a super ∆-point?
+Let us give some comments on this question. On the one hand, it may look that the answer is
+positive by Bourgain’s lemma (Lemma 2.2), but this lemma does not say that, in general, given an
+element x of a relative weak open subset W of BX, there is a convex combination of slices of BX
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+43
+contained in W and containing x. The later happens when x ∈ co(pre-ext (BX)) (see Remark 2.3) so,
+the answer to Question 7.1 is positive in this case. On the other hand, a possible counterexample to
+this problem could be the molecules in Examples 4.16 or 4.17, which are known to be ccs ∆-points
+and are extreme points but not preserved extreme points (hence they do not belong to the convex hull
+of the set of preserved extreme points). A way to show that these molecules are not super ∆-points
+would be to investigate whether RNP spaces may contains super ∆-points.
+Theorem 3.19 states that real Banach spaces with a one-unconditional basis do neither contain
+super ∆-points nor ccs ∆-points. It is likely that such result also holds true in the complex setting
+since we do believe that the preliminary results from [6] are also valid for complex scalars, provided
+that one works with the suitable notion of one-uncondional bases (for which [6, Proposition 2.3] holds).
+Also, we also expect that the results there can be easily extended to one-uncondional FDDs. Yet,
+since a sharper version of this result was obtained in Proposition 3.20 for super ∆-points in a very
+general setting, it is natural to ask whether improved results could be simultaneously obtained in both
+directions for ccs ∆-points by proving an analogue to Proposition 3.20. So let us ask the following.
+Question 7.2. Let X be a Banach space, and let us assume that there exists a subset A ⊆ F(X, X)
+satisfying that sup
+�
+∥Id − T∥ : T ∈ A
+�
+< 2 and that for every ε > 0 and every x ∈ X, there exists
+T ∈ A such that ∥x − Tx∥ < ε. Can X contain a ccs ∆-point?
+A negative answer to this question would be interesting, since it would provide an example of a ccs
+∆-point that is not a super ∆-point, hence a negative answer to Question 7.1.
+Another interesting question could be if a point of continuity could be a ccs ∆-point.
+Let us
+formalize the questions.
+Question 7.3. Let X be a Banach space.
+(1) Does X fail the RNP (or even the CPCP) if contains a super ∆-point or a super Daugavet
+point?
+(2) Is it possible for a point of continuity being a ccs ∆-point?
+The surprising examples given in Section 4 shows that the mere existence of some diametral notions
+(but ccs Daugavet points) on a Banach space does not imply that the whole space has any diameter
+two property nor the Daugavet property. Our question here is how many diametral points has to
+contain a Banach space to have any diameter two property or the Daugavet property or fails to have
+the RNP or one-unconditional basis.
+Question 7.4. How big can be the set of Daugavet points, super Daugavet points, ∆-points, super
+∆-points, or ccs ∆-points in a Banach space with the Radon-Nikod´ym property, or with the CPCP,
+or being strongly regular, or having one-unconditional basis?
+Concerning isometric consequences of the existence of diametral points, there are some recent results
+showing that a Banach space containing a ∆-point cannot be uniformly non-square [5] or even locally
+uniformly non-square [37], or asymptotic uniformly smooth [5, 52].
+Also, a Banach space having
+an unconditional basis with suppression-unconditional constant less that 2 cannot contains super ∆-
+points and a Banach space containing a ccs Daugavet point has the SD2P. Taking into account that
+it is not known if there exists an stictly convex Banach space with the Daugavet property (see [33,
+Section 5]), the following question makes sense. Recall that Paragraph 4.3.2 shows an example of an
+strictly convex Banach space in which every norm-one element is a ccs ∆-point and a super ∆-point,
+but it does not contain any Daugavet point by the way in which it is constructed.
+
+44
+MART´IN, PERREAU, AND RUEDA ZOCA
+Question 7.5. Is there an strictly convex Banach space containing a Daugavet point?
+In view of Proposition 3.13 and of Theorem 4.14, the following question makes sense.
+Question 7.6. Let X be a Banach space. Suppose that x ∈ ext (BX) is a ∆-point, does this imply
+that x is a ccs ∆-point or a super ∆-point?
+By now, the only isomorphic restriction which is known for a Banach space to contain ∆-points
+or even Daugavet points is that it cannot be finite-dimensional. It would be interesting to find some
+more.
+Question 7.7. Find isomorphic restrictions for a Banach space to contain ∆-points or any of the
+other diametral notions. In particular, is it possible for a reflexive or even super-reflexive Banach
+space to contain ∆-, super ∆-, ccs ∆-, Daugavet or super Daugavet points?
+The results about absolute sums in Subsection 3.2 are not complete in the case of super Daugavet
+points and they are even less clear in the case of ccs notions. Here are two possible questions.
+Question 7.8. Let X, Y be Banach spaces and let N be an absolute sum.
+(1) If N is A-octahedral, x ∈ SX and y ∈ SY are super Daugavet points, is (ax, by) a super
+Daugavet point in X ⊕N Y when a, b satisfy the conditions in the definition of A-octahedrality?
+(2) If N is the ℓ∞-sum, x ∈ SX and y ∈ SY are ccs ∆-points, are the elements of the form (ax, by)
+ccs ∆-points in X ⊕∞ Y for a, b ∈ [0, 1] with max{a, b} = 1?
+It would be also desirable to study the reversed results to those in Subsection 3.2 as it is done in
+[45] for ∆-points and Daugavet points (see the tables in pages 86 and 87 of [45]).
+Question 7.9. Let X, Y be Banach spaces, let N be an absolute sum, x ∈ SX, y ∈ SY , and a, b ⩾ 0
+such that N(a, b) = 1. Discuss what happens with x and y supposing that (ax, by) satisfies any of the
+six diametral notions.
+It maybe the case that some of the arguments given in Subsections 4.1 and 4.2 can be adapted to
+other classes of Banach spaces. We propose some possibilities.
+Question 7.10. Characterize the six diametral notions in uniform algebras, in Lorentz spaces and
+their isometric preduals, and in some vector-valued function spaces as C(K, X) or L∞(µ, X) spaces.
+The relations between the weak-star versions of the diametral points (see Remark 2.6) are not yet
+clear. For instance, the following questions arises.
+Question 7.11. Let X be a Banach space and x ∈ SX.
+(1) Is JX(x) a ccs ∆-point in X∗∗ if x is a ccs ∆-point?
+(2) Is there any relationship between the DD2P in X and the weak-star super ∆-points in SX∗?
+As commented in Remark 4.19, a Banach space X containing a sequence (yn) of super ∆-points
+such that the distance of yn to the set of strongly exposed points of BX is going to zero. But the
+following question remains open.
+Question 7.12. Can a super ∆-point (or even a ∆-point) belong to the closure of the set of denting
+points?
+
+DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE
+45
+The answer to the next question on the behaviour of ∆- and super ∆-points in rays is still unknown,
+as we commented in Section 5.
+Question 7.13. Let X be a Banach space and let x ∈ SX.
+(1) If rx is a ∆-point for some 0 < r < 1, does this imply that x is a ∆-point?
+(2) If rx is a super ∆-point for some 0 < r < 1, does this imply that x is a auper ∆-point?
+(3) If x is a super ∆-point, does this imply that rx is a super ∆-point for all 0 < r < 1?
+As we proved in Section 6, every relative weakly open subset which contains a super ∆-point
+(respectively, a ccw ∆-point) has Kuratowski measure 2. Our proofs do not seem to work for convex
+combination of slices, so let us ask the following.
+Question 7.14. If a ccs of the unit ball contains a ccs ∆-point, does it necessarily have maximal
+Kuratowski measure?
+Acknowledgments
+Part of this work was done during the visit of the second named author at the University of Granada
+in September 2022. He wishes to thank his colleagues for the warm welcome he received, and to thank
+all the people that made his visit possible.
+The authors thank Gin´es L´opez-P´erez for fruitful conversations on the topic of the paper, specially
+for providing enlightening ideas in connection with the example of Subsection 4.6.
+The authors also thank Trond A. Abrahamsen, Andr´e Martiny, and Vegard Lima for valuable discus-
+sions on the topic of the paper, and in particular for pointing out the ccs version of [6, Proposition 2.12]
+that is presented in Subsection 3.1.
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+(Mart´ın) Universidad de Granada, Facultad de Ciencias. Departamento de An´alisis Matem´atico, 18071
+Granada, Spain
+ORCID: 0000-0003-4502-798X
+Email address: mmartins@ugr.es
+URL: https://www.ugr.es/local/mmartins
+(Perreau) University of Tartu, Institute of Mathematics and Statistics, Narva mnt 18, 51009 Tartu
+linn, Estonia
+ORCID: 0000-0002-2609-5509
+Email address: yoel.perreau@ut.ee
+(Rueda Zoca) Universidad de Granada, Facultad de Ciencias. Departamento de An´alisis Matem´atico,
+18071 Granada, Spain
+ORCID: 0000-0003-0718-1353
+Email address: abrahamrueda@ugr.es
+URL: https://arzenglish.wordpress.com
+
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+page_content='DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A  BANACH SPACE  MIGUEL MART´IN, YO¨EL PERREAU, AND ABRAHAM RUEDA ZOCA  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We introduce extensions of ∆-points and Daugavet points in which slices are replaced by  relative weakly open subsets (super ∆-points and super Daugavet points) or by convex combinations of  slices (ccs ∆-points and ccs Daugavet points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' These notions represent the extreme opposite to denting  points, points of continuity, and strongly regular points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We first give a general overview on these  new concepts and provide some isometric consequences on the spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As examples: if a Banach space  contains a super ∆-point, then it does not admit an unconditional FDD (in particular, unconditional  basis) with suppression constant smaller than two;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' if a real Banach space contains a ccs ∆-point, then  it does not admit a one-unconditional basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' if a Banach space contains a ccs Daugavet point, then  every convex combination of slices of its unit ball has diameter two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We next characterize the notions in  some classes of Banach spaces showing, for instance, that all the notions coincide in L1-predual spaces  and that all the notions but ccs Daugavet points coincide in L1-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We next remark on some  examples which have previously appeared in the literature and provide some new intriguing examples:  examples of super ∆-points which are as closed as desired to strongly exposed points (hence failing  to be Daugavet points in an extreme way);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' an example of a super ∆-point which is strongly regular  (hence failing to be a ccs ∆-point in the strongest way);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' a super Daugavet point which fails to be a ccs  ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The extensions of the diametral notions to point in the open unit ball and the consequences  on the spaces are also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Last, we investigate the Kuratowski measure of relative weakly open  subsets and of convex combinations of slices in the presence of super ∆-points or ccs ∆-points, as well  as for spaces enjoying diameter 2 properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We conclude the paper with a section on open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Introduction  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Notation and preliminary results  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Characterisations of diametral-notions and implications on the geometry of the ambient  space  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Spaces with a one-unconditional basis and beyond  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Absolute sums  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Examples and counterexamples of diametral elements  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Characterization in C(K)-spaces, L1-preduals, and M¨untz spaces  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Characterization in L1-spaces  22  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Remarks on some examples from the literature  24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A super ∆-point which fails to be a Daugavet point in an extreme way  26  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A super ∆-point which is a strongly regular point  27  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A super Daugavet point which is not ccs ∆-point  28  Date: January 11th, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The first and third named authors were supported by grant PID2021-122126NB-C31 funded by MCIN/AEI/  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13039/501100011033 and “ERDF A way of making Europe”, by Junta de Andaluc´ıa I+D+i grants P20 00255  and FQM-185,  and by “Maria de Maeztu” Excellence Unit IMAG, reference CEX2020-001105-M funded by  MCIN/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13039/501100011033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The second named author was supported by the Estonian Research Council grant  SJD58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='04433v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='FA]  11 Jan 2023  2  MART´IN, PERREAU, AND RUEDA ZOCA  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A summary of relations between the properties  35  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Diametral-properties for elements of the open unit ball  35  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Kuratowski measure and large diameters  38  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Kuratowski measure and diameter two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  39  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Kuratowski measure and ∆-notions  41  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Commented open questions  42  Acknowledgments  45  References  45  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Introduction  It is fair to say that one of the most studied properties of Banach spaces is the Radon-Nikod´ym  property (RNP) because it has shown to be very useful;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' due to the large amount of its geometric,  analytic, and measure theoretic characterisations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' in several fields of Banach space theory such as  representation of bounded linear operators, representation of dual spaces or representation of certain  tensor product spaces (see [18, 21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A famous geometric characterization of the Radon-Nikod´ym property is related to the size of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Recall that a slice of a bounded non-empty subset C of a Banach space X is simply the (nonempty)  intersection of C with a half-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A Banach space X has the RNP if and only if every non-empty  closed and bounded subset of X admits slices of arbitrarily small diameter (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A closely related and equally important geometric property of Banach spaces is the point of con-  tinuity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a Banach space X has the point of continuity property (PCP) if every  non-empty closed and bounded subset of X admits non-empty relatively weakly open subsets of ar-  bitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us emphasize here as an example the striking equivalence between the  Radon-Nikod´ym property and the weak∗ version of the point of continuity property for dual spaces,  and the related characterization of Asplund spaces as preduals of RNP spaces (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In his  proof of the determination of the Radon-Nikod´ym property by subspaces with a finite dimensional  decomposition (FDD) in [17], Bourgain also introduced an important weakening of the point of conti-  nuity property, that he called property “(∗)”, and that is nowadays referred to as the convex point of  continuity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a Banach space X has the convex point of continuity property (CPCP)  if every non-empty closed, convex and bounded subset of X admits non-empty relatively weakly open  subsets of arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In fact, Bourgain implicitly used in his work the notion of strong regularity which, as he showed,  is implied by the CPCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a Banach space X is strongly regular (SR) if every non-empty  closed, convex and bounded subset of X contains convex combinations of slices of arbitrarily small  diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that the convexity of the subset is required in this definition in order to guarantee  that it contains all the convex combinations of its slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It later turned out that strong regularity  had important applications to the famous (still open) question of the equivalence between the Radon-  Nikod´ym property and the Krein-Milman property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a Banach space X has the Krein-  Milman property (KMP) if every non-empty closed, convex and bounded subset C of X admits an  extreme point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The RNP implies the KMP (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [18, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6]), and it follows from [48] that  every strongly regular space with the KMP has the RNP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also recall that it was proved in [29] the  RNP and the KMP are equivalent in dual spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  3  From the definitions it follows that RNP⇒PCP⇒CPCP and it is also known that CPCP⇒SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  None of the above implications reverse (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g [49] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In order to show that  strong regularity is implied by the CPCP, Bourgain made an important geometric observation, namely  that in every non-empty bounded and convex subset of a Banach space X, every non-empty relatively  weakly open subset contains a convex combination of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We will discuss this “Bougain Lemma”  and its applications to the subject of the present paper in more details in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Another classical refinement of the above characterization of the Radon-Nikod´ym property is related  to the notion of denting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a point x0 of a bounded subset C of X is a denting point  of C if there are slices of C containing x0 of arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A Banach space X has the  RNP if and only if every closed, convex and bounded subset contains a denting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Actually, every  nonempty closed, convex and bounded subset C of a Banach space X with the RNP is equal to the  closure of the convex hull of the set of its denting points (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [18, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For the PCP and the CPCP, a similar role is played by points of weak-to-norm continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Given  a bounded subset C of X, we say that a point x0 ∈ C is a point of weak-to-norm continuity (point  of continuity in short) if the identity mapping i: (C, w) −→ (C, τ) is continuous at the point x0 or,  equivalently, if x0 belongs to relatively weakly open subsets of C of arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Note  that a classical result by Lin-Lin-Troyanski [39] establishes that a point x0 ∈ C is a denting point if,  and only if, x0 is simultaneously a point of continuity and a extreme point of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In a space with the  PCP every non-empty closed and bounded subset contains a point of continuity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' and the set of all  points of continuity of a given closed, convex and bounded subset C of a Banach space X with the  CPCP is weakly dense in C (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In relation to strong regularity, a point x0 of a bounded, convex subset C of X is a point of strong  regularity if there are convex combinations of slices of C containing x0 of arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Then the set of all points of strong regularity of a given closed, convex and bounded subset C of a  strongly regular Banach space X is norm dense in C (see [25, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us observe that  points of strong regularity may be in the interior of a set, while denting points (and points of continuity  in the infinite-dimensional case) belong always to the border of the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In [3] the extreme opposite notion to denting point of the unit ball was introduced in the following  sense: an element x in the unit sphere of a Banach space X is a ∆-point if we can find in every slice  of BX containing x points which are at distance from x as close as we wish to the maximal possible  distance in the ball (distance 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A similar yet stronger notion appeared simultaneously in relation to  another quite famous property of Banach spaces, the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a Banach space  X has the Daugavet property (DPr) if the Daugavet equation  (DE)  ∥Id + T∥ = 1 + ∥T∥  holds for every rank-one operator T : X −→ X, where Id denotes the identity operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In this  case, all weakly compact operators also satisfy (DE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We refer the reader to the seminal paper [35]  for background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recent results can be found in [42] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The Daugavet property  admits a beautiful geometric characterization involving slices related to the notion of Daugavet points:  an element x on the unit sphere of a Banach space X is a Daugavet point if in every slice of BX (not  necessarily containing the point x) there are points which are at distance from x as close as we wish to  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' With this definition in mind, [35, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] states that X has the DPr if and only if all elements  in SX are Daugavet points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us comment that the Daugavet property imposes severe restriction  on the Banach space: if X is a Banach space with the DPr, then it fails the RNP and it has no  unconditional basis (actually, it cannot be embedded into a Banach space with unconditional basis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  4  MART´IN, PERREAU, AND RUEDA ZOCA  On the other hand, ∆- and Daugavet points have proved to be far more flexible than the global  properties that they define.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For example, there exists a Banach space with the RNP and a Daugavet  point [51] (see paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1), there exists a Banach space with a one-unconditional basis and a large  subset of Daugavet points [6] (see paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3), and there is an MLUR Banach space for which all  elements in its unit sphere are ∆-points, which contains convex combinations of slices of arbitrarily  small diameter, but satisfying that every convex combination of slices intersecting its unit sphere has  diameter two [2] (see paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Nonetheless, it has been recently proved that ∆-points have  some influence on the isometric structure of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For example, it is shown in [5] that uniformly  non-square spaces do not contain ∆-points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' actually, it has been very recently proved in [37] that  a ∆-point cannot be a locally uniformly non-square point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, combining the results from [5] and  [52], asymptotic uniformly smooth spaces and their duals do not contain ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However, it is still  an important open problem to understand whether ∆- or Daugavet point have any influence on the  isomorphic structure of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In this paper, our main aim is to study natural strengthening of the notions of Daugavet- and  ∆-points obtained by replacing slices by non-empty relatively weakly open subsets (“super points”)  or convex combination of slices (“ccs points”) in order to provide new diametral notions which are  extreme opposites to points of continuity and to strongly regular points, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' See Definitions  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Our main goal will be to understand the influence, for a given Banach space,  of the existence of such points on its geometry, and to study the different diametral notions in several  families of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A particular emphasis will be put on trying to distinguish between all the  various formally different notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us end this section by giving a brief description about the organization of the paper and the  main results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Section 2 contains the necessary notation (which is standard, anyway), needed  definitions, and some preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We include in Section 3 some characterizations of the newer  diametral point notions and some necessary conditions on the existence of such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular,  we study the existence of super ∆-points and ccs ∆-points in spaces with a one-unconditional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We first give an analogue for ccs ∆-points to a result from [6] which implicitly states that such spaces  contain no super ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Second, we provide sharper and improved versions of this super ∆ result  in the context of unconditional FDDs with a small unconditional constant, and more generally in the  context of spaces in which special families of operators are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The section finishes with the  study of the behaviour of super ∆-points and super Daugavet points with respect to absolute sums  somehow analogous to the known one for ∆-points and Daugavet points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' however, not all the results  extend to ccs ∆-points and ccs Daugavet points, but we also give some partial results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Section 4 is  devoted to examples and counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We first characterize the diametral notions in some families  of classical Banach spaces: we show that all notions are equivalent in L1-preduals and M¨untz spaces  (Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' all notions but ccs Daugavet points also coincide in L1-spaces (Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We next give in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 some remarks on examples which have previously appeared in the  journal literature, discussing the new diametral notions on them, and showing that they may help to  distinguish between the diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The most complicated and tricky examples are produced  in the last three subsection of this section: super ∆-points which are as closed as desired to strongly  exposed points (hence failing to be Daugavet points in an extreme way) in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' a super ∆-  point which is strongly regular (hence failing to be ccs ∆-point in an extreme way) in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  super Daugavet points which belong to convex combinations of slices of diameter as small as desired  (hence failing to be ccs ∆-points in an extreme way).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We finish this subsection with a summary of  relations between all the diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The idea in Section 5 is to generalize the diametral notions  to elements of the open unit ball, and use these notions to characterize some geometric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  5  particular, we properly localize the result by Kadets that the DSD2P is equivalent to the Daugavet  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Section 6 deals with Kuratowki index of non-compactness of slices, relative weakly open  subsets, and convex combinations of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We get that every relative weakly open subset (respectively,  every convex combination of slices) in a space with the diameter 2 property (respectively, with the  strong diameter 2 property) has Kuratowski measure 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' these results extends the analogous result for  slices and the the local diameter 2 property proved in [20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, we show that every  relative weakly open subset that contains a super ∆-point has Kuratowski measure 2, and a similar  result is obtained with convex combinations of relative weakly open subsets containing a ccw ∆-point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  these results extend [52, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally, Section 7 is devoted to collect some interesting open  questions and some remarks on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Notation and preliminary results  We will use standard notation as in the books [8], [23], and [24], for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Given a Banach  space X, BX (respectively, SX) stands for the closed unit ball (respectively, the unit sphere) of  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We denote by X∗ the topological dual of X and we write JX : X −→ X∗∗ for the canonical  injection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We denote by dent (BX) and ext (BX) the sets of all denting points of BX and of all  extreme points of BX, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The set of preserved extreme points of BX (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' those x ∈ BX such  that JX(x) ∈ ext (BX∗∗)) is denoted by pre-ext (BX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For Banach spaces X and Y , L(X, Y ), F(X, Y ),  K(X, Y ) denote, respectively, the set of all (bounded linear) operator, the finite-rank operators, and  the compact operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The properties in which we are interested only deal with the real structure of  the involved Banach spaces, but we do not restrict the study to real spaces in order to consider real or  complex examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We will use the notation K to denote either R or C, Re(z) to denote the real part  of z (which is just the identity when dealing with a real space), and T to represent the set of scalars  of modulus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Given a non-empty subset C of X, we will denote by co(C) the convex hull of C and by span(C)  the linear hull of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also we denote by co(C) (respectively, span(C)) the norm closure of the convex  hull (respectively, of the linear hull) of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By a slice of C we will mean any subset of C of the form  S(x∗, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' C) := {x ∈ C : Re x∗(x) > M − δ}  where x∗ ∈ X∗ is a continuous linear functional on X, δ > 0 is a positive real number, and M :=  supx∈C Re x∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For slices of the unit ball we will simply write S(x∗, δ) := S(x∗, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  By a  relatively weakly open subset of C we mean as usual any subset of C obtained as the (non-empty)  intersection of C with an open set of X in the weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  If C is assumed to be convex we will mean by a convex combination of slices of C (ccs of C in  short) any subset of C of the form  �n  i=1 λiSi,  where λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , λn ∈ (0, 1] are such that �n  i=1 λi = 1 and Si is a slice of C for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that convex combinations of slices are convex sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We define in the same way convex com-  binations of relatively weakly open subsets of C (ccw of C in short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The following lemma from [30] is a very useful tool when working with ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 ([30, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, and let x∗ ∈ SX∗ and α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every  x ∈ S(x∗, α) and every 0 < β < α there exists y∗ ∈ SX∗ such that  x ∈ S(y∗, β) ⊆ S(x∗, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  6  MART´IN, PERREAU, AND RUEDA ZOCA  We also often rely on the following result, due to Bourgain, and that we already mentioned in the  introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We provide a proof below, following the one from [25, Lemma II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1], for the sake of  completeness and for further discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 (Bourgain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let C be a bounded convex closed subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Then, every non-empty relatively weakly open subset W of C contains a convex combination of slices  of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Assume with no loss of generality that W :=  m�  i=1  S(fi, αi, C), write �C = JX(C)  w∗  ⊂ X∗∗, and  W ∗∗ :=  m  �  i=1  S  �  JX∗(fi), αi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' �C  �  ,  which is a non-empty relatively weak∗ open subset of �C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By the Krein-Milman theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [24,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='37]), it follows that �C = co(ext (BX∗∗))  w∗  , so co(ext (BX∗∗)) ∩ W ∗∗ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Pick a convex  combination of extreme points �n  i=1 λie∗∗  i  contained in W ∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By the continuity of the sum we can find,  for every 1 ⩽ i ⩽ n, a weak-star open subset W ∗∗  i  with e∗∗  i  ∈ W ∗∗  i  and such that �n  i=1 λiW ∗∗  i  ⊂ W ∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, since each e∗∗  i  is an extreme point of �C, we have by Choquet’s lemma (see [24, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='40],  for instance) that there are weak-star slices S  �  JX∗(gi), βi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' �C  �  with e∗∗  i  ∈ S  �  JX∗(gi), βi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' �C  �  ⊆ W ∗∗  i  for  every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Henceforth, �n  i=1 λiS  �  JX∗(gi), βi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' �C  �  ⊆ �n  i=1 λiW ∗∗  i  ⊆ W ∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, if we take  U :=  n  �  i=1  λiS(gi, βi, C)  it is not difficult to prove that U ⊆ W, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that, in general, it is unclear from the above proof whether or not, if we fix  x ∈ W, we can guarantee that there exists a convex combination of slices U of C such that x ∈ U ⊆ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  On the other hand, the result holds true if x ∈ W ∩ co(pre-ext (C)) in view of the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Indeed, if such situation, if we write x = �n  i=1 λixi ∈ W satisfying that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , xn ∈ pre-ext (C)  and λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , λn ∈ (0, 1] with �n  i=1 λi = 1, by the weak continuity of the sum, we can find, for every  1 ⩽ i ⩽ n, a non-empty relatively weakly open subset Vi with xi ∈ Vi for every i and such that  x = �n  i=1 λixi ∈ �n  i=1 λiVi ⊆ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, observe that, since each xi is a preserved extreme point of C,  slices of C containing xi are a neighbourhood basis for xi in the weak topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Hence, we can find,  for 1 ⩽ i ⩽ n, a slice Si of C with xi ∈ Si ⊆ Vi, and so x = �n  i=1 λixi ∈ �n  i=1 λiSi ⊆ �n  i=1 λiVi ⊆ W,  so U := �n  i=1 λiSi is the desired convex combination of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Throughout the text, we will often be discussing various “diameter 2 properties”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We use the  notation introduced in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A Banach space X has the local or slice diameter 2 property (LD2P) if  every slice of BX has diameter 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' X has the diameter two property (D2P) if every non-empty relatively  weakly open subset of BX has diameter 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' finally, X has the strong diameter 2 property (SD2P)  whenever every ccs of BX has diameter 2 (and then, every ccw has diameter 2 due to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For definitions and for examples concerning those properties, we refer to [2, 14, 15, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular,  let us comment that the three properties are different, a result which was not easy to show, see [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Our paper is closely related to the diametral versions of those properties which have been implicitly  studied for a long time in the literature, but whose formal definitions and names where fixed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A Banach space X has the diametral local diameter 2 property (DLD2P) if for every slice S of BX  and every x ∈ S ∩ SX, supy∈S ∥x − y∥ = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' if slices are replaced by non-empty relatively weakly  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  7  open subsets of BX, we obtain the diametral diameter 2 property (DD2P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is immediate that these  properties are not satisfied by any finite-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Clearly, DLD2P implies LD2P, DD2P  implies D2P (and none of these implications reverses, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' X = c0), and DD2P implies DLD2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is  unknown whether the DLD2P and the DD2P are equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' in fact it is even unknown whether the  DLD2P implies the D2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For the analogous definition using ccs, we have to discuss a little bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Even  for an infinite-dimensional space X, it is not true that every ccs of BX intersects SX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' actually, this  happens if and only if X has a property stronger than the SD2P (see [41, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Thus, the  definition of the diametral strong diameter 2 property (DSD2P) given in [13] deals with all points in  BX as follows: for every ccs C and every x ∈ C, supy∈C ∥x − y∥ = ∥x∥ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This definition allows to  show that DSD2P implies the SD2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' But, actually, it has been recently shown by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Kadets [34] that  the DSD2P is equivalent to the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We will discuss this in detail in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the  other hand, we will use the following property which is weaker than the DSD2P: a Banach space X  has the restricted DSD2P if for every ccs C and every x ∈ C ∩ SX, supy∈C ∥x − y∥ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This property  is strictly weaker than the DSD2P, see Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us now introduce all the notions of diametral points that we will consider in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us  start with the more closely related ones to the definitions above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We say that  (1) [3] x is a ∆-point if supy∈S ∥x − y∥ = 2 for every slice S of BX containing x,  (2) x is a super ∆-point if supy∈V ∥x − y∥ = 2 for every non-empty relatively weakly open subset  V of BX containing x,  (3) x is a ccs ∆-point if supy∈C ∥x − y∥ = 2 for every slice ccs C of BX containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  ∆-points were introduced in [3] as a natural localization of the DLD2P (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' X has the DLD2P if  and only if every element of SX is a ∆-point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The other two definitions are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Clearly, super  ∆-points are the natural localization of the DD2P: X has the DD2P if and only if every element of  SX is a super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Besides, ccs ∆-points are the localization of the restricted DSD2P: X has the  restricted DSD2P if and only if every element of SX is a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In relation with the Daugavet property, we have the following notions for points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We say that  (1) [3] x is a Daugavet point if supy∈S ∥x − y∥ = 2 for every slice S of BX,  (2) x is a super Daugavet point if supy∈V ∥x − y∥ = 2 for every non-empty relatively weakly open  subset V of BX,  (3) x is a ccs Daugavet point if supy∈C ∥x − y∥ = 2 for every ccs C of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us recall that Daugavet points were introduced in [3] as a natural localization of the Daugavet  property in the sense that a Banach space X has the Daugavet property if and only if every point in SX  is a Daugavet point ([35, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' From the geometric characterization given in [50, Lemma 3]  and the implicit result contained in its proof, it follows that super Daugavet points as well as ccs  Daugavet points are also natural localizations of the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since every slice of BX is relatively weakly open, and since by Bourgain’s lemma (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2)  every non-empty relatively weakly open subset of BX contains a ccs of BX, we clearly have the diagram  of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We will show throughout the text that none of the above implications reverses, see Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7  for a description of all the relations and the counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However, let us point out right away  8  MART´IN, PERREAU, AND RUEDA ZOCA  ccs Daugavet  ccs ∆  super Daugavet  super ∆  ∆  Daugavet  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Relations between the diametral notions  that we do not know whether there exists ccs ∆-points which are not super ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In view of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3  such examples may exist since Bourgain’s lemma is not localizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also let us point out that it follows  again from Bourgain’s lemma that a ccs Daugavet point x ∈ SX also satisfies supy∈D ∥x − y∥ = 2 for  every ccw D of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Again this is not clear for ccs ∆-points and we could thus naturally distinguish  between ccs ∆-points and “ccw ∆-points”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since we do not have concrete examples at hand, we will  focus on convex combination of slices and specifically point out any available ccw behavior throughout  the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us also comment that it is clear that if every ccs of the unit ball of a given Banach space is weakly  open (respectively, has non-empty relative weak interior), then every super ∆-point (respectively, every  super Daugavet point) in this space is a ccs ∆-point (respectively, a ccs Daugavet point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Several  properties of this kind where introduced and studied in [1], [4], and [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We refer to those papers for  some background and for examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' There are natural weak∗ versions in dual spaces of all the notions of diametral-points  introduced in the present section where slices and relatively weakly open subsets are respectively  replaced with weak∗ slices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' slices defined by elements of the predual) and relatively weak∗ open  subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' With obvious terminology, it then follows from [35, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] and from [50, Lemma 3] that  a Banach space X has the Daugavet property if and only if every element in SX∗ is a weak∗ Daugavet  point if and only if every element in SX∗ is a weak∗ ccs Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It also follows from [2,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6] that X has the DLD2P if and only if every point in SX∗ is a weak∗ ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However,  the relationship between the DD2P in X and weak∗ super ∆-points in SX∗ is currently unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that a direct consequence of those results is that weak∗ diametral points and their weak  counterparts might differ in a very strong way since, for instance, the unit ball of the space C[0, 1]∗  admits denting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Yet clearly all the results from the following sections concerning the different  notions of diametral-points admit obvious analogues for their weak∗ counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We leave the details  to the reader to avoid unnecessary repetitions, but let us still point out that it follows from Goldstine’s  theorem and from the lower weak∗ semicontinuity of the norm in dual spaces that there is a natural  correspondence between diametral-properties of points in SX and weak∗ properties of their image in  the bidual under the canonical embedding JX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Namely:  (1) x ∈ SX is a Daugavet point (respectively, a ccs Daugavet point) if and only if JX(x) is a weak∗  Daugavet point (respectively, a weak∗ ccs Daugavet point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  9  (2) x ∈ SX is a super Daugavet point if and only if JX(x) is a weak∗ super Daugavet point if  and only if for every y ∈ BX there exists a net (y∗∗  s ) in BX∗∗ which converges to JX(y) in the  weak∗ topology and such that ∥πX(x) − y∗∗  s ∥ −→ 2 (see Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) x ∈ SX is a ∆-point (respectively, a super ∆-point) if and only if JX(x) is a ∆-point (respec-  tively, a super ∆-point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (4) x ∈ SX is a ccs ∆-point if and only if JX(x) is a weak∗ ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us point out that (3) essentially follows from the obvious fact that ∆-points and super ∆-points  naturally pass to superspaces, that is if Y is a subspace of X and if x ∈ SY if a ∆-point (respectively,  a super ∆-point) in Y , then x is a ∆-point (respectively, a super ∆-point) in X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This property is  unclear for ccs ∆-points, so the assertion (4) is not analogous to assertion (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Characterisations of diametral-notions and implications on the geometry of the  ambient space  In view of the definitions of diametral-points, it is natural to expect that the presence of any kind  of Daugavet- or ∆-element in a given Banach space will affect, by the severe restrictions it inflicts  on the nature of the considered point, its global isometric geometry or even its topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  However, previous studies in the context have shown that the situation is much more complicated  than one could expect at first sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For example, let us comment that a Banach space X with  the RNP and admitting a Daugavet point, and a Banach space with a one-unconditional basis and  admitting a weakly dense subset of Daugavet points, were respectively constructed in [51] and in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In this section, we provide useful characterizations of the new diametral notions, and investigate the  immediate effect of the presence of such points on the geometry of the considered space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We start by an intuitive but not completely trivial observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By definition, it is clear that super ∆-points do not exist in finite dimensional  spaces because the weak and norm topology coincide in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Also, it was proved in [5,  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4] that finite dimensional spaces do also fail to contain ∆-points (hence ccs ∆-points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In fact they fail to contain them in a stronger way, see [5, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consequently, the study  of diametral-notions only makes sense in infinite dimension, and from now on we will assume unless  otherwise stated that all the Banach spaces we consider are infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us next prove a bunch of characterisations for super Daugavet- and super ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every x ∈ SX and for every ε > 0, let us define  ∆ε(x) := {y ∈ BX : ∥x − y∥ > 2 − ε}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We recall the following characterization of Daugavet- and ∆-points from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 ([3, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) An element x ∈ SX is a Daugavet point if and only if BX = co ∆ε(x) for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) An element x ∈ SX is a ∆-point if and only if x ∈ co ∆ε(x) for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We have similar characterisations for super points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) An element x ∈ SX is a super Daugavet point if and only if BX = ∆ε(x)  w for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) An element x ∈ SX is a super ∆-point if and only if x ∈ ∆ε(x)  w for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  10  MART´IN, PERREAU, AND RUEDA ZOCA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that for given x ∈ SX, y ∈ BX, and ε > 0, we have that y belongs to the weak  closure of the set ∆ε(x) if and only if ∆ε(x) has non-empty intersection with any neighborhood of  y in the relative weak topology of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Thus y belongs to ∆ε(x)  w for every ε > 0 if and only if  supz∈V ∥x − z∥ = 2 for every relatively weakly open subset V of BX containing y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The conclusion  easily follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  For any given x ∈ BX, we denote by V(x) the set of all neighborhoods of x for the relative weak  topology of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We can provide characterizations of super points using nets which is just a localization  of [13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be an infinite-dimensional Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) An element x ∈ SX is a super Daugavet point if and only if for every y ∈ BX there exists a  net (ys) in BX which converges weakly to y and such that ∥x − ys∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) An element x ∈ SX is a super ∆-point if and only if there exists a net (xs) in BX which  converges weakly to x and such that ∥x − xs∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In both cases we can moreover force the nets to be in SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us fix x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Given any y ∈ BX, it is clear that if there exists a net (ys) in BX which  converges weakly to y and such that ∥x − ys∥ −→ 2, then y belongs to the weak closure of ∆ε(x) for  every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Conversely let us pick y ∈ BX satisfying this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We turn S := V(y) × (0, ∞) into  a directed set by (V, ε) ⩽ (V ′, ε′) if and only if V ′ ⊂ V and ε′ ⩽ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By the assumptions we have that  V ∩ ∆ε(x) is a non-empty subset of BX for every couple s := (V, ε) in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Picking any ys in this set  will then provide the desired net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Finally observe that for x ∈ BX and ε > 0, we have that BX\\∆ε(x) = {y ∈ BX : ∥x − y∥ ⩽ 2 − ε}  is weakly closed by the lower semi-continuity of the norm, so that ∆ε(x) is a relatively weakly open  subset of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Thus we have that V ∩ ∆ε(x) is a non-empty relatively weakly open subset of BX for  every couple s := (V, ε) in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since X is infinite dimensional, this set has to intersect SX, and we can  actually pick ys in V ∩ ∆ε(x) ∩ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In [36] an example of a Banach space satisfying simultaneously the Daugavet property  and the Schur property was provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Such example shows that there is no hope to get a version of  the above result involving sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that the following result, similar to [32, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2], is included in the preceding  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) If x is a super Daugavet point, then for every ε > 0 and every non-empty relatively weakly  open subset V of BX we can find a non-empty relatively weakly open subset U of BX which is  contained in V and such that ∥x − y∥ > 2 − ε for every y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) If x is a super ∆-point, then for every ε > 0 and every non-empty relatively weakly open subset  V of BX containing x we can find a non-empty relatively weakly open subset U of BX which  is contained in V and such that ∥x − y∥ > 2 − ε for every y ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Fix any x ∈ SX and any y ∈ BX which belongs to the weak closure of ∆ε(x) for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Then, for every V ∈ V(y) and every ε > 0, we have that U := V ∩ ∆ε(x) is a non-empty relatively  weakly open subset of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  11  It is clear from the definition that denting points of BX cannot be ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also it was first  observed in [32, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] that every Daugavet point in a Banach space X has to be at distance  2 from every denting point of the unit ball of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This elementary observation turned out to play  an important role in the study of Daugavet points in Lipschitz-free spaces in [32] and [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We have  similar observations for super points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If x is a super ∆-point, then x cannot be  a point of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If, moreover, x is a super Daugavet point, then x has to be at distance 2 from  every point of continuity of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If and element y of BX is a point of continuity, then it is contained in relatively weakly open  subsets of BX of arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Clearly no super ∆-point can have this property, and any  super Daugavet point has to be at distance 2 from any such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  This lemma provides quite a few examples of Banach spaces which fail to contain super points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Following [26] let us recall that X has the Kadets property if the norm topology and the weak topology  coincide on SX, and that X has the Kadets-Klee property if weakly convergent sequences in SX are  norm convergent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us also recall that any LUR space has the Kadets-Klee property, and that any  space with the Kadets-Klee property which fails to contain ℓ1 has the Kadets property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 we clearly have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If X has the Kadets property, then X fails to contain super ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As a corollary we obtain the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a Banach space is asymptotic uniformly convex  (AUC in short) [31] if its modulus of asymptotic uniform convexity  δX(t) := inf  x∈SX  sup  dim X/Y <∞  inf  y∈SY ∥x + ty∥ − 1  is strictly positive for every t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, X fails to contain super ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In an AUC space, every element of the unit sphere is a point of continuity of BX, see [31,  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It was proved in [5, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4] that any reflexive AUC space fails to contain ∆-  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, combining the observations from [5, End of Section 4] about weak∗ quasi-denting points  in the unit ball of AUC∗ duals and [52, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4] about the maximality of the Kuratowski index  of weak∗ slices containing weak∗ ∆-points, we have that every AUC∗ dual space fails to contain weak∗  ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However, note that it is currently unknown whether non-reflexive AUC spaces (and, in  particular, whether the dual of the James tree spaces JT∗) may contain Daugavet- or ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It turns out that Daugavet points are characterized by this distance to denting points in RNP  spaces (because the unit ball of an RNP space X can be written as the closed convex hull of the set  of its denting points) as well as in Lipschitz-free spaces ([32, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2] for compact metric spaces  and [51, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] for a general statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In the same way we can characterize super Daugavet  points in terms of this distance to points of continuity of BX is spaces with the CPCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If a Banach space X has the CPCP, then a point x ∈ SX is a super Daugavet  point if and only if it is at distance 2 from any point of continuity of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  12  MART´IN, PERREAU, AND RUEDA ZOCA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If X has the CPCP, then the set of all points of continuity of BX is weakly dense in BX (see for  example [22, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9]), that is, every non-empty relatively weakly open subset of BX contains  a point of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The conclusion follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  For ccs points, the situation is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Indeed, although ccs ∆-points can clearly not be  points of strong regularity, we have by [41, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] that X has the SD2P if and only if every  convex combination of slices of BX contains elements of norm arbitrarily close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It readily follows  that any space X which contains a ccs Daugavet point satisfies the SD2P, so it is very far from being  strongly regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We will provide more details on this topic in Section 5, but for later reference let us  state the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If X contains a ccs Daugavet point, then it has the  SD2P (it fails to be strongly regular).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Next, we show that extreme points have a nice behaviour with respect to diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) If x ∈ pre-ext (BX) and it is a ∆-point, then x is a super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) If x ∈ ext (BX) and it is a super ∆-point, then x is a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) In particular, if x ∈ pre-ext (BX) is a ∆-point, then x is a super ∆-point as well as a ccs  ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It follows from Choquet’s lemma (see for example [23, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='69]) that slices form neighbor-  hood bases in the relative weak topology of the unit ball of a Banach space for its preserved extreme  points, so (1) immediately follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For (2), if x is extreme and belongs to a ccs C := �n  i=1 λiSi of BX  then x ∈ �n  i=1 Si, which is a relatively weakly open subset of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that, in fact, any extreme super ∆-point is “ccw ∆-point” as we discussed in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, Choquet’s lemma implies that every extreme weak∗ ∆-point in a dual space is weak∗  ccw ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Spaces with a one-unconditional basis and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In [6], it was proved that no real  Banach space with a subsymmetric basis contains a ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the other hand, an example of a  Banach space with a one-unconditional basis that contains a ∆-point was provided, and a more involved  example of a Banach space with a one-unconditional basis that contains many Daugavet-points was  constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We will discuss this second example in detail in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In the process, it was also implicitly shown that real Banach spaces with a one-unconditional basis  cannot contain super ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In the present subsection, we prove that the same goes for ccs ∆-  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, we provide sharper and more general versions of [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In the first part  of this section, we follow [6] and restrict ourselves to real Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let X be a real Banach space with a Schauder basis (ei)i⩾1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We denote by (e∗  i )i⩾1 the corresponding  sequence of biorthogonal functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Recall that (ei)i⩾1 is said to be unconditional if the series  �  i⩾1 e∗  i (x)ei converges unconditionally for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Also, recall that an unconditional basis  (ei)i⩾1 is said to be one-unconditional if  �����  �  i⩾1  θie∗  i (x)ei  ����� =  �����  �  i⩾1  e∗  i (x)ei  �����  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  13  for every (θi)i⩾1 ∈ {−1, 1}N and for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, if  �����  �  i⩾1  θie∗  i (x)eni  ����� =  �����  �  i⩾1  e∗  i (x)ei  �����  for every (θi)i⩾1 ∈ {−1, 1}N, for every x ∈ X, and for every strictly increasing sequence (ni)i⩾1 in N,  then the basis is called subsymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that for spaces with a one-unconditional basis, it is enough, in order to study the various  Daugavet- and ∆-notions, to work in the positive sphere  S+  X := {x ∈ SX : e∗  i (x) ⩾ 0 ∀i}  of the space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, the following result is well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a real Banach space with a one-unconditional basis (ei)i⩾1, and let (ai)i⩾1  and (bi)i⩾1 be sequences of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If the series �  i⩾1 biei converges, and if |ai| ⩽ |bi| for every  i, then �  i⩾1 aiei converges as well, and we have  �����  �  i⩾1  aiei  ����� ⩽  �����  �  i⩾1  biei  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us now recall a few notation and preliminary results from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a real Banach space  with a normalized one-unconditional basis (ei)i⩾1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every subset A of N, we denote by PA the  projection on span{ei, i ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then for every x ∈ X, we define  M(x) := {A ⊂ N: ∥PA(x)∥ = ∥x∥ , and  ��PA(x) − e∗  j(x)ej  �� < ∥x∥ ∀j ∈ A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The set M(x) can be seen as the set of all minimal norm-giving subsets of the support of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We denote  respectively by MF(x) and M∞(x) the subsets of all finite and infinite elements of M(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It follows  from [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7] that the set M(x) is never empty, and from [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='15] that no element  x ∈ SX satisfying M∞(x) = ∅ can be a ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For every non-empty ordered subset A := {a1 < a2 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' } of N, and for every n ∈ N smaller than  or equal to |A|, we denote by A(n) := {a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , an} the subset consisting of the n first elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We will implicitly assume in the following that the elements of M(x) are ordered subsets of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The  next two results were proved in [6, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8 and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a real Banach space with a normalized one-unconditional basis (ei)i⩾1 and  let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every n ∈ N, the sets  �  A ∈ M(x): |A| ⩽ n  �  and  �  A(n): A ∈ M(x), and |A| > n  �  are both finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a real Banach space with a normalized one-unconditional basis and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For every subset E of N such that E ∩ A ̸= ∅ for every A ∈ M(x), we have ∥x − PE(x)∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  With those tools at hand, we can now prove an analogue to [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13] for convex combi-  nation of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space with a normalized one-unconditional basis and x ∈ S+  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Then, there exists δ > 0 and a ccs C of BX containing x such that supy∈C ∥x − y∥ ⩽ 2 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  14  MART´IN, PERREAU, AND RUEDA ZOCA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let x ∈ S+  X, and define E = �  A∈M(x) A(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' From Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='16 and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='17, we have  that E is a finite subset of N and that ∥x − PE(x)∥ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, there exists γ > 0 such that  ∥x − PE(x)∥ ⩽ 1 − γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every i ∈ E, we define  Si := S  �  e∗  i , 1 − e∗  i (x)  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Then we consider the ccs  C :=  1  |E|  �  i∈E  Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since x ∈ S+  X, we clearly have that x ∈ �  i∈E Si and, in particular, that x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  So let us pick y :=  1  |E|  �  i∈E yi in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Then we have e∗  i (yi) >  e∗  i (x)  2  for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In particular,  e∗  i (yi) ⩾ 0, and  ��e∗  i (yi) − e∗  i (x)  �� ⩽ e∗  i (yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Indeed, for any given non-negative real numbers α and β  with β ⩾ α  2 , we have  |β − α| = β − α ⩽ β  if β ⩾ α, and  |β − α| = α − β ⩽ α − α  2 = α  2 ⩽ β  if β ⩽ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So in either case, |β − α| ⩽ β as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It then follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='15 that  ��yi − e∗  i (x)ei  �� ⩽  ��yi�� ⩽ 1 and, finally,  ∥x − y∥ ⩽  ����x − x  |E|  ���� +  ����  x  |E| − PE(x)  |E|  ���� +  ����  PE(x)  |E|  − y  ����  ⩽ 1 − 1  |E| + 1 − γ  |E|  + 1  |E|  �  i∈E  ��e∗  i (x)ei − yi�� ⩽ 2 − γ  |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The conclusion follows with δ :=  γ  |E|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, note that since x belongs to the relative weakly  open set �  i∈E Si ⊂ C, we also get that x is not super ∆, recovering the result from [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  So combining [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13] and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='18, we immediately get that spaces with a  normalized one-unconditional basis fail to contain super ∆-points and ccs ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So let us state  the following here for future reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a real Banach space with a normalized one-unconditional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then X  does not contain super ∆-points, and X does not contain ccs ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In the rest of the subsection, we aim at providing sharper and improved versions of [6, Proposi-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular we will go back to working with either real or complex Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The  main result of this study is the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, and let us assume that there exists a subset A ⊆ F(X, X)  satisfying that sup  �  ∥Id − T∥ : T ∈ A  �  < 2 and that for every ε > 0 and every x ∈ X, there exists  T ∈ A such that ∥x − Tx∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, X contains no super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us provide a lemma which is a localization of the above result from which its proof is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If there exists a finite-rank operator T on  X such that ∥x − Tx∥ + ∥Id − T∥ < 2, then x is not a super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consider ε > 0 such that K := ∥x − Tx∥ + ∥Id − T∥ + ε < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since T has finite rank, we can  find N ⩾ 1, w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , wN ∈ SX and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , fN ∈ X∗ such that T(z) = �N  n=1 fn(z)wn for every z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us consider  W :=  �  y ∈ BX : |fn(x − y)| <  ε  2n+1 ∀n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , N}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  W is a neighborhood of x in the relative weak topology of BX, and for every y ∈ W, we have  ∥x − y∥ ⩽ ∥x − Tx∥ + ∥Tx − Ty∥ + ∥y − Ty∥  ⩽ ∥x − Tx∥ + ∥Id − T∥ +  N  �  n=1  |fn(x − y)| ∥wn∥  ⩽ ∥x − Tx∥ + ∥Id − T∥ + ε  N  �  n=1  1  2n+1 ⩽ K < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is unclear whether an analogue to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='21 can be given for ccs ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So we do not  know whether Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='20 extends to this notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As particular cases of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='20, we have the following ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that a sequence (En)n⩾1  of finite dimensional subspaces of a given Banach space X is called a finite dimensional decomposition  (FDD) for X if every element x ∈ X can be represented in a unique way as a series x := �  n⩾1 xn  with xn ∈ En for every n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Such an FDD is said to be unconditional if the above series converges  unconditionally for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In this case, it is well known that the family (PA)A⊂N, where PA is the  projection given by PA(x) := �  n∈A xn, is uniformly bounded, and the constant KS := supA⊂N ∥PA∥ is  called the suppression-unconditional constant of the FDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We refer to [40, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g] for the details  and to [8, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] for the particular case of unconditional bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A Banach space X fails to have super ∆-points provided one of the following condi-  tions is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) There exists a family A ⊆ F(X, X) satisfying that sup  �  ∥Id − T∥ : T ∈ A  �  < 2 and that the  identity mapping belongs to its strong operator topology (SOT) closure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) There exists a family {Pλ}λ∈Λ of finite rank projections on X such that X = �  λ∈Λ Pλ(X),  and such that supλ∈Λ ∥Id − Pλ∥ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) The space X admits a FDD with suppression-unconditional constant less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular,  if X admits an unconditional basis with suppression-unconditional constant less than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us observe that the value 2 in the above results is sharp in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) The space C[0, 1] admits a monotone Schauder basis, so there exists a sequence  {Pn}n⩾1 of norm one finite rank projections on this space which converges to Id in SOT  topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As C[0, 1] has the Daugavet property, all elements in SX are super Daugavet points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that ∥Id − Pn∥ = 2 for every n ⩾ 1 by the DPr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) Let X be an arbitrary Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every x ∈ SX choose fx ∈ SX∗ such that fx(x) = 1,  and define Px(z) = fx(z)x for every z ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then {Px : x ∈ SX} is a family of norm one  rank-one projections on X, X = �  x∈SX Px(X), and ∥Id − Px∥ ⩽ 2 for every x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) The space c admits ccs Daugavet points (hence super Daugavet points), see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2, but  it is easy to check that its usual basis is 3-unconditional and 2-suppression unconditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (4) It is shown in [30] that a Banach space has the DLD2P if and only if ∥Id − P∥ ⩾ 2 for every  rank-one projection P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It follows that the suppression constant of an unconditional basis on  a Banach space with the DLD2P has to be greater than or equal to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us mention here  16  MART´IN, PERREAU, AND RUEDA ZOCA  that there is no local version of this result, as there are Banach spaces with one-unconditional  basis and containing many Daugavet points [6] (see Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Absolute sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In this subsection we look at the transfer of the diametral points through  absolute sums of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us first recall the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A norm N on R2 is absolute if N(a, b) = N(|a| , |b|) for every (a, b) ∈ R2 and  normalized if N(0, 1) = N(1, 0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  If X and Y are Banach spaces, and if N is an absolute normalized norm on R2, we denote by  X ⊕N Y the product space X × Y endowed with the norm ∥(x, y)∥ = N(∥x∥ , ∥y∥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is easy to check  that X ⊕N Y is a Banach space, and that its dual can be expressed as (X ⊕N Y )∗ ≡ X∗ ⊕N∗ Y ∗ where  N∗ is the absolute norm given by the formula N∗(c, d) = maxN(a,b)=1 |ac|+|bd|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Classical examples of  absolute normalized norms on R2 are the ℓp norms for p ∈ [1, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Information on absolute norms can  be found in [16, §21] and [43] and references therein, for instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us recall that for every absolute  normalized sum N, given non-negative a, b, c, d in R with a ⩽ b and c ⩽ d we have N(a, b) ⩽ N(c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In particular, ∥·∥∞ ⩽ N ⩽ ∥·∥1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Similar to the DD2P (see [13, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11]) and to ∆-points [28], super ∆-points transfer very  well through absolute sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X and Y be Banach spaces, and let N be an absolute normalized norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) If x ∈ SX and y ∈ SY are super ∆-points, then (ax, by) is a super ∆-point in X ⊕N Y for  every (a, b) ∈ R2 with N(a, b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) If x ∈ SX is a super ∆-point, then (x, 0) is a super ∆-point in X ⊕N Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If y ∈ SY is a super  ∆-point, then (0, y) is a super ∆-point in X ⊕N Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We can find two nets (xs)s∈S and (yt)t∈T respectively in SX and SY such that xs  w  −→ x,  yt  w  −→ y, and ∥x − xs∥ , ∥y − yt∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, if we take (a, b) ∈ R2 with N(a, b) = 1 we clearly have  (axs, byt)  w  −→  (s,t)∈S×T (ax, by) and ∥(ax, by) − (axs, byt)∥ = N (a ∥x − xs∥ , b ∥y − yt∥) −→ 2N(a, b) = 2,  so (ax, by) is a super ∆-point in X ⊕N Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For (2), we just repeat the previous proof with a = 1 and  b = 0 or with a = 0 and b = 1 and so we only need one of the points to be super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  For super Daugavet points the situation is more complicated and we need to distinguish between  different kinds of absolute norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The following definitions can be found, for instance, in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let N be an absolute normalized norm on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) N has property (α) if for every a, b ∈ R+ with N(a, b) = 1 we can find a neighborhood W of  (a, b) in R2 with sup(c,d)∈W c < 1 or sup(c,d)∈W d < 1 and such that any couple (c, d) ∈ R2  +  satisfying N(c, d) = 1 and N ((a, b) + (c, d)) = 2 belongs to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) N is A-octahedral if there exists a, b ∈ R+ such that N(a, b) = 1 and N ((a, b) + (c, d)) = 2 for  c = max{e ∈ R+ : N(e, 1) = 1}  and  d = max{f ∈ R+ : N(1, f) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) N is positively octahedral if there exists a, b ∈ R+ such that N(a, b) = 1 and  N ((a, b) + (0, 1)) = N ((a, b) + (1, 0)) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Positively octahedral norms where introduced in [27] in order to characterize the absolute norms for  which the corresponding absolute sum is octahedral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is clear that property (α) and A-octhaedrality  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  17  exclude each other and that every positively octahedral absolute normalized norm is A-octahedral  (while there clearly exists absolute A-octahedral norms which are not positively octahedral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover  it was proved in [28, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5] that every absolute normalized norm on R2 must either satisfy  property (α) or be A-octahedral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For ℓp-norms, we have that ∥·∥1 and ∥·∥∞ are both positively  octahedral, and that ∥·∥p satisfies property (α) for every p ∈ (1, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that if an absolute normalized norm N on R2 is positively octahedral, and if (a, b) is as in  the above definition, then the intersection of the unit sphere of N with the positive quadrant of R2 is  equal to the union of the segments [(1, 0), (a, b)] and [(0, 1), (a, b)] (see [45, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] for pictures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In particular, it follows that N((a, b)+(c, d)) = 2 for every non-negative c, d with N(c, d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Similar  to the results from [3, Section 4] concerning Daugavet points, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X and Y be Banach spaces, and let N be an absolute normalized norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) [3, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6] If N has property (α), then X ⊕N Y has no Daugavet point (hence, in  particular, no super Daugavet points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) If N is positively octahedral and if x ∈ SX and y ∈ SY are super Daugavet points, then (ax, by)  is a super Daugavet point in X ⊕N Y for every (a, b) ∈ R2  + as in the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Assume that N is positively octahedral, take (a, b) ∈ R2  + as in the definition, and let  x ∈ SX and y ∈ SY be super Daugavet points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For any given (u, v) ∈ X ⊕N Y of norm ∥(u, v)∥ = 1 we  can find two nets (us)s∈S and (vt)t∈T respectively in SX and SY such that ∥u∥us  w  −→ u, ∥v∥vt  w  −→ v,  and ∥x − us∥ , ∥y − vt∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then (∥u∥ us, ∥v∥ vt)  w  −→  (s,t)∈S×T (u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since  ∥ax − ∥u∥ us∥ = ∥(x − us) − [(1 − a)x − (1 − ∥u∥)us]∥  ⩾ ∥x − us∥ − (1 − a + 1 − ∥u∥),  = a + ∥u∥ − (2 − ∥x − us∥),  and, in the same way,  ∥by − ∥v∥ vt∥ ⩾ b + ∥v∥ − (2 − ∥y − vt∥),  we have  ∥(ax − ∥u∥ us, by − ∥v∥ vt)∥ = N (∥ax − ∥u∥ us∥ , ∥by − ∥v∥ vt∥)  ⩾ N (a + ∥u∥ − (2 − ∥x − us∥), b + ∥v∥ − (2 − ∥y − vt∥))  −→ N ((a + ∥u∥ , b + ∥v∥) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  This shows that (ax, by) is a super Daugavet point in X ⊕N Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Note that if (a, b) = (1, 0) (respectively, (a, b) = (0, 1)) in the previous statement (for  example, when N = ∥·∥1), then we only need to assume that x (respectively, y) is super Daugavet  in order to get that (x, 0) (respectively, (0, y)) is super Daugavet in X ⊕N Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, if N = ∥·∥∞,  then we only need to assume that x (respectively, y) is super Daugavet in order to obtain that (x, βy)  (respectively, (αx, y)) is super Daugavet in X ⊕N Y for every β ∈ [0, 1] (respectively, α ∈ [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In [28, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2] it is proved that regular Daugavet points do also transfer through A-octahedral  sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We do not know if a similar result can be obtained for super Daugavet points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Indeed, observe  that if N is an A-octahedral norm, and if c, d, and (a, b) are as in the above definition, then the  intersection of the unit sphere of N with the positive quadrant of R2 is equal to the union of the  segments [(1, 0), (1, d)], [(1, d), (a, b)], [(0, 1), (c, 1)] and [(c, 1), (a, b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, N((a, b)+(e, f)) =  2 for every couple (e, f) on the segments [(1, d), (a, b)] and [(c, 1), (a, b)], but this is no longer true  18  MART´IN, PERREAU, AND RUEDA ZOCA  on the segments [(1, 0), (1, d)] and [(0, 1), (c, 1)] and the argument in the above proof does not work  anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The situation for ccs ∆-points and ccs Daugavet point is not clear and the proofs of the above results  do not seem to admit easy extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For instance, it follows from the next result that Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='28  is not valid for ccs Daugavet points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be an arbitrary Banach space, let Y be a Banach space containing an  strongly exposed point y0 ∈ SY , and let E := X ⊕1 Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, there are convex combinations of slices of  BE around 0 of arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, E fails to contain ccs Daugavet points and  also fails to have the SD2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let y∗  0 ∈ SY ∗ strongly exposes y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Given ε > 0, there is 0 < δ < ε such that ∥y − y0∥ < ε  whenever y ∈ BY satisfies Re y∗  0(y) > 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consider f = (0, y∗  0) ∈ SE∗ and write  C := 1  2 (S(f, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BE) + S(−f, δ, BE))  Take u := 1  2(u1 + u2) ∈ C with u1 ∈ S(f, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BE) and u2 ∈ S(−f, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So if write u1 := (x1, y1) and  u2 := (x2, y2), we have  Re y∗  0(y1) = Re f(x1, y1) > 1 − δ and  Re y∗  0(y2) = Re f(x2, y2) < −1 + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  On the one hand, it follows that ∥y1−y0∥ < ε and ∥y2+y0∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the other hand, ∥y1∥, ∥y2∥ > 1−δ,  hence ∥x1∥ < δ < ε and ∥x2∥ < δ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Summarizing, we have  ∥u∥ = 1  2  �  ∥x1 + x2∥ + ∥y1 + y2∥  �  ⩽ 1  2(2ε + 2ε) = 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is straightforward to adapt the previous proof to ℓp-sums for 1 < p < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  However, note that the situation is very different for ℓ∞-sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X and Y be Banach spaces, and let E := X ⊕∞ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If x ∈ SX is a ccs Daugavet  point, then (x, y) ∈ SE is a ccs Daugavet point for every y ∈ BY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let C := �n  i=1 λiSi be a ccs of BE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n}, we can write Si := S(fi, δi) with  fi := (x∗  i , y∗  i ) ∈ SE∗ satisfying 1 = ∥fi∥ = ∥x∗  i ∥ + ∥y∗  i ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consider on the one side  ˜Si :=  �  s ∈ BX : Re x∗  i (s) > ∥x∗  i ∥ − δi  2  �  ,  and pick on the other side any ti ∈ BY such that Re y∗  i (ti) > ∥y∗  i ∥ − δi  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since ˜C := �n  i=1 λi ˜Si is a ccs  of BX, we can find for every ε > 0 an element s := �n  i=1 λisi in ˜C such that ∥x − s∥ > 2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then,  if we let t := �n  i=1 λiti, we get (si, ti) ∈ BE and  Re fi(si, ti) = Re x∗  i (si) + y∗  i (ti) > ∥x∗  i ∥ + ∥y∗  i ∥ − δi = 1 − δi  for every i, so that (si, ti) ∈ Si, and (s, t) = �n  i=1 λi(si, ti) ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally,  ∥(x, y) − (s, t)∥ ⩾ ∥x − s∥ > 2 − ε,  so (x, y) is a ccs Daugavet point as stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  19  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Examples and counterexamples of diametral elements  In this section we aim to include a number of examples and counterexamples of diametral elements  on the unit sphere of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We first characterize the notion in some spaces which have  natural relations with the Daugavet property, such as L1-preduals spaces, M¨untz spaces, and L1-  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Next, we will remark on some examples which have previously appear in the literature,  including some improvements in some cases (as for Lipschitz free spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally, we will include  some complicated examples which will be needed to see that no implication in Figure 1 in page 8  reverses and also to negate some other possible implications between the notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A summary of all  the relations between properties will be included in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Characterization in C(K)-spaces, L1-preduals, and M¨untz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It was shown in  [3, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7] that the notions of ∆-point and Daugavet point coincide for L1-preduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The authors first characterize the ∆-points in C(K) spaces and then get the result for L1-preduals by  using the principle of local reflexivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Later on, a characterization of ∆-points (equivalently, Daugavet  points) of L1-preduals was provided in [42, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2] which implicitly prove that actually ∆-points,  and super Daugavet points coincides in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us state this result here for further reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us observe that the authors of [3] works with real Banach spaces, but it is immediate that the  proof of [3, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4] works in the complex case as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' the paper [42] works in both the real  and the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 ([3, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7], [42, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be an L1-predual and let x ∈  SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) x is a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) x is a ∆-point  (3) For every δ > 0, the weak∗ slice S(JX(x), δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BX∗) contain infinitely many pairwise linearly  independent extreme points of BX∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (4) For every element y ∈ BX, there exists a sequence (x∗∗  n ) in BX∗∗ such that ∥x − x∗∗  n ∥ −→ 2  and  �����  �  n⩾1  an(y − x∗∗  n )  ����� ⩽ 2 ∥a∥∞  for every a := (an) ∈ c00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (5) For every element y ∈ BX, there exists a sequence (x∗∗  n ) in BX∗∗ which converges weak∗ to y  and such that ∥x − x∗∗  n ∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In the case that X = C(K) for a Hausdorff topological space K, the above is also equivalent to:  (6) x attains its norm at an accumulation point of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We will show that, in fact, ∆-points also coincide with the ccs versions for L1 preduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Our  approach will be analogous to the one used in [3] for ∆-points and Daugavet points: we first prove the  result for C(K) spaces and then deduce it for all L1-preduals using that the bidual of an L1-predual  is a C(K)-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In the case of C(K) spaces, we first prove a sufficient condition for ccs Daugavet  points which, for the same price, can be proved for vector-valued spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that given a compact  Hausdorff topological space K and a Banach space X, C(K, X) denotes the Banach space of those  continuous functions from K to X endowed with the supremum norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  20  MART´IN, PERREAU, AND RUEDA ZOCA  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let K be a compact Hausdorff topological space, X a Banach space, and let t0 be an  accumulation point of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If a function f ∈ SC(K,X) satisfies ∥f(t0)∥ = 1, then f is a ccs Daugavet  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Pick x∗ ∈ SX∗ such that Re x∗(f(t0)) = ∥f(t0)∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let C := �L  i=1 λiSi be a convex  combination of slices of BC(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , L}, pick a function gi ∈ Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since K is compact  and t0 is an accumulation point of K we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' There exists a sequence (Un)n⩾0 of open neighborhoods of t0 such that:  (1) U0 = K,  (2) Un+1 is a proper subset of Un for every n ⩾ 0,  (3) Re(x∗ ◦ f)|Un ⩾ 1 − 1  n and  ���gi|Un − gi(t0)  ��� ⩽ 1  n for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , L} and every n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Indeed, we construct the sequence inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let U0 := K and assume that U0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , Un are con-  structed for some n ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since K is normal, we can find an open subset U of Un such that  t0 ∈ U ⊂ U ⊂ Un.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Also since t0 is an accumulation point of K and since K is Hausdorff, we  can find an open subset V of U such that V is a proper subset of U (pick any point in U distinct from  t0 and separate the two points with open sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By continuity of f and of the finitely many g′  is, we  can then find an open subset W of V such that  Re(x∗ ◦ f)|W > 1 −  1  n + 1  and  ���gi|W − gi(t0)  ��� <  1  n + 1  for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The set Un+1 := W does the job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, let us pick (Un)n⩾0 as in the claim and let us define Fn := Un\\Un+1 for every n ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By  construction, the F ′  ns are closed non-empty subsets of K and cover K\\  ��  n⩾0 Un  �  , and each Fn may  only intersects its neighbors Fn−1 and Fn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By Urysohn’s lemma, for every n ⩾ 1 we can find a  function pn ∈ C(K) satisfying:  (1) 0 ⩽ pn ⩽ 1,  (2) pn|Fn+1 = 1,  (3) pn|F0∪···∪Fn−1∪Un+3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The sequence (pn) is normalized and converges pointwise to 0, so it converges weakly to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover,  observe that  ∥gi − (1 + gi(t0))pn∥∞ ⩽ 1 + 1  n  for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , L} since  ���gi|Un − gi(t0)  ��� ⩽  1  n and pn|(K\\Un) = 0 by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So all the  functions  gi,n :=  n  n + 1 (gi − (1 + gi(t0))pn)  belong to BC(K) and the sequences (gi,n)n∈N converges weakly to gi for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since  the finitely many S′  is are all weakly open, we may thus find some N ⩾ 1 such that gi,n ∈ Si for every  i and every n ⩾ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, the function  gn :=  L  �  i=1  λigi,n  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  21  belongs to C for every n ⩾ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' To conclude, fix t ∈ Fn+1 ⊂ Un+1 ⊂ Un and observe that  Re x∗f(t) ⩾ 1 − 1  n  and that  Re x∗gi,n(t) =  n  n + 1 Re x∗ (gi(t) − (1 + gi(t0))  ⩽  n  n + 1 Re x∗  �  gi(t0) + 1  n − (1 + gi(t0))  �  = −1 +  2  n + 1  for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Hence,  ∥f − gn∥∞ ⩾ Re x∗  �  f(t) −  L  �  i=1  λi Re(gi,n(t))  �  ⩾ 2 − 1  n −  2  n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Combining the previous result with Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1, we get the promised characterization of diame-  tral points in C(K) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let K be a Hausdorff topological compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then the six concepts of diametral  points are equivalent in C(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For vector-valued spaces, the situation is not that easy, but we may provide with some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that, clearly, if t0 is an isolated point of a compact Hausdorff topological space K and X is  a Banach space, then C(K, X) = C(K \\ {t0}, X) ⊕∞ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let K be a Hausdorff topological compact space, let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' and let  f ∈ C(K, X) be a function with ∥f∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) If f ∈ C(K, X) with ∥f∥ = 1 attains its norm at an accumulation point of K, then f is a ccs  Daugavet point (by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2) and hence, f satisfies the six diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) If f ∈ C(K, X) with ∥f∥ = 1 attains its norm at an isolated point t0 and f(t0) is a Dau-  gavet (respectively, super Daugavet, ccs Daugavet) point, then f is a Daugavet (respectively,  super Daugavet, ccs Daugavet) point (by [3, Section 4], Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='28, and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='31,  respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) Suppose that K contains an isolated point t0, let x0 ∈ SX, and let f ∈ C(K, X) be given by  f(t0) = x0 and f(t) = 0 for every t ∈ K \\ {t0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1) If x0 is a ∆- (respectively, super ∆-) point of X, then f is a ∆- (respectively, super ∆-)  point of C(K, X) (by [3, Section 4] and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='25, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2) If x0 is a Daugavet (respectively, super Daugavet, ccs Daugavet) point of X, then f  is a Daugavet (respectively, super Daugavet, ccs Daugavet) point of C(K, X) (by [45,  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11], Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='28 Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='31, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3) If f is a ∆- (respectively, Daugavet) point of C(K, X), then x0 is a ∆- (respectively,  Daugavet) point of X (by [45, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4], [45, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13], respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (4) It is now easy to show that the six diametral notions do not coincide in C(K, X) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Indeed, let K be a compact Hausdorff topological space containing an isolated point t0, let X  a Banach space containing a ∆-point x0 which is not a Daugavet point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' any x0 in the unit  sphere of X = C[0, 1] ⊕2 C[0, 1]), see Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='25 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='27), and consider the function  f ∈ C(K, X) given by f(t0) = x0 and f(t) = 0 for every t ∈ K \\ {t0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, f is a ∆-point by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1) but it is not a Daugavet point by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We are now ready to extend Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 to general L1-predual spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  22  MART´IN, PERREAU, AND RUEDA ZOCA  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be an L1-predual and let x ∈ SX be a ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, x is a ccs Daugavet  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Hence the six diametral notions are equivalent for L1-preduals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If x is a ∆-point in X, then as mentioned in item (3) of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6, we have that JX(x) is  a ∆-point in X∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, X∗∗ is isometric to a C(K) space so Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 gives that JX(x) is a ccs  Daugavet point in X∗∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, using now item (4) of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6 (or using a straightforward argument  based on the principle of local reflexivity as in [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7]), we get that x is a ccs Daugavet point  in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Let us observe that the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 also works for M¨untz spaces (by using [3, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='10]  to provide suitable replacements for the functions pn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We recall that given an an increasing sequence  Λ = (λn)∞  n=0 of non-negative real numbers with λ0 = 0 such that �∞  i=1  1  λi < ∞, then the real Banach  space  M(Λ) := span{tλn : n ⩾ 0} ⊆ C[0, 1]  is called the M¨untz space associated with Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Excluding the constant functions form M(Λ), we have  the subspace M0(Λ) := span{tλn : n ⩾ 1} of M(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  So, adapting the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 to M¨untz spaces (for real scalar-valued functions attaining  its norm at 1 ∈ [0, 1]) and also using [3, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12], we get the following result analogous to  Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X = M(Λ) or X = M0(Λ) for an increasing sequence Λ of non-negative real  numbers with λ0 = 0 such that �∞  i=1  1  λi < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, every ∆-point of X is a ccs Daugavet point (and  hence the six diametral notions are equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Characterization in L1-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] the equivalence between the notions of  Daugavet point and ∆-point was obtained for elements of σ-finite L1-spaces in the real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Actually,  it is not complicated to extend the results to arbitrary measures and also to the complex case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7 ([3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] for the σ-finite real case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let (Ω, Σ, µ) be a measure space, and  let f be a norm one element in L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, the following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) f is a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) f is a ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) The support of the function f contains no atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that (1) implies (2) is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For (2) implies (3), suppose that f is a ∆-point and let  A be an atom of finite measure (the only ones that can be contained in the support of an integrable  function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, we clearly have that L1(µ) = L1(µ|Ω\\A) ⊕1 K (as integrable functions are constant  on atoms), and we may write f = (f1, c) for suitable f1 ∈ L1(µ|Ω\\A) and c = f(A) ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If c ̸= 0, then  ∥f1∥ ̸= 1 and it follows from [45, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4] that 1 ∈ K is a ∆-point, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This shows  that the support of f does not contain any atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  To get that (3) implies (1), we actually prove the following more general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that given a  measured space (Ω, Σ, µ) and a Banach space X, L1(µ, X) denotes the Banach space of all B¨ochner-  integrable functions from Ω to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let (Ω, Σ, µ) be a measured space, let X be a Banach space, and let f be a norm one  element in L1(µ, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If the support of the function f contains no atom, then f is a super Daugavet  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  23  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us write S := supp f which contains no atom by hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us first prove that f is  a super Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since S contains no atoms, we have that L1(µ|S, X) satisfies the Daugavet  property (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [54, Example in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' 81]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, f is a super Daugavet point in this space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since L1(µ, X) = L1(µ|S, X) ⊕1 L1(µ|Σ\\S, X), we get that f is a super Daugavet point in L1(µ, X) by  the transfer results from Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Our next goal is to discuss the relationship with the ccs diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For real L1(µ)-spaces,  and using a result from [1], we may actually get that real-valued integrable functions with atomless  support are ccs ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let (Ω, Σ, µ) be a measured space and let f be a norm one element in the real space  L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If the support of the function f contains no atom, then f is a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take ε > 0 and D := �n  i=1 λiSi a ccs of BL1(µ) containing f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Write f := �n  i=1 λigi with gi ∈ Si  for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consider the measurable subset ˜S := supp f ∪ �n  i=1 supp gi of Ω and let ˜µ be the σ-finite  measure ˜µ := µ| ˜S on ( ˜S, Σ| ˜S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then D induces a ccs ˜D of BL1(˜µ) by restriction of the support which  contains the function ˜f which is just f viewed as an element of L1(˜µ) and hence, the support of ˜f  does not contains atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since ˜f belongs to the unit sphere of the real space L1(˜µ), we have by [1,  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5] that ˜f is an interior point of ˜D for the relative weak topology of BL1(˜µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As we have  already shown that ˜f is a super Daugavet point in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8 (and hence a super ∆-point), we can  find ˜g ∈ ˜D such that  �� ˜f − ˜g  �� > 2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By just considering the extension g of ˜g to the whole Ω by 0,  we get that g ∈ D and that ∥f − g∥ =  �� ˜f − ˜g  �� > 2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Let us comment that it is not clear whether ccs ∆-points transfer through absolute sums, but we  have used specific geometric properties of L1-spaces in the previous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that since [1, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5] is also valid for convex combination of relative  weakly open subsets of BL1(µ), we in fact have that every ∆-point in a real L1(µ) space is actually a  ccw ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Putting together Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8, and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9, we get the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let (Ω, Σ, µ) be a measured space and let f be a norm one element in L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then,  the following notions are equivalent for f: ∆-points, Daugavet point, super ∆-point, and super Dau-  gavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, in the real case, the previous four notions are also equivalent to being ccs  ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We now deal with ccs Daugavet points in L1(µ)-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that if Ω admits an atom A of  finite measure, then we have L1(µ) ≡ L1(µ|Ω\\A) ⊕1 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, in this case L1(µ) fails to have  ccs Daugavet points by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We then have the following characterization of the presence  of a ccs Daugavet point in an L1-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let (Ω, Σ, µ) be a measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, the following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) L1(µ) has the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) L1(µ) contains a ccs Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) L1(µ) has the SD2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (4) µ admits no atom of finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  24  MART´IN, PERREAU, AND RUEDA ZOCA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (1)⇔(4) is well known (see [54, Section 2, Example (b)]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (1)⇒(2) is also known;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (2)⇒(3) is  contained in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally, (3)⇒(4) follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='29 and the comment before  the statement of this proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Remarks on some examples from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Two examples in Lipschitz-free spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In [51], Veeorg constructed a surprising example of a  space satisfying the Radon-Nikod´ym property and containing a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We slightly improve  this result by showing that this point is also a ccs ∆-point by proving a general fact about extreme  ∆-molecules in Lipschitz-free spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For the necessary definitions we refer to the cited paper [51] and  to [9, 10, 32];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' for further background on Lipschitz-free spaces, we refer to the book [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For this purpose, we start by recalling the following characterization of molecules which are ∆-points  on Lipschitz-free spaces from [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13 ([32, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let M be a pointed metric space and let x ̸= y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The  molecule mx,y is a ∆-point if and only if every slice S of BF(M) containing mx,y also contains for  every ε > 0 a molecule mu,v with u ̸= v ∈ M satisfying d(u, v) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In the case in which the molecule is an extreme point, we have the following improved result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let M be a pointed metric space, and let x ̸= y ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If the molecule mx,y is an  extreme point and a ∆-point, then mx,y is a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that this result cannot be obtained from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13: molecules of Lipschitz-free  spaces which are preserved extreme points are denting points, hence very far from being ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  To give the proof of the theorem, we need a result which is just an equivalent reformulation of a  result in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='15 ([32, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let M be a pointed metric space, and let µ ∈ SF(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every  ε > 0, there exists δ > 0 such that given u ̸= v ∈ M with d(u, v) < δ we have ∥µ ± mu,v∥ > 2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Using this result and a homogeneity argument similar to the one from [15, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3], we can  provide the pending proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let C := �n  i=1 λiSi be a ccs of BF(M) containing mx,y and let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since  mx,y is extreme, we have that mx,y ∈ �n  i=1 Si, and by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13 every Si contains molecules  of F(M) supported at arbitrarily close points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='15, we construct inductively for every  η > 0 a finite sequence (mui,vi)n  i=1 of molecules in F(M) such that  (1) mui,vi ∈ Si for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2)  ���mx,y − �k  i=1 λimui,vi  ��� > 1 + �k  i=1 λi − kε  n for every k ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Indeed, since S1 contains molecules of F(M) supported at arbitrarily close points, we can find by  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='15 u1 ̸= v1 ∈ M such that mu1,v1 ∈ S1 and ∥mx,y − mu1,v1∥ > 2 − ε  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It follows that  ∥mx,y − λ1mu1,v1∥ ⩾ ∥mx,y − mu1,v1∥−(1−λ1) > 1+λ1− ε  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us assume that mu1,v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , muk,vk are  constructed as desired for a given k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since Sk+1 contains molecules of F(M) supported  at arbitrarily close points, we can find by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='15 uk+1 ̸= vk+1 ∈ M such that muk+1,vk+1 ∈ Sk+1  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  25  and  ������  mx,y − �k  i=1 λimui,vi  ���mx,y − �k  i=1 λimui,vi  ���  − muk+1,vk+1  ������  > 2 −  ε  n  ���mx,y − �k  i=1 λimui,vi  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Then,  ������  mx,y − �k+1  i=1 λimui,vi  ���mx,y − �k  i=1 λimui,vi  ���  ������  ⩾  ������  mx,y − �k  i=1 λimui,vi  ���mx,y − �k  i=1 λimui,vi  ���  − muk+1,vk+1  ������  −  �  �1 −  λk+1  ���mx,y − �k  i=1 λimui,vi  ���  �  �  > 1 +  λk+1  ���mx,y − �k  i=1 λimui,vi  ���  −  ε  n  ���mx,y − �k  i=1 λimui,vi  ���  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  By the assumption,  �����mx,y −  k+1  �  i=1  λimui,vi  ����� >  �����mx,y −  k  �  i=1  λimui,vi  ����� + λk+1 − ε  n > 1 +  k+1  �  i=1  λi − (k + 1)ε  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As a consequence, µ := �n  i=1 λimui,vi belongs to C and satisfies ∥mx,y − µ∥ > 2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  In particular, we have, as announced, that the molecule mx,y in the example from [51] is a ccs  ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Note that it cannot be a ccs Daugavet point by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12 since the space has the  RNP, but we do not know whether it is a super ∆-point or even a super Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us state  the result for further reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let M be the metric space constructed in [51, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] and let x, y be the points  described there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, F(M) has the RNP, the molecule mx,y is an extreme point of the unit ball of  F(M) which is a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Hence, by our Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14, mx,y is a ccs-∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Another interesting example in the Lipschitz-free space setting is the following one which uses a  metric space constructed by Aliaga, Noˆus, Petitjean, and Proch´azka [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let M be the metric space from [10, Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then one can check that the  molecule m0,q is an extreme point of BF(M) and, since the points 0 and q are discretely connectable, it  follows from an easy adjustment of [32, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2] that this molecule is a ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular,  it follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14 that this molecule is also a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However, it is not difficult to  show that there exists denting points in BF(M) that are at distance strictly less than 2 to m0,q (take  any among the molecules mxn  i ,xn  i+1), so this molecule is not a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also observe that this  space has the RNP since the metric space M is countable and complete [9, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us finally remark that the spaces F(M) of Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='16 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='17 have the RNP, so they  are strongly regular and hence strongly regular points are norm dense, but both examples have ccs  ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' They cannot contain ccs Daugavet points by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us also comment that the use of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14 above cannot be omitted, as the molecule m0,q  is not a preserved extreme point, hence Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13 is again not applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  26  MART´IN, PERREAU, AND RUEDA ZOCA  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' An example of a Banach space with the DD2P, the restricted DSD2P, but containing ccs of  arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In [2, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12], Abrahamsen, H´ajek, Nygaard, Talponen, and Troy-  anski constructed a space X which has the DLD2P, which is midpoint locally uniformly rotund (in  particular, satisfying that pre-ext (BX) = SX), and such that BX contains convex combinations of  slices of arbitrarily small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It then follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13 that every element of SX is  actually a super ∆-point and a ccs ∆-point (that is, X has the DD2P and the restricted DSD2P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' But  containing ccs of arbitrarily small diameter, X fails the SD2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The obvious explanation for the failure  of the SD2P and the fact that every element in the unit sphere is a ccs ∆-point is that none of the  convex combinations of slices of diameter strictly smaller than 2 intersects the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the  other hand, the space X is constructed as the ℓ2-sum of spaces, and so X does not contain Daugavet  points by [3, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6] (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe further that X has the restricted DSD2P and the DD2P, but fails the DSD2P (which is  equivalent to the Daugavet property by [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' An example in a space with one-unconditional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Abrahamsen, Lima, Martiny, and Troy-  anski constructed in [6, Section 4] a Banach space XM with one-unconditional basis which contains a  subset DB ⊆ SXM satisfying:  Every element in DB is both a Daugavet point and a point of continuity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  BXM = co(DB);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DB is weakly dense in the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that no element of DB is a super ∆-point (it is exactly the opposite!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='19, no  element of DB is a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A super ∆-point which fails to be a Daugavet point in an extreme way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In order to  put into a context the following result, let us recall that Daugavet points are at distance 2 from any  denting point (see [32, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' With this in mind, the following result can be interpreted as  the existence of super ∆-points which fail to be Daugavet points in an extreme way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space with the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, for every ε > 0, there  exists an equivalent norm | · | and two points x, y ∈ B(X,|·|) such that  (1) y is a super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) x is strongly exposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) |x − y| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take a subspace Y ⊆ X with dim(X/Y ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that Y has the Daugavet property (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [50, Theorem 6 (a)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take x ∈ SX with 0 < d(x, Y ) < ε (this can be settled taking a non-zero  element v ∈ X/Y with quotient norm smaller than ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, we can find an element y ∈ SY such that  ∥x − y∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By the Hahn-Banach theorem, we can take f ∈ SX∗ with Re f(x) > 0 and f = 0 on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  This means that x belongs to the slice T := {z ∈ BX : Re f(z) > α} for some α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take δ > 0 such  that ∥x−y∥  1−δ  < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 we can find x∗ ∈ SX∗ such that x ∈ S(x∗, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BX) ⊆ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By the above  inclusion we conclude that S(x∗, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BX) ∩ BY = ∅ or, in other words, that Re x∗(z) ⩽ 1 − δ for every  z ∈ BY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Set  B := co(BY ∪ (1 − δ)BX ∪ {±x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  27  B is the unit ball of an equivalent norm |·| which satisfies, in view of the inclusions (1−δ)BX ⊆ B ⊆ BX,  that  ∥x∥ ⩽ |x| ⩽  1  1 − δ∥x∥  for every x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us prove that | · |, x and y satisfies our requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' First, observe that  |x − y| ⩽ ∥x − y∥  1 − δ  < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Next, we claim that y is a super ∆ point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Indeed, since Y has the Daugavet property we can find a net  {ys} ⊆ BY with {ys} −→ y weakly and ∥y − ys∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Notice that the weak convergence {ys} −→ y  is still guaranteed on X because i: (Y, ∥ · ∥) −→ (X, | · |) is weak to weak continuous as ∥ · ∥ and | · |  are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, notice that ys ∈ BY ⊆ B for every s, so |ys| ⩽ 1 for every s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally,  |ys − y| ⩾ ∥ys − y∥ −→ 2,  and since y ∈ BY ⊆ B, we conclude |ys − y| −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' From there, y is clearly a super ∆-point for the  norm | · |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It remains to prove that x is strongly exposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Indeed, we will prove that Re x∗ strongly exposes  B at x, for which it is enough to prove that Re x∗ strongly exposes co(BY ∪ (1 − δ)BX ∪ {±x}) at  x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take z := αu + β(1 − δ)v + (γ − ω)x ∈ co(BY ∪ (1 − δ)BX ∪ {±x}) with α + β + γ + ω = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that 1 − δ < Re x∗(x) ⩽ |x∗| ⩽ ∥x∗∥ due to the inclusion B ⊆ BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Taking into account that  Re x∗(u) ⩽ 1 − δ since u ∈ BY as BY ∩ S(x∗, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BX) = ∅, we conclude  Re x∗(z) ⩽ (1 − δ)(α + β) + (γ − ω) Re x∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since Re x∗(x) > 1 − δ, we get that sup  �  Re x∗(z): z ∈ co(BY ∪ (1 − δ)BX ∪ {±x})  �  = Re x∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If we  take a sequence  zn := αnun + βn(1 − δ)vn + (γn − ωn)x ∈ co(BY ∪ (1 − δ)BX ∪ {±x})  with αn + βn + γn + ωn = 1 such that Re x∗(zn) −→ Re x∗(x), it follows from the previous argument  that αn → 0, βn → 0, ωn → 0 and γn → 1, which means zn → x in norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Using the previous theorem and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='25 it is easy to construct (considering  ℓ2-sums, for instance) a Banach space X containing a sequence of super ∆-points (yn) such that the  distance from yn to the set of strongly exposed points is going to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A super ∆-point which is a strongly regular point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In the present subsection, as well  as in the next, we aim to distinguish the super and ccs notions of ∆- and Daugavet points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The  following result shows that there are plenty of examples of spaces containing super ∆-points which are  strongly regular points (hence far from being ccs ∆-points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We do the construction in for real spaces  for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Every real Banach space with the Daugavet property can be equivalently renormed  so that the new unit ball has a point which is simultaneously super-∆ and a point of strong regularity  (hence, far away of being ccs ∆-point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We will use the following immediate result which follows from the fact that a convex combination  of ccs is again a ccs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let C be a closed, convex, bounded subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then  the set of strongly regular points of C is a convex set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  28  MART´IN, PERREAU, AND RUEDA ZOCA  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space with the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take a 1-codimensional  subspace Y of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since Y is complemented in X then X = Y ⊕ R, so we will see X in such way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take  r > 0, y0 ∈ SY and f ∈ SX∗ such that f(y0) = 1, and consider on X = Y ⊕ R the equivalent norm  | · | whose unit ball is B := co  �  BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It readily follows that | · | agrees  with the original norm ∥ · ∥ on the elements of the form (y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We claim that (y0, 0) satisfies our requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' First of all, let us prove that (y0, 0) is a super-∆  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since Y is one-codimensional, it has the Daugavet property (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' [50, Theorem 6 (a)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Consequently, there exists a net (ys) −→ y0 weakly in BY such that ∥y0 −ys∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, (ys, 0) −→  (y0, 0) weakly in (X, | · |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, it is clear that (ys, 0) ∈ B for every s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally,  |(ys, 0) − (y0, 0)| = |(ys − y0, 0)| = ∥ys − y0∥ −→ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us now prove that (y0, 0) is a point of strong regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  To do so, it is enough, in view of  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='21, to show that (y0, ±r) is a strongly exposed point (we will prove that for (y0, r), being  the other case completely analogous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us prove that Re(f, 1) strongly exposes (y0, r) in the set  BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the one hand, we have  Re(f, 1)(y0, r) = Re f(y0) + r = 1 + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  On the other hand, given (y, 0) ∈ BY × 0 we have Re(f, 1)(y, 0) = f(y) ⩽ 1 < 1 + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover,  Re(f, 1)(y0, −r) = 1 − r and Re(f, 1)(−y0, ±r) = −1 ± r < 1 + r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consequently,  sup{Re(f, 1)(a, b): (a, b) ∈ BY × {0} ∪ {±(y0, r)} ∪ {±(y0, −r)}, (a, b) ̸= (y0, r)}  ⩽ 1 < 1 + r = Re(f, 1)(y0, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  This is enough to guarantee that Re(f, 1) strongly exposes (y0, r) in B, so we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A super Daugavet point which is not ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The previous example shows that we  can distinguish the notion of super ∆-point and the one of ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It seems natural then that  we should be able to distinguish the notions of super Daugavet point and the one of ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In  order to do so, we need to consider an involved construction but, as a consequence, we will prove  that there are super Daugavet points which are contained in convex combinations of slices of small  diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The construction will be very similar to that of [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4], with a slight variation  which makes the resulting norm with a stronger Daugavet flavour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As in the previous subsection, we  will only work with real spaces here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In order to do so, let us recall a construction from Argyros, Odell, and Rosenthal [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Pick a  nonincreasing null sequence {εn} in R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We construct an increasing sequence of closed, bounded and  convex subsets {Kn} in the real space c0 and a sequence {gn} in c0 as follows: First define K1 = {e1},  g1 = e1 and K2 = co(e1, e1 + e2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Choose l2 > 1 and g2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , gl2 ∈ K2 an ε2-net in K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Assume that  n ⩾ 2 and that mn, ln, Kn, and {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , gln} have been constructed, with Kn ⊆ Bspan{e1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=',emn} and  gi ∈ Kn for every 1 ⩽ i ⩽ ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Define Kn+1 as  Kn+1 = co(Kn ∪ {gi + emn+i : 1 ⩽ i ⩽ ln}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Consider mn+1 = mn + ln and choose {gln+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , gln+1} ∈ Kn+1 so that {g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , gln+1} is an εn+1-net  in Kn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally, we define K0 = ∪nKn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then it follows that K0 is a non-empty closed, bounded  and convex subset of c0 such that x(n) ⩾ 0 for every n ∈ N and ∥x∥∞ ⩽ 1 for every x ∈ K0 and so  diam (K0) ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, for a fixed i, we have from the construction that {gi + emn+i}n is a sequence in K0 (for n  large enough) which is weakly convergent to gi, and ∥(gi − emn+i) − gi∥ = ∥emn+i∥ = 1 holds for every  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  29  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then diam (K0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We will freely use the set K0 and the above construction throughout the  subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that, from the above construction, it follows that  K0 = {gi : i ∈ N}  w = {gi : i ∈ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe finally that, by the inductive construction, gi has finite support for every i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  By [11, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2] we have that K0 contains convex combinations of slices of arbitrarily small  diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However, all the points in K0 are “super Daugavet points” in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For every x0 ∈ K0, every ε > 0, and every non-empty weakly open subset W of  K0, there exists y ∈ W satisfying that ∥x0 − y∥ > 1 − ε = diam (K0) − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take ε > 0 and a non-empty relatively weakly open subset of K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By a density argument, we  can find i ∈ N satisfying that ∥x0 − gi∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Again by a density argument there exists gk ∈ W for  certain k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As we explained above, by the definition of K0 we have that the sequence gk +emn+k ∈ K0 for every  n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since  �  gk + emn+k  �  n∈N −→ gk weakly, we can find n ∈ N large enough so that gk + emn+k ∈ W  and mn + k /∈ supp(gi) ∪ supp(gk) (this is possible because the previous set is finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  So taking  y = gk + emn+k, we get y(mm + k) = 1 and so  ∥gi − y∥ ⩾ y(mn + k) − gi(mn + k) = 1 − 0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As a consequence, ∥x0 − y∥ ⩾ ∥gi − y∥ − ∥gi − x0∥ > 1 − ε, and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  It is time to construct the announced renorming of C[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take a sequence of non-empty pairwise  disjoint open subsets Vn of [0, 1] satisfying that 0 /∈ �  n∈N  Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By Urysohn lemma, we can find, for  every n ∈ N, a function hn ∈ SC[0,1] with 0 ⩽ hn ⩽ 1 and such that supp(hn) ⊆ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If we consider  Z := span{hn : n ∈ N}, we get that Z is lattice isometrically isomorphic to c0 (indeed, the mapping  en �−→ hn is an isometric Banach lattice isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consequently, we can consider the set K0  constructed in Z, obtaining that K0 ⊆ BC[0,1] is a set of positive functions (because the latter linear  isometry preserves the lattice structure) which contains convex combination of slices of arbitrarily small  diameter but enjoying the property exhibited in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, by the construction of  the functions hn, f(0) = 0 for every f ∈ Z so, in particular, f(0) = 0 for every f ∈ K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, take 0 < ε < 1 and write  Bε := co  �  2  �  K0 − 1  2  �  ∪ 2  �  −K0 + 1  2  �  ∪ ((1 − ε)BC[0,1] + εBker(δ0))  �  ,  where 1 stands for the constant function 1 in C[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Consider ∥·∥ε the norm on (the real version of) C[0, 1] whose unit ball is Bε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As we have indicated,  the renorming technique follows the scheme of the renorming given in [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4] with the  difference that we use Bker(δ0) instead of Bc0 in the last term because ker(δ0) is a Banach space with  the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The space (X, ∥ · ∥ε) satisfies that:  (1) Every element of 2(K0 − 1  2 ) is a super Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) For every η > 0 there exists a convex combination of slices D of Bε with D ∩ 2(K0 − 1  2 ) ̸= ∅  and such that diam (D) < η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  30  MART´IN, PERREAU, AND RUEDA ZOCA  In particular, there are super Daugavet points which are not ccs−∆ points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take a ∈ K0, and let us prove that 2a − 1 is a super Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In order to do so,  pick a non-empty relatively weakly open subset W of Bε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Write  A := 2(K0 − 1  2 ) and B := (1 − ε)BC[0,1] + εBker(δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since Bε = co(A ∪ −A ∪ B) we have that W has non-empty intersection with co(A ∪ −A ∪ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now  observe that A−A  2  = K0 −K0 ⊆ Bker(δ0) ⊆ B so that co(A∪−A∪B) = co(A∪B)∪co(−A∪B) by [14,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consequently, either W ∩ co(A ∪ B) or W ∩ co(−A ∪ B) is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us distinguish  by cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Assume first that W ∩ co(A ∪ B) is non-empty, so find a′ ∈ K0, f ∈ BC[0,1], g ∈ Bker(δ0), and  α, β ∈ [0, 1] with α + β = 1 satisfying that  α(2a′ − 1) + β((1 − ε)f + εg) ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Take η > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='22, there exists a net (as) −→ a′ weakly with as ∈ K0 for every s and  satisfying that ∥a − as∥ −→ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since (2as − 1) −→ 2a′ − 1 weakly, we can find s large enough so that  α(2as − 1) + β((1 − ε)f + εg) ∈ W  and  ∥(2a − 1) − (2as − 1)∥ = 2∥a − as∥ > 2 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that 2a − 1 and 2as − 1 are functions in BC[0,1] since a, as are positive functions of norm at  most one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since ∥(2a − 1) − (2as − 1)∥ > 2 − η, there exists t0 ∈ [0, 1] and θ ∈ {−1, 1} such that  θ(2a − 1)(t0) > 1 − η and θ(2as − 1)(t0) < −1 + η (observe that t0 ̸= 0 since a(t0) = as(t0) = 0 by  construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consequently, the set  U := {t ∈ [0, 1]: θ(2a − 1)(t) > 1 − η and θ(2as − 1)(t) < −1 + η}  is a non-empty open subset of [0, 1], and we can construct a sequence of non-empty pairwise disjoint  open sets Wn ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that 0 /∈ �  n∈N Wn since 0 /∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take pn ∈ Wn for every n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We  can construct, for every n ∈ N, two functions fn and gn in the unit ball of C[0, 1] satisfying fn = f  and gn = g in [0, 1] \\ Wn and fn(pn) = gn(pn) = −θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that the sequence of functions (f − fn)  have pairwise disjoint supports, so (f − fn) −→ 0 weakly or, in other words, (fn) −→ f weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A  similar argument shows that (gn) −→ g weakly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Notice also that, given n ∈ N, since 0 /∈ Wn then  gn(0) = g(0) = 0, so (gn) ⊆ ker(δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Henceforth α(2as − 1) + β((1 − ε)fn + εgn) is a sequence in Bε  which converges in n weakly to α(2as − 1) + β((1 − ε)f + εg) ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consequently, we can find n large  enough such that α(2as −1)+β((1−ε)fn +εgn) ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Finally, observe that the inclusion Bε ⊆ BC[0,1]  implies that ∥z∥ ⩽ ∥z∥ε, so  ��(2a − 1) − α(2as − 1) − β((1 − ε)fn + εgn)  ��  ε ⩾  ��(2a − 1) − α(2as − 1) − β((1 − ε)fn + εgn)  ��  ⩾ θ((2a − 1) − α(2as − 1) − β((1 − ε)fn)(pn)  = θ(2a − 1)(pn) − θα(2as − 1)(pn)  − θβ((1 − ε)fn(pn) + θεgn(pn))  > 1 − η − α(−1 + η) − β(−1)  = 1 + α + β − (1 + α)η = 2 − 2η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since η > 0 was arbitrary this finishes the case W ∩ co(A ∪ B) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  31  For the case W ∩ co(−A ∪ B) ̸= ∅, find a′ ∈ K0, f ∈ BC[0,1], g ∈ Bker(δ0), and α, β ∈ [0, 1] with  α + β = 1 satisfying that  α(−2a′ + 1) + β((1 − ε)f + εg) ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  This case is simpler because ∥(2a − 1) − (−2a′ + 1)∥ ⩾ (2a − 1) − (−2a′ + 1)(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, an  approximation argument for fn and gn similar to that of the above case (working on a non-empty  open subset of (0, 1) in order to get gn(0) = 0) finishes this case and, consequently, the proof of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The first part of the proof will be a repetition of the argument of [14, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Fix γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  From [11, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2] there exist slices S1, · · · , Sn of K0 such that  diam  �  1  n  n  �  i=1  Si  �  < 1  4(1 − ε)γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We can assume that Si = {x ∈ K0 : x∗  i (x) > 1 − �δ} where 0 < �δ < 1, x∗  i ∈ C[0, 1]∗ and sup x∗  i (K0) = 1  holds for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is clear that  sup x∗  i  �  2(K0 − 1  2 )  �  = 2(1 − x∗  i (1  2 )),  for all i = 1, · · · , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We put ρ, δ > 0 such that 1  2ρ∥x∗  i ∥ + δ < �δ, 2ρ < ε, ρ∥x∗  i ∥ < 4δ, and (7−2ε)ρ  (1−ε)  < γ,  for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We consider the relatively weakly open set of Bε given by  Ui :=  �  x ∈ Bε : x∗  i (x) > 2  �  1 − δ − x∗  i  �1  2  ��  + 1  2ρ∥x∗  i ∥, x(0) = δ0(x) < −1 + ρ2  �  for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is clear that ∥x∗  i ∥ε ⩽ ∥x∗  i ∥ for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n and ∥δ0∥ε = ∥δ0∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since ρ∥x∗  i ∥ < 4δ, we have that 2(1 − x∗  i ( 1  2 )) > 2(1 − δ − x∗  i ( 1  2)) + 1  2ρ∥x∗  i ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, we have that  sup x∗  i (2(K0 − 1  2 )) = 2(1 − x∗  i ( 1  2)), then there exists x ∈ K0 such that  x∗  i (2(x − 1  2 )) > 2(1 − δ − x∗  i (1  2)) + 1  2ρ∥x∗  i ∥ and δ0(2(x − 1  2)) = −1 < −1 + ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  This implies that Ui ̸= ∅ for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In order to estimate the diameter of 1  n  �n  i=1 Ui, it is  enough to compute the diameter of  1  n  n  �  i=1  Ui ∩ co  �  2  �  K0 − 1  2  �  ∪ −2  �  K0 − 1  2  �  ∪ [(1 − ε)BX + εBker(δ0)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since 2(K0 − 1  2 ) and (1 − ε)BC[0,1] + εBker(δ0) are convex subsets of Bε, given x ∈ Bε, we can assume  that x = λ12(a − 1  2 ) + λ22(−b + 1  2 ) + λ3[(1 − ε)x0 + εy0], where λi ∈ [0, 1] with �3  i=1 λi = 1 and  a, b ∈ K0, x0 ∈ BC[0,1], and y0 ∈ Bker(δ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  So given x, y ∈ 1  n  �n  i=1 Ui, for i = 1, · · · , n, there exist ai, a′  i, bi, b′  i ∈ K0, λ(i,j), λ′  (i,j) ∈ [0, 1] with  j = 1, 2, 3 and, xi, x′  i ∈ BC[0,1], and yi, y′  i ∈ BKer(δ0), such that  ui := 2λ(i,1)  �  ai − 1  2  �  + 2λ(i,2)  �  −bi + 1  2  �  + λ(i,3)[(1 − ε)xi + εyi]  u′  i := 2λ′  (i,1)  �  a′  i − 1  2  �  + 2λ′  (i,2)  �  −b′  i + 1  2  �  + λ′  (i,3)[(1 − ε)x′  i + εy′  i]  belong to Ui for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n}, and such that  x = 1  n  n  �  i=1  ui and y = 1  n  n  �  i=1  u′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  32  MART´IN, PERREAU, AND RUEDA ZOCA  For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n} we have that ui ∈ Ui so  δ0(ui) = δ0  �  2λ(i,1)  �  ai − 1  2  �  + 2λ(i,2)  �  −bi + 1  2  �  + λ(i,3)[(1 − ε)xi + εyi]  �  < −1 + ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that, by construction,  δ0  �  ai − 1  2  �  = −1  2, δ0  �  −bi + 1  2  �  = 1  2 and δ0((1 − ε)xi + εyi) = δ0((1 − ε)xi) ⩾ −(1 − ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  This implies that  2λ(i,2) + λ(i,3)ε − 1 = −λ(i,1) + λ(i,2) − λ(i,3)(1 − ε) < −1 + ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since 2ρ < ε, we deduce that λ(i,2) + λ(i,3) < 1  2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As a consequence we get that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1)  λ(i,1) > 1 − 1  2ρ,  and, similarly, we get that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2)  λ′  (i,1) > 1 − 1  2ρ,  for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' the previous inequalities imply that  ∥x − y∥ε ⩽ 1  n  �����  n  �  i=1  2λ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1)  �  ai − 1  2  �  − 2λ′  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1)  �  a′  i − 1  2  ������  ε  + 1  n  n  �  i=1  ����2λ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2)  �  −bi + 1  2  �����  ε  + 1  n  n  �  i=1  ����2λ′  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2)  �  −b′  i + 1  2  �����  ε  + 1  n  n  �  i=1  ∥λ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3)[(1 − ε)xi + εyi]∥ε + 1  n  n  �  i=1  ∥λ′  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3)[(1 − ε)x′  i + εy′  i]∥ε  ⩽ 1  n  �����  n  �  i=1  2λ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1)  �  ai − 1  2  �  − 2λ′  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1)  �  a′  i − 1  2  ������  ε  + 1  n  n  �  i=1  �  λ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2) + λ(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3)  �  + 1  n  n  �  i=1  �  λ′  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2) + λ′  (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3)  �  and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' by using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1),(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2),  ⩽ 1  n  �����  n  �  i=1  2λ(i,1)  �  ai − 1  2  �  − 2λ′  (i,1)  �  a′  i − 1  2  ������  ε  + ρ  ⩽ 2  n  �����  n  �  i=1  λ(i,1)ai − λ′  (i,1)a′  i  �����  ε  + 1  n  n  �  i=1  |λ(i,1) − λ′  (i,1)|∥1∥ε + ρ  ⩽ 2  n  �����  n  �  i=1  λ(i,1)ai − λ′  (i,1)a′  i  �����  ε  + (3 − 2ε)  2(1 − ε)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  33  Now,  �����  n  �  i=1  λ(i,1)ai − λ′  (i,1)a′  i  �����  ε  ⩽  �����  n  �  i=1  (λ(i,1) − 1)ai  �����  ε  +  �����  n  �  i=1  ai − a′  i  �����  ε  +  �����  n  �  i=1  (λ′  (i,1) − 1)a′  i  �����  ε  ⩽  1  1 − ε  �����  n  �  i=1  ai − a′  i  ����� +  n  �  i=1  1  1 − ε|λ(i,1) − 1|∥ai∥ +  n  �  i=1  1  1 − ε|λ′  (i,1) − 1|∥a′  i∥  ⩽  1  1 − ε  �����  n  �  i=1  ai − a′  i  ����� +  1  1 − εnρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (In the previous estimate observe that ∥ai∥ ⩽ 1 and ∥a′  i∥ ⩽ 1 since ai, a′  i ∈ K0 ⊆ Bker(δ0) ⊆ Bε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Hence,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3)  ∥x − y∥ε ⩽  2  1 − ε  �����  1  n  n  �  i=1  ai − a′  i  ����� + (7 − 2ε)  2(1 − ε)ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, in order to prove that the previous norm is small we will prove that both elements 1  n  �n  i=1 ai, 1  n  �n  i=1 a′  i  are elements of 1  n  �n  i=1 Si, which has small diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' To this end, note that  x∗  i  �  2λ(i,1)  �  ai − 1  2  �  + 2λ(i,2)  �  −bi + 1  2  �  + λ(i,3)[(1 − ε)xi + εyi]  �  > 2  �  1 − δ − x∗  i  �1  2  ��  + ρ  2∥x∗  i ∥,  for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then,  x∗  i  �  2λ(i,1)  �  ai − 1  2  ��  + 1  2ρ∥x∗  i ∥ ⩾ x∗  i  �  2λ(i,1)  �  ai − 1  2  ��  + λ(i,2)∥x∗  i ∥ + λ(i,3)∥x∗  i ∥  ⩾ x∗  i  �  2λ(i,1)  �  ai − 1  2  ��  + λ(i,2)∥x∗  i ∥ε + λ(i,3)∥x∗  i ∥ε  ⩾ x∗  i  �  2λ(i,1)  �  ai − 1  2  �  + 2λ(i,2)  �  −bi + 1  2  �  + λ(i,3)[(1 − ε)xi + εyi]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We have that  x∗  i  �  2λ(i,1)  �  ai − 1  2  ��  > 2  �  1 − δ − x∗  i  �1  2  ��  ,  and hence  x∗  i (λ(i,1)ai) > 1 − δ − (1 − λ(i,1))x∗  i  �1  2  �  ⩾ 1 − δ − 1  2ρ∥x∗  i ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We recall that δ+ 1  2ρ∥x∗  i ∥ < �δ, so x∗  i (λ(i,1)ai) > 1−�δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It follows that x∗  i (ai) > 1−�δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, ai ∈ K0∩Si  and, similarly, we get that a′  i ∈ K0 ∩ Si, for every i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Therefore,  1  n  n  �  i=1  ai, 1  n  n  �  i=1  a′  i ∈ 1  n  n  �  i=1  Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  34  MART´IN, PERREAU, AND RUEDA ZOCA  Since the diameter of 1  n  �n  i=1 Si is less than 1  4(1 − ε)γ, we deduce that 1  n∥ �n  i=1 ai − a′  i∥ < 1  4(1 − ε)γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Finally, we conclude from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3) and the above estimate that ∥x − y∥ε ⩽ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Hence, the set C :=  1  n  �n  i=1 Ui has diameter at most γ for the norm ∥ · ∥ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, Bourgain’s lemma (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2) ensures the existence of a convex combination of slices  �pi  j=1 αijTij ⊆ Ui for every 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Using this fact, we will find a convex combination of slices of  B of diameter smaller than γ +  4ρ2  (1− ρ2  ε )ε and such that every slice contains points of 2(K0 − 1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since  ρ and γ can be taken as small as we wish, we will be done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In order to do so, fix 1 ⩽ i ⩽ n and define  Ai :=  �  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , pi}: Tij ∩  �  2K0 − 1  2  �  = ∅  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Bi := {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , pi} \\ Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Given xij ∈ Tij we have that, for j ∈ Ai, that δ0(xij) ⩾ −1 + ε by the definition of the unit ball Bε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since �pi  j=1 αijxij ∈ �pi  j=1 αijTij ⊆ Ui we derive −1 + ρ2 > δ0  ��pi  j=1 αijxij  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Hence  −1 + ρ2 >  �  j∈Ai  αijδ0(xij) +  �  i∈Bi  αijδ0(xij) ⩾ (−1 + ε)  �  j∈Ai  αij −  �  j∈Bi  αij = −1 + ε  �  j∈Ai  αij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  From the above inequality we infer that �  j∈Ai αij < ρ2  ε holds for every 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, we set  Λi := �  j∈Bi λij, which belongs to the interval [1 − ρ2  ε , 1] for 1 ⩽ i ⩽ n and set  D := 1  n  n  �  i=1  �  j∈Bi  αij  ΛI  Tij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that D is a convex combination of slices of Bε since every Tij is a slice of Bε and since  1  n  n  �  i=1  �  j∈BI  αij  Λi  αij = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We claim that D ⊆ C +  2  1− ρ2  ε  ρ2  ε Bε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This is enough to finish the proof because the above condition  implies that  diam (D) ⩽ diam (C) +  4  1 − ρ2  ε  ρ2  ε ⩽ γ +  4  1 − ρ2  ε  ρ2  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  So let us prove the above inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take z := 1  n  �n  i=1  �  j∈Bi  αij  Λi xij ∈ D for certain xij ∈ Tij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Write  z′ := 1  n  �n  i=1  �  j∈Bi αijxij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then  |z − z′| ⩽ 1  n  n  �  i=1  �  j∈Bij  ����1 − 1  Λi  ���� αij|xij| <  1  1 − ρ2  ε  ρ2  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  On the other hand, for 1 ⩽ i ⩽ n and j ∈ Ai take xij ∈ Tij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Define  z′′ := 1  n  n  �  i=1  pi  �  j=1  αijxij ∈ 1  n  n  �  i=1  pi  �  j=1  αijTij ⊆ 1  n  n  �  i=1  Ui = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Moreover, we have  |z′ − z′′| ⩽ 1  n  n  �  i=1  �  j∈Ai  αij < ρ2  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  35  Consequently z = z′′ + (z − z′′) ∈ C +  2  1− ρ2  ε  ρ2  ε Bε since  |z − z′′| ⩽ |z − z′| + |z′ − z′′| <  1  1 − ρ2  ε  ρ2  ε + ρ2  ε <  2  1 − ρ2  ε  ρ2  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A summary of relations between the properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Figure 2 below is an scheme which  complements Figure 1 with the counterexamples following from known results and from the results in  this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  ccs Daugavet  ccs ∆  super Daugavet  super ∆  ∆  Daugavet  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  /  (d)  /  (a)  /  (b)  /  (e)  /  (c)  /  (f)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Scheme of all relations between the diametral notions  Let us list the corresponding counterexamples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (a) The example in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (b) The example in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5 negates this implication in the strongest possible way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (c) Any of the elements in DB in Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' They also show directly that ∆-points are not  necessarily ccs ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (d) Every element of the unit sphere of the space X given in Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 is ccs ∆-point but  not Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Another example is the molecule m0,q of Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (e) In X = C[0, 1]⊕2 C[0, 1], every element in the unit sphere is super ∆-point (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  but X contains no Daugavet point (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also, every element of the unit sphere  of the space X given in Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 is super ∆-point but not Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (f) Any of the elements in DB in Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Diametral-properties for elements of the open unit ball  As mentioned in Section 2, the DSD2P is equivalent to the Daugavet property by [34], but the  ccs ∆-points on the unit sphere of a Banach space do not characterize the DSD2P, but the restricted  DSD2P, which is not equivalent to the Daugavet property (see Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Actually, the elements  in the open unit ball play a decisive role in the proof in [34] of the equivalence between the DSD2P  and the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Our objective here is to introduce and study the diametral notions for  interior points, providing interesting applications, and to investigate the behavior of Daugavet- and  ∆-elements on rays in the unit ball of a given Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The definition of the Daugavet notions for elements in the open unit ball is the natural extension  of the definitions for elements of norm one given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  36  MART´IN, PERREAU, AND RUEDA ZOCA  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We say that  (1) x is a Daugavet point if supy∈S ∥x − y∥ = ∥x∥ + 1 for every slice S of BX,  (2) x is a super Daugavet point if supy∈V ∥x − y∥ = ∥x∥ + 1 for every non-empty relatively weakly  open subset V of BX,  (3) x is a ccs Daugavet point if supy∈C ∥x − y∥ = ∥x∥ + 1 for every ccs C of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It turns out that the existence of a non-zero Daugavet kind element actually forces the whole ray  to which it belongs to be composed of similar elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) x is a Daugavet- (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' super Daugavet-, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' ccs Daugavet-) point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) rx is a Daugavet- (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' super Daugavet-, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' ccs Daugavet-) point for every r ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) rx is a Daugavet- (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' super Daugavet- , resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' ccs Daugavet-) point for some r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us recall the following elementary but very useful result from [34] due to Kadets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 ([34, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a normed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If x, y ∈ X and ε > 0 satisfies that  ∥x + y∥ > ∥x∥ + ∥y∥ − ε,  then for every a, b > 0, it is satisfied that  ∥ax + by∥ > a ∥x∥ + b ∥y∥ − max{a, b}ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We will only do the proof for Daugavet points, being the other cases com-  pletely analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So let us first assume that x is a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Take r ∈ [0, 1], ε > 0, and S  a slice of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, there exists y ∈ S such that ∥x − y∥ > 2 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, ∥y∥ > 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As  ∥y∥ ⩽ 1,  ∥x − y∥ > 2 − ε ⩾ ∥x∥ + ∥y∥ − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 that  ∥rx − y∥ > ∥rx∥ + ∥y∥ − ε > ∥rx∥ + 1 − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Hence, rx is also a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, let us assume that rx is a Daugavet point for some r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Again take ε > 0 and S slice  of BX, and pick y ∈ S such that ∥rx − y∥ > ∥rx∥ + 1 − rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, ∥y∥ > 1 − rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' As ∥y∥ ⩽ 1,  we have  ∥rx − y∥ > ∥rx∥ + ∥y∥ − rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Hence, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3, we get that  ∥x − y∥ > ∥x∥ + ∥y∥ − ε > 2 − (1 + r)ε  and so x is a Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  As mentioned in the discussion preceding Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12, the presence of a ccs Daugavet point in  a given Banach space forces the space to satisfy the SD2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This can now be viewed as a consequence  of the previous proposition and the following immediate reformulation of [41, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 ([41, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, X has the SD2P if and only  if 0 is a ccs Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  37  Note that c0 has the SD2P but has no non-zero Daugavet points (use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1, for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Compare Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 with the following obvious remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A Banach space X is infinite-dimensional if and only if 0 is a super Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  We also mention that 0 is always a Daugavet point (in finite or infinite dimension), as every slice  of the unit ball has to intersect the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us also point out that [41, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1] admits the following scaled version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, and let r ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, the following assertions are  equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) Every ccs of BX has diameter greater than or equal to 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) sup{∥x∥: x ∈ C} ⩾ r for every ccs C of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) sup{∥x∥: x ∈ D} ⩾ r for every symmetric ccs D of BX (so containing 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Suppose (1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, for any given ccs C of BX, and for any fixed ε > 0, there exists  x, y ∈ C such that ∥x − y∥ > 2r − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, it follows that ∥x∥ > r − ε or ∥y∥ > r − ε, giving  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (2)⇒(3) is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Suppose that (1) fails, that is, that there exists a ccs C of BX and ε > 0  such that diam (C) ⩽ 2r − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We consider the ccs D of BX given by D := 1  2(C − C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then D is  symmetric, and for every u := x−y  2  and u′ := x′−y′  2  in D we have ∥u − u′∥ =  ��� x+y′  2  − x′+y  2  ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now,  x, x′, y, y′ belong to C, and C is convex, so x+y′  2  and x′+y  2  do also belong to C, so ∥u − u′∥ ⩽ 2r − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Being D symmetric, it implies that D ⊂ (r − ε)BX, hence (3) fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  The following is a nice consequence of the proposition above outside the diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, BX contains ccs of arbitrarily small diameter if and  only if 0 is a strongly regular point of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now let us consider the ∆ notions for points of the open unit ball which are just the adaptation of  the notions given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We say that  (1) x is a ∆-point if supy∈S ∥x − y∥ = ∥x∥ + 1 for every slice S of BX containing x,  (2) x is a super ∆-point if supy∈V ∥x − y∥ = ∥x∥ + 1 for every non-empty relatively weakly open  subset V of BX containing x,  (3) x is a ccs ∆-point if supy∈C ∥x − y∥ = ∥x∥ + 1 for every slice ccs C of BX containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  With this definitions in hands, we may get an improvement of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, X has the SD2P if and only if 0 is ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Compare the previous corollary with the following obvious remark which is analogous to Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A Banach space X is infinite-dimensional if and only if 0 is a super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that the definition of ccs ∆-points for elements in BX gives a localization of the DSD2P,  that is, X has the DSD2P (and hence the Daugavet property [34]) if and only if all the elements of  BX are ccs ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that the DSD2P is not equivalent to the restricted DSD2P (meaning that  all points in SX are ccs ∆-points), see Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  38  MART´IN, PERREAU, AND RUEDA ZOCA  The following result is a localization of Kadets’ theorem [34] on the equivalence of the DSD2P and  the DPr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If rx is a ccs ∆-point for every r ∈ (0, 1),  then x is a ccs Daugavet point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, it is enough that inf{r ∈ (0, 1): rx is a ccs ∆-point} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Fix a ccs C of BX and ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since ˜C := 1  2(C − C) is also a ccs of BX and since 0 ∈ ˜C is a  norm interior point of ˜C by [41, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1], we have that rx belongs to ˜C for every r ∈ (0, δ)  for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By hypothesis, there is r > 0 such that rx is a ccs ∆-point and rx ∈ ˜C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So there  exists y ∈ ˜C such that ∥rx − y∥ > r + 1 − rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then if we write y := y1 − y2 with y1, y2 ∈ C, we have  ∥rx − y1∥ > r +1−rε or ∥rx − y2∥ > r +1−rε by the triangle inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' in particular, ∥y1∥ > 1−rε  and ∥y2∥ > 1 − rε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In both cases, we have that there is y ∈ C such that ∥rx − y∥ > r∥x∥ + ∥y∥ − rε  and it follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 that  ∥x − y∥ > ∥x∥ + ∥y∥ − ε > 2 − (1 + r)ε > 2 − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  At this point, it is natural to ask whether an equivalent formulation of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 is valid for  some of the various ∆-notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For ccs ∆-points, the answer is negative as follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11  and, for instance, the example in Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Another, maybe simpler, example showing that is  the following one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us assume that a positive measure µ admits an atom of finite measure and also  has a non-empty non-atomic part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, the real space L1(µ) contains no ccs Daugavet point by  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However, it contains elements in the unit sphere which are ccs ∆-points and super  Daugavet points by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, as a consequence of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11  and of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2, there must exist f in the unit sphere which is a ccs ∆-point and t ∈ (0, 1) such  that tf is not a ccs ∆-point but it is a super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For ∆-points, we have the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If x is a ∆-point, then rx is a ∆-point  for every r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us assume that x is a ∆-point and let us fix r ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Take ε > 0 and a slice S of  BX containing rx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now, either x belongs to S or −x belongs to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In the first case, we can find  y ∈ S such that ∥x − y∥ ⩾ 2 − ε, and using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3, we get ∥rx − y∥ ⩾ ∥rx∥ + 1 − 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Else,  ∥rx − (−x)∥ = r + 1 = ∥rx∥ + 1 and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  For super ∆-points, it is currently quite obscure whether they behave like ∆-points up on rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Kuratowski measure and large diameters  Let M be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The Kuratowski measure of non-compactness α(A) of a non-empty  bounded subset A of M is defined as the infimum of all real numbers ε > 0 such that A can be covered  by a finite number of subsets of M of diameter smaller than or equal to ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  From the definition, we clearly have α(A) = 0 if and only if A is totally bounded (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' precompact).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It follows that every complete subset A of M with α-measure 0 is compact, and in particular, if M is  a complete metric space, that α(A) = 0 if and only if A is compact, where A stands for the closure of  the set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The α-measure can be thus seen as a way to measure how far a given (non-empty) bounded  and closed subset of M is from being a compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It was introduced by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Kuratowski in [38] in  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  39  order to provide a generalization of the famous intersection theorem from Cantor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A general theory  on measures of non-compactness was later developed, and it turned out to provide important results  in metric fixed point theory, and in particular to have applications in functional equations or optimal  control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We refer e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' to [12] for an introduction to the topic and for more precise applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that A ⊆ B implies α(A) ⩽ α(B), and that α(A) = α(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Also note that α(A ∪ B) =  max{α(A), α(B)} for every non-empty bounded subsets A, B of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Furthermore, if M = X is a  normed space, then α is known to enjoy additional useful properties: it is symmetric, translation  invariant, positively homogeneous, sub-additive, and satisfies α(co A) = α(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The α-measure has  proved to be a powerful tool for the study of the geometry of Banach spaces and we refer e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' to the  works [46], [47] and [44] in connection with property (α), with drop property, and with an isomorphic  characterization of reflexive Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  From the definition it is clear that the Kuratowski measure of A is smaller than or equal to its  diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Obviously, equality does not always hold, but a fruitful relationship between the notion  of ∆-points and the Kuratowski measure of slices was discovered in [5] and completed in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In  particular, the following result was obtained (see [52, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If x is a ∆-point, then α(S) = 2 for every  slice S of BX containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Besides, α(S(x, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BX∗)) = 2 in BX∗ for every δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that the converse does not hold in general, as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consider X := L1([0, 1]) ⊕∞ ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It follows that both X and X∗ enjoy the SD2P [15,  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6], so Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 below implies that given any slice S = S(x∗, δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' BX) we have α(S) = 2  and the same holds for the slices in the dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' However, there are points which are not ∆-points because  X fails the DLD2P [30, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2] since ℓ1 fails it, so it remains to take any point x ∈ SX which  is not a ∆-point to get the desired counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that the connection between having big slices in diameter and having big slices in Kura-  towski index goes beyond Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The following result was first pointed out in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 ([20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let β ∈ (0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The following  assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) Every slice of BX has diameter greater than or equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) Every slice of BX has Kuratowski measure greater than or equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In this section, we aim to prove analogues to this result for relative weakly open subsets, as well  as for convex combinations of slices or of weakly open sets, and to extend Veeorg’s result to super  ∆-points and ccw ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Kuratowski measure and diameter two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The analogue to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 for  non-empty weakly open subsets is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let β ∈ (0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) Every non-empty relatively weakly open subset of BX has diameter greater than or equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) Every non-empty relatively weakly open subset of BX has Kuratowski measure greater than or  equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (2)⇒(1) is immediate, so let us prove (1)⇒(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' To this end, fix β ∈ (0, 2] and assume that  every non-empty relatively weakly open subset of BX has diameter greater than or equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then  40  MART´IN, PERREAU, AND RUEDA ZOCA  pick ε > 0, and let us prove by induction on n that for every non-empty relatively weakly open subset  W of BX and for every finite collection C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , Cn of subsets of X with diam (Ci) ⩽ β − ε for every  i, we have that W ̸⊂  n�  i=1  Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For n = 1, it is clear since by assumption diam (W) ⩾ β > β − ε for every non-empty relatively  weakly open subset W of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  So assume that the result is true for every non-empty relatively weakly open subset W of BX  and for every collection of n sets, and let us prove the result for collections of n + 1 sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' To this  end, consider C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , Cn, Cn+1 be subsets of X with diam (Ci) ⩽ β − ε for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that  diam (Ci) = diam (Ci  w) ⩽ β − ε by w-lower semicontinuity of the norm of X, so that we may and do  assume that Ci is weakly closed for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that by the case n = 1 we have that W ̸⊂ Cn+1, which means that W \\ Cn+1 is non-  empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, it is a weakly open subset of BX since Cn+1 is assumed to be weakly closed, and by  induction hypothesis we conclude that W\\Cn+1 ̸⊂  n�  i=1  Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular W ̸⊂  n+1  �  i=1  Ci and the theorem  is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Next, let us establish the analogue Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3 for convex combinations of slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' To this end,  observe that by Bourgain lemma (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2) every convex combination of non-empty relatively  weakly open subsets of BX contains a convex combination of slices of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This assertion makes valid  the following lemma which allows us to focus our attention in convex combination of weakly open  subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) The following are equivalent:  (a) Every convex combination of slices of BX has diameter greater than or equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (b) Every convex combination of non-empty relatively weakly open subsets of BX has diameter  greater than or equal to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) The following are equivalent:  (a) α(C) ⩾ r holds for every convex combination C of slices of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (b) α(D) ⩾ r holds for every convex combination D of non-empty relatively weakly open  subsets of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now we are able to give the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let β ∈ (0, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The following assertions are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) Every convex combination of slices of BX has diameter greater than or equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) Every convex combination of slices of BX has Kuratowski measure greater than or equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' (2)⇒(1) is immediate, so let us prove (1)⇒(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' To this end, fix β ∈ (0, 2] and assume that every  convex combination of non-empty relatively weakly open subsets of BX has diameter greater than or  equal to β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then pick ε > 0, and let us prove by induction on n that for every D convex combination  of non-empty relatively weakly open subsets of BX and for every finite collection C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , Cn of subsets  of X with diam (Ci) ⩽ β − ε for every i, we have that D ̸⊂  n�  i=1  Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  For n = 1 it is clear since by assumption diam (D) ⩾ β > β − ε for every convex combination of  non-empty relatively weakly open subsets of D of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  41  Assume by inductive step that the result stands for n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now pick D convex combination of non-empty relatively weakly open subsets of BX and a finite  collection C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , Cn+1 of subsets of X with diam (Ci) ⩽ β − ε for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We can assume as in the  proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4 that every Ci is weakly closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Write D = �k  i=1 λiWi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that by the case  n = 1 we have that D ̸⊆ Cn+1, so there exists z ∈ D \\ Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since z ∈ D we can write z = �k  i=1 λixi  where xi ∈ Wi holds for every 1 ⩽ i ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, since z = �k  i=1 λixi /∈ Cn+1, this means that  z = �k  i=1 λixi ∈ X \\ Cn+1, and the latter is a weakly open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By a weak-continuity argument of  the sum we can find weakly open subsets Vi of BX, with xi ∈ Vi for every 1 ⩽ i ⩽ k, satisfying that  z = �k  i=1 λixi ∈ �n  i=1 λiVi ⊆ X \\ Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Up to taking smaller Vi, we can assume Vi ⊆ Wi for every  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Now call ˜D := �k  i=1 λiVi, which is a convex combination of weakly open subsets of BX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By the  inductive step we get that ˜D ̸⊆  n�  j=1  Cj, so there exists y ∈ ˜D with y /∈ Cj for 1 ⩽ j ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Observe that  the condition Vi ⊆ Wi implies ˜D ⊆ D, so y ∈ D indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Moreover, ˜D ⊆ X \\Cn+1 implies in particular  y /∈ Cn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' This implies that y ∈ D \\  n+1  �  i=1  Ci, which is precisely what we wanted to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Kuratowski measure and ∆-notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We now prove an analogue to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 for super  ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX be a super ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then every non-empty  relatively weakly open subset W of BX containing x satisfies that α(W) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The proof will be an obvious consequence of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, x ∈ SX be a super ∆ point, and W be a weakly open  subset of BX such that x ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, for every ε > 0, there exists a sequence {xn} ⊆ W such that  ∥xi − xj∥ > 2 − ε holds for every i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Set ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us construct by induction a sequence {xn} satisfying that ∥x − xi∥ > 2 − ε  2 and  such that ∥xi − xj∥ > 2 − ε for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Using that x is a super ∆ point select, by the definition of super ∆, a point x1 ∈ W with ∥x−x1∥ >  2 − ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now, assume that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , xn have been constructed and let us construct xn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' By the properties  defining the sequence observe that, given 1 ⩽ i ⩽ n, we have ∥x−xi∥ > 2− ε  2, so we can find gi ∈ SX∗  with Re gi(x − xi) > 2 − ε  2, which implies Re gi(x) > 1 − ε  2 and Re gi(xi) < −1 + ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Consequently  x ∈ V := W ∩  n�  i=1  S  �  gi, ε  2  �  ,  which is a weakly open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Since x is super ∆ we can find xn+1 ∈ V such that ∥x−xn+1∥ > 2− ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In  order to finish the construction we only must to prove that ∥xi−xn+1∥ > 2−ε holds for every 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  But this is clear because, given 1 ⩽ i ⩽ n, the condition xn+1 ∈ V implies that Re gi(xn+1) > 1 − ε  2,  so  ∥xn+1 − xi∥ ⩾ Re gi(xn+1 − xi) > 1 − ε  2 + 1 − ε  2 = 2 − ε,  and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  Note that a similar statement than Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7 can be established for ccw ∆ points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  42  MART´IN, PERREAU, AND RUEDA ZOCA  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX be a ccw ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then every non-empty  convex combination D of relatively weakly open subsets of BX containing x satisfies that α(D) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As in the previous case, the proof follows directly from the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, x ∈ SX be a ccw ∆ point, and D a ccw of BX such  that x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Then, for every ε > 0, there exists a sequence {xn} ⊆ D such that ∥xi − xj∥ > 2 − ε holds  for every i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Set ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Write D := �k  i=1 λiWi with λi ̸= 0 for every i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Set δ :=  ε  2 min1⩽i⩽k λi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let us construct  by induction a sequence {xn} ⊆ D satisfying that ∥x − xi∥ > 2 − δ and such that ∥xi − xj∥ > 2 − ε  for i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Using that x is a ccw ∆ point select, by the definition of ccw ∆, a point x1 ∈ D with  ∥x − x1∥ > 2 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Now assume that x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' , xn have been constructed and let us construct xn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We can write x =  �k  j=1 λjxj and xi := �k  j=1 λjxi  j as being elements of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  By the properties defining the sequence, observe that, given 1 ⩽ i ⩽ n we have ∥x − xi∥ > 2 − δ, so  we can find gi ∈ SX∗ with  Re gi(x − xi) =  k  �  j=1  λj Re gi(xj − xi  j) > 2 − δ = 2 −  ε  2 min1⩽j⩽n λj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A convexity argument implies that Re gi(xj − xi  j) > 2 − ε  2 holds for every 1 ⩽ j ⩽ k, which implies  that  Re gi(xj) > 1 − ε  2 and  Re gi(xi  j) < −1 + ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Observe that  xj ∈ Vi := Wi ∩  n�  i=1  S  �  gi, ε  2  �  ,  which is a weakly open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Since x is a ccw ∆-point and x ∈ �k  j=1 λjVj, we can find a point  xn+1 = �k  j=1 λjzj ∈ �k  j=1 λjVj ⊆ D such that ∥x − xn+1∥ > 2 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In order to finish the construction  we only must to prove that ∥xi − xn+1∥ > 2 − ε holds for every 1 ⩽ i ⩽ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Given 1 ⩽ j ⩽ k, the  condition zj ∈ Vj implies Re gi(zj) > 1 − ε  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the other hand, Re gi(xi  j) < −1 + ε  2, so  ∥xn+1 − xi∥ ⩾ Re gi(x − xi) =  k  �  j=1  λj Re gi(zj − xi  j) > (2 − ε)  k  �  j=1  λj = 2 − ε,  and the proof is finished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Commented open questions  The only implications between properties which is not known to hold or not is the following one  (see Figure 2 in page 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX be a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Is x a super ∆-point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us give some comments on this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the one hand, it may look that the answer is  positive by Bourgain’s lemma (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2), but this lemma does not say that, in general, given an  element x of a relative weak open subset W of BX, there is a convex combination of slices of BX  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  43  contained in W and containing x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' The later happens when x ∈ co(pre-ext (BX)) (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3) so,  the answer to Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 is positive in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' On the other hand, a possible counterexample to  this problem could be the molecules in Examples 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='16 or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='17, which are known to be ccs ∆-points  and are extreme points but not preserved extreme points (hence they do not belong to the convex hull  of the set of preserved extreme points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A way to show that these molecules are not super ∆-points  would be to investigate whether RNP spaces may contains super ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='19 states that real Banach spaces with a one-unconditional basis do neither contain  super ∆-points nor ccs ∆-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It is likely that such result also holds true in the complex setting  since we do believe that the preliminary results from [6] are also valid for complex scalars, provided  that one works with the suitable notion of one-uncondional bases (for which [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3] holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Also, we also expect that the results there can be easily extended to one-uncondional FDDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Yet,  since a sharper version of this result was obtained in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='20 for super ∆-points in a very  general setting, it is natural to ask whether improved results could be simultaneously obtained in both  directions for ccs ∆-points by proving an analogue to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' So let us ask the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space, and let us assume that there exists a subset A ⊆ F(X, X)  satisfying that sup  �  ∥Id − T∥ : T ∈ A  �  < 2 and that for every ε > 0 and every x ∈ X, there exists  T ∈ A such that ∥x − Tx∥ < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Can X contain a ccs ∆-point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  A negative answer to this question would be interesting, since it would provide an example of a ccs  ∆-point that is not a super ∆-point, hence a negative answer to Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Another interesting question could be if a point of continuity could be a ccs ∆-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Let us  formalize the questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) Does X fail the RNP (or even the CPCP) if contains a super ∆-point or a super Daugavet  point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) Is it possible for a point of continuity being a ccs ∆-point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The surprising examples given in Section 4 shows that the mere existence of some diametral notions  (but ccs Daugavet points) on a Banach space does not imply that the whole space has any diameter  two property nor the Daugavet property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Our question here is how many diametral points has to  contain a Banach space to have any diameter two property or the Daugavet property or fails to have  the RNP or one-unconditional basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' How big can be the set of Daugavet points, super Daugavet points, ∆-points, super  ∆-points, or ccs ∆-points in a Banach space with the Radon-Nikod´ym property, or with the CPCP,  or being strongly regular, or having one-unconditional basis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Concerning isometric consequences of the existence of diametral points, there are some recent results  showing that a Banach space containing a ∆-point cannot be uniformly non-square [5] or even locally  uniformly non-square [37], or asymptotic uniformly smooth [5, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Also, a Banach space having  an unconditional basis with suppression-unconditional constant less that 2 cannot contains super ∆-  points and a Banach space containing a ccs Daugavet point has the SD2P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Taking into account that  it is not known if there exists an stictly convex Banach space with the Daugavet property (see [33,  Section 5]), the following question makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Recall that Paragraph 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 shows an example of an  strictly convex Banach space in which every norm-one element is a ccs ∆-point and a super ∆-point,  but it does not contain any Daugavet point by the way in which it is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  44  MART´IN, PERREAU, AND RUEDA ZOCA  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Is there an strictly convex Banach space containing a Daugavet point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  In view of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13 and of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14, the following question makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Suppose that x ∈ ext (BX) is a ∆-point, does this imply  that x is a ccs ∆-point or a super ∆-point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  By now, the only isomorphic restriction which is known for a Banach space to contain ∆-points  or even Daugavet points is that it cannot be finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' It would be interesting to find some  more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Find isomorphic restrictions for a Banach space to contain ∆-points or any of the  other diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' In particular, is it possible for a reflexive or even super-reflexive Banach  space to contain ∆-, super ∆-, ccs ∆-, Daugavet or super Daugavet points?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The results about absolute sums in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 are not complete in the case of super Daugavet  points and they are even less clear in the case of ccs notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Here are two possible questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X, Y be Banach spaces and let N be an absolute sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) If N is A-octahedral, x ∈ SX and y ∈ SY are super Daugavet points, is (ax, by) a super  Daugavet point in X ⊕N Y when a, b satisfy the conditions in the definition of A-octahedrality?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) If N is the ℓ∞-sum, x ∈ SX and y ∈ SY are ccs ∆-points, are the elements of the form (ax, by)  ccs ∆-points in X ⊕∞ Y for a, b ∈ [0, 1] with max{a, b} = 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It would be also desirable to study the reversed results to those in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 as it is done in  [45] for ∆-points and Daugavet points (see the tables in pages 86 and 87 of [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X, Y be Banach spaces, let N be an absolute sum, x ∈ SX, y ∈ SY , and a, b ⩾ 0  such that N(a, b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Discuss what happens with x and y supposing that (ax, by) satisfies any of the  six diametral notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  It maybe the case that some of the arguments given in Subsections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='2 can be adapted to  other classes of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' We propose some possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Characterize the six diametral notions in uniform algebras, in Lorentz spaces and  their isometric preduals, and in some vector-valued function spaces as C(K, X) or L∞(µ, X) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The relations between the weak-star versions of the diametral points (see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6) are not yet  clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' For instance, the following questions arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) Is JX(x) a ccs ∆-point in X∗∗ if x is a ccs ∆-point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) Is there any relationship between the DD2P in X and the weak-star super ∆-points in SX∗?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As commented in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='19, a Banach space X containing a sequence (yn) of super ∆-points  such that the distance of yn to the set of strongly exposed points of BX is going to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' But the  following question remains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Can a super ∆-point (or even a ∆-point) belong to the closure of the set of denting  points?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  DIAMETRAL NOTIONS FOR ELEMENTS OF THE UNIT BALL OF A BANACH SPACE  45  The answer to the next question on the behaviour of ∆- and super ∆-points in rays is still unknown,  as we commented in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Let X be a Banach space and let x ∈ SX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (1) If rx is a ∆-point for some 0 < r < 1, does this imply that x is a ∆-point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (2) If rx is a super ∆-point for some 0 < r < 1, does this imply that x is a auper ∆-point?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  (3) If x is a super ∆-point, does this imply that rx is a super ∆-point for all 0 < r < 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  As we proved in Section 6, every relative weakly open subset which contains a super ∆-point  (respectively, a ccw ∆-point) has Kuratowski measure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Our proofs do not seem to work for convex  combination of slices, so let us ask the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' If a ccs of the unit ball contains a ccs ∆-point, does it necessarily have maximal  Kuratowski measure?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Acknowledgments  Part of this work was done during the visit of the second named author at the University of Granada  in September 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' He wishes to thank his colleagues for the warm welcome he received, and to thank  all the people that made his visit possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The authors thank Gin´es L´opez-P´erez for fruitful conversations on the topic of the paper, specially  for providing enlightening ideas in connection with the example of Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  The authors also thank Trond A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Abrahamsen, Andr´e Martiny, and Vegard Lima for valuable discus-  sions on the topic of the paper, and in particular for pointing out the ccs version of [6, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='12]  that is presented in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  References  [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Abrahamsen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Becerra Guerrero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Haller, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Lima, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' P¨oldvere, Banach spaces where convex  combinations of relatively weakly open subsets of the unit ball are relatively weakly open, Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Abrahamsen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' H´ajek, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Nygaard, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Talponen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Troyanski, Diameter 2 properties and convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Studia Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' 232 (2016), 227–242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  [3] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Abrahamsen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Haller, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Lima, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Pirk, Delta- and Daugavet points in Banach spaces, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Edinb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' 63 (2020), 475–496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  [4] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Abrahamsen and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Lima, Relatively weakly open convex combinations of slices, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  146 (2018), 4421–4427.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  [5] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Abrahamsen, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Lima, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Martiny, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Perreau, Asymptotic geometry and Delta-points, Banach J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' 16 (2022), article 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='  [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
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+page_content=' Departamento de An´alisis Matem´atico, 18071  Granada, Spain  ORCID: 0000-0003-4502-798X  Email address: mmartins@ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='es  URL: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
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+page_content='es/local/mmartins  (Perreau) University of Tartu, Institute of Mathematics and Statistics, Narva mnt 18, 51009 Tartu  linn, Estonia  ORCID: 0000-0002-2609-5509  Email address: yoel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='perreau@ut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='ee  (Rueda Zoca) Universidad de Granada, Facultad de Ciencias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content=' Departamento de An´alisis Matem´atico,  18071 Granada, Spain  ORCID: 0000-0003-0718-1353  Email address: abrahamrueda@ugr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='es  URL: https://arzenglish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
+page_content='wordpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfSglq/content/2301.04433v1.pdf'}
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+Springer Nature 2021 LATEX template
+Multilingual Entity and Relation Extraction
+from Unified to Language-specific Training
+Zixiang Wang1, Jian Yang1, Tongliang Li1, Jiaheng
+Liu1, Ying Mo1, Jiaqi Bai1, Longtao He2 and Zhoujun Li1*
+1State Key Lab of Software Development Environment, Beihang
+University, Beijing, Beijing, China.
+2National Computer Network Emergency Response Technical
+Team/Coordination Center of China, Beijing, Beijing, China.
+*Corresponding author(s). E-mail(s): lizj@buaa.edu.cn;
+Contributing authors: wangzixiang@buaa.edu.cn;
+jiaya@buaa.edu.cn; tonyliangli@buaa.edu.cn;
+liujiaheng@buaa.edu.cn; moying@buaa.edu.cn; bjq@buaa.edu.cn;
+hlt@cert.org.cn;
+Abstract
+Entity and relation extraction is a key task in information extraction,
+where the output can be used for downstream NLP tasks. Existing
+approaches for entity and relation extraction tasks mainly focus on
+the English corpora and ignore other languages. Thus, it is critical to
+improving performance in a multilingual setting. Meanwhile, multilingual
+training is usually used to boost cross-lingual performance by trans-
+ferring knowledge from languages (e.g., high-resource) to other (e.g.,
+low-resource) languages. However, language interference usually exists
+in multilingual tasks as the model parameters are shared among all
+languages. In this paper, we propose a two-stage multilingual train-
+ing method and a joint model called Multilingual Entity and Relation
+Extraction framework (mERE) to mitigate language interference across
+languages. Specifically, we randomly concatenate sentences in differ-
+ent languages to train a Language-universal Aggregator (LA), which
+narrows the distance of embedding representations by obtaining the
+unified language representation. Then, we separate parameters to mit-
+igate interference via tuning a Language-specific Switcher (LS), which
+includes several independent sub-modules to refine the language-specific
+1
+arXiv:2301.04434v1  [cs.CL]  11 Jan 2023
+
+Springer Nature 2021 LATEX template
+2
+Article Title
+feature representation. After that, to enhance the relational triple extrac-
+tion, the sentence representations concatenated with the relation feature
+are used to recognize the entities. Extensive experimental results show
+that our method outperforms both the monolingual and multilingual
+baseline methods. Besides, we also perform detailed analysis to show
+that mERE is lightweight but effective on relational triple extraction
+and mERE is easy to transfer to other backbone models of multi-field
+tasks, which further demonstrates the effectiveness of our method.
+Keywords: Joint extraction, Information extraction, Multilingual entity and
+relation extraction, Relational triple
+1 Introduction
+Entity and relation extraction (ERE) contains two sub-tasks called named
+entity recognition (NER) [1–4] and relation classification (RC) [5, 6], which is
+the fundamental step of automatic knowledge graphs (KGs) [7] construction,
+knowledge discovery and intelligent question answering system. The results of
+ERE are typically described as a relational triple (h, r, t), where h and t are
+the head entity and the tail entity, respectively, and r denotes the relation
+between them. For example, for the sentence “Big Ben is in UK.” with a
+predefined relation called “Locate in”, an ideal relational triple of this sentence
+is expressed as (Big Ben, Locate in, UK).
+As a large amount of data is available from different languages on the
+Internet, it is important to utilize such valuable resources and develop multilin-
+gual entity and relation extraction models, which can operate across language
+barriers. However, most existing methods propose to solve ERE on English
+corpora, which can only deal with the monolingual extraction task. The main
+reason is that many languages suffer from the scarcity of corpora in ERE.
+Thus, multilingual training is proposed to help each other in a shared model,
+where the well-trained knowledge of high-resource languages can be trans-
+ferred to low-resource languages with a small amount of data. Recently, [8]
+propose a multilingual dataset called SMiLER, which is the first work to apply
+both monolingual and multilingual training. The authors in [8] introduce the
+multilingual entity and relation extraction model (i.e., HERBERTa) without
+considering interference across languages. However, such language interfer-
+ence is prevalent in multilingual tasks because of parameter sharing [9–11].
+As shown in Figure 1, to mitigate interference among languages, we propose
+to extract the feature representation of the corresponding language sentence.
+First, to facilitate the cross-lingual transfer among different languages, mul-
+tilingual representations are supposed to be closed under similar semantics
+using cross-lingual sentence-level concatenation. Then, based on the shared
+multilingual parameters, the language-specific representations derived from
+the independent modules can mitigate interference among multiple languages.
+
+Springer Nature 2021 LATEX template
+Article Title
+3
+Specifically, we propose a two-stage multilingual training method and an
+我爱吃苹果。
+amo le mele.
+I love apples.
+J'adore les pommes.
+…  
+Unified 
+Feature
+Chinese 
+Feature
+Italian 
+Feature
+English
+Feature
+French 
+Feature
+Fig. 1 This example includes 4 sentences from different languages, which express the same
+meaning. The four arrows represent four independent sentence representations extracted
+from different languages.
+effective model called multilingual Entity and Relation Extraction framework
+(mERE) to address the multilingual ERE task. In the first stage, we utilize a
+cross-lingual encoder to encode different language sentences and extract rela-
+tions directly. Then, we train the joint model with our Language-universal
+Aggregator (LA) to generate the unified language feature, which narrows the
+distance of similar semantic representation across languages. LA consists of
+a self-attention layer and is trained by random multi-sentences concatena-
+tion, which is used to learn semantic similarities in multilingual training. In
+the second stage, to alleviate the interference among languages, we freeze the
+parameters of LA and cross-lingual encoder in the first stage and optimize the
+independent parameters via fine-tuning the model with a Language-specific
+Switcher (LS), which consists of several independent sub-modules to produce
+the specific language features. Meanwhile, a selection mechanism is applied
+to choose the optimal group of sub-modules from LS, which enables the sub-
+module to share the same parameters with a certain group of languages. Such
+an automatic sub-module selection mechanism saves many model parameters
+when the number of languages is large. After that, each token representation
+is concatenated with the relation representation to enhance the recognition
+of the positions of entities in a sentence. Finally, in mERE, we adopt joint
+training to mitigate the error propagation problem.
+We conduct extensive experiments on the SMiLER benchmark of 14 lan-
+guages with 36 relations (including no relation) in total. The experimental
+results demonstrate that our method outperforms previous monolingual and
+multilingual ERE baseline methods by a large margin across languages, which
+demonstrates that our method can effectively mitigate language interference
+by improving representation quality among languages. Besides, we conduct
+detailed experiments to analyze how our method affects relational triple extrac-
+tion. Moreover, our method is simple but effective, and it is also easy to transfer
+to different backbone models of multi-field tasks with lightweight modules.
+
+Springer Nature 2021 LATEX template
+4
+Article Title
+2 Related Work
+Information Extraction Information extraction mainly focuses on extract-
+ing knowledge from unstructured text. A well-known system called Never-
+Ending Language Learner was reading the Web for almost 10 years to collect
+new instances of pre-defined relations and entity types [12]. Instead of the pre-
+defined entity and relation types, Open Information Extraction (OpenIE) has
+also attracted much attention during the past decade. A notable example is
+TextRunner [13], which utilizes a syntactic parser to extract triples from the
+Internet automatically. Many systems have been proposed subsequently, such
+as rule-based systems [14–16] and clause based systems [17, 18]. Recent super-
+vised methods are divided into three categories based on different architectures:
+(1) Generation-based models are typically sequence-to-sequence structure [19–
+21]. (2) Sequence labeling-based models using Begin Inside Outside (BIO)
+or Subject Relation Object None (SRON) to label every word in a sentence
+[22, 23]. (3) Span-based model takes advantage of span level feature which can
+be sufficiently exploited [24].
+Entity and Relation Extraction Early entity and relation extraction tasks
+use a pipeline approach, which are two separate subtasks including named
+entity recognition and relation classification. [25] first works on Recurrent Neu-
+ral Network (RNN) based model for extraction, capturing the semantics of the
+entity and its adjacent phrases through parsing trees. While [26] uses a syntac-
+tic tree-based RNN model to add weights to the important phrases. [27] first
+used a Convolutional Neural Network (CNN) structure to fuse the extracted
+word and sentence level features for extraction work. [28] uses a CNN structure
+based on a dependency tree to improve the performance. However, the pipeline
+approach has inevitable deficiencies: (1) The architecture ignores the interac-
+tions between entities and relations, causing the error propagation problem. (2)
+Some of the extracted entities are redundant in the named entity recognition
+phase, resulting in a degradation of performance in the relation classification
+phase.
+Most studies focus on the joint approach, which models entity recognition
+and relation classification in the same network and naturally relieves error
+propagation problem. The initial joint models are feature-based methods that
+heavily rely on NLP tools and manual efforts [29–32]. Recent joint models are
+typically neural network-based methods, which benefit from their excellent fea-
+ture learning capability. SPTree [33] is the first joint model based on the neural
+network method. Due to the two subtasks decoding with independent decoders
+but sharing parameters of the same encoding layers, this architecture also is
+known as parameters sharing. Following such kind of structure, [34] proposed
+an LSTM-based network that decodes entities and a CNN network to classify
+relations. [35, 36] employ CRF to improve performance of entity recognition.
+[37–40] use a pre-trained model called bidirectional encoder representation
+from transformers (BERT) to improve the accuracy of entity recognition. [41]
+proposes a multi-feature fusion sentence representation and decoder sequence
+annotation to handle the overlapping triples which are overlapped with one
+
+Springer Nature 2021 LATEX template
+Article Title
+5
+or two entities. Another architecture is joint decoding, which extracts entity
+pairs and corresponding relations simultaneously in one stage. NovelTagging
+[42] first proposes a tagging scheme to implement a joint decoding manner.
+But it cannot figure out the overlapping problem. The sequence-to-sequence
+scheme [43–46] models relational triples as a sequence, which can naturally
+deal with the nested entity and overlapping problem.
+Multilingual Models Multilingual models are a type of model that per-
+forms cross-lingual transfer among different languages, such as multilingual
+pre-training [47–51] and machine translation [11, 52–54]. Specifically, mBERT
+pre-trained on 104 languages in Wikipedia has a strong ability for cross-
+lingual transfer. Multilingual neural machine translation (MNMT) trains a
+single NMT model in multiple language pairs supporting translation direc-
+tions between multiple languages by sharing parameters [55–58]. Early studies
+mainly utilize high-resource languages to help low-resource languages and
+even perform zero-shot transfer translation [59, 60]. Recent studies focus on
+designing language-specific components to mitigate the language interference
+in shared parameters, especially on high-resource pairs [11, 61, 62]. Our method
+boosts the sentence representation quality from superior unified representation
+to further language-specific representation.
+Multilingual Entity and Relation Extraction Existing entity and rela-
+tion extraction datasets are insufficient in diversity and size. English is always
+used to be training corpora. [8] presents a new, large and diversified dataset
+Samsung MultiLingual Entity and Relation Extraction (SMiLER) dataset to
+entity and relation extraction both for English and multilingual setting. This
+is currently the most comprehensive and largest multilingual dataset.
+In this paper, we propose a multilingual entity and relation extraction
+framework called mERE with two-stage training strategies. In the first stage,
+we concatenate random sentences and use the self-attention mechanism [63]
+to learn the unified representation across languages. Inspired by MoE [64],
+we use several sub-modules with a selection mechanism to learn the specific
+representation of each language in the second stage. Such two-stage learning
+greatly improves the performance of relational triple extraction.
+3 Methodology
+In this section, we introduce the details of our training method for the multi-
+lingual joint extraction model as shown in Figure 2. We propose a two-stage
+training strategy. In the first stage, we train a Language-universal Aggrega-
+tor (LA) for learning the unified representations among multiple languages. In
+the second stage, we freeze the parameters and fine-tune the Language-specific
+Switcher (LS), which is applied to select specific feature representations of
+various languages.
+
+Springer Nature 2021 LATEX template
+6
+Article Title
+FR: Tour Eiffel à Paris.
+EN: Big Ben is in UK. 
+ES: España en Europa.
+IT: Torre pendente di Pisa in Italia.
+….
+Cross-lingual Pretrained Encoder
+Embeddings
+Classifier
+Relation
+[CLS]
+[CLS]
+𝜃1
+𝜃2
+𝜃3
+𝜃4
+Language-universal Aggregator
+Language-specific
+Switcher
+Entity1
+Entity2
+Big
+Ben
+UK
+Weighted sum
+NER
+Concatenate
+Encoder
+RC
+Switcher-based 
+Tuning
+NER
+LA
+Encoder
+RC
+NER
+LA
+LS
+Freeze
+Multilingual 
+Training
+Selection 
+Distribution
+Fig. 2 The left part shows the two-stage training strategy. The right part is our frame-
+work with Language-universal Aggregator (LA) for unified representation generation and
+Language-specific Switcher (LS) for language-specific feature extraction. We first train the
+LA with a concatenation of 2 random sentence representations, which are denoted as the
+green boxes (English) and yellow boxes (Italian) below the figure. Note that each sentence
+representation is directly regarded as input of LA during the evaluation stage. Then, we
+freeze part of the parameters and fine-tune the LS with all sub-modules during the training
+stage. The figure illustrates 4 sub-modules of LS with a top-2 strategy during evaluation.
+3.1 Task Formulation
+The goal of multilingual joint entity and relation extraction aims to identify all
+possible relational triples from sentences in different languages. Formally, given
+a sentence X from multilingual corpora D = {Dn}N
+n=1, where N represents
+the number of the all languages Lall = {Ln}N
+n=1. The probability of the target
+triple Y = {s, r, o} is defined as below:
+P(Y | X) = p(r | X; φ)p(s, o | X, r; ϕ),
+(1)
+where r denotes relation, s and o are subject (head entity) and object (tail
+entity), respectively. p(r | X; φ) means relation is only related to sentence X,
+and p(s, o | X, r; ϕ) means the entity pair (s, o) is related to both sentence X
+and the relation r that they shared.
+3.2 Language-aggregation Training
+We train the model with Language-universal Aggregator (LA) to learn the
+unified representation, which effectively narrows the distance of semantic
+representations across different languages. To obtain context representations
+of each token from the multilingual sentences, we utilize the cross-lingual
+pre-trained encoder for building a multilingual model. Given the sentence
+XLn = {xLn
+1 , . . . , xLn
+i
+, . . . , xLn
+m } with m tokens (including [CLS], [SEP] and
+
+Springer Nature 2021 LATEX template
+Article Title
+7
+[PAD]), xLn
+i
+∈ Rd is the i-th token embedding and d is the embedding size.
+The whole sentence is encoded by the cross-lingual pre-trained encoder:
+hLn = H(XLn; φ),
+(2)
+where hLn = {hLn
+1 , . . . , hLn
+i
+, . . . , hLn
+m } ∈ Rm×d represents the encoded rep-
+resentation and d is the hidden size. H denotes the cross-lingual pre-trained
+encoder. Meanwhile, a relation classifier W r ∈ Rd×U is used to project pooled
+output vector hp (from the [CLS] token) to the relation rc, where U is the
+number of relation types. The relation extraction is defined as:
+rc = hpW r,
+(3)
+To better learn the unified semantic representation among multiple lan-
+guages, we randomly sample s sentences of different languages from the
+training corpora to generate the cross-lingual representations using Equation 2
+and concatenate them to obtain hcat = [h
+LX1
+1
+, . . . , h
+LXi
+i
+, . . . , hLXs
+s
+], where LXi
+denotes the language symbol of the i-th sentence. Considering that each token
+needs to capture the dependency of inner-sentence and acquire semantic sim-
+ilarity representation of inter-sentence among languages, we train LA which
+applies the self-attention mechanism for fusing the information of the given
+concatenated representation:
+ˆhcat = SF(QKT
+√ϵ )V
+(4)
+where Q = hcatWq, K = hcatWk and V = hcatWv. SF represents the softmax
+operation. The three-parameter matrices Wq, Wk, and Wv are trainable. The
+term 1/√ϵ is the scaling factor. ˆhcat = {ˆh
+LX1
+1
+, . . . , ˆh
+LXi
+i
+, . . . , ˆhLXs
+s
+} and ˆh
+LXi
+i
+is
+i-th element. Instead of using language-specific features generated via Equation
+8, we directly utilize each element representation in ˆh
+LXi
+i
+to train the model
+via Equation 9.
+3.3 Language-specific Training
+To acquire features of a specific language, we freeze the parameters of language
+aggregation and cross-lingual encoder in the first training stage and fine-tune
+the model with LS. After obtaining the unified representation via LA, we
+extract the language-specific features via the LS with the selection mechanism
+from the unified representations.
+Given the language symbol Ln ∈ Lall(1 ≤ n ≤ N) and our LS θ =
+{θt}T
+t=1(1 ≤ t ≤ T , 1 ≤ T ≤ N), our selection mechanism is used to select
+corresponding sub-modules θf(Ln), in which f(·) is a function that maps a lan-
+guage to corresponding LS modules. To design an appropriate map function for
+our selection mechanism, each sentence is prefixed to the corresponding lan-
+guage symbol, which enables the model to correctly route sentences. Besides,
+
+Springer Nature 2021 LATEX template
+8
+Article Title
+all sub-modules from LS attend to the selection procedure during the training
+stage, which solves the undifferentiability problem. Specifically, the function
+ft(·) indicates the probability of selection of sub-module θt:
+ft (Ln) =
+exp
+�
+eLn
+t
+�
+�T
+i=1 exp
+�
+eLn
+i
+�
+(5)
+where eLn
+i
+is i-th element of the probability vector eLn = El[n]Wf. El ∈
+RN×d denotes the look-up table for all language prefix embeddings. The router
+matrix Wf ∈ Rd×T is used to project eLn which are normalized via a softmax
+distribution over the total T modules.
+For each sub-module θt from θ, we utilize Eθt(·) to transform unified feature
+representation ˆhLn into language-specific feature branch ˜hLn
+θt :
+˜hLn
+θt = Eθt(ˆhLn)
+(6)
+Eθt(ˆhLn) = LN
+�
+σ(ˆhLnWu)Wd + ˆhLn�
+(7)
+where ˆhLn ∈ Rm×d is an element of ˆhcat. Wu ∈ Rd×b and Wd ∈ Rb×d are
+projection matrices (b > d). σ is the ReLU activation function and LN(·) is
+the layer normalization function. The right part of Figure 2 corresponds to
+Equation 7.
+To ensure gradients are propagated to all sub-modules of LS {θt}T
+t=1, we
+apply the weighted average for obtaining the language-specific feature:
+˜hLn =
+T
+�
+t=1
+ft(Ln)Eθt
+�
+ˆhLn�
+(8)
+Note that for the whole process, function ft(Ln) in Equation 8 permits
+differentiability of the router.
+In the evaluation stage, it is necessary to prune several sub-module branches
+with the lowest selection probabilities to obtain the best performance. There-
+fore, we use the top-K strategy to select the best k(1 ≤ k ≤ T ) sub-modules
+with the highest probabilities to generate the language-specific representation.
+When k = T indicates all sub-modules involved in the calculation which means
+the selection mechanism is the same as the training stage. The mapping pro-
+cess is described as: Ln −→ {πLn
+1 , . . . , πLn
+i
+, . . . , πLn
+k } ∈ Π(k), where πLn
+i
+is
+one of the sub-module index that corresponds to language Ln and Π(k) is the
+space of all k-length combinations of Ck
+T in total.
+After obtaining the language-specific representation from LS, we create
+four matrices to recognize the head and tail positions of two named entities. To
+enhance the accuracy of recognition, we add a relation feature that constrains
+the extracted entities that are only related to the relevant relation. Formally,
+given a language-specific representation ˜hLn ∈ Rm×d of the m-length sentence
+
+Springer Nature 2021 LATEX template
+Article Title
+9
+and the relation vector re retrieved from relation embedding table Er ∈ RI×d,
+where I is the number of relations, the two entities are recognized as followed:
+entityx = (η((˜hLn ⊕ re)Wy))Uy
+(9)
+where the symbol collection entity={head, tail}, x={start,end} and y =
+{hs, he, ts, te}. We concatenate the relation vector with each token rep-
+resentation to enhance the recognition of entities, namely ˜hLn ⊕ re
+=
+{[˜hLn
+1 , re], . . . , [˜hLn
+i
+, re], . . . , [˜hLn
+m , re]} ∈ Rm×2d. Wy ∈ R2d×d are four down
+projection matrices and Uy ∈ Rd×1 are four index projection matrices. η
+denotes tanh activation function. Note that we use ground-truth relation as
+input in training entity recognition, which conforms to the joint training
+method in our architecture.
+3.4 Training Objective
+Our model presented in Figure 2 is trained jointly on multilingual ERE cor-
+pora. We first train the model only using a multilingual training strategy for
+our Language-universal Aggregator. Based on the unified language representa-
+tion, we fine-tune the model with Language-specific Switcher for learning the
+language-specific feature in the next step. The objective is to minimize the two
+training loss functions which are defined below:
+LLAT =
+M
+�
+m=1
+E(x,y)∼Dm[Lere(x, y; Θ)]
+(10)
+LLST =
+M
+�
+m=1
+E(x,y)∼Dm[Lere(x, y; Θ, θ)]
+(11)
+where D means multilingual entity and relation extraction training corpora
+and M denotes the number of the samples. Θ indicates shared parameters and
+θ is parameters in LS with selection mechanism. Lere is the loss function for
+entity and relation extraction, which is defined as below:
+Lere = α
+2 (Lstart
+h
++ L end
+h
++ L start
+t
++ L end
+t
+) + βLrel
+(12)
+where each L with any superscript is a cross-entropy loss. The subscripts with
+h and t indicate the head entity and tail entity respectively. The start and end
+of superscripts denote the first token index and last token index of an entity
+separately. Lrel is the loss function for relation classification. α and β are two
+weights on entity recognition loss and relation classification loss respectively.
+
+Springer Nature 2021 LATEX template
+10
+Article Title
+4 Experiments
+4.1 Datasets
+We evaluate our model on the dataset SMiLER [8], which is the largest
+and most diversified multilingual dataset for multilingual entity and relation
+extraction tasks with 14 languages from 36 relation types. The SMiLER con-
+sists of about 1.1M annotated sentences from Wikipedia and DBpedia, which
+includes English (En), Korean (Ko), Italian (It), French (Fr), German (De),
+Portuguese (Pt), Nederlands (Nl), Polish (Pl), Spanish (Es), Arabic (Ar), Rus-
+sian (Ru), Swedish (Sv), Farsi (Fa), Ukrainian (Uk). The relation types belong
+to roughly nine domains: location, organization, person, animal, art, device,
+measurement, event, and no relation. The statistics of SMiLER are shown in
+Table 1. As the development set in SMiLER is not publicly available, we only
+randomly extract the sentences from the training set to create new files with
+the same split ratio as the original paper.
+Table 1 The statistics of SMiLER dataset. English corpora include full-size, middle-size,
+and small-size. The languages are ordered from high-resource languages (left) to
+low-resource languages (right).
+Languages
+EN-full
+EN-mid
+It
+Fr
+De
+Pt
+Nl
+En-small
+Ko
+Pl
+Es
+Ar
+Ru
+Sv
+Fa
+Uk
+sentences num.
+748k
+269k
+76k
+62k
+53k
+45k
+40k
+35k
+20k
+17k
+12k
+9k
+7k
+5k
+3k
+1k
+relation types
+36
+36
+22
+22
+22
+22
+22
+32
+28
+22
+22
+9
+8
+22
+8
+7
+4.2 Implementation Details
+We conduct experiments on SMiLER, 14 languages in total. EN-small is
+treated as our English corpora. We utilize mBERT as our cross-lingual encoder.
+We train our model with AdamW, the learning rate is 3e-5 and weight decay
+is 0.1. The batch size is set to 16 on Tesla V100 GPU. The hidden size d is 768
+and dimension b of projection matrices Wu and Wd is 1024. The max sequence
+length is 256 and we concatenate 2 sentences during the first training stage. For
+the second training stage, we freeze most parameters in the first stage except
+the relation classifier and 8 matrices used to predict entities from Equation 9.
+The sub-module number T of LS is set to 6 (2 layers for 3 sub-modules and 1
+layer for the other). The epoch is set to 5 at the first stage. The max epoch of
+the second stage is set to 8 with an early stopping mechanism. The loss weights
+are set to 2 in named entity recognition and 1 in relation classification.
+In the evaluation stage, we set k = 3 in the top-K strategy to select the
+sub-modules in LS. We adopt standard micro-F1 metric to calculate scores on
+the models. The extracted entity pair is regarded as correct if the predictions
+of the head entity and tail entity are both the same as the ground truth. A
+triple is treated as correct if the entity pair and the corresponding relation
+type are all correct. no relation type is included in relation prediction. We
+also add a mask for the relation that is not absent in a language.
+
+Springer Nature 2021 LATEX template
+Article Title
+11
+4.3 Baselines
+As far as we know, the SMiLER is a new dataset and thus only an existing
+method for multilingual ERE without publishing source code. The relevant
+task is cross-lingual relation classification, which is also few in studies. There-
+fore, we reproduce the following competitive baselines to compare with our
+proposed approach for a fair comparison:
+• HEBERTa [8]: A multilingual entity and relation extraction framework
+called Hybrid Entity and Relation extraction BERT, which achieves the
+state-of-the-art performance on SMiLER. HERBERTa uses a pipeline train-
+ing manner that combines two independent BERT models. The first
+sub-model classifies the input sequence as one of 36 pre-defined relations
+(including no relation). The relation generated from the first sub-model is
+then fed to the second BERT and concatenated with the same input sequence
+as the input of the second model for entity recognition.
+• mBERT [65]: A cross-lingual model first uses the mBERT as a backbone
+for RC, which is trained on 104 languages with the corresponding Wikipedia
+dumps. We reproduce the results with the code shared at https://github.
+com/boun-tabi/RELX
+• MTMB [65]: A multilingual pre-training scheme called Matching the Mul-
+tilingual Blanks (MTMB). The framework shows several advantages against
+the mBERT on monolingual tasks and achieves significant improvements in
+cross-lingual transfer. Note that this framework is only designed for RC and
+not adapted to entity and relation extraction. Therefore, we simply modified
+the output layer of the baseline to conduct the ERE task.
+In addition to the above baselines, we also build a simplified multilin-
+gual joint entity and relation extraction framework called mERE-LS-LA as a
+basic structure which is concatenated relation representation with the sentence
+representation to enhance the extraction performance.
+Table 2 The F1 scores of different models. * denotes the model is reproduced by us on
+our experiment settings. - denotes that the language data is not involved both in the
+training and the evaluation stage. MONO, EURO, and SVO mean training data in 3
+different language groups. The languages are ordered from high-resource languages (left) to
+low-resource languages (right). The bold font number is the best score in each language.
+Test Sets
+AVG
+It
+Fr
+De
+Pt
+Nl
+En
+Ko
+Pl
+Es
+Ar
+Ru
+Sv
+Fa
+Uk
+HERBERTa*
+75.5
+83.9
+68.7
+71.5
+72.1
+78.5
+60.9
+80.4
+83.1
+60.0
+88.4
+79.4
+84.8
+79.6
+65.0
+mBERT*1
+75.2
+81.5
+68.2
+70.7
+71.0
+77.6
+59.9
+78.5
+81.1
+61.3
+89.5
+81.7
+81.5
+79.6
+70.0
+MTMB*1
+75.6
+80.9
+67.8
+70.9
+70.3
+79.1
+58.3
+79.3
+82.2
+58.2
+91.1
+74.1
+83.7
+77.8
+85.0
+mERE
+77.9
+81.7
+70.3
+73.4
+74.3
+81.1
+62.3
+82.7
+81.6
+64.7
+91.6
+83.1
+83.7
+79.6
+80.0
+mERE (EURO)
+70.9
+81.4
+70.2
+72.1
+74.2
+-
+62.2
+-
+-
+65.2
+-
+-
+-
+-
+-
+mERE (SVO)
+75.7
+81.3
+70.0
+72.9
+73.3
+80.6
+62.1
+-
+81.0
+64.7
+-
+83.1
+83.7
+-
+80.0
+mERE-LS
+77.2
+80.9
+69.7
+72.0
+73.5
+80.4
+62.2
+80.4
+81.6
+62.1
+91.6
+83.1
+84.8
+77.8
+80.0
+mERE-LS-LA (MONO)
+70.9
+81.2
+68.3
+67.1
+68.4
+77.9
+58.6
+79.3
+79.0
+48.4
+90.0
+72.5
+80.4
+66.7
+55.0
+mERE-LS-LA
+76.5
+81.3
+69.0
+71.9
+71.4
+80.3
+60.3
+76.4
+84.2
+60.7
+90.0
+83.9
+83.7
+77.8
+80.0
+1We modified the output layer to implement the entity recognition to accommodate the
+ERE task. We train the model in the joint training method.
+
+Springer Nature 2021 LATEX template
+12
+Article Title
+4.4 Models and Languages Comparison
+The results presented from the Tables are rounded to one decimal place. From
+Table 2, our method improves multilingual baselines by a large margin over pre-
+vious baselines. There is a 2.3% improvement on averaged F1 score compared
+with the previous strongest baseline MTMB which outperforms HERBERTa
+due to its strong multilingual pre-training scheme. Our mERE achieves the
+best scores on 8 out of 14 languages, especially on high-resource languages.
+The other 5 out of 6 languages achieve the second-best scores. Surprisingly,
+even our baseline mERE-LS-LA has 0.9% improvement over the MTMB. It
+seems that our basic structure is more effective on multilingual entity and rela-
+tion extraction tasks. Compared with mERE-LS that only uses LA, our full
+model mERE has nearly 0.7% F1 value improvement on average and yields
+similar or higher results on 13 languages except for Sv. The improvement can
+be attributed to our switcher-based language-specific training strategy, which
+finally extracts accurate information for entity recognition in each language.
+Compared with our baseline mERE-LS-LA, our full model mERE has nearly
+1.4% F1 value improvement on average which means mERE-LS also has nearly
+0.7% F1 value improvement on average. All such impressive results demon-
+strate that our full model mERE truly enhances the representation quality
+and mitigates language interference to a certain extent.
+We set several language groups to analyze the impact of different languages:
+(1)MONO: 14 languages in monolingual training. (2)EURO: It, Fr, Pt, De,
+Es, En. (3)SVO2: EURO, Ru, Sv, Nl, Pl, Uk. The default is all languages
+in multilingual training from Table 2. Compared with mERE-LS-LA training
+in multilingual corpora, we can observe that multilingual training achieves
+much higher results than mERE-LS-LA (MONO) monolingual training from
+Table 2, especially on low-resource languages. Such as improvements of Uk
+(25%), Fa (11.1%), and Ru (11.4%). It demonstrates that languages with less
+training data can benefit most from high-resource languages in multilingual
+training including ERE tasks. The results of the EURO family group are close
+to mERE. It is worth noting that Es achieves the best score in the EURO
+group. We conclude that Es benefits a lot from similarities of languages that
+are in the same language family even with less training data. In the SVO
+group, we can also visualize that most languages in EURO decrease slightly
+with the interference of other non-EURO languages. The different language
+families, or the languages with a big difference in syntactic structures might
+be the main interference among languages. However, compared with mERE
+(SVO), mERE yields the same results on low-resource languages and somewhat
+higher results on high-resource languages even the three non-SVO (Fa, Ar, and
+Ko) data involved during the training stage. We suppose that these non-SVO
+languages which are big different from others and are all low-resource may
+facilitate distinguishing high-resource languages in learning language-specific
+2SVO stands for the relative position of the Subject, Verb, and Object in the typical affirmative
+sentence. We treat Korean, Farsi, and Arabic as non-SVO languages. Arabic is VSO, while Korean
+and Farsi are SOV.
+
+Springer Nature 2021 LATEX template
+Article Title
+13
+features due to each sub-module from LS being independent, without sharing
+parameters in the same space. Lastly, we also observe some duplicated F1
+scores across low-resource languages. This phenomenon is caused by a small
+number of sentences in test sets.
+4.5 Entity and Relation Analysis
+Figure 3 shows F1 scores of relation and entity pair of mERE and mERE-LS-
+LA. We can observe that the relation classification seems to be easier than the
+named entity recognition. The correctness of entity pair extraction is the main
+bottleneck of the model performance. With the help of our LA and LS, mERE
+achieves higher results on entity pair recognition compared with mERE-LS-
+LA in general. Surprisingly, we can visualize that the performance of relation
+classification also has a slight improvement in mERE. We conclude that the
+improvement of the named entity recognition facilitates relation classification.
+Since information interaction between two sub-tasks can benefit each other in
+the joint training architecture.
+50
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Total
+It
+Fr
+De
+Pt
+Nl
+En
+Ko
+Pl
+Es
+Ar
+Ru
+Sv
+Fa
+Uk
+Relation(mERE)
+Entity Pair(mERE)
+Relation(mERE-LS-LA)
+Entity Pair(mERE-LS-LA)
+Fig. 3 The F1 scores of relations and entity pairs on all languages.
+F1 scores of detailed relation labels are shown in Figure 4. Most of the
+relations achieve higher F1 scores across languages, such as “no relation” and
+“has-type”. Part of relations differs widely across languages, such as relation
+“has-child”(F1 = 100 on Nl, F1 = 33 on De, F1 = 0 on Es). The big difference
+is caused by the number of relations of training data in each language. For
+some relations that occur F1 = 0 scores, we find out the relations (e.g won-
+award on Nl. has-parent on Pl. has-child on Es) are only one test sample.
+Such low results for some languages could be explained by a smaller number
+of relations in the test set.
+
+Springer Nature 2021 LATEX template
+14
+Article Title
+En
+It
+Fr
+Pt
+Es
+De
+Ar
+Nl
+Ru
+Pl
+Uk
+Sv
+Fa
+Ko
+no_relation
+is-where
+birth-place
+has-type
+movie-has-director
+has-occupation
+from-country
+has-genre
+has-author
+has-population
+headquarters
+is-member-of
+org-has-member
+has-parent
+org-has-founder
+has-spouse
+won-award
+has-nationality
+org-leader
+starring
+has-edu
+has-child
+event-year
+has-sibling
+has-length
+invented-when
+has-tourist-attraction
+has-lifespan
+first-product
+has-height
+has-highest-mountain
+invented-by
+has-weight
+post-code
+loc-leader
+eats
+85
+100
+100
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+100
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+100
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+100
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+100
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+100
+100
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+0
+100
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+67
+100
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+0
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+100
+0
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+100
+100
+100
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+83
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+100
+100
+100
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+100
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+Fig. 4 The F1 scores of all relation labels on all languages. The darker color means a higher
+F1 score, while the lighter color means a lower F1 score.
+Figure 5 shows F1 scores of head entities and tail entities. We can observe
+that F1 scores of head entities are much higher than tail entities among most
+languages. It seems that head entities are easier to be recognized than tail
+entities. It is because the head entity always occurs at the beginning position
+of the sentence and thus the model probably memorizes the position, while the
+tail entity does not have any consistent position which is hard to predict.
+4.6 Ablation Study
+Sentences Concatenation To validate the effect of the number of sentences
+for learning the unified features among different languages, we conduct sev-
+eral experiments on the different numbers of sentences in concatenation. We
+learn from Figure 6 that there are evident F1 improvements with LA on dif-
+ferent concatenation numbers of sentences over only one sentence encoding.
+The multilingual model obtains the best performance when concatenating with
+the sentence pair. The increasing number of concatenated sentences has a
+slight decrease in performance. We conjecture that increasing the number of
+sentences may also bring somewhat interference.
+
+Springer Nature 2021 LATEX template
+Article Title
+15
+50
+55
+60
+65
+70
+75
+80
+85
+90
+95
+100
+Total It
+Fr
+De
+Pt
+Nl
+En
+Ko
+Pl
+Es
+Ar
+Ru
+Sv
+Fa
+Uk
+head entity
+tail entity
+Fig. 5 The performance of head entities (blue bar) and tail entities (orange bar) on different
+languages.
+1
+2
+3
+4
+Languages
+76.6
+76.7
+76.8
+76.9
+77.0
+77.1
+F1 score
+mERE-LS
+Fig. 6 The performance of sentences concatenation in the first training stage.
+Selection Mechanism To observe how the selection mechanism affects our
+model performance, we also train one-to-one sub-modules of LS called mERE14
+without using the selection mechanism in the second training stage. Each inde-
+pendent sub-module corresponds to a language and each sentence is routed via
+a language prefix which represents the number of sub-module. We can visualize
+from Figure 7 that increasing the number of parameters also improves obvi-
+ously over mERE-LS-LA. Nonetheless, the mERE14 will suffer from the sharp
+increasing training time and inference time, and big space consumption when
+the number of languages is large enough. Instead of increasing parameters,
+our Language-specific Switcher can effectively ameliorate extraction quality
+with only slight extra parameters and less time consumption. Since similar
+languages tend to select the same sub-modules from our LS. The mERE saves
+
+Springer Nature 2021 LATEX template
+16
+Article Title
+nearly 700M model capacity in our statistics and achieves better performance
+among most languages compared with mERE14. It is obvious that mERE is
+light and easy to transfer to other multi-field tasks. Selection Distribution
+It
+Fr
+De
+Pt
+Nl
+En
+Ko
+Pl
+Es
+Ar
+Ru
+Sv
+Fa
+Uk
+Languages
+60
+62
+64
+66
+68
+70
+72
+74
+76
+78
+80
+82
+84
+86
+88
+90
+92
+94
+F1 scores
+mERE-LS-LA
+mERE14
+mERE
+Fig. 7 The performance of three models on 14 languages. mERE14 utilizes 14 one-to-one
+sub-modules of LS without the selection mechanism. Each sub-module corresponds to a
+language.
+Figure 8 illustrates the heatmap of selection probability on 6 sub-modules from
+LS for each language. For each sub-module from top to bottom in Figure 8,
+we can visualize θ1 pays more attention to low-resource languages while θ4,
+θ5, and θ6 pay more attention to languages from the EURO family, which
+are mostly high-resource languages. The θ2 and θ3 seem to be more balanced
+on parameters sharing of languages except for 2 or 3 prominent languages.
+We conclude that some sub-modules are mainly used to extract features from
+similar languages and others are used to assist the specific languages.
+For each language from left to right, we can visualize that the selected
+sub-modules with higher probabilities are easy to distinguish in high-resource
+languages. In contrast, the selection probabilities across all sub-modules are
+relatively similar on low-resource languages in total. We conclude that train-
+ing data is rich enough to determine which way to route on high-resource
+languages and a more balanced selection decision is made on less training
+data. It is learned from Figure 8 that there are nearly 3 out of 6 prominently
+higher selection probabilities on the high-resource languages, and so do the
+low-resource languages with careful observation. It proves that only 3 sub-
+modules play the dominant role in refining the language-specific feature for
+each language. To avoid interference from the other irrelevant sub-modules,
+we adopt a top-K strategy to filter out 6 − k sub-modules with lower selec-
+tion probabilities in the evaluation stage. The top-6 strategy means selecting
+all sub-modules, which is the same as the training stage and the performance
+is relatively low (77.74) on average, while our mERE achieves the best perfor-
+mance (77.87) when adopting the top-3 strategy. It demonstrates that filtering
+out the least important sub-modules is necessary to enhance the prediction
+quality, which also reduces the redundant parameters in the evaluation. The
+
+Springer Nature 2021 LATEX template
+Article Title
+17
+top-1 achieves the worst performance (77.60), which demonstrates the part of
+sub-modules are also helpful for the task. Therefore, the best performance is
+obtained when the k value is balanced in all languages. Layer Number of
+It
+Fr
+De
+Pt
+Nl
+En
+Ko
+Pl
+Es
+Ar
+Ru
+Sv
+Fa
+Uk
+ 
+1 
+ 
+2 
+ 
+3 
+ 
+4 
+ 
+5 
+ 
+6 
+Fig. 8 The selection probability distributions of 6 sub-modules from LS on 14 languages.
+The sub-modules {θ}6
+1 are numbered from 1 to 6. The languages are ordered from high-
+resource languages (left) to low-resource languages (right). The darker color means a higher
+selection probability to the corresponding sub-module and a lower probability to select a
+certain sub-module when the color is lighter.
+Language-specific Switcher Table 3 used to evaluate the effect of the layer
+number of LS. We divide the 6 sub-modules into 2 groups (each group has the
+same layer number) with different combinations of layer numbers to accommo-
+date the scenarios, such as high- and low-resource language feature extraction.
+From Table 3, we can observe that the combination 1-2 achieves the best F1
+score on average. The combinations which are set to 1-1 and 4-4 also achieve
+better performance. With the increase or decrease of the layer number to a cer-
+tain degree, the performances are almost the same, which maintains relatively
+low averaged F1 scores. The full layer number combination 4-4 is an exception
+in the case, which demonstrates the performance still can be improved when
+the model capacity is large enough. According to the outcomes from Table 3,
+we conclude that the layer number of LS obviously impacts the results, with
+the best results attained when a balance is reached.
+5 Conclusion
+In this paper, we introduce a two-stage training method and a robust frame-
+work called mERE for multilingual entity and relation extraction, which
+ameliorates the sentence representation quality and mitigates the language
+interference among multiple languages. Specifically, we first learn the gener-
+alities across all languages to obtain the unified language representation via
+the Language-universal Aggregator and then learn the specialties of each lan-
+guage via the Language-specific Switcher. Experimental results demonstrate
+
+Springer Nature 2021 LATEX template
+18
+Article Title
+Table 3 The different layer numbers of sub-modules. Every 3 sub-modules in a group has
+the same layer numbers. Layer Num.01 and Layer Num.02 denote the layer number of the
+first group and second group respectively.
+Layer Num.01
+Layer Num.02
+AVG
+IT
+FR
+DE
+PT
+NL
+EN
+KO
+PL
+ES
+AR
+RU
+SV
+FA
+UK
+1
+1
+77.4
+81.3
+69.1
+72.1
+73.4
+80.4
+63.1
+81.9
+81.3
+63.4
+91.1
+85.4
+83.7
+77.8
+80.0
+1
+2
+77.9
+81.7
+70.3
+73.4
+74.3
+81.1
+62.3
+82.7
+81.6
+64.7
+91.6
+83.1
+83.7
+79.6
+80.0
+1
+3
+77.5
+81.5
+69.4
+72.8
+73.8
+81.1
+62.2
+81.9
+81.3
+64.7
+90.5
+83.1
+83.7
+79.6
+80.0
+1
+4
+77.5
+81.0
+70.2
+72.5
+74.1
+80.8
+62.2
+82.2
+81.3
+65.6
+90.5
+83.1
+83.7
+77.8
+80.0
+2
+2
+77.8
+81.7
+70.1
+73.1
+74.1
+81.0
+62.3
+81.9
+81.6
+66.1
+90.5
+83.1
+83.7
+79.6
+80.0
+2
+3
+77.4
+81.1
+70.2
+72.7
+74.1
+80.9
+62.1
+82.5
+81.3
+65.2
+90.5
+81.5
+83.7
+77.8
+80.0
+2
+4
+77.4
+81.3
+70.2
+72.2
+74.2
+81.0
+62.2
+81.4
+81.0
+64.7
+90.5
+81.5
+83.7
+79.6
+80.0
+3
+3
+77.4
+81.6
+70.4
+72.5
+73.3
+81.0
+62.1
+82.7
+81.0
+64.7
+91.1
+81.5
+83.7
+77.8
+80.0
+3
+4
+77.5
+81.5
+70.2
+73.3
+73.9
+81.4
+62.7
+83.0
+81.0
+64.3
+90.5
+82.3
+83.7
+77.8
+80.0
+4
+4
+77.7
+81.1
+70.2
+73.1
+74.2
+80.9
+62.3
+81.9
+81.6
+65.6
+90.5
+83.1
+83.7
+79.6
+80.0
+that our method significantly outperforms both monolingual and multilingual
+ERE baselines, which demonstrates that our framework can extract relational
+triples among various languages well. Moreover, our framework is also light
+and easy to transfer to other backbone models of multi-field tasks.
+In the future, we will pay more attention to complex multilingual relational
+triple extraction, such as overlapping relational triples or multiple relational
+triples. Besides, we will also do further research on a better contextual repre-
+sentation among multiple languages. Although there is a long way to experience
+in multilingual entity and relation extraction tasks, it is important to inves-
+tigate the valuable structured information in many other languages for the
+downstream NLP tasks.
+Acknowledgments.
+This work was supported in part by the National Nat-
+ural Science Foundation of China (Grant Nos. 62276017, U1636211, 61672081),
+the 2022 Tencent Big Travel Rhino-Bird Special Research Program, and the
+Fund of the State Key Laboratory of Software Development Environment
+(Grant No. SKLSDE-2021ZX-18).
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+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' bjq@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  hlt@cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Abstract  Entity and relation extraction is a key task in information extraction,  where the output can be used for downstream NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Existing  approaches for entity and relation extraction tasks mainly focus on  the English corpora and ignore other languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Thus, it is critical to  improving performance in a multilingual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Meanwhile, multilingual  training is usually used to boost cross-lingual performance by trans-  ferring knowledge from languages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', high-resource) to other (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=',  low-resource) languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' However, language interference usually exists  in multilingual tasks as the model parameters are shared among all  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In this paper, we propose a two-stage multilingual train-  ing method and a joint model called Multilingual Entity and Relation  Extraction framework (mERE) to mitigate language interference across  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Specifically, we randomly concatenate sentences in differ-  ent languages to train a Language-universal Aggregator (LA), which  narrows the distance of embedding representations by obtaining the  unified language representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Then, we separate parameters to mit-  igate interference via tuning a Language-specific Switcher (LS), which  includes several independent sub-modules to refine the language-specific  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='04434v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='CL]  11 Jan 2023  Springer Nature 2021 LATEX template  2  Article Title  feature representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' After that, to enhance the relational triple extrac-  tion, the sentence representations concatenated with the relation feature  are used to recognize the entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Extensive experimental results show  that our method outperforms both the monolingual and multilingual  baseline methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Besides, we also perform detailed analysis to show  that mERE is lightweight but effective on relational triple extraction  and mERE is easy to transfer to other backbone models of multi-field  tasks, which further demonstrates the effectiveness of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Keywords: Joint extraction, Information extraction, Multilingual entity and  relation extraction, Relational triple  1 Introduction  Entity and relation extraction (ERE) contains two sub-tasks called named  entity recognition (NER) [1–4] and relation classification (RC) [5, 6], which is  the fundamental step of automatic knowledge graphs (KGs) [7] construction,  knowledge discovery and intelligent question answering system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The results of  ERE are typically described as a relational triple (h, r, t), where h and t are  the head entity and the tail entity, respectively, and r denotes the relation  between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' For example, for the sentence “Big Ben is in UK.” with a  predefined relation called “Locate in”, an ideal relational triple of this sentence  is expressed as (Big Ben, Locate in, UK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  As a large amount of data is available from different languages on the  Internet, it is important to utilize such valuable resources and develop multilin-  gual entity and relation extraction models, which can operate across language  barriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' However, most existing methods propose to solve ERE on English  corpora, which can only deal with the monolingual extraction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The main  reason is that many languages suffer from the scarcity of corpora in ERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Thus, multilingual training is proposed to help each other in a shared model,  where the well-trained knowledge of high-resource languages can be trans-  ferred to low-resource languages with a small amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Recently, [8]  propose a multilingual dataset called SMiLER, which is the first work to apply  both monolingual and multilingual training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The authors in [8] introduce the  multilingual entity and relation extraction model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', HERBERTa) without  considering interference across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' However, such language interfer-  ence is prevalent in multilingual tasks because of parameter sharing [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  As shown in Figure 1, to mitigate interference among languages, we propose  to extract the feature representation of the corresponding language sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  First, to facilitate the cross-lingual transfer among different languages, mul-  tilingual representations are supposed to be closed under similar semantics  using cross-lingual sentence-level concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Then, based on the shared  multilingual parameters, the language-specific representations derived from  the independent modules can mitigate interference among multiple languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Article Title  3  Specifically, we propose a two-stage multilingual training method and an  我爱吃苹果。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  amo le mele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  I love apples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content="  J'adore les pommes." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  …  Unified  Feature  Chinese  Feature  Italian  Feature  English  Feature  French  Feature  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 1 This example includes 4 sentences from different languages, which express the same  meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The four arrows represent four independent sentence representations extracted  from different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  effective model called multilingual Entity and Relation Extraction framework  (mERE) to address the multilingual ERE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In the first stage, we utilize a  cross-lingual encoder to encode different language sentences and extract rela-  tions directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Then, we train the joint model with our Language-universal  Aggregator (LA) to generate the unified language feature, which narrows the  distance of similar semantic representation across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' LA consists of  a self-attention layer and is trained by random multi-sentences concatena-  tion, which is used to learn semantic similarities in multilingual training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In  the second stage, to alleviate the interference among languages, we freeze the  parameters of LA and cross-lingual encoder in the first stage and optimize the  independent parameters via fine-tuning the model with a Language-specific  Switcher (LS), which consists of several independent sub-modules to produce  the specific language features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Meanwhile, a selection mechanism is applied  to choose the optimal group of sub-modules from LS, which enables the sub-  module to share the same parameters with a certain group of languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Such  an automatic sub-module selection mechanism saves many model parameters  when the number of languages is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' After that, each token representation  is concatenated with the relation representation to enhance the recognition  of the positions of entities in a sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Finally, in mERE, we adopt joint  training to mitigate the error propagation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  We conduct extensive experiments on the SMiLER benchmark of 14 lan-  guages with 36 relations (including no relation) in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The experimental  results demonstrate that our method outperforms previous monolingual and  multilingual ERE baseline methods by a large margin across languages, which  demonstrates that our method can effectively mitigate language interference  by improving representation quality among languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Besides, we conduct  detailed experiments to analyze how our method affects relational triple extrac-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Moreover, our method is simple but effective, and it is also easy to transfer  to different backbone models of multi-field tasks with lightweight modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  4  Article Title  2 Related Work  Information Extraction Information extraction mainly focuses on extract-  ing knowledge from unstructured text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' A well-known system called Never-  Ending Language Learner was reading the Web for almost 10 years to collect  new instances of pre-defined relations and entity types [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Instead of the pre-  defined entity and relation types, Open Information Extraction (OpenIE) has  also attracted much attention during the past decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' A notable example is  TextRunner [13], which utilizes a syntactic parser to extract triples from the  Internet automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Many systems have been proposed subsequently, such  as rule-based systems [14–16] and clause based systems [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Recent super-  vised methods are divided into three categories based on different architectures:  (1) Generation-based models are typically sequence-to-sequence structure [19–  21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' (2) Sequence labeling-based models using Begin Inside Outside (BIO)  or Subject Relation Object None (SRON) to label every word in a sentence  [22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' (3) Span-based model takes advantage of span level feature which can  be sufficiently exploited [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Entity and Relation Extraction Early entity and relation extraction tasks  use a pipeline approach, which are two separate subtasks including named  entity recognition and relation classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' [25] first works on Recurrent Neu-  ral Network (RNN) based model for extraction, capturing the semantics of the  entity and its adjacent phrases through parsing trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' While [26] uses a syntac-  tic tree-based RNN model to add weights to the important phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' [27] first  used a Convolutional Neural Network (CNN) structure to fuse the extracted  word and sentence level features for extraction work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' [28] uses a CNN structure  based on a dependency tree to improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' However, the pipeline  approach has inevitable deficiencies: (1) The architecture ignores the interac-  tions between entities and relations, causing the error propagation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' (2)  Some of the extracted entities are redundant in the named entity recognition  phase, resulting in a degradation of performance in the relation classification  phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Most studies focus on the joint approach, which models entity recognition  and relation classification in the same network and naturally relieves error  propagation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The initial joint models are feature-based methods that  heavily rely on NLP tools and manual efforts [29–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Recent joint models are  typically neural network-based methods, which benefit from their excellent fea-  ture learning capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' SPTree [33] is the first joint model based on the neural  network method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Due to the two subtasks decoding with independent decoders  but sharing parameters of the same encoding layers, this architecture also is  known as parameters sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Following such kind of structure, [34] proposed  an LSTM-based network that decodes entities and a CNN network to classify  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' [35, 36] employ CRF to improve performance of entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  [37–40] use a pre-trained model called bidirectional encoder representation  from transformers (BERT) to improve the accuracy of entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' [41]  proposes a multi-feature fusion sentence representation and decoder sequence  annotation to handle the overlapping triples which are overlapped with one  Springer Nature 2021 LATEX template  Article Title  5  or two entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Another architecture is joint decoding, which extracts entity  pairs and corresponding relations simultaneously in one stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' NovelTagging  [42] first proposes a tagging scheme to implement a joint decoding manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  But it cannot figure out the overlapping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The sequence-to-sequence  scheme [43–46] models relational triples as a sequence, which can naturally  deal with the nested entity and overlapping problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Multilingual Models Multilingual models are a type of model that per-  forms cross-lingual transfer among different languages, such as multilingual  pre-training [47–51] and machine translation [11, 52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Specifically, mBERT  pre-trained on 104 languages in Wikipedia has a strong ability for cross-  lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Multilingual neural machine translation (MNMT) trains a  single NMT model in multiple language pairs supporting translation direc-  tions between multiple languages by sharing parameters [55–58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Early studies  mainly utilize high-resource languages to help low-resource languages and  even perform zero-shot transfer translation [59, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Recent studies focus on  designing language-specific components to mitigate the language interference  in shared parameters, especially on high-resource pairs [11, 61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Our method  boosts the sentence representation quality from superior unified representation  to further language-specific representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Multilingual Entity and Relation Extraction Existing entity and rela-  tion extraction datasets are insufficient in diversity and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' English is always  used to be training corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' [8] presents a new, large and diversified dataset  Samsung MultiLingual Entity and Relation Extraction (SMiLER) dataset to  entity and relation extraction both for English and multilingual setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' This  is currently the most comprehensive and largest multilingual dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  In this paper, we propose a multilingual entity and relation extraction  framework called mERE with two-stage training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In the first stage,  we concatenate random sentences and use the self-attention mechanism [63]  to learn the unified representation across languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Inspired by MoE [64],  we use several sub-modules with a selection mechanism to learn the specific  representation of each language in the second stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Such two-stage learning  greatly improves the performance of relational triple extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  3 Methodology  In this section, we introduce the details of our training method for the multi-  lingual joint extraction model as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We propose a two-stage  training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In the first stage, we train a Language-universal Aggrega-  tor (LA) for learning the unified representations among multiple languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In  the second stage, we freeze the parameters and fine-tune the Language-specific  Switcher (LS), which is applied to select specific feature representations of  various languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  6  Article Title  FR: Tour Eiffel à Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  EN: Big Ben is in UK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  ES: España en Europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  IT: Torre pendente di Pisa in Italia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  ….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Cross-lingual Pretrained Encoder  Embeddings  Classifier  Relation  [CLS]  [CLS]  𝜃1  𝜃2  𝜃3  𝜃4  Language-universal Aggregator  Language-specific  Switcher  Entity1  Entity2  Big  Ben  UK  Weighted sum  NER  Concatenate  Encoder  RC  Switcher-based  Tuning  NER  LA  Encoder  RC  NER  LA  LS  Freeze  Multilingual  Training  Selection  Distribution  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 2 The left part shows the two-stage training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The right part is our frame-  work with Language-universal Aggregator (LA) for unified representation generation and  Language-specific Switcher (LS) for language-specific feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We first train the  LA with a concatenation of 2 random sentence representations, which are denoted as the  green boxes (English) and yellow boxes (Italian) below the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Note that each sentence  representation is directly regarded as input of LA during the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Then, we  freeze part of the parameters and fine-tune the LS with all sub-modules during the training  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The figure illustrates 4 sub-modules of LS with a top-2 strategy during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1 Task Formulation  The goal of multilingual joint entity and relation extraction aims to identify all  possible relational triples from sentences in different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Formally, given  a sentence X from multilingual corpora D = {Dn}N  n=1, where N represents  the number of the all languages Lall = {Ln}N  n=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The probability of the target  triple Y = {s, r, o} is defined as below:  P(Y | X) = p(r | X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' φ)p(s, o | X, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' ϕ),  (1)  where r denotes relation, s and o are subject (head entity) and object (tail  entity), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' p(r | X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' φ) means relation is only related to sentence X,  and p(s, o | X, r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' ϕ) means the entity pair (s, o) is related to both sentence X  and the relation r that they shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='2 Language-aggregation Training  We train the model with Language-universal Aggregator (LA) to learn the  unified representation, which effectively narrows the distance of semantic  representations across different languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' To obtain context representations  of each token from the multilingual sentences, we utilize the cross-lingual  pre-trained encoder for building a multilingual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Given the sentence  XLn = {xLn  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , xLn  i  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , xLn  m } with m tokens (including [CLS], [SEP] and  Springer Nature 2021 LATEX template  Article Title  7  [PAD]), xLn  i  ∈ Rd is the i-th token embedding and d is the embedding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  The whole sentence is encoded by the cross-lingual pre-trained encoder:  hLn = H(XLn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' φ),  (2)  where hLn = {hLn  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , hLn  i  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , hLn  m } ∈ Rm×d represents the encoded rep-  resentation and d is the hidden size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' H denotes the cross-lingual pre-trained  encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Meanwhile, a relation classifier W r ∈ Rd×U is used to project pooled  output vector hp (from the [CLS] token) to the relation rc, where U is the  number of relation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The relation extraction is defined as:  rc = hpW r,  (3)  To better learn the unified semantic representation among multiple lan-  guages, we randomly sample s sentences of different languages from the  training corpora to generate the cross-lingual representations using Equation 2  and concatenate them to obtain hcat = [h  LX1  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , h  LXi  i  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , hLXs  s  ], where LXi  denotes the language symbol of the i-th sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Considering that each token  needs to capture the dependency of inner-sentence and acquire semantic sim-  ilarity representation of inter-sentence among languages, we train LA which  applies the self-attention mechanism for fusing the information of the given  concatenated representation:  ˆhcat = SF(QKT  √ϵ )V  (4)  where Q = hcatWq, K = hcatWk and V = hcatWv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' SF represents the softmax  operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The three-parameter matrices Wq, Wk, and Wv are trainable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The  term 1/√ϵ is the scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' ˆhcat = {ˆh  LX1  1  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , ˆh  LXi  i  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , ˆhLXs  s  } and ˆh  LXi  i  is  i-th element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Instead of using language-specific features generated via Equation  8, we directly utilize each element representation in ˆh  LXi  i  to train the model  via Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='3 Language-specific Training  To acquire features of a specific language, we freeze the parameters of language  aggregation and cross-lingual encoder in the first training stage and fine-tune  the model with LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' After obtaining the unified representation via LA, we  extract the language-specific features via the LS with the selection mechanism  from the unified representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Given the language symbol Ln ∈ Lall(1 ≤ n ≤ N) and our LS θ =  {θt}T  t=1(1 ≤ t ≤ T , 1 ≤ T ≤ N), our selection mechanism is used to select  corresponding sub-modules θf(Ln), in which f(·) is a function that maps a lan-  guage to corresponding LS modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' To design an appropriate map function for  our selection mechanism, each sentence is prefixed to the corresponding lan-  guage symbol, which enables the model to correctly route sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Besides,  Springer Nature 2021 LATEX template  8  Article Title  all sub-modules from LS attend to the selection procedure during the training  stage, which solves the undifferentiability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Specifically, the function  ft(·) indicates the probability of selection of sub-module θt:  ft (Ln) =  exp  �  eLn  t  �  �T  i=1 exp  �  eLn  i  �  (5)  where eLn  i  is i-th element of the probability vector eLn = El[n]Wf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' El ∈  RN×d denotes the look-up table for all language prefix embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The router  matrix Wf ∈ Rd×T is used to project eLn which are normalized via a softmax  distribution over the total T modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  For each sub-module θt from θ, we utilize Eθt(·) to transform unified feature  representation ˆhLn into language-specific feature branch ˜hLn  θt :  ˜hLn  θt = Eθt(ˆhLn)  (6)  Eθt(ˆhLn) = LN  �  σ(ˆhLnWu)Wd + ˆhLn�  (7)  where ˆhLn ∈ Rm×d is an element of ˆhcat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Wu ∈ Rd×b and Wd ∈ Rb×d are  projection matrices (b > d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' σ is the ReLU activation function and LN(·) is  the layer normalization function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The right part of Figure 2 corresponds to  Equation 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  To ensure gradients are propagated to all sub-modules of LS {θt}T  t=1, we  apply the weighted average for obtaining the language-specific feature:  ˜hLn =  T  �  t=1  ft(Ln)Eθt  �  ˆhLn�  (8)  Note that for the whole process, function ft(Ln) in Equation 8 permits  differentiability of the router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  In the evaluation stage, it is necessary to prune several sub-module branches  with the lowest selection probabilities to obtain the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' There-  fore, we use the top-K strategy to select the best k(1 ≤ k ≤ T ) sub-modules  with the highest probabilities to generate the language-specific representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  When k = T indicates all sub-modules involved in the calculation which means  the selection mechanism is the same as the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The mapping pro-  cess is described as: Ln −→ {πLn  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , πLn  i  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , πLn  k } ∈ Π(k), where πLn  i  is  one of the sub-module index that corresponds to language Ln and Π(k) is the  space of all k-length combinations of Ck  T in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  After obtaining the language-specific representation from LS, we create  four matrices to recognize the head and tail positions of two named entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' To  enhance the accuracy of recognition, we add a relation feature that constrains  the extracted entities that are only related to the relevant relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Formally,  given a language-specific representation ˜hLn ∈ Rm×d of the m-length sentence  Springer Nature 2021 LATEX template  Article Title  9  and the relation vector re retrieved from relation embedding table Er ∈ RI×d,  where I is the number of relations, the two entities are recognized as followed:  entityx = (η((˜hLn ⊕ re)Wy))Uy  (9)  where the symbol collection entity={head, tail}, x={start,end} and y =  {hs, he, ts, te}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We concatenate the relation vector with each token rep-  resentation to enhance the recognition of entities, namely ˜hLn ⊕ re  =  {[˜hLn  1 , re], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , [˜hLn  i  , re], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' , [˜hLn  m , re]} ∈ Rm×2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Wy ∈ R2d×d are four down  projection matrices and Uy ∈ Rd×1 are four index projection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' η  denotes tanh activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Note that we use ground-truth relation as  input in training entity recognition, which conforms to the joint training  method in our architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='4 Training Objective  Our model presented in Figure 2 is trained jointly on multilingual ERE cor-  pora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We first train the model only using a multilingual training strategy for  our Language-universal Aggregator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Based on the unified language representa-  tion, we fine-tune the model with Language-specific Switcher for learning the  language-specific feature in the next step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The objective is to minimize the two  training loss functions which are defined below:  LLAT =  M  �  m=1  E(x,y)∼Dm[Lere(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Θ)]  (10)  LLST =  M  �  m=1  E(x,y)∼Dm[Lere(x, y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Θ, θ)]  (11)  where D means multilingual entity and relation extraction training corpora  and M denotes the number of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Θ indicates shared parameters and  θ is parameters in LS with selection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Lere is the loss function for  entity and relation extraction, which is defined as below:  Lere = α  2 (Lstart  h  + L end  h  + L start  t  + L end  t  ) + βLrel  (12)  where each L with any superscript is a cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The subscripts with  h and t indicate the head entity and tail entity respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The start and end  of superscripts denote the first token index and last token index of an entity  separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Lrel is the loss function for relation classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' α and β are two  weights on entity recognition loss and relation classification loss respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  10  Article Title  4 Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1 Datasets  We evaluate our model on the dataset SMiLER [8], which is the largest  and most diversified multilingual dataset for multilingual entity and relation  extraction tasks with 14 languages from 36 relation types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The SMiLER con-  sists of about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1M annotated sentences from Wikipedia and DBpedia, which  includes English (En), Korean (Ko), Italian (It), French (Fr), German (De),  Portuguese (Pt), Nederlands (Nl), Polish (Pl), Spanish (Es), Arabic (Ar), Rus-  sian (Ru), Swedish (Sv), Farsi (Fa), Ukrainian (Uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The relation types belong  to roughly nine domains: location, organization, person, animal, art, device,  measurement, event, and no relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The statistics of SMiLER are shown in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' As the development set in SMiLER is not publicly available, we only  randomly extract the sentences from the training set to create new files with  the same split ratio as the original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Table 1 The statistics of SMiLER dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' English corpora include full-size, middle-size,  and small-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The languages are ordered from high-resource languages (left) to  low-resource languages (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Languages  EN-full  EN-mid  It  Fr  De  Pt  Nl  En-small  Ko  Pl  Es  Ar  Ru  Sv  Fa  Uk  sentences num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  748k  269k  76k  62k  53k  45k  40k  35k  20k  17k  12k  9k  7k  5k  3k  1k  relation types  36  36  22  22  22  22  22  32  28  22  22  9  8  22  8  7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='2 Implementation Details  We conduct experiments on SMiLER, 14 languages in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' EN-small is  treated as our English corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We utilize mBERT as our cross-lingual encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  We train our model with AdamW, the learning rate is 3e-5 and weight decay  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The batch size is set to 16 on Tesla V100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The hidden size d is 768  and dimension b of projection matrices Wu and Wd is 1024.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The max sequence  length is 256 and we concatenate 2 sentences during the first training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' For  the second training stage, we freeze most parameters in the first stage except  the relation classifier and 8 matrices used to predict entities from Equation 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  The sub-module number T of LS is set to 6 (2 layers for 3 sub-modules and 1  layer for the other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The epoch is set to 5 at the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The max epoch of  the second stage is set to 8 with an early stopping mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The loss weights  are set to 2 in named entity recognition and 1 in relation classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  In the evaluation stage, we set k = 3 in the top-K strategy to select the  sub-modules in LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We adopt standard micro-F1 metric to calculate scores on  the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The extracted entity pair is regarded as correct if the predictions  of the head entity and tail entity are both the same as the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' A  triple is treated as correct if the entity pair and the corresponding relation  type are all correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' no relation type is included in relation prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We  also add a mask for the relation that is not absent in a language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Article Title  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='3 Baselines  As far as we know, the SMiLER is a new dataset and thus only an existing  method for multilingual ERE without publishing source code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The relevant  task is cross-lingual relation classification, which is also few in studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' There-  fore, we reproduce the following competitive baselines to compare with our  proposed approach for a fair comparison:  HEBERTa [8]: A multilingual entity and relation extraction framework  called Hybrid Entity and Relation extraction BERT, which achieves the  state-of-the-art performance on SMiLER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' HERBERTa uses a pipeline train-  ing manner that combines two independent BERT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The first  sub-model classifies the input sequence as one of 36 pre-defined relations  (including no relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The relation generated from the first sub-model is  then fed to the second BERT and concatenated with the same input sequence  as the input of the second model for entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  mBERT [65]: A cross-lingual model first uses the mBERT as a backbone  for RC, which is trained on 104 languages with the corresponding Wikipedia  dumps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We reproduce the results with the code shared at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  com/boun-tabi/RELX  MTMB [65]: A multilingual pre-training scheme called Matching the Mul-  tilingual Blanks (MTMB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The framework shows several advantages against  the mBERT on monolingual tasks and achieves significant improvements in  cross-lingual transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Note that this framework is only designed for RC and  not adapted to entity and relation extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Therefore, we simply modified  the output layer of the baseline to conduct the ERE task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  In addition to the above baselines, we also build a simplified multilin-  gual joint entity and relation extraction framework called mERE-LS-LA as a  basic structure which is concatenated relation representation with the sentence  representation to enhance the extraction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Table 2 The F1 scores of different models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' * denotes the model is reproduced by us on  our experiment settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' - denotes that the language data is not involved both in the  training and the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' MONO, EURO, and SVO mean training data in 3  different language groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The languages are ordered from high-resource languages (left) to  low-resource languages (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The bold font number is the best score in each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Test Sets  AVG  It  Fr  De  Pt  Nl  En  Ko  Pl  Es  Ar  Ru  Sv  Fa  Uk  HERBERTa*  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='5  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='9  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='7  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='5  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='5  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='9  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='4  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='0  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' Surprisingly,  even our baseline mERE-LS-LA has 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='9% improvement over the MTMB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It  seems that our basic structure is more effective on multilingual entity and rela-  tion extraction tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Compared with mERE-LS that only uses LA, our full  model mERE has nearly 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='7% F1 value improvement on average and yields  similar or higher results on 13 languages except for Sv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The improvement can  be attributed to our switcher-based language-specific training strategy, which  finally extracts accurate information for entity recognition in each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Compared with our baseline mERE-LS-LA, our full model mERE has nearly  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='4% F1 value improvement on average which means mERE-LS also has nearly  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='7% F1 value improvement on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' All such impressive results demon-  strate that our full model mERE truly enhances the representation quality  and mitigates language interference to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  We set several language groups to analyze the impact of different languages:  (1)MONO: 14 languages in monolingual training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' (2)EURO: It, Fr, Pt, De,  Es, En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' (3)SVO2: EURO, Ru, Sv, Nl, Pl, Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The default is all languages  in multilingual training from Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Compared with mERE-LS-LA training  in multilingual corpora, we can observe that multilingual training achieves  much higher results than mERE-LS-LA (MONO) monolingual training from  Table 2, especially on low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Such as improvements of Uk  (25%), Fa (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1%), and Ru (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It demonstrates that languages with less  training data can benefit most from high-resource languages in multilingual  training including ERE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The results of the EURO family group are close  to mERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It is worth noting that Es achieves the best score in the EURO  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We conclude that Es benefits a lot from similarities of languages that  are in the same language family even with less training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In the SVO  group, we can also visualize that most languages in EURO decrease slightly  with the interference of other non-EURO languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The different language  families, or the languages with a big difference in syntactic structures might  be the main interference among languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' However, compared with mERE  (SVO), mERE yields the same results on low-resource languages and somewhat  higher results on high-resource languages even the three non-SVO (Fa, Ar, and  Ko) data involved during the training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We suppose that these non-SVO  languages which are big different from others and are all low-resource may  facilitate distinguishing high-resource languages in learning language-specific  2SVO stands for the relative position of the Subject, Verb, and Object in the typical affirmative  sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We treat Korean, Farsi, and Arabic as non-SVO languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Arabic is VSO, while Korean  and Farsi are SOV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Article Title  13  features due to each sub-module from LS being independent, without sharing  parameters in the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Lastly, we also observe some duplicated F1  scores across low-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' This phenomenon is caused by a small  number of sentences in test sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='5 Entity and Relation Analysis  Figure 3 shows F1 scores of relation and entity pair of mERE and mERE-LS-  LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We can observe that the relation classification seems to be easier than the  named entity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The correctness of entity pair extraction is the main  bottleneck of the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' With the help of our LA and LS, mERE  achieves higher results on entity pair recognition compared with mERE-LS-  LA in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Surprisingly, we can visualize that the performance of relation  classification also has a slight improvement in mERE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We conclude that the  improvement of the named entity recognition facilitates relation classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Since information interaction between two sub-tasks can benefit each other in  the joint training architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  50  55  60  65  70  75  80  85  90  95  100  Total  It  Fr  De  Pt  Nl  En  Ko  Pl  Es  Ar  Ru  Sv  Fa  Uk  Relation(mERE)  Entity Pair(mERE)  Relation(mERE-LS-LA)  Entity Pair(mERE-LS-LA)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 3 The F1 scores of relations and entity pairs on all languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  F1 scores of detailed relation labels are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Most of the  relations achieve higher F1 scores across languages, such as “no relation” and  “has-type”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Part of relations differs widely across languages, such as relation  “has-child”(F1 = 100 on Nl, F1 = 33 on De, F1 = 0 on Es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The big difference  is caused by the number of relations of training data in each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' For  some relations that occur F1 = 0 scores, we find out the relations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='g won-  award on Nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' has-parent on Pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' has-child on Es) are only one test sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Such low results for some languages could be explained by a smaller number  of relations in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='Springer Nature 2021 LATEX template  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='14  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='Article Title  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='En  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='It  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='Fr  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='Pt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='Es  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='De  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='is-where  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='birth-place  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 4 The F1 scores of all relation labels on all languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The darker color means a higher  F1 score, while the lighter color means a lower F1 score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Figure 5 shows F1 scores of head entities and tail entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We can observe  that F1 scores of head entities are much higher than tail entities among most  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It seems that head entities are easier to be recognized than tail  entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It is because the head entity always occurs at the beginning position  of the sentence and thus the model probably memorizes the position, while the  tail entity does not have any consistent position which is hard to predict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='6 Ablation Study  Sentences Concatenation To validate the effect of the number of sentences  for learning the unified features among different languages, we conduct sev-  eral experiments on the different numbers of sentences in concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We  learn from Figure 6 that there are evident F1 improvements with LA on dif-  ferent concatenation numbers of sentences over only one sentence encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  The multilingual model obtains the best performance when concatenating with  the sentence pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The increasing number of concatenated sentences has a  slight decrease in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We conjecture that increasing the number of  sentences may also bring somewhat interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Springer Nature 2021 LATEX template  Article Title  15  50  55  60  65  70  75  80  85  90  95  100  Total It  Fr  De  Pt  Nl  En  Ko  Pl  Es  Ar  Ru  Sv  Fa  Uk  head entity  tail entity  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 5 The performance of head entities (blue bar) and tail entities (orange bar) on different  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  1  2  3  4  Languages  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='6  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='7  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='8  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='9  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='0  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='1  F1 score  mERE-LS  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 6 The performance of sentences concatenation in the first training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Selection Mechanism To observe how the selection mechanism affects our  model performance, we also train one-to-one sub-modules of LS called mERE14  without using the selection mechanism in the second training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Each inde-  pendent sub-module corresponds to a language and each sentence is routed via  a language prefix which represents the number of sub-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We can visualize  from Figure 7 that increasing the number of parameters also improves obvi-  ously over mERE-LS-LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Nonetheless, the mERE14 will suffer from the sharp  increasing training time and inference time, and big space consumption when  the number of languages is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Instead of increasing parameters,  our Language-specific Switcher can effectively ameliorate extraction quality  with only slight extra parameters and less time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Since similar  languages tend to select the same sub-modules from our LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The mERE saves  Springer Nature 2021 LATEX template  16  Article Title  nearly 700M model capacity in our statistics and achieves better performance  among most languages compared with mERE14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It is obvious that mERE is  light and easy to transfer to other multi-field tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Selection Distribution  It  Fr  De  Pt  Nl  En  Ko  Pl  Es  Ar  Ru  Sv  Fa  Uk  Languages  60  62  64  66  68  70  72  74  76  78  80  82  84  86  88  90  92  94  F1 scores  mERE-LS-LA  mERE14  mERE  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 7 The performance of three models on 14 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' mERE14 utilizes 14 one-to-one  sub-modules of LS without the selection mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Each sub-module corresponds to a  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Figure 8 illustrates the heatmap of selection probability on 6 sub-modules from  LS for each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' For each sub-module from top to bottom in Figure 8,  we can visualize θ1 pays more attention to low-resource languages while θ4,  θ5, and θ6 pay more attention to languages from the EURO family, which  are mostly high-resource languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The θ2 and θ3 seem to be more balanced  on parameters sharing of languages except for 2 or 3 prominent languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  We conclude that some sub-modules are mainly used to extract features from  similar languages and others are used to assist the specific languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  For each language from left to right, we can visualize that the selected  sub-modules with higher probabilities are easy to distinguish in high-resource  languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In contrast, the selection probabilities across all sub-modules are  relatively similar on low-resource languages in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We conclude that train-  ing data is rich enough to determine which way to route on high-resource  languages and a more balanced selection decision is made on less training  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It is learned from Figure 8 that there are nearly 3 out of 6 prominently  higher selection probabilities on the high-resource languages, and so do the  low-resource languages with careful observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It proves that only 3 sub-  modules play the dominant role in refining the language-specific feature for  each language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' To avoid interference from the other irrelevant sub-modules,  we adopt a top-K strategy to filter out 6 − k sub-modules with lower selec-  tion probabilities in the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The top-6 strategy means selecting  all sub-modules, which is the same as the training stage and the performance  is relatively low (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='74) on average, while our mERE achieves the best perfor-  mance (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='87) when adopting the top-3 strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' It demonstrates that filtering  out the least important sub-modules is necessary to enhance the prediction  quality, which also reduces the redundant parameters in the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The  Springer Nature 2021 LATEX template  Article Title  17  top-1 achieves the worst performance (77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='60), which demonstrates the part of  sub-modules are also helpful for the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Therefore, the best performance is  obtained when the k value is balanced in all languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Layer Number of  It  Fr  De  Pt  Nl  En  Ko  Pl  Es  Ar  Ru  Sv  Fa  Uk  1  2  3  4  5  6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 8 The selection probability distributions of 6 sub-modules from LS on 14 languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  The sub-modules {θ}6  1 are numbered from 1 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The languages are ordered from high-  resource languages (left) to low-resource languages (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The darker color means a higher  selection probability to the corresponding sub-module and a lower probability to select a  certain sub-module when the color is lighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Language-specific Switcher Table 3 used to evaluate the effect of the layer  number of LS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' We divide the 6 sub-modules into 2 groups (each group has the  same layer number) with different combinations of layer numbers to accommo-  date the scenarios, such as high- and low-resource language feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  From Table 3, we can observe that the combination 1-2 achieves the best F1  score on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The combinations which are set to 1-1 and 4-4 also achieve  better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' With the increase or decrease of the layer number to a cer-  tain degree, the performances are almost the same, which maintains relatively  low averaged F1 scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The full layer number combination 4-4 is an exception  in the case, which demonstrates the performance still can be improved when  the model capacity is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' According to the outcomes from Table 3,  we conclude that the layer number of LS obviously impacts the results, with  the best results attained when a balance is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  5 Conclusion  In this paper, we introduce a two-stage training method and a robust frame-  work called mERE for multilingual entity and relation extraction, which  ameliorates the sentence representation quality and mitigates the language  interference among multiple languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Specifically, we first learn the gener-  alities across all languages to obtain the unified language representation via  the Language-universal Aggregator and then learn the specialties of each lan-  guage via the Language-specific Switcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Experimental results demonstrate  Springer Nature 2021 LATEX template  18  Article Title  Table 3 The different layer numbers of sub-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Every 3 sub-modules in a group has  the same layer numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Layer Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='01 and Layer Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='02 denote the layer number of the  first group and second group respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='0  that our method significantly outperforms both monolingual and multilingual  ERE baselines, which demonstrates that our framework can extract relational  triples among various languages well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Moreover, our framework is also light  and easy to transfer to other backbone models of multi-field tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  In the future, we will pay more attention to complex multilingual relational  triple extraction, such as overlapping relational triples or multiple relational  triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Besides, we will also do further research on a better contextual repre-  sentation among multiple languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Although there is a long way to experience  in multilingual entity and relation extraction tasks, it is important to inves-  tigate the valuable structured information in many other languages for the  downstream NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  This work was supported in part by the National Nat-  ural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 62276017, U1636211, 61672081),  the 2022 Tencent Big Travel Rhino-Bird Special Research Program, and the  Fund of the State Key Laboratory of Software Development Environment  (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' SKLSDE-2021ZX-18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  References  [1] Huang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Xu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Yu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=': Bidirectional LSTM-CRF models for sequence  tagging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' CoRR abs/1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='01991 (2015) 1508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='01991  [2] Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=') Proceedings of the 28th International  Conference on Computational Linguistics, COLING 2020, Barcelona,  Spain (Online), December 8-13, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='  Fusion 90, 253–264 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='org/D12-  1110/  [26] Hashimoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=': Simple cus-  tomization of recursive neural networks for semantic relation classifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In: Proceedings of the 2013 Conference on Empirical Methods in  Natural Language Processing, EMNLP 2013, 18-21 October 2013, Grand  Hyatt Seattle, Seattle, Washington, USA, A Meeting of SIGDAT, a  Special Interest Group of The ACL, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' ACL, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  https://aclanthology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' ACL, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=') NAACL HLT 2016, The 2016 Conference  of the North American Chapter of the Association for Computa-  tional Linguistics: Human Language Technologies, San Diego California,  USA, June 12-17, 2016, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 866–875.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' The Association for Compu-  tational Linguistics, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Schuster, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Hughes, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Linguistics 5, 339–351 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=': Pre-training  multilingual neural machine translation by leveraging alignment informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=': Massively multilingual neural  machine translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=') Pro-  ceedings of the 2019 Conference of the North American Chapter of the  Association for Computational Linguistics: Human Language Technolo-  gies, NAACL-HLT 2019, Minneapolis, MN, USA, June 2-7, 2019, Volume  1 (Long and Short Papers), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' Association for Compu-  tational Linguistics, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=') Proceedings  of the 58th Annual Meeting of the Association for Computational Lin-  guistics, ACL 2020, Online, July 5-10, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Creutz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' In: Augenstein, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Gella,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Conneau, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Rei, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=') Proceedings of the 4th Workshop on Representation Learning for  NLP, RepL4NLP@ACL 2019, Florence, Italy, August 2, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='  Association for Computational Linguistics, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Berard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Besacier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=') Proceedings of the 2020 Conference on Empir-  ical Methods in Natural Language Processing, EMNLP 2020, Online,  November 16-20, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=': Attention is all you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In:  Guyon, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Luxburg, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=') Advances in Neural Information  Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='  https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Mirhoseini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Maziarz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Davis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=', Le, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content=', Dean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=': Outrageously large neural networks: The sparsely-gated  mixture-of-experts layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In: International Conference on Learning Rep-  resentations (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+page_content='id=B1ckMDqlg  [65] K¨oksal,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=',  ¨Ozg¨ur,  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=':  The  RELX  dataset  and  matching  the  multilingual blanks for cross-lingual relation classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content=' In: Find-  ings  of  the  Association  for  Computational  Linguistics:  EMNLP  2020,  pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  340–350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
+page_content='  Association  for  Computational  Linguistics,  Online  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE3T4oBgHgl3EQfSgnV/content/2301.04434v1.pdf'}
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+Differentially Private Distributed Bayesian
+Linear Regression with MCMC
+Barı¸s Alparslan1, Sinan Yıldırım1, 2, and S¸. ˙Ilker Birbil3
+1Faculty of Engineering and Natural Sciences, Sabancı University, ˙Istanbul, Turkey∗
+2Center of Excellence in Data Analytics (VER˙IM), Sabancı University, ˙Istanbul, Turkey
+3Department of Business Analytics, University of Amsterdam, Amsterdam, The Netherlands
+February 1, 2023
+Abstract
+We propose a novel Bayesian inference framework for distributed differentially private
+linear regression. We consider a distributed setting where multiple parties hold parts of the
+data and share certain summary statistics of their portions in privacy-preserving noise. We
+develop a novel generative statistical model for privately shared statistics, which exploits a
+useful distributional relation between the summary statistics of linear regression. Bayesian
+estimation of the regression coefficients is conducted mainly using Markov chain Monte Carlo
+algorithms, while we also provide a fast version to perform Bayesian estimation in one iteration.
+The proposed methods have computational advantages over their competitors. We provide
+numerical results on both real and simulated data, which demonstrate that the proposed
+algorithms provide well-rounded estimation and prediction.
+Keywords: Differential privacy, linear regression, distributed learning, MCMC
+1
+Introduction
+Linear regression is a mathematical method that lies at the core of statistical research. Many
+researchers have been working on linear regression since the 19th century, and hence, many
+well-known solution methods exist. On a separate note, privacy-preserving statistical learning has
+gained popularity and importance in recent years, with differential privacy prevailing as the most
+commonly used definition for privacy (Dwork, 2006; Dwork et al., 2014a; Dankar and El Emam,
+2013). As a result, there is a recent but growing interest in differentially private linear regression.
+Many works in the data privacy literature do not mainly focus on regression but are motivated by
+or can be applied to regression. As an example, differentially private empirical risk minimisation
+(Chaudhuri et al., 2009; Bassily et al., 2014; Abadi et al., 2016; Kuru et al., 2022) can be applied to
+regression once it is cast as a data-driven optimisation problem. Many general-purpose Bayesian
+differentially private estimation methods can also be used in regression problems. Williams and
+Mcsherry (2010) is one of the first works that considered a hierarchical model for the privatised
+data and Bayesian estimation for the model parameters. Zhang et al. (2016) analyse several
+differential privacy mechanisms for posterior sampling and suggest using these mechanisms also
+∗The study was funded by the Scientific and Technological Research Council of Turkey (T¨UB˙ITAK) ARDEB
+Grant No 120E534. Barı¸s Alparslan and Sinan Yıldırım were supported by the project.
+1
+arXiv:2301.13778v1  [stat.ML]  31 Jan 2023
+
+for linear regression. Dimitrakakis et al. (2017) developed a posterior sampling query algorithm
+to combine differential privacy and Bayesian inference. Contrary to those one-sample approaches,
+general-purpose differentially private Markov chain Monte Carlo (MCMC) algorithms, which aim
+to identify the posterior distribution via iterative sampling, can also be applied to regression
+(Wang et al., 2015; Foulds et al., 2016; Wang et al., 2015; Yıldırım and Ermi¸s, 2019; Heikkil¨a
+et al., 2019; Gong, 2022; Alparslan and Yıldırım, 2022; Ju et al., 2022).
+Several works in the literature are somewhat more directly related to differentially private regression.
+Zhang et al. (2012) suggested a functional mechanism method, which is based on perturbing
+polynomial objective functions with privacy-preserving noise. As an alternative, Dwork et al.
+(2014b); Wang (2018) considered perturbation of summary statistics. Alabi et al. (2022) provide
+a technical discussion on different point estimation methods for differentially private simple linear
+regression, that is when we have a single feature. Ferrando et al. (2022) present a method to
+compute confidence intervals for the coefficients of linear regression. Cai et al. (2021) study the
+rates of convergence for parameter estimation with differential privacy via output perturbation,
+where a non-private estimator is perturbed. All those works consider point estimation of the
+linear regression parameters.
+In this paper, we focus on differential private distributed Bayesian inference for the parameters of
+linear regression. We use a novel hierarchical model that relies on a distributional relationship
+(Proposition 1) between the summary statistics of linear regression, which, to the best of our
+knowledge, has not been exploited so far. We propose Bayesian inference algorithms that take
+perturbations of summary statistics as observations. The general inferential tool we pick in this
+paper is MCMC, a well-known framework for iterative sampling from posterior distributions. As
+we shall see, the proposed MCMC algorithms in this paper already have lower computational
+complexities per iteration than their closest competitors in Bernstein and Sheldon (2019). Addi-
+tionally, we also propose much faster Bayesian estimation methods that perform estimation in
+one iteration. Finally, we assume a distributed setting where the total dataset is shared among
+multiple parties (data nodes), who want to collaborate for the inference of a common parameter,
+see e.g., Heikkil¨a et al. (2017) for such a setting. The non-distributed setting is just a special
+case (single data holder) for our methodology.
+This paper has connections with several works in the literature, yet it has significant differences
+from each of those, as we shall explain below.
+For the privacy-preserving mechanism, we consider adding noise to summary statistics of linear
+regression, similarly to Wang (2018); Bernstein and Sheldon (2019). The adaSSP framework of
+Wang (2018) motivates the fast Bayesian estimation methods developed in this paper. However,
+adaSSP is a point estimation method while we aim for a posterior distribution. The latter work,
+Bernstein and Sheldon (2019), is particularly related to this paper as they also study Bayesian
+linear regression with differential privacy using perturbed statistics of data. However, there are
+some important differences between our work and that of Bernstein and Sheldon (2019). These
+differences stem from the choice of summary statistics and the consequent hierarchical structure
+used for modelling linear regression. Those modelling differences lead to significant differences in
+the inference methods as well as significant computational advantages for our methods. Specifically,
+the computational complexity of our methods is O(d3), where d is the number of features. This
+order is much less than the O(d6) of Bernstein and Sheldon (2019). Finally, neither Wang (2018)
+nor Bernstein and Sheldon (2019) has considered a distributed learning setting like we do in
+2
+
+this paper, although both works can be modified for the distributed setting after moderate
+modifications.
+Foulds et al. (2016); Heikkil¨a et al. (2017) are other differentially Bayesian inference methods
+that target posterior distributions of perturbed summary statistics of sensitive data. The one by
+Heikkil¨a et al. (2017) is particularly interesting because they consider a distributed setting and
+present linear regression as their showcase example. However, we differ from those works in the
+way we model the perturbed statistics and in the choice of inference methods. Specifically, Foulds
+et al. (2016); Heikkil¨a et al. (2017) treat the perturbed statistics as if not perturbed, while we
+incorporate the effect of perturbation in our model.
+Recently, Alparslan and Yıldırım (2022) and Ju et al. (2022) employ data augmentation for
+modelling sensitive and privatised data and propose MCMC for Bayesian inference, the latter work
+having linear regression as a major application. Their methods have O(n) complexity per iteration
+in general where n is the number of instances in the data set, which can be slow when n is large.
+In contrast, our methods are scalable in data size since their computational complexities do not
+depend on n. We note that Alparslan and Yıldırım (2022, Section 4.2) also present an MCMC
+method scalable with n that exploits the approximate normality of additive summary statistics.
+However, a direct application of that would lead to an algorithm with O(d6) computational
+complexity (per iteration), like in Bernstein and Sheldon (2019).
+The paper is organised as follows: In Section 2 we review differential privacy. In Section 3 we lay
+out the hierarchical model for differentially private distributed linear regression with perturbed
+summary statistics. In Section 4, we present and discuss the aspects of the proposed inference
+algorithms. Section 5, we provide numerical experiments. We conclude in Section 6.
+Notation:
+Matrices and vectors are shown in bold-face notation. For a matrix A, its transpose,
+trace, and determinant (whenever they exist) are AT , tr(A), and |A|, respectively. For any
+sequence {ai}i≥0, we let ai:j = (ai, . . . , aj). We write x ∼ P to mean the random variable x
+has distribution P. N(m, Σ) stands for the multivariate normal distribution with mean m and
+covariance Σ. Wishart and inverse-Wishart distributions with scale matrix Λ and κ degrees of
+freedom are shown as W(Λ, κ) and IW(Λ, κ), respectively. IG(a, b) stands for the inverse-gamma
+distribution with shape and scale parameters a and b. We augment those notations with x to
+denote the respective probability density functions (pdf), e.g., as N(x; m, Σ).
+2
+Differential Privacy
+Differential privacy (Dwork, 2006, 2008) concerns randomised algorithms that run on sensitive,
+or usually private, data. A randomised algorithm takes an input data set D ∈ D and returns a
+random output in O, where the randomness is intrinsic to the algorithm. A differentially private
+algorithm constrains the difference between the probability distributions of the outputs obtained
+from neighbouring data sets. We say two data sets are neighbours if they differ by one individual’s
+piece of data.
+Definition 1 (Differential privacy). A randomised algorithm M : D �→ O is (ϵ, δ)-differentially
+private (DP) if for any pair of neighbouring data sets D, D′ ∈ D and for any subset O ⊆ O of the
+of support domain, it satisfies
+P[M(D) ∈ O] ≤ eϵP[M(D′) ∈ O] + δ.
+3
+
+The definition implies that smaller (ϵ, δ) leads to more privacy.
+Privacy-preserving algorithms often use noise-adding mechanisms. A popular noise-adding mecha-
+nism is the Gaussian mechanism (Dwork et al., 2006), which perturbs a function f : D �→ Rk of
+the sensitive data, for some k ≥ 1, with a random noise drawn from the Gaussian distribution.
+The amount of the added noise depends on the L2-sensitivity of the function, given by
+∆f =
+max
+neighbourD1,D2∈D∥f(D1) − f(D2)∥2.
+An (ϵ, δ)-DP Gaussian mechanism returns
+f(D) + ∆fσ(ϵ, δ)v,
+v ∼ N(0, Ik)
+(1)
+upon taking D as the input, where the quantity σ(ϵ, δ) ensures (ϵ, δ)-DP. In this work, we take
+σ(ϵ, δ) as the analytical solution given in Balle and Wang (2018, Algorithm 1) due to its tightness.
+The Gaussian mechanism is also central to other forms of privacy, such as zero-concentrated DP
+(Bun and Steinke, 2016) and Gaussian DP (Dong et al., 2022).
+In this paper, we consider (ϵ, δ)-DP as the type of privacy and the Gaussian mechanism to generate
+noisy observations. Moreover, the proposed methods in this paper never use the sensitive data
+once given the noisy observations generated using the Gaussian mechanism, hence exploiting the
+post-processing property of differential privacy (Dwork and Roth, 2014).
+Theorem 1 (Post-processing). If M : D �→ O be (ϵ, δ)-DP and let f : O → O′ be another mapping
+independent of D given M(D). Then fM : D �→ O′ with fM(D) = f(M(D)) is (ϵ, δ)-DP.
+3
+Differentially Private Distributed Linear Regression
+In this section, we present a new hierarchical model for differentially private distributed linear
+regression. For ease of exposition, we first present a model with a single data holder, then
+generalise the model for the distributed setting.
+3.1
+Basic Model and Privacy Setup
+Suppose we have a sequence of random variables {(xi, yi) : i = 1, . . . , n}, where xi ∈ X ⊆ Rd×1
+are the feature vectors and yi ∈ Y ⊆ R is the i’th response variable. We consider the normal
+linear regression to model the dependency between xi and yi. Specifically,
+yi = xT
+i θ + ei,
+ei
+i.i.d.
+∼ N(0, σ2
+y),
+i = 1, . . . , n,
+where θ ∈ Rd is the vector of the linear regression coefficients. We assume that the feature vectors
+xi’s are i.i.d. with distribution Px. Below, we will particularly focus on the case when Px can be
+assumed to be a normal distribution. However, we will also present algorithms for general Px.
+In matrix notation, the above can shortly be expressed as
+y = Xθ + e,
+e ∼ N(0, σ2
+yIn),
+where X =
+�
+xT
+1
+. . .
+xT
+n
+�T is the so-called design matrix, y =
+�
+y1
+. . .
+yn
+�T . Additionally, we
+also define the summary statistics of X and y given by
+S := XT X,
+z := XT y,
+4
+
+respectively. We assume a setup where S and z are privately released as the noisy summary
+statistics ˆS and ˆz are constructed as
+ˆS = S + σsM,
+(2)
+ˆz = z + σzv,
+v ∼ N(0, Id),
+(3)
+where M is a d × d symmetric matrix with its upper triangular elements drawn from N(0, 1).
+Dwork et al. (2014b) arrange σs and σz so that both (2) and (3) are (ϵ/2, δ/2) differentially
+private, leading to (ϵ, δ)-DP overall. Differently than Dwork et al. (2014b), we set
+σs = σz = ∆szσ(ϵ, δ),
+where σ(ϵ, δ) is given in Balle and Wang (2018, Algorithm 1), and ∆sz is the overall L2 sensitivity
+of [S, z], given by
+∆sz =
+�
+∥X∥4 + ∥X∥2∥Y ∥2
+with ∥X∥ = maxx∈X ∥x∥2 and ∥Y ∥ = maxy∈Y |y|.
+Based on the above relations, we shall represent a hierarchical model that enables Bayesian
+inference of θ given ˆS and ˆz. One important element of our modelling approach is the following
+result that establishes the conditional distribution of z given S, θ, and σ2
+y.
+Proposition 1. For the normal linear regression model, we have
+z|S, θ, σ2
+y ∼ N(Sθ, Sσ2
+y).
+Proof. First, note that,
+E[z|X, θ, σ2
+y] = E[XT Xθ + XT e] = Sθ,
+(4)
+Cov(z|X, θ, σ2
+y) = XT Xσ2
+y = Sσ2
+y,
+(5)
+and observe that both moments depend on X through its statistic S. Therefore, the conditional
+density of z given S, θ, and σ2
+y is
+p(z|X, θ, σ2
+y) = N(z; Sθ, Sσ2
+y).
+Next, define the function f : Rn×d �→ [0, ∞) with f(X) = p(z|X, θ, σ2
+y) and let CS,θ,σ2y = {X :
+XT X = S}, Since the function f is constant over CS,θ,σ2y, we can write
+p(z|S) =
+�
+CS,θ,σ2y
+fdPx = N(z; Sθ, Sσ2
+y),
+where the second equation is by moment equations in (4) and (5) above. This concludes the
+proof.
+Finally, we assign prior distributions for θ, σ2
+y as
+θ ∼ N(m, C),
+σ2
+y ∼ IG(a, b).
+(6)
+5
+
+At this point, it is worth discussing some important modelling differences between our work and
+Bernstein and Sheldon (2019). In Bernstein and Sheldon (2019), the central limit theorem (CLT)
+is applied to
+�
+S, z, yT y
+�
+, leading to a normality assumption for the whole vector. In contrast,
+we use the exact conditional distribution p(z|S, θ, σ2) thanks to Proposition 1. Moreover, unlike
+Bernstein and Sheldon (2019), we do not require a noisy version yT y, hence have a slight advantage
+of using less privacy-preserving noise. In summary, our model has a different hierarchical structure
+and requires less privacy-preserving noise.
+3.2
+Distributed Setting
+Next, we extend our model to the distributed setting, where the total data are shared among
+J ≥ 1 data holders as
+(X, y) = {(Xj, yj); j = 1, . . . , J}.
+(7)
+We let ni be number of rows in each xi, so that n = n1 + . . . + nJ. Each data holder j shares
+their own summary statistics Sj = XT
+j Xj, zj = XT
+j yj with privacy-preserving noise
+ˆSj = Sj + σsMj,
+ˆzj = z + σzvj,
+vj ∼ N(0, Id).
+(8)
+Note that, to preserve a given (ϵ, δ)-DP overall, each party must provide that level of privacy
+for their data, hence σs and σz are the same as before. The hierarchical structure of the overall
+model (specified for normally distributed xi’s) is shown in Figure 1.
+Figure 1: Differentially private distributed linear regression model (specified for normally distributed xi’s.)
+The distributed setting deserves separate consideration than the single data holder case for a couple
+of reasons: Firstly, the node-specific observations ( ˆS1, ˆz1), . . . , ( ˆSJ, ˆzJ) are altogether statistically
+more informative on θ than their aggregates �J
+j=1 ˆSj and �J
+j=1 ˆzj. This is because the aggregate
+versions are not sufficient statistics of the node-specific observations ( ˆS1, ˆz1), . . . , ( ˆSJ, ˆzJ) with
+respect to θ (even when σ2
+y is known.) Therefore, when the node-specific observations are available,
+one should not, in principle, trivially aggregate them and apply an inference method designed for
+J = 1 using those aggregates.
+Secondly, the partitioning of data as in (7) can be relevant to data privacy applications even
+outside the distributed learning framework, rendering the methodology in Section 4 useful in a
+6
+
+broader sense. For example, batches of (x, y)-type of data may be donated to a common data
+collector as in (8). At this point, a particular and interesting relation exists with pan-privacy
+applications (Dwork et al., 2010). Imagine that sensitive data from individuals are collected
+sequentially in time, and the data holder is concerned about possible intrusions into the memory
+where the sensitive data are stored. Then, one possible way to ensure the privacy of the data
+against such possible intrusions, which is the promise of pan-privacy, is to store the noisy statistics
+of every new batch of data and erase the original sensitive data. Then, at any time the data
+collector has data of the form ( ˆS1, ˆz1), . . . , ( ˆSJ, ˆzJ), each pair corresponding to a batch. As a
+result, inference algorithms as in Section 4 can be applied.
+4
+Algorithms for Bayesian Inference
+Bayesian inference targets the posterior distribution of the latent variables of the model, in
+particular θ, given the observations ˆS1:J and ˆz1:J. We present several Bayesian inference algorithms
+for the hierarchical model described in the previous section. In addition to other concerns like
+computational budget, the choice among those approaches mainly depends on the specification of
+Px as the distribution of S directly depends on it. In this paper, we have considered the following
+two cases and devised algorithms for each of them:
+1. In some cases it may be adequate to specify Px = N(0, Σx). This leads to S|Σx ∼ W(Σx, n).
+Further, to account for the uncertainty about the covariance Σx, one can treat it as a random
+variable with Σx ∼ IW(Λ, κ). Figure 1 shows the hierarchical structure of the distributed
+setting with those specifications. We defer discussing the conflict between the normality and
+boundedness assumptions to Remark 1 towards the end of Section 4.1.
+2. As the second case, we assume a general (non-normal) Px. A normal approximation, based on
+the CLT, could be considered for the distribution S (Wilson and Ghahramani, 2011). However,
+this would require the knowledge (or accurate estimation) of up to the fourth moments of Px
+as well as expensive computations for sampling S. We circumvent those difficulties by plugging
+in a point estimate of S given ˆS and use it during the sampling process as if it is the true
+S itself. Then, we develop two different algorithms for inference of θ, one being an MCMC
+algorithm and the other providing a closed form-solution for the posterior of θ following a
+rough point-wise estimation of σ2
+y. Note that these algorithms with fixed S do not require a
+distribution for x.
+Next, we provide the details of our approaches and the resulting algorithms.
+4.1
+Normally Distributed Features
+In this section, we present an MCMC algorithm for Bayesian inference for the differentially private
+distributed linear regression model when Px = N(0, Σx) and Σx ∼ IW(Λ, κ). The latent variables
+involved in this variant are θ, Σx, σ2
+y, S1:J, z1:J. Their posterior distribution given ˆS1:J, ˆz1:J can
+be written as
+p(θ, σ2
+y, Σx, z1:J, S1:J|ˆz1:J, ˆS1:J) ∝ p(θ)p(σ2
+y)p(Σx)
+J
+�
+j=1
+p(zj|θ, σ2
+y, S)p(Sj|Σx)p( ˆSj|Sj)p(ˆzj|zj).
+(9)
+7
+
+One could design an MCMC algorithm for this posterior distribution that updates θ, σ2
+y, Σx, z1:J,
+S1:J in turn based on their full conditional distributions. However, such an algorithm suffers from
+poor convergence because of a high posterior correlation between θ and z1:J (as verified in our
+numerical studies). It is well known that highly correlated variables result in poor convergence
+if they are updated one conditional on the other. To alleviate that problem, we work with the
+reduced model where z1:J are integrated out. The reduced model has θ, Σx, σ2
+y as its latent
+variables, whose joint posterior distribution can be written as
+p(θ, σ2
+y,Σx, S|ˆz, ˆS) ∝ p(θ)p(σ2
+y)p(Σx)
+J
+�
+j=1
+p(Sj|Σx)p( ˆSj|Sj)p(ˆzj|Sj, θ, σ2
+y),
+(10)
+where p(ˆz|S, θ, σ2
+y) = N(ˆz; Sθ, σ2
+ySθ + σ2
+zId).
+We would like to sample from the posterior distribution in (10) via MCMC that updates θ, σ2
+y,
+Σx, S1:J in turn based on their full conditional distributions. The variables θ and Σx enjoy
+closed-form full conditional distributions (see Appendix A for the derivations):
+Σx|S1:J, ˆS1:J, ˆz1:J ∼ IW
+�
+�Λ +
+J
+�
+j=1
+Sj, κ + n
+�
+� ,
+(11)
+θ|σ2
+y, ˆz, S1:J ∼ N(mp, Σp),
+(12)
+where the posterior moments for θ are
+Σ−1
+p
+=
+J
+�
+j=1
+Sj(σ2
+ySj + σ2
+zI)−1Sj + C−1,
+mp = Σp
+�
+�
+J
+�
+j=1
+Sj(σ2
+ySj + σ2
+zI)−1 ˆzj + C−1m
+�
+� .
+The full-conditional distributions of S1:J and σ2
+y have no closed form; hence we design Metropolis-
+Hastings (MH) moves to update them. For σ2
+y, one can simply use a random-walk MH move
+targeting p(σ2
+y|θ, S1:J, ˆz1:J). For S1:J, their full conditional distribution can be factorised as
+p(S1:J| ˆS1:J, ˆz1:J, Σx, σ2
+y, θ) =
+J
+�
+j=1
+p(Sj| ˆSj, ˆzj, Σx, σ2
+y, θ),
+where each factor is given by
+p(Sj| ˆSj, ˆzj, Σx, σ2
+y, θ) ∝ p(ˆzj|Sj, θ, σ2
+y)p(Sj|Σx)p( ˆSj|Sj).
+Thanks to that factorised form, each Sj can be updated with an MH move independently and
+in parallel. For the MH algorithm to update one Sj, we propose a new value from a Wishart
+distribution as S′
+j ∼ W(Sj/α, α), which has mean Sj and variance determined by α. In our
+experiments, we adjust a using ideas from the adaptive MCMC framework (Andrieu and Thoms,
+2008) to target an acceptance rate of around 0.2.
+Algorithm 1 represents the overall MCMC algorithm for the hierarchical model for differentially
+Bayesian distributed linear regression when Px is a normal distribution with a random covariance
+matrix having an inverse-Wishart distribution. We call this algorithm MCMC-normalX.
+8
+
+Algorithm 1: MCMC-normalX - one iteration
+Input: Current values of S1:J, θ, σ2
+y, Σx; observations ˆS1:J,ˆz1:J; noise variances σ2
+s, σ2
+z;
+proposal parameters a, σ2
+q; hyperparameters a, b, κ, Λ, m, C.
+Output: New sample of Σx, S, σ2
+y, θ
+1 Sample Σx using (11).
+2 for j = 1, 2, . . . J do
+3
+Update Sj via an MH move targeting p(Sj|Σx, θ, ˆzj).
+4 Sample θ using (12).
+5 Update σ2
+y via an MH move targeting p(σ2
+y|θ, S1:J, ˆz1:J).
+Remark 1. Admittedly, a potential concern is a conflict between the normality and boundedness
+assumptions (both for x and y). However, we also note that the collected data often happen
+to have some natural boundaries (which can be exploited to determine the sensitivity of the
+shared statistics), and yet the normal distribution is still used for modelling and subsequent
+inference mainly for sake of tractability. With the normality assumption, one can implement
+computationally efficient algorithms at the expense of minor modelling inaccuracies. While we
+acknowledge the methodologies in Alparslan and Yıldırım (2022, Section 4.2) and Ju et al. (2022)
+that can correctly incorporate the effect of truncation into inference, we remark that those methods
+pay the price of exactness by having O(n) computational complexity per iteration.
+4.2
+Features with a General Distribution
+The normality assumption for xi’s in Section 2 may not be adequate for some data sets. Moreover,
+when d is large, updating Sj’s can be the bottleneck of MCMC-normalX in Algorithm 1 in terms of
+computation time and convergence. We propose two algorithms to address both of those concerns.
+As it turns out, those algorithms provide accurate estimations even for the case of normally
+distributed features; see Section 5.1.
+Our approach for xi’s with a general distribution is based on estimating Sj’s from the beginning,
+using some principled estimation method, and fixing Sj’s to those estimates during the whole
+course of the inference procedure. In that way, we obtain a faster MCMC algorithm at the expense
+of targeting an approximate posterior distribution. Moreover, we have observed in our experiments
+that this variant is quite competitive in terms of accuracy, especially when the total number of
+nodes J increases. We call this variant MCMC-fixedS and present it in Algorithm 2.
+As for estimating Sj’s, one could simply consider taking the privately shared ˆSj as an estimator
+for Sj, but ˆSj is not necessarily a positive (semi-)definite matrix. Instead, we propose the nearest
+positive semi-definite matrix of to ˆSj as the estimator of Sj in terms of the Frobenius norm. (The
+nearest positive definite matrix to ˆSj does not exist.) To find the nearest positive semi-definite
+matrix, we follow Higham (1988) and apply the following procedure for each j = 1, . . . , J: (i)
+Calculate the eigendecomposition ˆSj = EDET , where E is a matrix of eigenvectors, and D is a
+diagonal matrix consisting of the eigenvalues λi. (ii) The nearest symmetric positive semi-definite
+matrix is �Sj = ED+ET , where D+ is a diagonal matrix with D+(i, i) = max{D(i, i), 0}.
+Note that �Sj found above is the maximum likelihood estimator of Sj given ˆSj (over the set
+of positive semi-definite matrices) since the conditional distribution of ˆSj given Sj is a normal
+9
+
+Algorithm 2: MCMC-fixedS - one iteration
+Input: Current values of θ, σ2
+y; estimates ˆS1:J, observations ˆz1:J; noise variance σ2
+z, and
+hyperparameters a, b, m, C.
+Output: New sample of σ2
+y, θ.
+1 Use S1:J = �S1:J throughout.
+2 Sample θ using (12).
+3 Update σ2
+y via an MH move targeting p(σ2
+y|θ, S1:J, ˆz1:J).
+Algorithm 3: Bayes-fixedS-fast
+Input: ˆS1:J, ˆz1:J; noise variance: σ2
+z; estimate ˜σ2
+y of σ2
+y; hyperparameters: m, C.
+Output: Estimate ˆθ.
+1 for j = 1, 2, . . . J do
+2
+Calculate the estimate �Sj for Sj using ˆSj.
+3
+Calculate Σj = �Sj(˜σ2
+y �Sj + σ2
+zI)−1 �Sj.
+4
+Calculate mj = �Sj(˜σ2
+y �Sj + σ2
+zI)−1 ˆzj.
+5 return Posterior moments of θ: Σ−1
+post = �J
+j=1 Σj + C−1,
+mpost = Σpost
+�
+C−1m + �J
+j=1 mj
+�
+.
+distribution with mean Sj.
+MCMC-fixedS in Algorithm 2 is faster than MCMC-normalX in Algorithm 1, since it avoids the step
+to update Sj’s, which constitutes the main computational burden on Algorithm 1. However,
+MCMC-fixedS can be made even faster by fixing σ2
+y also. As a crude estimator, we used ˜σ2
+y = ∥Y∥/3
+throughout the experiments. When σ2
+y is fixed in addition to S1:J, we end up with a non-iterative
+method where the posterior distribution of θ is calculated in closed form. We call the resulting
+algorithm Bayes-fixedS-fast and present it in Algorithm 3. Algorithm 3 does nothing but
+returns the moments of the posterior distribution of θ given �Sj’s, ˆzj’s, ˜σ2
+y, and the prior parameters
+for θ.
+4.3
+Computational Cost
+All our methods described in this section require O(d3) computation (per iteration for the iterative
+ones in Algorithms 1 and 2, or as a whole for the fast version in Algorithm 3) since they deal with
+d × d matrices. In contrast, as Bernstein and Sheldon (2019) apply CLT to the vector [S, z, yT y],
+their methods deal with covariance matrices of size (d2 + d + 1) explicitly, which leads to O(d6)
+computation per MCMC iteration. For even moderate d, this computational difference becomes
+dramatic and the latter may be prohibitive. Moreover, the complexity of our methods does not
+depend on n. This is in contrast to the O(n) complexity of general-purpose methods, such as
+Alparslan and Yıldırım (2022, Section 4.3) and Ju et al. (2022), that can be applied to linear
+regression.
+10
+
+4.4
+Extensions
+We mention two other variants of our methodology, deferring the details to Appendix B.
+Another solution for dealing with non-normal Px could be to average the feature vectors in X
+(and the corresponding response variables in y), so that the averaged rows of X can be modelled
+as approximately normal, due to CLT. This enables using the methods devised for normally
+distributed features. For the details of this approach, see Appendix B.1.
+Secondly, if the features are normally distributed but the data are not centred, we need to
+include the intercept parameter, which corresponds to appending xi with a one from the left, and
+MCMC-normalX does not directly apply. In that case, we can modify the hierarchical model that
+accommodates the non-centralised features and the intercept parameter and still benefit from
+the sampling techniques involved in MCMC-normalX in Algorithm 1. Appendix B.2 contains the
+details of the modified hierarchical model.
+5
+Numerical Experiments
+We present several numerical evaluations of the proposed methods, MCMC-normalX, MCMC-fixedS,
+and Bayes-fixedS-fast with simulated and real data. We compare our algorithms with two
+methods: adaSSP of Wang (2018) and the MCMC method of Bernstein and Sheldon (2019) for
+differentially private linear regression that we call MCMC-B&S. Note that adaSSP and MCMC-B&S
+are originally proposed for the non-distributed setting, that is, J = 1. For a comprehensive
+comparison, we have implemented their extensions for J ≥ 1. The details of those extensions are
+provided in Appendix C. In particular, we have carefully generalised the model in Bernstein and
+Sheldon (2019) for J ≥ 1 similarly as we have done for our model in Section 3.2. What we call
+MCMC-B&S is the adaptation of Bernstein and Sheldon (2019, Algorithm 1) for this generalised
+model (and (ϵ, δ)-DP). The code to replicate all of the experiments in this section can be found at
+https://github.com/sinanyildirim/Bayesian_DP_dist_LR.git.
+5.1
+Experiments with Simulated Data
+We have considered two different configurations, (n = 105, d = 2) and (n = 105, d = 5), for
+the problem size. For each (n, d), we have simulated the data as follows: We have generated
+θ ∼ N(0, Id), xi ∼ N(0, Σx) where Σx ∼ IW(Λ, κ) with κ = d + 1 and selected the scale matrix
+randomly as Λ = V T V , where V is a d × d matrix of i.i.d. variables from N(0, 1). The response
+variables y have been generated with σ2
+y = 1. For inference, we have used the same Λ, κ as above
+and a = 20, b = 0.5, m = 0d×1, C = (a − 1)/bId for the other hyperparameters.
+We have evaluated the methods at all combinations of J ∈ {1, 5, 10} and ϵ ∈ {0.1, 0.2, 0.5, 1, 2, 5, 10}.
+All the MCMC algorithms have been run for 104 iterations. For each (J, ϵ) pair, we have tried
+each method 50 times (each with different noisy observations) to obtain average performances.
+For performance metrics, we have looked at the mean squared errors (MSE) of (i) the estimates ˆθ,
+and (ii) the predictions ˆy(xtest) generated by the methods. For the Bayesian methods, ˆθ is taken as
+the mean posterior, which can be numerically estimated for the MCMC algorithms. For prediction
+performance, we have calculated E[ˆy(xtest) − ytest]2. For the Bayesian methods, ˆy(xtest) is the
+posterior predictive expectation of ytest at xtest. For adaSSP, we simply take ˆy(xtest) = xT
+test ˆθ.
+11
+
+The results are summarised in Figure 2. We observe that MCMC-fixedS and Bayes-fixedS-fast
+outperform adaSSP and MCMC-B&S in almost all cases both in terms of estimation and prediction.
+Comparing the full-scale algorithms MCMC-normalX and MCMC-B&S (that involve updates of S), we
+observe a clear advantage of MCMC-normalX at d = 2, but MCMC-B&S becomes more competitive at
+d = 5. This can be attributed to the fact that MCMC-B&S requires the extra statistic yT y, unlike
+MCMC-normalX, which causes MCMC-B&S to use more noisy statistics. This difference becomes more
+significant at small d, where the relative effect of the presence of yT y on the sensitivity is more
+significant. Finally, all methods improve as ϵ grows, which is expected.
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-9.5
+-9
+-8.5
+-8
+-7.5
+(log-)MSE: prediction, J = 1
+MCMC-normalX
+MCMC-fixedS
+Bayes-fixedS-fast
+MCMC-B&S
+adaSSP
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-9
+-8
+-7
+-6
+(log-)MSE: prediction, J = 5
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-9
+-8
+-7
+-6
+-5
+(log-)MSE: prediction, J = 10
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-10.5
+-10
+-9.5
+-9
+(log-)MSE: estimation J = 1
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-10.5
+-10
+-9.5
+-9
+-8.5
+-8
+-7.5
+(log-)MSE: estimation J = 5
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-10
+-9
+-8
+-7
+-6
+(log-)MSE: estimation J = 10
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-6
+-5
+-4
+-3
+-2
+(log-)MSE: prediction, J = 1
+MCMC-normalX
+MCMC-fixedS
+Bayes-fixedS-fast
+MCMC-B&S
+adaSSP
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-5
+-4
+-3
+-2
+-1
+(log-)MSE: prediction, J = 5
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-4
+-3
+-2
+-1
+0
+(log-)MSE: prediction, J = 10
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-3
+-2.5
+-2
+-1.5
+-1
+(log-)MSE: estimation J = 1
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-3
+-2.5
+-2
+-1.5
+-1
+(log-)MSE: estimation J = 5
+0.1
+0.2
+0.5
+1  
+2  
+5  
+10 
+0
+-2.5
+-2
+-1.5
+-1
+-0.5
+(log-)MSE: estimation J = 10
+Figure 2: Averaged prediction and estimation performances (over 50 runs). Top row: n = 105, d = 2, Bottom row:
+n = 105, d = 5.
+0
+10
+20
+d
+0
+2
+4
+6
+8 #10-3
+J = 1
+MCMC-normalX
+MCMC-fixedS
+MCMC-B&S
+0
+10
+20
+d
+0
+0.005
+0.01
+0.015
+0.02
+J = 5
+0
+10
+20
+d
+0
+0.01
+0.02
+0.03
+0.04
+J = 10
+Figure 3: Run times per iteration for MCMC algorithms
+We also compare the computation times of the MCMC algorithms MCMC-normalX, MCMC-fixedS,
+and MCMC-B&S1. Figure 3 shows the run-times of the algorithms vs d. The drastic difference in
+computational loads explained in Section 4.3 is also visible in the figure. While MCMC-B&S may be
+improved in terms of accuracy as d increases, the O(d6) dramatically slows it down.
+1The algorithms were run in MATLAB 2021b on an Apple M1 chip with 8 cores and 16 GB LPDDR4 memory.
+12
+
+5.2
+Experiments with Real Data
+For the real data case, we have used four different data sets from the UCI Machine Learning
+Repository. We have disregarded the columns including string data or key values (ID, name,
+date, etc.), and we have considered the most right-hand column as y. The finalised data sets are
+summarised below.
+data set
+n
+d
+hyperlinks
+power plant energy
+7655
+4
+view link
+bike sharing
+13904
+14
+view link
+air quality
+7486
+12
+view link
+3d road
+347900
+3
+view link
+For prediction, we have taken 80% of the data for training and the rest for testing. We present the
+average prediction performances (out of 50 runs) in Table 1 for each dataset and J with ϵ = 1. We
+observe that the prediction performances of the compared methods are close, while MCMC-fixed-S
+and Bayes-fixed-S are arguably the most stable ones. When J > 1 (the distributed data setting),
+those two methods beat adaSSP and MCMC-B&S more satisfactorily.
+Table 1: Averaged prediction performances (over 50 runs) for the real datasets - ϵ = 1
+J
+data sets
+MCMC-normalX
+MCMC-fixedS
+Bayes-fixedS-fast
+MCMC-B&S
+adaSSP
+J = 1
+PowerPlant
+0.0129
+0.0129
+0.0129
+0.0128
+0.0139
+BikeSharing
+0.0024
+0.0021
+0.0021
+0.0020
+0.0107
+AirQuality
+0.0060
+0.0057
+0.0057
+0.0062
+0.0066
+3droad
+0.0229
+0.0229
+0.0229
+0.0229
+0.0229
+J = 5
+PowerPlant
+0.0133
+0.0134
+0.0134
+0.0136
+0.0235
+BikeSharing
+0.0174
+0.0045
+0.0045
+0.0086
+0.0382
+AirQuality
+0.0142
+0.0100
+0.0099
+0.0130
+0.0227
+3droad
+0.0229
+0.0229
+0.0229
+0.0229
+0.0229
+J = 10
+PowerPlant
+0.0142
+0.0143
+0.0143
+0.0143
+0.0351
+BikeSharing
+0.0812
+0.0082
+0.0082
+0.0137
+0.0526
+AirQuality
+0.0985
+0.0117
+0.0117
+0.0216
+0.0314
+3droad
+0.0229
+0.0229
+0.0229
+0.0229
+0.0229
+6
+Conclusion
+We propose a novel Bayesian inference framework, with MCMC being its main workhorse, for a
+differentially private distributed linear regression setting where the data is partitioned among the
+data holders. We provide several Bayesian inference algorithms suited to the developed hierarchical
+model for linear regression. Those algorithms can be preferred one over the other depending on
+the computational budget, model specifics, or how much we know about the underlying statistical
+facts of the data. We exploit the conditional structure between the summary statistics of linear
+regression, as given in Proposition 1, which leads to feasible algorithms with computational
+advantages over their competitors. The numerical experiments show that the proposed methods
+are competitive with their state-of-the-art alternatives in terms of accuracy.
+The extensions mentioned in Section 4.4 indicate potential future directions. There is also room
+13
+
+for improvement of MCMC-normalX. We chose the most common MH moves to update σ2
+y and
+Sj’s, without paying much attention to their efficiencies. Especially for large d, more advanced
+techniques, such as those stemming from Hamiltonian Monte Carlo (Neal, 2001) or pseudo-marginal
+MCMC (Andrieu and Roberts, 2009), may be employed to facilitate the mixing of the algorithm.
+7
+Acknowledgement
+The study was funded by the Scientific and Technological Research Council of Turkey (T¨UB˙ITAK)
+ARDEB Grant No 120E534.
+Supplementary material:
+The code to replicate the experiments in Section 5 can be found at
+https://github.com/sinanyildirim/Bayesian_DP_dist_LR.git.
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+16
+
+A
+Derivations for MCMC-normalX
+We reserve this section for the derivations required for our algorithm MCMC-normalX.
+Full Conditional Distribution of Σx:
+We note that
+p(Σx|S1:J, ˆS1:J, ˆz1:J) ∝ p(Σx)
+J
+�
+j=1
+p(Sj|Σx)
+=
+|Λ|dκ/2
+2dk/2Γd( κ
+2)|Σx|−(d+κ+1)/2e− 1
+2 tr(ΛΣ−1
+x )
+J
+�
+j=1
+|Sj|(nj−d−1)/2e− 1
+2 tr(Σ−1
+x Sj)
+2njd/2|Σx|nj/2Γd(nj/2)
+∝ |Σx|− n
+2 − (d+κ+1)
+2
+e− 1
+2 (� tr(Σ−1
+x Sj)+tr(ΛΣ−1
+x ))
+∝ |Σx|− (d+κ+n+1)
+2
+e− 1
+2 tr((� Sj+Λ)Σ−1
+x ).
+Therefore, we have
+Σx|S1:J, ˆS1:J, ˆz1:J ∼ IW
+�
+�Λ +
+J
+�
+j=1
+Sj, κ + n
+�
+� .
+Full Conditional Distribution of θ:
+The posterior of θ is proportional to
+p(θ|S1:J, σ2
+y, ˆz1:J) ∝ N(θ; m, C)p(ˆz1:J|S1:J, θ, σ2
+y).
+For the second factor, we have
+p(ˆz1:J|S1:J, θ, σ2
+y) ∝
+J
+�
+i=1
+p(ˆzj|Sj, θ, σ2
+y) =
+J
+�
+i=1
+N
+� ˆzj; Sjθ, σ2
+ySj + σ2
+zI
+�
+∝
+J
+�
+i=1
+exp
+�
+−1
+2(ˆzj − Sjθ)T (σ2
+ySj + σ2
+zI)−1(ˆzj − Sjθ)
+�
+∝ exp
+�
+�
+�−1
+2
+�
+�θT
+�
+��
+j
+Sj(σ2
+ySj + σ2
+zI)−1Sj
+�
+� θ − 2θT
+�
+��
+j
+Sj(σ2
+ySj + σ2
+zI)−1
+�
+� ˆzj
+�
+�
+�
+�
+� .
+Reorganising the terms, we end up with
+p(θ|S1:J, σ2
+y, ˆz1:J) ∝ exp
+�
+−1
+2
+�
+θT Σ−1
+postθ − 2θT Σ−1
+postmpost
+��
+,
+where Σ−1
+post = �
+j Sj(σ2
+ySj + σ2
+ZI)−1Sj + C−1 and mpost = Σpost[�
+j Sj(σ2
+ySj + σ2
+zI)−1)ˆzj +
+C−1m]. Therefore, θ|S1:J, σ2
+y, ˆz1:J ∼ N(mpost, Σpost).
+Acceptance Ratio for the MH Update of Sj:
+We drop the index j from Sj for simplicity.
+When S′ ∼ W(S/α, α), the proposal density is
+q(S′|S) = |S′|(α−d−1)/2e−tr[αS−1S′]/2
+|S/α|α/22αd/2Γd( α
+2 )
+= |S′|(α−d−1)/2e−tr[αS−1S′]/2
+|S|α/22αd/2Γd( α
+2 )
+αα/2.
+17
+
+Therefore, the acceptance ratio corresponding to this proposal is
+min
+�
+1, q(S|S′)
+q(S′|S)
+W(S′; njΣx, κ)p( ˆS| ˆS′)N(ˆz; S′θ, σ2
+ySθ + σ2
+zId)
+W(S; njΣx, κ)p( ˆS| ˆS)N(ˆz; Sθ, σ2ySθ + σ2zId)
+�
+,
+where the ratio of proposals becomes
+q(S|S′)
+q(S′|S) = |S|(α−d−1)/2|S|α/2e−tr[aS′−1S]/2
+|S′|(α−d−1)/2|S′|α/2e−tr[αS−1S′]/2 =
+� |S|
+|S′|
+�α−(d+1)/2
+eα(tr[S−1S′]−tr[S′−1S])/2.
+Acceptance Ratio for the MH Update of σ2
+y:
+To update σ2
+y, we use a random walk proposal
+σ2′
+y ∼ N(σ2
+y, σ2
+q). The resulting acceptance ratio is
+min
+�
+1,
+IG(σ2′
+y ; a, b) �J
+j=1 N(ˆzj; Sjθ, σ2′
+y Sjθ + σ2
+zId)
+IG(σ2y; a, b) �J
+j=1 N(ˆzj; Sjθ, σ2ySjθ + σ2zId)
+�
+B
+Other Variants
+This appendix is reserved for the details of the other variants mentioned in Section 4.4. For
+simplicity, we will assume a single data holder, i.e., J = 1; the extension to J > 1 should be
+straightforward.
+B.1
+Approximating Normality by Averaging
+When xi, i = 1, . . . , n are not normal, another approach that we propose is based on modifying
+the data to such that the rows of the modified feature matrix, called Xav, are averages of k > 1
+original features in X, and thus approximately normal, by the CLT. Specifically, let n be divisible
+by k so that m = n/k is an integer. Consider the m × n matrix
+A =
+1
+√
+k
+�
+����
+11×k
+01×k
+. . .
+01×k
+01×k
+11×k
+. . .
+01×k
+...
+...
+...
+...
+01×k
+01×k
+. . .
+11×k
+�
+����
+m×n
+,
+Then the matrix Xav = AX corresponds to constructing a shorter m × d matrix whose i’th
+column is the average of the rows (i − 1)k + 1, . . . , ik of X (scaled by 1/
+√
+k the preserve the
+norm). When k is large enough, we can make normality assumptions for the rows of Xav. Further,
+we consider
+yav := Ay = Xavθ + Ae,
+whose mean is Xavθ and covariance AAT σ2
+y. But, we have AAT = Im, so the covariance is σ2
+yIm.
+Therefore, the same hierarchical model in Figure 1 can be used for X′, y′ with their respective
+summary statistics
+zav = (Xav)T yav,
+Sav = (Xav)T Xav,
+as well as the noisy versions of those summary statistics to provide a given level of privacy. Note
+that Sav and zav have the same sensitivities as S and z, hence the same noise variances are
+needed for privacy. However, there is less information in Sav and zav due to averaging.
+18
+
+B.2
+Including the Intercept
+If we include the intercept parameter, which corresponds to appending xi with a 1 from the left,
+the design matrix will be changed from S to S0 =
+� n
+n¯xT
+n¯x
+S
+�
+, where ¯x = 1
+n
+�n
+i=1 xi. Also, note
+that S = (n − 1)�Σx + n¯x¯xT where �Σx is the sample covariance. Under the normality assumption
+for xi’s, ¯x ∼ N(m, Σx/n) and (n − 1)�Σx ∼ W(n − 1, Σx) are independent and have known
+distributions. Therefore, we can write a model that includes b = ¯x, ˆ
+Σx, and S0 where S0 replaces
+S in the standard model. More specifically, we have the following hierarchical model:
+θ ∼ N(m, C),
+Σx ∼ IW(Λ, κ),
+ˆ
+Σx|Σx ∼ W(n − 1, Σx),
+b|Σx ∼ N(µ, Σx/n),
+z|θ, Σ2
+y, ˆΣ, b ∼ N(S0θ, S0σ2
+y),
+ˆS| ˆΣ, b = N(S0, σ2
+sI),
+ˆz|z = N(z, σ2
+zI)
+with S0 =
+� n
+nbT
+nb
+(n − 1) ˆΣ + nbbT
+�
+.
+C
+Compared Methods
+Here, we provide the details of the methods which we compare with the proposed methods in
+this paper. Those methods are originally proposed for J = 1. However, for comparison, we
+implemented their natural extensions to the general (distributed) case J ≥ 1. The implementations
+of those methods can also be found in the code package provided for this paper.
+C.1
+MCMC-B&S Adapted to the Distributed Setting
+In Bernstein and Sheldon (2019), only J = 1 is considered, and the vector ss = [vec(S), z =
+XT y, u = yT y] is perturbed with privacy-preserving noise to generate the observations of
+the model. For J ≥ 1, we consider the following natural extension for generating perturbed
+observations ˆss = [vec( ˆSj), ˆzj, ˆuj] along with
+ˆSj = Sj + σdpMj,
+ˆzj = zj + vj,
+vj ∼ N(0, σ2
+dpId),
+ˆuj = uj + wj,
+wj ∼ N(0, σ2
+dp), (13)
+where σdp = σ(ϵ, δ)∆ss with ∆ss =
+�
+∥X∥4 + ∥X∥2∥Y∥2 + ∥Y∥4.
+For completeness, we provide the further specifics of the model: We take (θ, σ2
+y) ∼ NIG(a0, b0, m, Λ0)
+where Λ0 = C−1 and Px = N(0, Σx) with Σx ∼ IW(Λ, κ).
+During the comparisons, we set a0, b0, m, C, Λ, κ to the same values for both this model and our
+proposed model that assumes normally distributed features, i.e., Px = N(0, Σx). Then, we apply
+an extension of Bernstein and Sheldon (2019, Algorithm 1) suited to those observations. One
+iteration of that algorithm includes the following steps in order:
+• Calculate the D × 1 mean vector and D × D covariance matrix
+µss = E[ss],
+Σss = Cov[ss].
+This step requires the fourth moments N(0, Σx).
+• Sample ssj ∼ N(µ(j)
+post,ss, Σ(j)
+post,ss) with
+Σ(j)
+post,ss = (njΣss(θ)−1 + (1/σ2
+dp)I)−1,
+and
+µ(j)
+post,ss = Σ(j)
+post,ss(Σss(θ)−1µss + ˆssj/σ2
+dp).
+19
+
+• Sample Σx ∼ IW
+�
+Λ + �J
+j=1 Sj, n + κ
+�
+.
+• Sample (θ, σ2
+y) ∼ NIG(an, bn, mn, Λn) by sampling σ2
+y ∼ IG(an, bn), followed by sampling
+θ ∼ N(µn, σ2
+yΛ−1
+n ) with an = a0 + n/2, bn = 0.5u + mT C−1m − mT
+nΛnmn, and
+Λn = Λ0 +
+J
+�
+j=1
+Sj,
+mn = Λ−1
+n
+�
+�
+J
+�
+j=1
+zj + Λ0m
+�
+� ,
+.
+C.2
+A Variant of adaSSP for the Distributed Setting
+The adaSSP algorithm of (Wang, 2018) is originally designed for a single data holder, i.e., J = 1.
+In adaSSP, a differentially private estimate of θ is released as
+ˆθ = ( ˆS + λI)−1 ˆz.
+(14)
+Here, ˆS and ˆz are the privatised versions of S and z as in (2) and (3), except that ϵ and δ must be
+changed to 2ϵ/3 and 2δ/3 in those equations to provide (ϵ, δ) differential privacy. This is because
+adaSSP uses another parameter λ, which is also calculated from the sensitive data and a third of
+the privacy budget is spent for privatising that calculation. With v ∼ N(0, 1), λ is specifically
+calculated as
+λ = max{0, σ
+�
+d ln(6/δ) ln(2d2/ρ) − ˜λmin}
+with σ = ∥X∥2/(ϵ/3), λmin = min(eig(S)), and
+˜λmin = max{λmin +
+�
+ln(6/δ)σv − ln(6/δ)σv, 0}.
+We consider an extension of (Wang, 2018) for J ≥ 1. To perform the extension, we reflect on its
+tendency to approximate a (regularised) least square solution and consider the following estimate
+ˆθ =
+�
+�
+J
+�
+j=1
+ˆSj + I
+J
+�
+j=1
+λj
+�
+�
+−1 �
+�
+J
+�
+j=1
+ˆzj
+�
+� .
+(15)
+Here, ˆSj, ˆzj and λj are calculated in data node j separately from the other nodes. The estimation
+procedure in (15) does not properly account for the Bayesian paradigm but aggregates the shared
+ˆSj’s and ˆzj’s to approximate the (regularised) least squares solution.
+20
+
diff --git a/0tFST4oBgHgl3EQfVziF/content/tmp_files/load_file.txt b/0tFST4oBgHgl3EQfVziF/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..425c8d5d0751ff2b267392152dccd6b32a163c57
--- /dev/null
+++ b/0tFST4oBgHgl3EQfVziF/content/tmp_files/load_file.txt
@@ -0,0 +1,943 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf,len=942
+page_content='Differentially Private Distributed Bayesian  Linear Regression with MCMC  Barı¸s Alparslan1, Sinan Yıldırım1, 2, and S¸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' ˙Ilker Birbil3  1Faculty of Engineering and Natural Sciences, Sabancı University, ˙Istanbul, Turkey∗  2Center of Excellence in Data Analytics (VER˙IM), Sabancı University, ˙Istanbul, Turkey  3Department of Business Analytics, University of Amsterdam, Amsterdam, The Netherlands  February 1, 2023  Abstract  We propose a novel Bayesian inference framework for distributed differentially private  linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We consider a distributed setting where multiple parties hold parts of the  data and share certain summary statistics of their portions in privacy-preserving noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We  develop a novel generative statistical model for privately shared statistics, which exploits a  useful distributional relation between the summary statistics of linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Bayesian  estimation of the regression coefficients is conducted mainly using Markov chain Monte Carlo  algorithms, while we also provide a fast version to perform Bayesian estimation in one iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  The proposed methods have computational advantages over their competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We provide  numerical results on both real and simulated data, which demonstrate that the proposed  algorithms provide well-rounded estimation and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Keywords: Differential privacy, linear regression, distributed learning, MCMC  1  Introduction  Linear regression is a mathematical method that lies at the core of statistical research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Many  researchers have been working on linear regression since the 19th century, and hence, many  well-known solution methods exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' On a separate note, privacy-preserving statistical learning has  gained popularity and importance in recent years, with differential privacy prevailing as the most  commonly used definition for privacy (Dwork, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Dankar and El Emam,  2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' As a result, there is a recent but growing interest in differentially private linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Many works in the data privacy literature do not mainly focus on regression but are motivated by  or can be applied to regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' As an example, differentially private empirical risk minimisation  (Chaudhuri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Bassily et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Abadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Kuru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2022) can be applied to  regression once it is cast as a data-driven optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Many general-purpose Bayesian  differentially private estimation methods can also be used in regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Williams and  Mcsherry (2010) is one of the first works that considered a hierarchical model for the privatised  data and Bayesian estimation for the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2016) analyse several  differential privacy mechanisms for posterior sampling and suggest using these mechanisms also  ∗The study was funded by the Scientific and Technological Research Council of Turkey (T¨UB˙ITAK) ARDEB  Grant No 120E534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Barı¸s Alparslan and Sinan Yıldırım were supported by the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='13778v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='ML]  31 Jan 2023  for linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Dimitrakakis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2017) developed a posterior sampling query algorithm  to combine differential privacy and Bayesian inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Contrary to those one-sample approaches,  general-purpose differentially private Markov chain Monte Carlo (MCMC) algorithms, which aim  to identify the posterior distribution via iterative sampling, can also be applied to regression  (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Foulds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Yıldırım and Ermi¸s, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Heikkil¨a  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Gong, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Alparslan and Yıldırım, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Several works in the literature are somewhat more directly related to differentially private regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2012) suggested a functional mechanism method, which is based on perturbing  polynomial objective functions with privacy-preserving noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' As an alternative, Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  (2014b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Wang (2018) considered perturbation of summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Alabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2022) provide  a technical discussion on different point estimation methods for differentially private simple linear  regression, that is when we have a single feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Ferrando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2022) present a method to  compute confidence intervals for the coefficients of linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2021) study the  rates of convergence for parameter estimation with differential privacy via output perturbation,  where a non-private estimator is perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' All those works consider point estimation of the  linear regression parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  In this paper, we focus on differential private distributed Bayesian inference for the parameters of  linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We use a novel hierarchical model that relies on a distributional relationship  (Proposition 1) between the summary statistics of linear regression, which, to the best of our  knowledge, has not been exploited so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We propose Bayesian inference algorithms that take  perturbations of summary statistics as observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The general inferential tool we pick in this  paper is MCMC, a well-known framework for iterative sampling from posterior distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' As  we shall see, the proposed MCMC algorithms in this paper already have lower computational  complexities per iteration than their closest competitors in Bernstein and Sheldon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Addi-  tionally, we also propose much faster Bayesian estimation methods that perform estimation in  one iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Finally, we assume a distributed setting where the total dataset is shared among  multiple parties (data nodes), who want to collaborate for the inference of a common parameter,  see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', Heikkil¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2017) for such a setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The non-distributed setting is just a special  case (single data holder) for our methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  This paper has connections with several works in the literature, yet it has significant differences  from each of those, as we shall explain below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  For the privacy-preserving mechanism, we consider adding noise to summary statistics of linear  regression, similarly to Wang (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Bernstein and Sheldon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The adaSSP framework of  Wang (2018) motivates the fast Bayesian estimation methods developed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However,  adaSSP is a point estimation method while we aim for a posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The latter work,  Bernstein and Sheldon (2019), is particularly related to this paper as they also study Bayesian  linear regression with differential privacy using perturbed statistics of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However, there are  some important differences between our work and that of Bernstein and Sheldon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' These  differences stem from the choice of summary statistics and the consequent hierarchical structure  used for modelling linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Those modelling differences lead to significant differences in  the inference methods as well as significant computational advantages for our methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Specifically,  the computational complexity of our methods is O(d3), where d is the number of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This  order is much less than the O(d6) of Bernstein and Sheldon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Finally, neither Wang (2018)  nor Bernstein and Sheldon (2019) has considered a distributed learning setting like we do in  2  this paper, although both works can be modified for the distributed setting after moderate  modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Foulds et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Heikkil¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2017) are other differentially Bayesian inference methods  that target posterior distributions of perturbed summary statistics of sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The one by  Heikkil¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2017) is particularly interesting because they consider a distributed setting and  present linear regression as their showcase example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However, we differ from those works in the  way we model the perturbed statistics and in the choice of inference methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Specifically, Foulds  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2016);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Heikkil¨a et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2017) treat the perturbed statistics as if not perturbed, while we  incorporate the effect of perturbation in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Recently, Alparslan and Yıldırım (2022) and Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2022) employ data augmentation for  modelling sensitive and privatised data and propose MCMC for Bayesian inference, the latter work  having linear regression as a major application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Their methods have O(n) complexity per iteration  in general where n is the number of instances in the data set, which can be slow when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  In contrast, our methods are scalable in data size since their computational complexities do not  depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We note that Alparslan and Yıldırım (2022, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2) also present an MCMC  method scalable with n that exploits the approximate normality of additive summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  However, a direct application of that would lead to an algorithm with O(d6) computational  complexity (per iteration), like in Bernstein and Sheldon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  The paper is organised as follows: In Section 2 we review differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In Section 3 we lay  out the hierarchical model for differentially private distributed linear regression with perturbed  summary statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In Section 4, we present and discuss the aspects of the proposed inference  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Section 5, we provide numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Notation:  Matrices and vectors are shown in bold-face notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For a matrix A, its transpose,  trace, and determinant (whenever they exist) are AT , tr(A), and |A|, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For any  sequence {ai}i≥0, we let ai:j = (ai, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , aj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We write x ∼ P to mean the random variable x  has distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' N(m, Σ) stands for the multivariate normal distribution with mean m and  covariance Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Wishart and inverse-Wishart distributions with scale matrix Λ and κ degrees of  freedom are shown as W(Λ, κ) and IW(Λ, κ), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' IG(a, b) stands for the inverse-gamma  distribution with shape and scale parameters a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We augment those notations with x to  denote the respective probability density functions (pdf), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', as N(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' m, Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  2  Differential Privacy  Differential privacy (Dwork, 2006, 2008) concerns randomised algorithms that run on sensitive,  or usually private, data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' A randomised algorithm takes an input data set D ∈ D and returns a  random output in O, where the randomness is intrinsic to the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' A differentially private  algorithm constrains the difference between the probability distributions of the outputs obtained  from neighbouring data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We say two data sets are neighbours if they differ by one individual’s  piece of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Definition 1 (Differential privacy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' A randomised algorithm M : D �→ O is (ϵ, δ)-differentially  private (DP) if for any pair of neighbouring data sets D, D′ ∈ D and for any subset O ⊆ O of the  of support domain, it satisfies  P[M(D) ∈ O] ≤ eϵP[M(D′) ∈ O] + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  3  The definition implies that smaller (ϵ, δ) leads to more privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Privacy-preserving algorithms often use noise-adding mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' A popular noise-adding mecha-  nism is the Gaussian mechanism (Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2006), which perturbs a function f : D �→ Rk of  the sensitive data, for some k ≥ 1, with a random noise drawn from the Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  The amount of the added noise depends on the L2-sensitivity of the function, given by  ∆f =  max  neighbourD1,D2∈D∥f(D1) − f(D2)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  An (ϵ, δ)-DP Gaussian mechanism returns  f(D) + ∆fσ(ϵ, δ)v,  v ∼ N(0, Ik)  (1)  upon taking D as the input, where the quantity σ(ϵ, δ) ensures (ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In this work, we take  σ(ϵ, δ) as the analytical solution given in Balle and Wang (2018, Algorithm 1) due to its tightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  The Gaussian mechanism is also central to other forms of privacy, such as zero-concentrated DP  (Bun and Steinke, 2016) and Gaussian DP (Dong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  In this paper, we consider (ϵ, δ)-DP as the type of privacy and the Gaussian mechanism to generate  noisy observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Moreover, the proposed methods in this paper never use the sensitive data  once given the noisy observations generated using the Gaussian mechanism, hence exploiting the  post-processing property of differential privacy (Dwork and Roth, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Theorem 1 (Post-processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' If M : D �→ O be (ϵ, δ)-DP and let f : O → O′ be another mapping  independent of D given M(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Then fM : D �→ O′ with fM(D) = f(M(D)) is (ϵ, δ)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  3  Differentially Private Distributed Linear Regression  In this section, we present a new hierarchical model for differentially private distributed linear  regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For ease of exposition, we first present a model with a single data holder, then  generalise the model for the distributed setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1  Basic Model and Privacy Setup  Suppose we have a sequence of random variables {(xi, yi) : i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , n}, where xi ∈ X ⊆ Rd×1  are the feature vectors and yi ∈ Y ⊆ R is the i’th response variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We consider the normal  linear regression to model the dependency between xi and yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Specifically,  yi = xT  i θ + ei,  ei  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  ∼ N(0, σ2  y),  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , n,  where θ ∈ Rd is the vector of the linear regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We assume that the feature vectors  xi’s are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' with distribution Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Below, we will particularly focus on the case when Px can be  assumed to be a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However, we will also present algorithms for general Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  In matrix notation, the above can shortly be expressed as  y = Xθ + e,  e ∼ N(0, σ2  yIn),  where X =  �  xT  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  xT  n  �T is the so-called design matrix, y =  �  y1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  yn  �T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Additionally, we  also define the summary statistics of X and y given by  S := XT X,  z := XT y,  4  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We assume a setup where S and z are privately released as the noisy summary  statistics ˆS and ˆz are constructed as  ˆS = S + σsM,  (2)  ˆz = z + σzv,  v ∼ N(0, Id),  (3)  where M is a d × d symmetric matrix with its upper triangular elements drawn from N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2014b) arrange σs and σz so that both (2) and (3) are (ϵ/2, δ/2) differentially  private, leading to (ϵ, δ)-DP overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Differently than Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2014b), we set  σs = σz = ∆szσ(ϵ, δ),  where σ(ϵ, δ) is given in Balle and Wang (2018, Algorithm 1), and ∆sz is the overall L2 sensitivity  of [S, z], given by  ∆sz =  �  ∥X∥4 + ∥X∥2∥Y ∥2  with ∥X∥ = maxx∈X ∥x∥2 and ∥Y ∥ = maxy∈Y |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Based on the above relations, we shall represent a hierarchical model that enables Bayesian  inference of θ given ˆS and ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' One important element of our modelling approach is the following  result that establishes the conditional distribution of z given S, θ, and σ2  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For the normal linear regression model, we have  z|S, θ, σ2  y ∼ N(Sθ, Sσ2  y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' First, note that,  E[z|X, θ, σ2  y] = E[XT Xθ + XT e] = Sθ,  (4)  Cov(z|X, θ, σ2  y) = XT Xσ2  y = Sσ2  y,  (5)  and observe that both moments depend on X through its statistic S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Therefore, the conditional  density of z given S, θ, and σ2  y is  p(z|X, θ, σ2  y) = N(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Sθ, Sσ2  y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Next, define the function f : Rn×d �→ [0, ∞) with f(X) = p(z|X, θ, σ2  y) and let CS,θ,σ2y = {X :  XT X = S}, Since the function f is constant over CS,θ,σ2y, we can write  p(z|S) =  �  CS,θ,σ2y  fdPx = N(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Sθ, Sσ2  y),  where the second equation is by moment equations in (4) and (5) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This concludes the  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Finally, we assign prior distributions for θ, σ2  y as  θ ∼ N(m, C),  σ2  y ∼ IG(a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  (6)  5  At this point, it is worth discussing some important modelling differences between our work and  Bernstein and Sheldon (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In Bernstein and Sheldon (2019), the central limit theorem (CLT)  is applied to  �  S, z, yT y  �  , leading to a normality assumption for the whole vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In contrast,  we use the exact conditional distribution p(z|S, θ, σ2) thanks to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Moreover, unlike  Bernstein and Sheldon (2019), we do not require a noisy version yT y, hence have a slight advantage  of using less privacy-preserving noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In summary, our model has a different hierarchical structure  and requires less privacy-preserving noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2  Distributed Setting  Next, we extend our model to the distributed setting, where the total data are shared among  J ≥ 1 data holders as  (X, y) = {(Xj, yj);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  (7)  We let ni be number of rows in each xi, so that n = n1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' + nJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Each data holder j shares  their own summary statistics Sj = XT  j Xj, zj = XT  j yj with privacy-preserving noise  ˆSj = Sj + σsMj,  ˆzj = z + σzvj,  vj ∼ N(0, Id).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  (8)  Note that, to preserve a given (ϵ, δ)-DP overall, each party must provide that level of privacy  for their data, hence σs and σz are the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The hierarchical structure of the overall  model (specified for normally distributed xi’s) is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Figure 1: Differentially private distributed linear regression model (specified for normally distributed xi’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=')  The distributed setting deserves separate consideration than the single data holder case for a couple  of reasons: Firstly, the node-specific observations ( ˆS1, ˆz1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , ( ˆSJ, ˆzJ) are altogether statistically  more informative on θ than their aggregates �J  j=1 ˆSj and �J  j=1 ˆzj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This is because the aggregate  versions are not sufficient statistics of the node-specific observations ( ˆS1, ˆz1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , ( ˆSJ, ˆzJ) with  respect to θ (even when σ2  y is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=') Therefore, when the node-specific observations are available,  one should not, in principle, trivially aggregate them and apply an inference method designed for  J = 1 using those aggregates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Secondly, the partitioning of data as in (7) can be relevant to data privacy applications even  outside the distributed learning framework, rendering the methodology in Section 4 useful in a  6  broader sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For example, batches of (x, y)-type of data may be donated to a common data  collector as in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' At this point, a particular and interesting relation exists with pan-privacy  applications (Dwork et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Imagine that sensitive data from individuals are collected  sequentially in time, and the data holder is concerned about possible intrusions into the memory  where the sensitive data are stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Then, one possible way to ensure the privacy of the data  against such possible intrusions, which is the promise of pan-privacy, is to store the noisy statistics  of every new batch of data and erase the original sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Then, at any time the data  collector has data of the form ( ˆS1, ˆz1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , ( ˆSJ, ˆzJ), each pair corresponding to a batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' As a  result, inference algorithms as in Section 4 can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  4  Algorithms for Bayesian Inference  Bayesian inference targets the posterior distribution of the latent variables of the model, in  particular θ, given the observations ˆS1:J and ˆz1:J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We present several Bayesian inference algorithms  for the hierarchical model described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In addition to other concerns like  computational budget, the choice among those approaches mainly depends on the specification of  Px as the distribution of S directly depends on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In this paper, we have considered the following  two cases and devised algorithms for each of them:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In some cases it may be adequate to specify Px = N(0, Σx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This leads to S|Σx ∼ W(Σx, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Further, to account for the uncertainty about the covariance Σx, one can treat it as a random  variable with Σx ∼ IW(Λ, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Figure 1 shows the hierarchical structure of the distributed  setting with those specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We defer discussing the conflict between the normality and  boundedness assumptions to Remark 1 towards the end of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' As the second case, we assume a general (non-normal) Px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' A normal approximation, based on  the CLT, could be considered for the distribution S (Wilson and Ghahramani, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However,  this would require the knowledge (or accurate estimation) of up to the fourth moments of Px  as well as expensive computations for sampling S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We circumvent those difficulties by plugging  in a point estimate of S given ˆS and use it during the sampling process as if it is the true  S itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Then, we develop two different algorithms for inference of θ, one being an MCMC  algorithm and the other providing a closed form-solution for the posterior of θ following a  rough point-wise estimation of σ2  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Note that these algorithms with fixed S do not require a  distribution for x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Next, we provide the details of our approaches and the resulting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1  Normally Distributed Features  In this section, we present an MCMC algorithm for Bayesian inference for the differentially private  distributed linear regression model when Px = N(0, Σx) and Σx ∼ IW(Λ, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The latent variables  involved in this variant are θ, Σx, σ2  y, S1:J, z1:J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Their posterior distribution given ˆS1:J, ˆz1:J can  be written as  p(θ, σ2  y, Σx, z1:J, S1:J|ˆz1:J, ˆS1:J) ∝ p(θ)p(σ2  y)p(Σx)  J  �  j=1  p(zj|θ, σ2  y, S)p(Sj|Σx)p( ˆSj|Sj)p(ˆzj|zj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  (9)  7  One could design an MCMC algorithm for this posterior distribution that updates θ, σ2  y, Σx, z1:J,  S1:J in turn based on their full conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However, such an algorithm suffers from  poor convergence because of a high posterior correlation between θ and z1:J (as verified in our  numerical studies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' It is well known that highly correlated variables result in poor convergence  if they are updated one conditional on the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' To alleviate that problem, we work with the  reduced model where z1:J are integrated out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The reduced model has θ, Σx, σ2  y as its latent  variables, whose joint posterior distribution can be written as  p(θ, σ2  y,Σx, S|ˆz, ˆS) ∝ p(θ)p(σ2  y)p(Σx)  J  �  j=1  p(Sj|Σx)p( ˆSj|Sj)p(ˆzj|Sj, θ, σ2  y),  (10)  where p(ˆz|S, θ, σ2  y) = N(ˆz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Sθ, σ2  ySθ + σ2  zId).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  We would like to sample from the posterior distribution in (10) via MCMC that updates θ, σ2  y,  Σx, S1:J in turn based on their full conditional distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The variables θ and Σx enjoy  closed-form full conditional distributions (see Appendix A for the derivations):  Σx|S1:J, ˆS1:J, ˆz1:J ∼ IW  �  �Λ +  J  �  j=1  Sj, κ + n  �  � ,  (11)  θ|σ2  y, ˆz, S1:J ∼ N(mp, Σp),  (12)  where the posterior moments for θ are  Σ−1  p  =  J  �  j=1  Sj(σ2  ySj + σ2  zI)−1Sj + C−1,  mp = Σp  �  �  J  �  j=1  Sj(σ2  ySj + σ2  zI)−1 ˆzj + C−1m  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  The full-conditional distributions of S1:J and σ2  y have no closed form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' hence we design Metropolis-  Hastings (MH) moves to update them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For σ2  y, one can simply use a random-walk MH move  targeting p(σ2  y|θ, S1:J, ˆz1:J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For S1:J, their full conditional distribution can be factorised as  p(S1:J| ˆS1:J, ˆz1:J, Σx, σ2  y, θ) =  J  �  j=1  p(Sj| ˆSj, ˆzj, Σx, σ2  y, θ),  where each factor is given by  p(Sj| ˆSj, ˆzj, Σx, σ2  y, θ) ∝ p(ˆzj|Sj, θ, σ2  y)p(Sj|Σx)p( ˆSj|Sj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Thanks to that factorised form, each Sj can be updated with an MH move independently and  in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For the MH algorithm to update one Sj, we propose a new value from a Wishart  distribution as S′  j ∼ W(Sj/α, α), which has mean Sj and variance determined by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In our  experiments, we adjust a using ideas from the adaptive MCMC framework (Andrieu and Thoms,  2008) to target an acceptance rate of around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Algorithm 1 represents the overall MCMC algorithm for the hierarchical model for differentially  Bayesian distributed linear regression when Px is a normal distribution with a random covariance  matrix having an inverse-Wishart distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We call this algorithm MCMC-normalX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  8  Algorithm 1: MCMC-normalX - one iteration  Input: Current values of S1:J, θ, σ2  y, Σx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' observations ˆS1:J,ˆz1:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' noise variances σ2  s, σ2  z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  proposal parameters a, σ2  q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' hyperparameters a, b, κ, Λ, m, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Output: New sample of Σx, S, σ2  y, θ  1 Sample Σx using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  2 for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' J do  3  Update Sj via an MH move targeting p(Sj|Σx, θ, ˆzj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  4 Sample θ using (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  5 Update σ2  y via an MH move targeting p(σ2  y|θ, S1:J, ˆz1:J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Admittedly, a potential concern is a conflict between the normality and boundedness  assumptions (both for x and y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However, we also note that the collected data often happen  to have some natural boundaries (which can be exploited to determine the sensitivity of the  shared statistics), and yet the normal distribution is still used for modelling and subsequent  inference mainly for sake of tractability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' With the normality assumption, one can implement  computationally efficient algorithms at the expense of minor modelling inaccuracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' While we  acknowledge the methodologies in Alparslan and Yıldırım (2022, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2) and Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2022)  that can correctly incorporate the effect of truncation into inference, we remark that those methods  pay the price of exactness by having O(n) computational complexity per iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2  Features with a General Distribution  The normality assumption for xi’s in Section 2 may not be adequate for some data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Moreover,  when d is large, updating Sj’s can be the bottleneck of MCMC-normalX in Algorithm 1 in terms of  computation time and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We propose two algorithms to address both of those concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  As it turns out, those algorithms provide accurate estimations even for the case of normally  distributed features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Our approach for xi’s with a general distribution is based on estimating Sj’s from the beginning,  using some principled estimation method, and fixing Sj’s to those estimates during the whole  course of the inference procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In that way, we obtain a faster MCMC algorithm at the expense  of targeting an approximate posterior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Moreover, we have observed in our experiments  that this variant is quite competitive in terms of accuracy, especially when the total number of  nodes J increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We call this variant MCMC-fixedS and present it in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  As for estimating Sj’s, one could simply consider taking the privately shared ˆSj as an estimator  for Sj, but ˆSj is not necessarily a positive (semi-)definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Instead, we propose the nearest  positive semi-definite matrix of to ˆSj as the estimator of Sj in terms of the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (The  nearest positive definite matrix to ˆSj does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=') To find the nearest positive semi-definite  matrix, we follow Higham (1988) and apply the following procedure for each j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , J: (i)  Calculate the eigendecomposition ˆSj = EDET , where E is a matrix of eigenvectors, and D is a  diagonal matrix consisting of the eigenvalues λi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (ii) The nearest symmetric positive semi-definite  matrix is �Sj = ED+ET , where D+ is a diagonal matrix with D+(i, i) = max{D(i, i), 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Note that �Sj found above is the maximum likelihood estimator of Sj given ˆSj (over the set  of positive semi-definite matrices) since the conditional distribution of ˆSj given Sj is a normal  9  Algorithm 2: MCMC-fixedS - one iteration  Input: Current values of θ, σ2  y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' estimates ˆS1:J, observations ˆz1:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' noise variance σ2  z, and  hyperparameters a, b, m, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Output: New sample of σ2  y, θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  1 Use S1:J = �S1:J throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  2 Sample θ using (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  3 Update σ2  y via an MH move targeting p(σ2  y|θ, S1:J, ˆz1:J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Algorithm 3: Bayes-fixedS-fast  Input: ˆS1:J, ˆz1:J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' noise variance: σ2  z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' estimate ˜σ2  y of σ2  y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' hyperparameters: m, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Output: Estimate ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  1 for j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' J do  2  Calculate the estimate �Sj for Sj using ˆSj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  3  Calculate Σj = �Sj(˜σ2  y �Sj + σ2  zI)−1 �Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  4  Calculate mj = �Sj(˜σ2  y �Sj + σ2  zI)−1 ˆzj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  5 return Posterior moments of θ: Σ−1  post = �J  j=1 Σj + C−1,  mpost = Σpost  �  C−1m + �J  j=1 mj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  distribution with mean Sj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  MCMC-fixedS in Algorithm 2 is faster than MCMC-normalX in Algorithm 1, since it avoids the step  to update Sj’s, which constitutes the main computational burden on Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However,  MCMC-fixedS can be made even faster by fixing σ2  y also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' As a crude estimator, we used ˜σ2  y = ∥Y∥/3  throughout the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' When σ2  y is fixed in addition to S1:J, we end up with a non-iterative  method where the posterior distribution of θ is calculated in closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We call the resulting  algorithm Bayes-fixedS-fast and present it in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Algorithm 3 does nothing but  returns the moments of the posterior distribution of θ given �Sj’s, ˆzj’s, ˜σ2  y, and the prior parameters  for θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='3  Computational Cost  All our methods described in this section require O(d3) computation (per iteration for the iterative  ones in Algorithms 1 and 2, or as a whole for the fast version in Algorithm 3) since they deal with  d × d matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In contrast, as Bernstein and Sheldon (2019) apply CLT to the vector [S, z, yT y],  their methods deal with covariance matrices of size (d2 + d + 1) explicitly, which leads to O(d6)  computation per MCMC iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For even moderate d, this computational difference becomes  dramatic and the latter may be prohibitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Moreover, the complexity of our methods does not  depend on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This is in contrast to the O(n) complexity of general-purpose methods, such as  Alparslan and Yıldırım (2022, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='3) and Ju et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2022), that can be applied to linear  regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='4  Extensions  We mention two other variants of our methodology, deferring the details to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Another solution for dealing with non-normal Px could be to average the feature vectors in X  (and the corresponding response variables in y), so that the averaged rows of X can be modelled  as approximately normal, due to CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This enables using the methods devised for normally  distributed features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For the details of this approach, see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Secondly, if the features are normally distributed but the data are not centred, we need to  include the intercept parameter, which corresponds to appending xi with a one from the left, and  MCMC-normalX does not directly apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In that case, we can modify the hierarchical model that  accommodates the non-centralised features and the intercept parameter and still benefit from  the sampling techniques involved in MCMC-normalX in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2 contains the  details of the modified hierarchical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  5  Numerical Experiments  We present several numerical evaluations of the proposed methods, MCMC-normalX, MCMC-fixedS,  and Bayes-fixedS-fast with simulated and real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We compare our algorithms with two  methods: adaSSP of Wang (2018) and the MCMC method of Bernstein and Sheldon (2019) for  differentially private linear regression that we call MCMC-B&S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Note that adaSSP and MCMC-B&S  are originally proposed for the non-distributed setting, that is, J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For a comprehensive  comparison, we have implemented their extensions for J ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The details of those extensions are  provided in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In particular, we have carefully generalised the model in Bernstein and  Sheldon (2019) for J ≥ 1 similarly as we have done for our model in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' What we call  MCMC-B&S is the adaptation of Bernstein and Sheldon (2019, Algorithm 1) for this generalised  model (and (ϵ, δ)-DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The code to replicate all of the experiments in this section can be found at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='com/sinanyildirim/Bayesian_DP_dist_LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1  Experiments with Simulated Data  We have considered two different configurations, (n = 105, d = 2) and (n = 105, d = 5), for  the problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For each (n, d), we have simulated the data as follows: We have generated  θ ∼ N(0, Id), xi ∼ N(0, Σx) where Σx ∼ IW(Λ, κ) with κ = d + 1 and selected the scale matrix  randomly as Λ = V T V , where V is a d × d matrix of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' variables from N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The response  variables y have been generated with σ2  y = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For inference, we have used the same Λ, κ as above  and a = 20, b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='5, m = 0d×1, C = (a − 1)/bId for the other hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  We have evaluated the methods at all combinations of J ∈ {1, 5, 10} and ϵ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='5, 1, 2, 5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  All the MCMC algorithms have been run for 104 iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For each (J, ϵ) pair, we have tried  each method 50 times (each with different noisy observations) to obtain average performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  For performance metrics, we have looked at the mean squared errors (MSE) of (i) the estimates ˆθ,  and (ii) the predictions ˆy(xtest) generated by the methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For the Bayesian methods, ˆθ is taken as  the mean posterior, which can be numerically estimated for the MCMC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For prediction  performance, we have calculated E[ˆy(xtest) − ytest]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For the Bayesian methods, ˆy(xtest) is the  posterior predictive expectation of ytest at xtest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For adaSSP, we simply take ˆy(xtest) = xT  test ˆθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  11  The results are summarised in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We observe that MCMC-fixedS and Bayes-fixedS-fast  outperform adaSSP and MCMC-B&S in almost all cases both in terms of estimation and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Comparing the full-scale algorithms MCMC-normalX and MCMC-B&S (that involve updates of S), we  observe a clear advantage of MCMC-normalX at d = 2, but MCMC-B&S becomes more competitive at  d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This can be attributed to the fact that MCMC-B&S requires the extra statistic yT y, unlike  MCMC-normalX, which causes MCMC-B&S to use more noisy statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This difference becomes more  significant at small d, where the relative effect of the presence of yT y on the sensitivity is more  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Finally, all methods improve as ϵ grows, which is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content='5  1  (log-)MSE: estimation J = 5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content='5  (log-)MSE: estimation J = 10  Figure 2: Averaged prediction and estimation performances (over 50 runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Top row: n = 105, d = 2, Bottom row:  n = 105, d = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  0  10  20  d  0  2  4  6  8 #10-3  J = 1  MCMC-normalX  MCMC-fixedS  MCMC-B&S  0  10  20  d  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content='02  J = 5  0  10  20  d  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='04  J = 10  Figure 3: Run times per iteration for MCMC algorithms  We also compare the computation times of the MCMC algorithms MCMC-normalX, MCMC-fixedS,  and MCMC-B&S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Figure 3 shows the run-times of the algorithms vs d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The drastic difference in  computational loads explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='3 is also visible in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' While MCMC-B&S may be  improved in terms of accuracy as d increases, the O(d6) dramatically slows it down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  1The algorithms were run in MATLAB 2021b on an Apple M1 chip with 8 cores and 16 GB LPDDR4 memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  12  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2  Experiments with Real Data  For the real data case, we have used four different data sets from the UCI Machine Learning  Repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We have disregarded the columns including string data or key values (ID, name,  date, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' ), and we have considered the most right-hand column as y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The finalised data sets are  summarised below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  data set  n  d  hyperlinks  power plant energy  7655  4  view link  bike sharing  13904  14  view link  air quality  7486  12  view link  3d road  347900  3  view link  For prediction, we have taken 80% of the data for training and the rest for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We present the  average prediction performances (out of 50 runs) in Table 1 for each dataset and J with ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We  observe that the prediction performances of the compared methods are close, while MCMC-fixed-S  and Bayes-fixed-S are arguably the most stable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' When J > 1 (the distributed data setting),  those two methods beat adaSSP and MCMC-B&S more satisfactorily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Table 1: Averaged prediction performances (over 50 runs) for the real datasets - ϵ = 1  J  data sets  MCMC-normalX  MCMC-fixedS  Bayes-fixedS-fast  MCMC-B&S  adaSSP  J = 1  PowerPlant  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content='0229  6  Conclusion  We propose a novel Bayesian inference framework, with MCMC being its main workhorse, for a  differentially private distributed linear regression setting where the data is partitioned among the  data holders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We provide several Bayesian inference algorithms suited to the developed hierarchical  model for linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Those algorithms can be preferred one over the other depending on  the computational budget, model specifics, or how much we know about the underlying statistical  facts of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We exploit the conditional structure between the summary statistics of linear  regression, as given in Proposition 1, which leads to feasible algorithms with computational  advantages over their competitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The numerical experiments show that the proposed methods  are competitive with their state-of-the-art alternatives in terms of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  The extensions mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='4 indicate potential future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' There is also room  13  for improvement of MCMC-normalX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' We chose the most common MH moves to update σ2  y and  Sj’s, without paying much attention to their efficiencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Especially for large d, more advanced  techniques, such as those stemming from Hamiltonian Monte Carlo (Neal, 2001) or pseudo-marginal  MCMC (Andrieu and Roberts, 2009), may be employed to facilitate the mixing of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  7  Acknowledgement  The study was funded by the Scientific and Technological Research Council of Turkey (T¨UB˙ITAK)  ARDEB Grant No 120E534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Supplementary material:  The code to replicate the experiments in Section 5 can be found at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='com/sinanyildirim/Bayesian_DP_dist_LR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=' Statistic selection and mcmc for differentially private  bayesian estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Statistics and Computing, 32(5):66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content='3  Andrieu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' and Roberts, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=' The pseudo-marginal approach for efficient Monte Carlo  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Annals of Statistics, 37(2):569–1078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' 6  Andrieu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' and Thoms, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=' A tutorial on adaptive mcmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Statistics and Computing,  18(4):343–373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1  Balle, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' and Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='-X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Improving the Gaussian mechanism for differential privacy:  Analytical calibration and optimal denoising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In Dy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' and Krause, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', editors, Proceedings of  the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine  Learning Research, pages 394–403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1  Bassily, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', Smith, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', and Thakurta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Private empirical risk minimization: Efficient  algorithms and tight error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In 2014 IEEE 55th Annual Symposium on Foundations of  Computer Science, pages 464–473.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' 1  Bernstein, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' and Sheldon, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=' Differentially private bayesian linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' In  Wallach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=", d'Alch´e-Buc, F." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=' In Proceedings, Part I, of the 14th International Conference on Theory of  Cryptography - Volume 9985, pages 635–658, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Springer-Verlag New York,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=' In Proceedings of the Forty-Sixth Annual ACM  Symposium on Theory of Computing, STOC ’14, page 11–20, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+page_content=', Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', and Winslett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Functional mechanism:  Regression analysis under differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' VLDB Endow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', 5(11):1364–1375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' 1  Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', Rubinstein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', and Dimitrakakis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' On the differential privacy of bayesian  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Proceedings of the AAAI Conference on Artificial Intelligence, 30(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' 1  16  A  Derivations for MCMC-normalX  We reserve this section for the derivations required for our algorithm MCMC-normalX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Full Conditional Distribution of Σx:  We note that  p(Σx|S1:J, ˆS1:J, ˆz1:J) ∝ p(Σx)  J  �  j=1  p(Sj|Σx)  =  |Λ|dκ/2  2dk/2Γd( κ  2)|Σx|−(d+κ+1)/2e− 1  2 tr(ΛΣ−1  x )  J  �  j=1  |Sj|(nj−d−1)/2e− 1  2 tr(Σ−1  x Sj)  2njd/2|Σx|nj/2Γd(nj/2)  ∝ |Σx|− n  2 − (d+κ+1)  2  e− 1  2 (� tr(Σ−1  x Sj)+tr(ΛΣ−1  x ))  ∝ |Σx|− (d+κ+n+1)  2  e− 1  2 tr((� Sj+Λ)Σ−1  x ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Therefore, we have  Σx|S1:J, ˆS1:J, ˆz1:J ∼ IW  �  �Λ +  J  �  j=1  Sj, κ + n  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Full Conditional Distribution of θ:  The posterior of θ is proportional to  p(θ|S1:J, σ2  y, ˆz1:J) ∝ N(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' m, C)p(ˆz1:J|S1:J, θ, σ2  y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  For the second factor, we have  p(ˆz1:J|S1:J, θ, σ2  y) ∝  J  �  i=1  p(ˆzj|Sj, θ, σ2  y) =  J  �  i=1  N  � ˆzj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Sjθ, σ2  ySj + σ2  zI  �  ∝  J  �  i=1  exp  �  −1  2(ˆzj − Sjθ)T (σ2  ySj + σ2  zI)−1(ˆzj − Sjθ)  �  ∝ exp  �  �  �−1  2  �  �θT  �  ��  j  Sj(σ2  ySj + σ2  zI)−1Sj  �  � θ − 2θT  �  ��  j  Sj(σ2  ySj + σ2  zI)−1  �  � ˆzj  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Reorganising the terms, we end up with  p(θ|S1:J, σ2  y, ˆz1:J) ∝ exp  �  −1  2  �  θT Σ−1  postθ − 2θT Σ−1  postmpost  ��  ,  where Σ−1  post = �  j Sj(σ2  ySj + σ2  ZI)−1Sj + C−1 and mpost = Σpost[�  j Sj(σ2  ySj + σ2  zI)−1)ˆzj +  C−1m].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Therefore, θ|S1:J, σ2  y, ˆz1:J ∼ N(mpost, Σpost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Acceptance Ratio for the MH Update of Sj:  We drop the index j from Sj for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  When S′ ∼ W(S/α, α), the proposal density is  q(S′|S) = |S′|(α−d−1)/2e−tr[αS−1S′]/2  |S/α|α/22αd/2Γd( α  2 )  = |S′|(α−d−1)/2e−tr[αS−1S′]/2  |S|α/22αd/2Γd( α  2 )  αα/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  17  Therefore, the acceptance ratio corresponding to this proposal is  min  �  1, q(S|S′)  q(S′|S)  W(S′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' njΣx, κ)p( ˆS| ˆS′)N(ˆz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' S′θ, σ2  ySθ + σ2  zId)  W(S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' njΣx, κ)p( ˆS| ˆS)N(ˆz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Sθ, σ2ySθ + σ2zId)  �  ,  where the ratio of proposals becomes  q(S|S′)  q(S′|S) = |S|(α−d−1)/2|S|α/2e−tr[aS′−1S]/2  |S′|(α−d−1)/2|S′|α/2e−tr[αS−1S′]/2 =  � |S|  |S′|  �α−(d+1)/2  eα(tr[S−1S′]−tr[S′−1S])/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Acceptance Ratio for the MH Update of σ2  y:  To update σ2  y, we use a random walk proposal  σ2′  y ∼ N(σ2  y, σ2  q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The resulting acceptance ratio is  min  �  1,  IG(σ2′  y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' a, b) �J  j=1 N(ˆzj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Sjθ, σ2′  y Sjθ + σ2  zId)  IG(σ2y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' a, b) �J  j=1 N(ˆzj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Sjθ, σ2ySjθ + σ2zId)  �  B  Other Variants  This appendix is reserved for the details of the other variants mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For  simplicity, we will assume a single data holder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', J = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' the extension to J > 1 should be  straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1  Approximating Normality by Averaging  When xi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , n are not normal, another approach that we propose is based on modifying  the data to such that the rows of the modified feature matrix, called Xav, are averages of k > 1  original features in X, and thus approximately normal, by the CLT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Specifically, let n be divisible  by k so that m = n/k is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Consider the m × n matrix  A =  1  √  k  �  ����  11×k  01×k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  01×k  01×k  11×k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  01×k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  01×k  01×k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  11×k  �  ����  m×n  ,  Then the matrix Xav = AX corresponds to constructing a shorter m × d matrix whose i’th  column is the average of the rows (i − 1)k + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' , ik of X (scaled by 1/  √  k the preserve the  norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' When k is large enough, we can make normality assumptions for the rows of Xav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Further,  we consider  yav := Ay = Xavθ + Ae,  whose mean is Xavθ and covariance AAT σ2  y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' But, we have AAT = Im, so the covariance is σ2  yIm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Therefore, the same hierarchical model in Figure 1 can be used for X′, y′ with their respective  summary statistics  zav = (Xav)T yav,  Sav = (Xav)T Xav,  as well as the noisy versions of those summary statistics to provide a given level of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Note  that Sav and zav have the same sensitivities as S and z, hence the same noise variances are  needed for privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However, there is less information in Sav and zav due to averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  18  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2  Including the Intercept  If we include the intercept parameter, which corresponds to appending xi with a 1 from the left,  the design matrix will be changed from S to S0 =  � n  n¯xT  n¯x  S  �  , where ¯x = 1  n  �n  i=1 xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Also, note  that S = (n − 1)�Σx + n¯x¯xT where �Σx is the sample covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Under the normality assumption  for xi’s, ¯x ∼ N(m, Σx/n) and (n − 1)�Σx ∼ W(n − 1, Σx) are independent and have known  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Therefore, we can write a model that includes b = ¯x, ˆ  Σx, and S0 where S0 replaces  S in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' More specifically, we have the following hierarchical model:  θ ∼ N(m, C),  Σx ∼ IW(Λ, κ),  ˆ  Σx|Σx ∼ W(n − 1, Σx),  b|Σx ∼ N(µ, Σx/n),  z|θ, Σ2  y, ˆΣ, b ∼ N(S0θ, S0σ2  y),  ˆS| ˆΣ, b = N(S0, σ2  sI),  ˆz|z = N(z, σ2  zI)  with S0 =  � n  nbT  nb  (n − 1) ˆΣ + nbbT  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  C  Compared Methods  Here, we provide the details of the methods which we compare with the proposed methods in  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Those methods are originally proposed for J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' However, for comparison, we  implemented their natural extensions to the general (distributed) case J ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The implementations  of those methods can also be found in the code package provided for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='1  MCMC-B&S Adapted to the Distributed Setting  In Bernstein and Sheldon (2019), only J = 1 is considered, and the vector ss = [vec(S), z =  XT y, u = yT y] is perturbed with privacy-preserving noise to generate the observations of  the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' For J ≥ 1, we consider the following natural extension for generating perturbed  observations ˆss = [vec( ˆSj), ˆzj, ˆuj] along with  ˆSj = Sj + σdpMj,  ˆzj = zj + vj,  vj ∼ N(0, σ2  dpId),  ˆuj = uj + wj,  wj ∼ N(0, σ2  dp), (13)  where σdp = σ(ϵ, δ)∆ss with ∆ss =  �  ∥X∥4 + ∥X∥2∥Y∥2 + ∥Y∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  For completeness, we provide the further specifics of the model: We take (θ, σ2  y) ∼ NIG(a0, b0, m, Λ0)  where Λ0 = C−1 and Px = N(0, Σx) with Σx ∼ IW(Λ, κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  During the comparisons, we set a0, b0, m, C, Λ, κ to the same values for both this model and our  proposed model that assumes normally distributed features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', Px = N(0, Σx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' Then, we apply  an extension of Bernstein and Sheldon (2019, Algorithm 1) suited to those observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' One  iteration of that algorithm includes the following steps in order:  Calculate the D × 1 mean vector and D × D covariance matrix  µss = E[ss],  Σss = Cov[ss].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  This step requires the fourth moments N(0, Σx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Sample ssj ∼ N(µ(j)  post,ss, Σ(j)  post,ss) with  Σ(j)  post,ss = (njΣss(θ)−1 + (1/σ2  dp)I)−1,  and  µ(j)  post,ss = Σ(j)  post,ss(Σss(θ)−1µss + ˆssj/σ2  dp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  19  Sample Σx ∼ IW  �  Λ + �J  j=1 Sj, n + κ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  Sample (θ, σ2  y) ∼ NIG(an, bn, mn, Λn) by sampling σ2  y ∼ IG(an, bn), followed by sampling  θ ∼ N(µn, σ2  yΛ−1  n ) with an = a0 + n/2, bn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='5u + mT C−1m − mT  nΛnmn, and  Λn = Λ0 +  J  �  j=1  Sj,  mn = Λ−1  n  �  �  J  �  j=1  zj + Λ0m  �  � ,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='2  A Variant of adaSSP for the Distributed Setting  The adaSSP algorithm of (Wang, 2018) is originally designed for a single data holder, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=', J = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  In adaSSP, a differentially private estimate of θ is released as  ˆθ = ( ˆS + λI)−1 ˆz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  (14)  Here, ˆS and ˆz are the privatised versions of S and z as in (2) and (3), except that ϵ and δ must be  changed to 2ϵ/3 and 2δ/3 in those equations to provide (ϵ, δ) differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' This is because  adaSSP uses another parameter λ, which is also calculated from the sensitive data and a third of  the privacy budget is spent for privatising that calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' With v ∼ N(0, 1), λ is specifically  calculated as  λ = max{0, σ  �  d ln(6/δ) ln(2d2/ρ) − ˜λmin}  with σ = ∥X∥2/(ϵ/3), λmin = min(eig(S)), and  ˜λmin = max{λmin +  �  ln(6/δ)σv − ln(6/δ)σv, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  We consider an extension of (Wang, 2018) for J ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' To perform the extension, we reflect on its  tendency to approximate a (regularised) least square solution and consider the following estimate  ˆθ =  �  �  J  �  j=1  ˆSj + I  J  �  j=1  λj  �  �  −1 �  �  J  �  j=1  ˆzj  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  (15)  Here, ˆSj, ˆzj and λj are calculated in data node j separately from the other nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content=' The estimation  procedure in (15) does not properly account for the Bayesian paradigm but aggregates the shared  ˆSj’s and ˆzj’s to approximate the (regularised) least squares solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
+page_content='  20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tFST4oBgHgl3EQfVziF/content/2301.13778v1.pdf'}
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+arXiv:2301.05274v1  [math.PR]  12 Jan 2023
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE
+CHAOS ON PHASE BOUNDARIES
+HUBERT LACOIN
+Abstract. The complex Gaussian Multiplicative Chaos (or complex GMC) is infor-
+mally defined as a random measure eγXdx where X is a log correlated Gaussian field on
+Rd and γ “ α ` iβ is a complex parameter. The correlation function of X is of the form
+Kpx, yq “ log
+1
+|x ´ y| ` Lpx, yq,
+where L is a continuous function. In the present paper, we consider the cases γ P PI{II
+and γ P P1
+II{III where
+PI{II :“ tα ` iβ : α, β P R ; |α| ą |β| ; |α| ` |β| “
+?
+2du,
+and
+P1
+II{III :“ tα ` iβ : α, β P R ; |α| “
+a
+d{2 ; |β| ą
+?
+2du,
+We prove that if X is replaced by an approximation Xε obtained via mollification, then
+eγXεdx, when properly rescaled, converges when ε Ñ 0. The limit does not depend on
+the mollification kernel. When γ P PI{II, the convergence holds in probability and in Lp
+for some value of p P r1,
+?
+2d{αq. When γ P P1
+II{III the convergence holds only in law.
+In this latter case, the limit can be described a complex Gaussian white noise with a
+random intensity given by a critical real GMC. The regions PI{II and P1
+II{III correspond to
+phase boundary between the three different regions of the complex GMC phase diagram.
+These results complete previous results obtained for the GMC in phase I [18] and III [16]
+and only leave as an open problem the question of convergence in phase II.
+2010 Mathematics Subject Classification: 60F99, 60G15, 82B99.
+Keywords: Random distributions, log-correlated fields, Gaussian Multiplicative Chaos.
+Contents
+1.
+Introduction
+2
+2.
+Main results
+5
+3.
+The martingale approximation for GMC
+8
+4.
+Proof of convergence results on for γ P PI{II
+13
+5.
+Proof of Proposition 3.5
+18
+6.
+Proof of Proposition 2.8
+27
+7.
+Proof of Theorem 3.6
+34
+Appendix A.
+Technical results and their proof
+36
+Appendix B.
+The convergence of Mγ
+ε as a distribution
+39
+Appendix C.
+Beyond star-scale invariance
+43
+Appendix D.
+Proof of Lemma 4.1
+47
+References
+49
+1
+
+2
+HUBERT LACOIN
+1. Introduction
+Let K : Rd ˆ Rd Ñ p´8, 8s be a positive definite kernel on Rd (d ě 1 is fixed) which
+admits a decomposition of the form
+Kpx, yq “ log
+1
+|x ´ y| ` Lpx, yq,
+(1.1)
+(with the convention logp1{0q “ 8) where L is a continuous function on R2d . A kernel
+K is positive definite if for ρ P CcpRdq (ρ continuous with compact support)
+ż
+R2d Kpx, yqρpxqρpyqdxdy ě 0.
+(1.2)
+Given a centered Gaussian field X with covariance K and γ “ α ` iβ a complex number
+(α, β P R) the complex Gaussian Multiplicative Chaos (or complex GMC) with parameter
+γ is the random distribution formally defined by the expression
+Mγpdxq “ eγXpxqdx.
+(1.3)
+A difficulty comes up when trying to give an interpretation to the r.h.s. of (1.3). A field
+X with a covariance given by (1.1) can be defined only as a random distribution. For a
+fixed x P Rd it is not possible to make sense of Xpxq.
+The problem of providing a mathematical construction of Mγ that gives a meaning to
+(1.3) was first considered by Kahane in [15] in the case where γ P R, we refer to [25, 27]
+for reviews on the subject. The case of γ P C was considered only more recently, see for
+instance [11, 12, 13, 16, 18, 19, 20] and references therein. The standard procedure to define
+the GMC is to use a sequence of approximation of the field X, consider the exponential
+of the approximation and then pass to the limit. Mostly two kinds of approximation of X
+have been considered in the literature:
+(A) A mollification of the field, Xε, via convolution with a smooth kernel on scale ε,
+(B) A martingale approximation, Xt, via an integral decomposition of the kernel K.
+In the present paper we present convergence results for the random distribution eγXεpxqdx
+and eγXtpxqdx and in a certain range of parameter γ. Before describing our results in more
+details and provide some motivation, we first rigorously introduce the setup.
+1.1. The mollification of a log-correlated field.
+Log-correlated fields defined as distributions. Since K is infinite on the diagonal, it is not
+possible to define a Gaussian field indexed by Rd with covariance function K. We consider
+instead a process indexed by test functions. We define pK, a bilinear form on CcpRdq (the
+set of compactly supported continuous functions) by
+pKpρ, ρ1q “
+ż
+R2d Kpx, yqρpxqρ1pyqdxdy.
+(1.4)
+Since pK is positive definite (in the usual sense: for any pρiqk
+i“1, the matrix pKpρi, ρjqk
+i,j“1 is
+positive definite), it is possible to define X “ xX, ρyρPCcpRdq a centered Gaussian process
+indexed by CcpRdq with covariance kernel given by pK.
+Remark 1.1. There exists a modification of the process X which take value in a distri-
+bution space (more specifically, such that X takes values in the Sobolev space Hs
+locpRdq for
+every s ă 0 (see the definition (B.3) in the appendix). For this reason (and although we
+will not use this fact) we refer to X as a random distribution.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 3
+Approximation of X via mollification. The random distribution X can be approximated
+by a sequence of functional fields - processes indexed by Rd - by the mean of mollification
+by a smooth kernel. Consider θ a nonnegative function in C8
+c pRdq (the set of infinitely
+differentiable functions in CcpRdq) whose compact support is included in Bp0, 1q (for the
+remainder of the paper Bpx, rq denotes the closed Euclidean ball of center x and radius r)
+and which satisfies
+ş
+Bp0,1q θpxqdx “ 1. We define for ε ą 0, θε :“ ε´dθpε´1¨q and consider
+pXεpxqqxPRd, the mollified version of X, that is
+Xεpxq :“ xX, θεpx ´ ¨qy
+(1.5)
+From (1.4), the field Xεp¨q has covariance
+Kεpx, yq :“ ErXεpxqXεpyqs “
+ż
+R2d θεpx ´ z1qθεpy ´ z2qKpz1, z2qdz1dz2.
+(1.6)
+We set Kεpxq :“ Kεpx, xq and extend this convention to other functions of two variables
+in Rd.
+Since Kε is infinitely differentiable - thus in particular is H¨older continuous -
+by Kolmogorov’s Continuity Theorem (e.g. [21, Theorem 2.9]) there exists a continuous
+modification of Xεp¨q. In the remainder of the paper, we always consider the continuous
+modification of a process when it exists.
+This ensures that integrals such as the one
+appearing in (1.7) are well defined. We define the distribution Mγ
+ε by setting for f P CcpRdq
+Mγ
+ε pfq :“
+ż
+Rd fpxqeγXεpxq´ γ2
+2 Kεpxqdx.
+(1.7)
+The question of interest in the present paper is the convergence of Mγ
+ε when ε Ñ 0.
+Remark 1.2. Note that, even if we have chosen to omit this dependence in the notation,
+Xε and Mγ
+ε both depend on the particular convolution kernel θ. An important feature of
+our results is that the limits obtained for Mγ
+ε pfq do not depend on θ.
+1.2. Star-scale invariance and our assumption on K. On top of assuming that K
+admits a decomposition like (1.1), we also assume that it has an almost star-scale invariant
+part (see the definition (1.8)-(1.9)). This assumption might seem at first quite restrictive,
+but it has been shown in [12] that it is locally satisfied as soon as the function L in
+(1.1) is sufficiently regular. In Appendix C we provides details concerning the regularity
+assumption for L and explain how to extend the validity of our results to all sufficiently
+regular log-correlated kernels using the ideas in [12].
+Following a terminology introduced in [12], we say that a the kernel K defined on Rd is
+almost star-scale invariant if it can be written in the form
+@x, y P Rd, Kpx, yq “
+ż 8
+0
+p1 ´ η1e´η2tqκpetpx ´ yqqdt,
+(1.8)
+where η1 P r0, 1s and η2 ą 0 are constants and the function κ P C8
+c pRdq is radial, nonneg-
+ative and definite positive. More precisely we assume the following:
+(i) κ P C8
+c pRdq and there exists rκ : R` Ñ r0, 8q such that κpxq :“ rκp|x|q,
+(ii) rκp0q “ 1 and rκprq “ 0 for r ě 1,
+(iii) The mapping px, yq ÞÑ κpx ´ yq defines a positive definite kernel on Rd ˆ Rd.
+We say furthermore that a kernel K has an almost star-scale invariant part, if
+@x, y P Rd, Kpx, yq “ K0px, yq ` Kpx, yq
+(1.9)
+where Kpx, yq is an almost star-scale invariant kernel and K0 is H¨older continuous on R2d
+and positive definite.
+
+4
+HUBERT LACOIN
+1.3. Phase transitions and phase diagrams for GMC. Our main results concerns
+the asymptotic behavior of Mγ
+ε in the specific range of γ given in the abstract. In order
+to properly motivate and present these results, it is necessary to introduce some context,
+and recall known facts about the phase diagram of the complex GMC.
+Phase transition at |α| “
+?
+2d for the real valued GMC. The question of the existence and
+identification of the limit
+lim
+εÑ0 Mα
+ε p¨q,
+has first been considered in the work of Kahane in the eighties [15], in the case when
+α P R. The obtained limit in that case crucially depends on α: when |α| ă
+?
+2d - referred
+to as the subcritical case - then Mα
+ε converges in probability to a non-trivial limiting
+distribution (see for instance [2, Theorem 1.1] for a short and self contained proof, we
+refer to the introduction in [2] for a detailed chronological account of results obtained for
+the subcritical case).
+When |α| ě
+?
+2d, we have limεÑ0 Mα
+ε pfq “ 0 and a rescaling procedure is needed in
+order to obtain a non-trivial limit. The phenomenology is however different according to
+whether |α| “
+?
+2d (α critical) or |α| ą
+?
+2d (α supercritical).
+In the critical case (α “ ˘
+?
+2d), is has been shown, under fairly mild assumptions (see
+[4, 5, 10, 26] and Theorem A below) that
+a
+log p1{εqMα
+ε converges in probability to a
+non-trivial limit called the critical GMC.
+When |α| ą
+?
+2d, the results are less complete. So far the convergence has not been
+proved for Mα
+ε but only for an approximating martingale sequence Mα
+t (see (3.6)) in [24].
+Besides this technical point, the most important differences with the case |α| ď
+?
+2d
+concerns the type of the convergence and the nature of limiting object. The convergence
+only holds only in law, and the limit is a purely atomic measure (a measure supported by
+a countable set) see [24, Corollary 2.3].
+Phase diagram for complex GMC. When γ is allowed to assume complex value, the phase
+diagram becomes more intricate. The complex plane can be divided in three open regions
+with intersecting boundaries
+PI :“
+␣
+α ` iβ : α2 ` β2 ă d
+(
+Y
+!
+α ` iβ : α P p
+a
+d{2,
+?
+2dq ; |α| ` |β| ă
+?
+2d
+)
+,
+PII :“
+!
+α ` iβ : |α| ` |β| ą
+?
+2d ; |α| ą
+a
+d{2
+)
+,
+PIII :“
+!
+α ` iβ : α2 ` β2 ą d ; |α| ă
+a
+d{2
+)
+.
+(1.10)
+This diagram first appeared in the context of complex Gaussian multiplicative cascade
+[3], and also serves to describe the behavior of other related models such as the complex
+REM [14] or complex branching Brownian Motion [6, 7, 23].
+The region PI corresponds to the subcritical phase. For γ P PI it has been proved
+[12, 18] that Mγ
+ε converges to a limit that does not depend on the mollifier θ.
+The region PII corresponds to the supercritical phase, in which it is believed that Mγ
+ε
+- after proper renormalization - converges only in law to a purely atomic random distri-
+bution. This conjecture is supported by rigorous results obtained in the case of Complex
+Branching Brownian Motion [6, 23].
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 5
+PSfrag replacements
+β
+α
+?
+d
+a
+d{2
+?
+2d
+´
+?
+2d
+PI
+PII
+PIII
+Figure 1. The phase diagram of the complex GMC in the complex plane. Each region
+correspond to a different limiting behavior for M γ
+ε in terms of renormalization factor,
+type of convergence and properties of the limit. In the present paper, we prove results
+concerning the asymptotic behavior on frontier of PI Y PIII with PII. Results concerning
+convergence in PI Y PIII where proved in [12, 18] (for PI) and [16] (for PIII Y PI{III). The
+convergence in the region PII remains a challenging conjecture.
+Finally the region PIII corresponds to yet another asymptotic behavior for Mγ
+ε . Like in
+PII, Mγ
+ε - properly rescaled - only converges in law. The limit is given by a white noise
+whose intensity is random and is given by the real valued GMC with parameter 2α (which
+is subcritical, according to the definition of PIII). A similar convergence result holds on
+the boundary between PII and PIII that is
+PII{III :“
+!
+α ` iβ : α2 ` β2 “ d ; |α| ă
+a
+d{2
+)
+.
+These convergence statements for γ P PIII Y PI{III are proved in [16].
+The present contribution. The aim of the present paper is to come closer to a completition
+of the phase diagram by stating and proving convergence results for Mγ
+ε on the phase
+transition curves PI{II and PI{III as well as at the triple points PI{II{III.
+In each case,
+the limit obtained does not depend on the regularization kernel θ. We leave as an open
+problem the challenging task of proving a convergence result in the frozen phase PII.
+2. Main results
+For simplicity of notation, we consider, for the remainder of the paper and without loss
+of generality that γ is in the upper-right quarterplane of C, that is α, β ě 0.
+2.1. The boundary between phase I and II. Our first result concerns the case when
+γ lies on the boundary between regions I and II
+PI{II :“ tα ` iβ : α ą β ą 0 ; α ` β “
+?
+2du.
+(2.1)
+
+6
+HUBERT LACOIN
+Note that our definition of PI{II excludes one point of the boundary which correspond to
+Critical Gaussian multiplicative chaos γ “
+?
+2d (see Section 2.2 below).
+Theorem 2.1. If X is a centered Gaussian field whose covariance kernel K has an almost
+star-scale invariant part, γ P PI{II, and f P CcpRdq then there exists a complex valued
+random variable Mγ
+8pfq such that for any choice of mollifier θ the following convergence
+holds in Lp if p P
+”
+1,
+?
+2d{α
+¯
+.
+lim
+εÑ0 Mγ
+ε pfq “ Mγ
+8pfq.
+(2.2)
+The above result extends [18, Theorem 2.2] which established convergence for γ P PI.
+The method which we use to prove it however, completely differs from the one employed in
+[18]. In fact the method of proof that we employ in Section 4 balso provides an alternative
+and much shorter proof of [18, Theorem 2.2], with the additional benefit of establishing
+convergence in Lp for an optimal range of p.
+Remark 2.2. We have chosen to denote the limit by Mγ
+8 rather than Mγ
+0 . While the
+latter may seem a more natural choice, it is already in use for the initial condition of the
+martingale GMC approximation introduced in Section 3 (see for instance (4.2)).
+Remark 2.3. We have chosen to put the emphasis on the proof of the convergence of
+Mγ
+ε pfq for all fixed f, but it is true also that Mγ
+ε p¨q converges as a random distribution.
+More precisely the convergence (in probability) of Mγ
+ε in a local Sobolev space of negative
+index can in fact be deduced from the estimates obtained in the proof of (2.2). We include
+the argument in Appendix B.
+2.2. The boundary between phase II and III, and the triple point. Our second
+result concerns the case when γ P P1
+II{III where
+PII{III :“ t
+a
+d{2 ` iβ : β ą
+a
+d{2u,
+P1
+II{III :“ PII{III Y t
+a
+d{2p1 ` iqu
+(2.3)
+In that case Mγ
+ε needs to be rescaled in order to obtain a non-trivial limit. The convergence
+holds only in law. To describe the limit we need to introduce two notions: Critical Gaussian
+Multiplicative Chaos, and Gaussian White Noise with a random intensity.
+Critical GMC. As explained in the introduction critical Gaussian Multiplicative Chaos
+is obtained as the limit of Mα
+ε when α “
+?
+2d. The value
+?
+2d represent a threshold
+for the convergence of Mα
+ε . The convergence result below follows from a combination of
+[5, Theorem 5] - which establishes the convergence for the martingale sequence Mα
+t (see
+Section 3) and [10, Theorems 1.1 and 4.4] which establish that the limit is the same for
+the exponential of the mollified field Mα
+ε . Alternative concise proofs of these results have
+been recently given in [17].
+Theorem A. Let X be a Gaussian random field with an almost-star scale invariant kernel.
+There exists a locally finite random measure M1 with dense support and no atoms such
+that for every f P CcpRdq the following convergence holds in probability
+lim
+εÑ0
+c
+π log p1{εq
+2
+M
+?
+2d
+ε
+pfq “ M1pfq.
+(2.4)
+Remark 2.4. Note that we have set different conventions and that our M1 differs from
+that in [5, Theorem 5] by a factor
+b
+2
+π.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 7
+Complex white noise with random intensity given by a Real GMC. For γ P P1
+II{III we
+define Mγ to be a complex white noise with intensity measure given by M1pe|γ|2L¨q It is a
+random linear form which is constructed jointly with X, on an extended probability space.
+Conditionally on X, for f P C8
+c pDq, Mγpfq is a complex Gaussian random variable, with
+independent real and imaginary parts, both with a variance equal to
+M1pe|γ|2Lf 2q “
+ż
+D
+e|γ|2Lpx,xqfpxq2M1pdxq.
+Formally, letting P and P denote respectively the law of X and the joint law of pX, Mγp¨qq,
+Mγp¨q is the random process indexed by CcpRdq which satisfies for any m, n ě 1, ρ1, . . . , ρm, f1, . . . , fn P
+CcpRdq and any bounded measurable function F on Cn`m
+E
+“
+F
+`
+pxX, ρiyqm
+i“1, pMγpfjqqn
+j“1
+˘‰
+“ E b En
+“
+F
+`
+pxX, ρiyqm
+i“1, Σrγ, X, pfjqn
+j“1s ¨ Nn
+˘‰
+(2.5)
+where under Pn, Nn is an n dimensional vector whose coordinate are IID standard com-
+plex Gaussian variables, and Σrγ, X, pfjqn
+j“1s is the positive definite square root of the
+Hermitian matrix
+´
+M1pe|γ|2Lfif jq
+¯n
+i,j“1 .
+The result. Let us define the function ℓθ on Rd, obtained by convoluting z ÞÑ log 1{|z|
+twice with θ, that is
+ℓθpzq :“
+ż
+Rd log
+ˆ
+1
+|z ` z1 ´ z2|
+˙
+θpz1qθpz2qdz1dz2,
+(2.6)
+and set (recall (2.3))
+vpε, θ, γq :“
+$
+’
+&
+’
+%
+p2π logp1{εqq´1{4ε
+d´|γ|2
+2
+´ş
+Rd e|γ|2ℓθpzqdz
+¯1{2
+if γ P PII{III,
+a
+Σd´1
+´
+2 logp1{εq
+π
+¯1{4
+if γ “
+a
+d{2pi ` 1q
+(2.7)
+where Σd is the volume of the d ´ 1 dimensional sphere. Note that limεÑ0 vpε, θ, γq “ 8
+in all cases.
+Theorem 2.5. Let X be a Gaussian random field with an almost star-scale covariance.
+Then given γ P P1
+II{III, we have the following joint convergence in law
+ˆ
+X,
+Mγ
+ε
+vpε, θ, γq
+˙
+εÑ0
+ñ pX, Mγq,
+(2.8)
+Remark 2.6. The convergence in (2.8) implies that vpε, θ, γq´1Mγ
+ε does not converge
+in probability. On the heuristic level, this can be explained as follows: The white noise
+that appears in the limit is the product of local fluctuations of Xε. These fluctuations are
+produced by high frequencies in the Fourier spectrum of X. The set of frequencies that
+produce the fluctuations diverges to infinity when ε Ñ 0. This means that the randomness
+that produces the white noise become asymptotically independent of X in the limit.
+Remark 2.7. The convergence (2.8) means that for any collection pρiqm
+i“1 and pfjqn
+j“1 we
+have the convergence in law of the Cm`n valued vector
+lim
+εÑ0
+˜
+pxX, ρiyqm
+i“1,
+ˆ Mγ
+ε pfjq
+vpε, θ, γq
+˙n
+j“1
+¸
+“
+´
+pxX, ρiyqm
+i“1, pMγpfjqqn
+j“1
+¯
+.
+(2.9)
+
+8
+HUBERT LACOIN
+The convergence can also be shown to hold in a space of distribution. More precisely, there
+exists a modification of the process Mγ taking values in the local Sobolev space H´u
+loc pRdq
+with u ą d{2 and Mγ
+ε pfjq
+vpε,θ,γq converges in law in that space. See Appendix B.
+Since both X and Mγ
+ε are linear forms, the convergence of finite dimensional marginals
+follows from that of one dimensional marginals (this can simply be checked using Fourier
+transform and L´evy Theorem). More precisely, we only need to prove the convergence for
+every f P CcpRdq and ω P r0, 2πq of the real valued variable (Re denotes the real part)
+Mγ
+ε pf, ωq :“ Re
+`
+e´iωMγ
+ε pfq
+˘
+(2.10)
+Hence Theorem 2.5 can be reduced to the proof of the following statement
+Proposition 2.8. Under the assumption of Theorem 2.5, given ρ, f P CcpRdq, ω P r0, 2πq,
+we have
+lim
+εÑ0 E
+„
+eixX,ρy`i Mγ
+ε pf,ωq
+vpε,θ,γq
+
+“ E
+„
+eixX,ρy´ 1
+2M1pe|γ|2L|f|2q
+
+.
+(2.11)
+The r.h.s. in (2.11) of corresponds to the Fourier transform of pX, Mγq (cf. (2.5))
+E
+„
+eixX,ρy´ 1
+2 M1pe|γ|2L|f|2q
+
+“ E
+”
+eixX,ρy`iMγpfqı
+,
+and the convergence of the Fourier transform implies that of finite dimensional marginals.
+More detailed justifications are exposed in [16, Section 1.2].
+3. The martingale approximation for GMC
+Before getting to the technical core of the paper, we need one more introductory section
+to present an essential tool which is used in the proof of both Theorem 2.1 and Theorem
+2.5: the martingale decomposition of the field X. Under the almost star-scale assumption
+for K, besides mollification, there is another natural way to approximate the log-correlated
+field X by a smooth field. Extending the probability space, one can define a martingale
+sequence of smooth fields pXtqtě0 that converges to X.
+This allows for another approach to the construction of GMC, considering the exponen-
+tial of the martingale approximation of X (see (3.7)) which we call Mγ
+t (see Remark 3.2
+concerning the conflict of notation). Convergence results for Mγ
+t which are a analogous
+to Theorem 2.1 and 2.5 are also presented in this section. In section 3.5 we introduce an
+important technical tool which is used to prove Theorem 2.5. The result (a central limit
+Theorem proving convergence to a Gaussian with random variance) may find applications
+in other context, so it is stated in a rather general setup.
+3.1. The martingale decomposition of X. Given K with an almost star-scale invariant
+part, and using the decomposition (1.8) for K, we set Qtpx, yq :“ κpet1px ´ yqq where t1 is
+defined as the unique positive solution of
+t1 ´ η1
+η2
+p1 ´ e´η2t1q “ t.
+(3.1)
+We set
+Ktpx, yq :“ K0px, yq `
+ż t
+0
+Qspx, yqds
+“ K0px, yq `
+ż t1
+0
+p1 ´ η1e´η2sqκpespx ´ yqqds “: K0px, yq ` Ktpx, yq.
+(3.2)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 9
+Note that we have limtÑ8 Ktpx, yq “ Kpx, yq. We define pXtpxqqxPRd,tě0 to be a centered
+Gaussian field with covariance given by (using the notation a ^ b :“ minpa, bq, a _ b :“
+maxpa, bq)
+ErXtpxqXspyqs “ Ks^tpx, yq.
+(3.3)
+Since ps, t, x, yq ÞÑ Ks^tpx, yq is H¨older continuous, the field admits a continuous modifi-
+cation. We let Ft :“ σ
+´
+pXspxqqxPRd,sPr0,ts
+¯
+denote the natural filtration associated with
+X¨p¨q. The process X indexed by CcpRdq and defined by xX, ρy “ limtÑ8
+ş
+Rd Xtpxqρpxqdx,
+is a centered Gaussian field with covariance, so that Xt is an approximation sequence for
+a log-correlated field with covariance K. We also define X¨ :“ X¨ ´ X0. Recalling (3.2)
+we have
+ErXtpxqXspyqs “ Ks^tpx, yq.
+(3.4)
+An important observation is that since Ktpxq :“ Ktpx, xq “ t, for any fixed x P Rd, the
+process pXtpxqqtě0 is a standard Brownian Motion. We also introduce the field Xt,ε which
+is the mollification of Xt, that is
+Xt,εpxq :“
+ż
+Rd θεpx ´ zqXtpzqdz “ E rXεpxq | Fts .
+We let Kt,εpx, yq denote the covariance of the field Xt,ε and Kt,ε,0px, yq the cross-covariance
+of Xt,ε and Xt
+Kt,εpx, yq :“ ErXt,εpxqXt,εpyqs “
+ż
+Rd θεpx ´ z1qθεpy ´ z2qKtpz1, z2qdz1dz2,
+Kt,ε,0px, yq :“ ErXt,εpxqXtpyqs “
+ż
+Rd θεpx ´ zqKtpz, yqdz.
+(3.5)
+The quantity Kt,ε is defined similarly and we use the notation Qt,ε and Qt,ε,0 the corre-
+sponding mollified versions of Qt.
+3.2. The martingale approximation for the GMC. We define the distribution Mγ
+t
+by setting for f P CcpRdq
+Mγ
+t pfq :“
+ż
+Rd fpxqeγXtpxq´ γ2
+2 Ktpxqdx.
+(3.6)
+Using the independence of the increments of X, it is elementary to check that Mtpfq is an
+pFtq-martingale. We also define
+Mγ
+t,εpfq :“
+ż
+Rd fpxqeγXt,εpxq´ γ2
+2 Kt,εpxqdx “ E rMγ
+ε pfq | Fts .
+(3.7)
+3.3. A few properties of the covariance kernels. We introduce some technical nota-
+tion and estimates that are going to be of use throughout the article. Let us first not that
+if a kernel K has an almost star-scale invariant part then it can be written in the form
+(1.1). Indeed, if K satisfies (1.8) then the function L defined for x ‰ y by
+Lpx, yq :“ Kpx, yq ` log |x ´ y|,
+(3.8)
+can be extended to a continuous function on R2d. Note that we have
+Lpxq “ lim
+yÑx pKpx, yq ` log |x ´ y|q “ K0pxq ´ j
+(3.9)
+
+10
+HUBERT LACOIN
+where the difference term j does not depend on x and can be computed explicitely
+j :“ lim
+zÑ0
+`
+logp1{|z|q ´ Kp0, zq
+˘
+“ η1
+η2
+`
+ż 8
+0
+`
+1 ´ rκpe´sq
+˘
+ds ă 8.
+(3.10)
+The above comes from the fact that
+logp1{|z|q ´ Kp0, zq “
+ż logp1{|z|q
+0
+p1 ´ Qlogp1{|z|q´up0, zqqdu
+and the fact that the integrand on the r.h.s. converges to 1 ´ rκpe
+η1
+η2 ´sq. Lastly one can
+observe that the following identity holds
+ℓpzq :“ lim
+tÑ8
+`
+Ktp0, e´tzq ´ t
+˘
+“ lim
+tÑ8
+ż t
+0
+pκpes1´tzq ´ 1qds “
+ż 8
+0
+pκpe
+η1
+η2 ´uzq ´ 1qdu, (3.11)
+where in the integral in s, s1 is related to s via (3.1). To obtain the third equality, one
+simply observe that s1 “ s ` η1{η2 ` op1q in the large s limit and make the change of
+variable u “ t ´ s. Note that ℓpzq is a continuous negative function and that for any
+|z| ě e´ η1
+η2 we have ℓpzq “ log 1
+|z| ´ j.
+To conclude this subsection, we gather in a a technical lemma a couple of useful estimates
+concerning Kt, Qt and their variant.
+Lemma 3.1. Given R ą 0, there exists a constant CR such that for any x, y P Bp0, Rq,
+t ą 0 and ε P r0, 1s
+ˇˇˇˇKt,εpx, yq ´ log
+ˆ
+1
+maxpe´t, ε, |x ´ y|q
+˙ˇˇˇˇ ď CR.
+(3.12)
+The bound (3.12) remains valid with Kt,ε replaced by Kt (with ε “ 0), Kt,ε,0, Kt,ε etc...
+We also have ż
+Rd Qtpx, yq “
+ż
+Rd Qt,εpx, yqdy “
+ż
+Rd Qt,ε,0px, yqdy ď Ce´dt,
+(3.13)
+and
+0 ď t ´ Kt,εpx, yq ď C
+`
+etp|x ´ y| ` εq
+˘2
+(3.14)
+The estimates above can be proved rather directly from the definition. A detailed proof
+of (3.12) is provided in [17, Appendix A.3]. The bound (3.13) follows directly from the
+definition of Qt given above (3.1) and the fact that |t´t1| is uniformy bounded. The upper
+bound in (3.14) can be obtained by integrating (in time and space) the inequality
+1 ´ Qtpz1, z2q ď Cret1|z1 ´ z2|s2
+which follows directly from the Taylor expansion at second order of κ.
+Remark 3.2. There is an obvious conflict of notation between Kt introduced above and
+Kε introduced in (1.6) and the same can be said about Xt and Mγ
+t . This should not cause
+any confusion since we keep using the letter ε for quantities related to the mollified field
+Xε and latin letters for quantities related to the martingale approximation Xt.
+3.4. Convergence results for the martingale approximation. An intermediate step
+to prove Theorem 2.1 and Theorem 2.5 is to show that similar results hold for the mar-
+tingale approximation Mγ
+t . These results present of course an interest in their own right.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES11
+The case of γ P PI{II.
+Proposition 3.3. When γ P PI{II, the martingale Mγ
+t pfq is bounded Lp for p P r1,
+?
+2d{|α|q.
+As a consequence the limit
+lim
+tÑ8 Mγ
+t pfq “: Mγ
+8pfq
+(3.15)
+exists almost surely. The convergence holds in Lp and the limit is non-trivial.
+Remark 3.4. The martingale limit in (3.15) is the same as the limit of Mγ
+ε appearing in
+Theorem 2.1 (this is the reason why we use the same notation), we have
+lim
+tÑ8 Mγ
+t pfq “ lim
+εÑ0 Mγ
+ε pfq.
+(3.16)
+This observation is important, since it establishes that the limit in (2.2) does not depend
+on the choice of the mollifier.
+The case γ P P1
+II{III. In order to state the convergence in law result for Mγ
+t , we need to
+introduce a normalization factor vpt, γq (analogous to (2.7) for the mollified case). Let us
+set
+vpt, γq “
+$
+&
+%
+e
+|γ2|j
+2
+` 1
+2πt
+˘1{4 e
+p|γ|2´dqt
+2
+´ş
+Rd e|γ|2ℓpzqdz
+¯1{2
+,
+if |γ|2 ą d
+a
+Σd´1
+` 2t
+π
+˘1{4
+if |γ|2 “ d.
+(3.17)
+and define
+Mγ
+t pf, ωq :“ Re
+`
+e´iωMγ
+t pfq
+˘
+.
+(3.18)
+The following analogue of Proposition 2.8 holds.
+Proposition 3.5. If X is an almost star-scale invariant field and γ P P1
+II{III we have for
+any ρ, f P CcpRdq
+lim
+tÑ0 E
+„
+eixX,ρy`i
+Mγ
+t pf,ωq
+vpt,γq
+
+“ E
+„
+eixX,ρy´ 1
+2 M1pe|γ|2L|f|2q
+
+.
+(3.19)
+As a consequence we have the following convergence in law (in the sense of finite dimen-
+sional marginals)
+ˆ
+X,
+Mγ
+t
+vpt, γq
+˙
+tÑ8
+ùñ
+pX, Mγq.
+(3.20)
+3.5. CLT towards a Gaussian with random variance. We conclude this section by
+introducing a technical results which is essential to prove the convergence of a sequence of
+variable towards a Gaussian with random intensity in Theorem 2.5. We provide the result
+and its proof in a reasonably high level of generality since it may find application in other
+contexts.
+Consider pFtqtě0 a filtration and pWnqně1 a sequence of real valued random variables
+in L1. We introduce for each n ě 1 the martingale
+Wn,t :“ E rWn | Fts .
+(3.21)
+We assume that the martingale Wn,t admits a modification which is continuous in t for
+every n ě 1 We prove that Wn converges to to a Gaussian with random variance if the
+quadratic variation of pWn,tqtě0 satisfy a law of large number and a couple of additional
+technical assumptions. The result generalizes a similar CLT established for a single mar-
+tingale process (see [9, Theorem 5.50, Chap. VIII-Section 5c] or [16, Theorem 2.5]).
+
+12
+HUBERT LACOIN
+Theorem 3.6. Let us assume that and that there exists a non-negative valued random-
+variable Z which is such that the three following convergences in probability hold
+lim
+nÑ8
+xWny8
+v2pnq “ Z,
+@t ě 0 lim
+nÑ8
+xWnyt
+v2pnq “ 0, and lim
+nÑ8
+Wn,0
+vpnq “ 0.
+(3.22)
+Then Xn{vpnq converges in distribution towards a random Gaussian with variance given
+by Z, that is to say that for any F8 bounded measurable H we have
+lim
+nÑ8 E
+”
+HeiξWn{vpnqı
+“ lim
+nÑ8 E
+„
+He´ ξ2Z
+2
+
+(3.23)
+This is equivalent to saying that for any F8 random variable Y we have the following
+convergence in law
+pY, Wnq ùñ pY,
+?
+ZNq
+where N is a standard Gaussian which is independent of Z and Y .
+Remark 3.7. We believe that with adequate assumption on the size of the jumps, the
+result may extend to the case where pWn,tqtě0 is a c`ad-l`ag martingale, with the quadratic
+variation is replaced by the predictable bracket. Since we have no application in that setup,
+we restricted ourselves to the continuous case where the proof is technically simpler.
+Remark 3.8. In Section 6 we apply Theorem 3.6 for a sequence of variables indexed by
+ε P p0, 1q (namely Mγ
+ε pf, ωq) in the limit when ε Ñ 0 rather than n ě 1 and n Ñ 8.
+These setups are equivalent.
+3.6. Organization of the paper. The remainder of the paper is organized as follows
+‚ In Section 4 we prove all the statements concerning convergence in PI{II. Section
+4.1 is devoted to the proof of Proposition 3.3. The more technical proof of Theorem
+2.1, which uses Proposition 3.3 as in imput is displayed in Section 4.2.
+‚ The statements concerning γ P P1
+II{III, namely Proposition 3.5 and Proposition
+2.8, while relying on relatively simple ideas, require a certain amount of technical
+computations. In Section 5 we prove Proposition 3.5, in Section 6 Proposition 2.8.
+‚ In Section 7, we present the proof of Theorem 3.6.
+A significant amount of material is presented in appendices.
+‚ In Appendix A, we prove a couple of auxilliary results used in Section 5/6.
+‚ In Appendix B, we present and prove an extension of our main results, that is,
+the convergence of Mγ
+ε p¨q as a distribution. After identifying the right topology,
+the proof mostly boils down to repeating the computation made in Section 4 (for
+γ P PI{II) and Section 6 (for γ P P1
+II{III).
+‚ In Appendix C, we explain how our results can be extended to the case of a
+(sufficiently regular) log-correlated Gaussian field defined on an arbitrary open
+domain D Ă Rd.
+‚ In Appendix D, we present a relatively short proof of Lemma 4.1 for the sake
+of completeness. It the same as the one presented in [20, Lemma 3.15], except
+that we include a short proof of Lemma D.2 instead of relying on the branching
+random walk literature where more general results have been shown, albeit with
+much longer proofs (see for instance [8, 22]).
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES13
+A comment on notation. Throughout the paper, we use the letter C for a generic positive
+constant when we need to compare two quantities. It may depend on some parameters
+(for instance on γ or on the kernel K) but never on the variable t or ε. The value of C
+might change from one equation to the other withing the same proof. We use C1 and C2
+if we need several constants in the same display.
+4. Proof of convergence results on for γ P PI{II
+In this section we prove Proposition 3.3 and Theorem 2.1. The first one is easier, recall
+that due to the martingale property of Mγ
+t , it is sufficient to show that the sequence
+is bounded in Lp to prove convergence. This is performed in Section 4.1. We rely on
+the Burkeholder-Davis-Gundy (BDG) inequality, compute the quadratic variation of the
+martingale and studying its moment of order p{2.
+In Section 4.2, we adapt the same method to estimate the Lp norm of Mγ
+ε ´ Mγ
+8.
+More precisely the BDG inequality for the martingale pMγ
+t,ε ´ Mγ
+t qtě0, and show that the
+moment of order p{2 of its quadratic variation is uniformly small in t.
+4.1. Proof of Proposition 3.3. Recalling that
+?
+2d{α ą 1, we are going to prove that
+Mγ
+t pfq is bounded in Lp for
+p P
+´?
+8d{p3αq _ 1,
+?
+2d{α
+¯
+.
+(4.1)
+In the whole paper, when Mt is a complex valued continuous martingale, we use the
+notation xMyt to denote the the bracket between M and M. It is the predictable process
+such that |Mt|2 ´ xMyt is a local martingale. Using Burkeholder-Davis-Gundy (BDG)
+inequality for Mγ
+t pfq, there exists a constant Cp such that for every t ą 0
+E
+“
+|Mγ
+t pfq|p‰
+ď Cp
+´
+ErxMγpfqyp{2
+t
+s ` E r|Mγ
+0 pfq|ps
+¯
+.
+(4.2)
+We have
+E
+“
+|Mγ
+0 pfq|2‰
+“
+ż
+R2d e|γ|2K0px,yqfpxqfpyqdxdy ă 8.
+(4.3)
+Since p ă 2 by assumption, Jensen’s inequality implies that E r|Mγ
+0 pfq|ps ă 8. Using Itˆo
+calculus, we obtain an explicit expression for the quadratic variation
+xMγpfqy8 “ |γ|2
+ż 8
+0
+Atdt
+(4.4)
+where
+At :“
+ż
+R2d fpxqfpyqQtpx, yqeγXtpxq`γXtpyq´ γ2
+2 Ktpxq´ γ2
+2 Ktpyqdxdy.
+(4.5)
+Note that At is real and positive. From (4.2), we deduce that Mγ
+t pfq is bounded in Lp if
+E
+“
+p
+ş8
+0 Atdtqp{2‰
+ă 8. To bound At from above, we take the modulus of the integrand in
+(4.5) and using the assumption that β “
+?
+2d ´ α (γ P PI{II) we obtain that
+At ď
+ż
+R2d |fpxqfpyq|Qtpx, yqeαpXtpxq`Xtpyqq` 2d´2
+?
+2dα
+2
+pKtpxq`Ktpyqqdxdy.
+(4.6)
+Then using the inequality ab ď a2
+2 ` b2
+2 with
+a “ |fpxq|eαXtpxq` 2d´2
+?
+2dα
+2
+Ktpxq
+and
+b “ |fpyq|eαXtpyq` 2d´2
+?
+2dα
+2
+Ktpyq
+
+14
+HUBERT LACOIN
+and symmetry in x and y, we have
+At ď
+ż
+R2d |fpxq|2Qtpx, yqe2αXtpxq`p2d´2
+?
+2dαqKtpxqdxdy.
+(4.7)
+We use (3.13) to integrate over y and (3.12) to replace replace Ktpxq by t (at the cost of
+multiplicative constant) and we have
+At ď Cedt
+ż
+Rd |fpxq|2e2αpXtpxq´
+?
+2dtqdx.
+(4.8)
+Now, as α ą
+a
+d{2, we have by p{2 ă 1 by assumption. We can use thus the following
+inequality (valid for an arbitrary collection of positive real numbers paiqiPI and q P p0, 1q)
+˜ÿ
+iPI
+ai
+¸q
+ď
+ÿ
+iPI
+aq
+i ,
+(4.9)
+with q “ p{2. In the remainder of the paper, we simply say “by subadditivity” when using
+(4.9). Using (4.9) and Jensen’s inequality we have
+E
+«ˆż 8
+0
+Atdt
+˙p{2ff
+ď
+ÿ
+ně0
+E
+«ˆż n`1
+n
+Atdt
+˙p{2ff
+ď
+ÿ
+ně0
+E
+«ˆż n`1
+n
+ErAs | Fnsds
+˙p{2ff
+.
+(4.10)
+Averaging with respect to pXs ´ Xnq we obtain from (4.8)
+ż n`1
+n
+E rAs | Fns ď Cedn
+ż
+R2d |fpxq|2e2αpXnpxq´
+?
+2dnqdx “: CBn.
+(4.11)
+As p ą
+?
+8d{3α by assumption, we can conclude using the estimate in Lemma 4.1 below
+for the fractional moments of Bn (the assumption on p makes the r.h.s. of (4.12) summable
+in n). More precisely, we deduce from (4.10),(4.11) and (4.12) that E
+”`ş8
+0 Atdt
+˘p{2ı
+ă 8.
+Lemma 4.1. For α ą
+a
+d{2 and p ă
+?
+2d{α we have
+E
+”
+Bp{2
+n
+ı
+ď Cn´ 3αp
+?
+8d plog nq6.
+(4.12)
+This result is a weaker version of [20, Lemma 3.15]. We provide, for the commodity of the
+reader a self-contained of Lemma 4.1 in Appendix D.
+4.2. Proof of Theorem 2.1. We prove Theorem 2.1 in the setup where our probability
+space contains a martingale approximation pXtqtě0 of the field X with covariance 3.3.
+More precisely we show that Mγ
+ε pfq converges to the same limit as Mγ
+t pfq. Working in
+an enlarged probability space entails by no mean a loss of generality since the validity of
+the statement “the sequence pMγ
+ε pfqqεPp0,1s is Cauchy in Lp” is entirely determined by the
+distribution of pXεpxqqxPRd,εPp0,1s.
+Proposition 4.2. Given γ P PI{II and p P r1,
+?
+2d{αq we have
+lim
+εÑ0 sup
+tą0
+E
+“
+|pMγ
+t ´ Mγ
+t,εqpfq|p‰
+“ 0.
+(4.13)
+As a consequence the following convergence holds in Lp
+lim
+εÑ0 Mγ
+ε pfq “ Mγ
+8pfq
+(4.14)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES15
+Proof. Let us first show indicate how (4.14) follows from (4.13). We observe that
+E
+“
+|pMγ
+t,ε ´ Mγ
+ε qpfq|2‰
+“
+ż 8
+0
+fpxqfpyq
+´
+e|γ|2Kεpx,yq ´ e|γ|2Kt,εpx,yq¯
+dxdy.
+(4.15)
+Since limtÑ8 Kt,εpx, yq “ Kεpx, yq, using dominated convergence the r.h.s. tends to 0 when
+t Ñ 8 and thus limtÑ8 Mγ
+t,εpfq “ Mγ
+ε pfq in L2, and hence also in Lp. Using Proposition
+3.3 we thus have the following convergence in Lp
+lim
+tÑ8pMγ
+t ´ Mγ
+t,εqpfq “ pMγ
+8 ´ Mγ
+ε qpfq.
+Taking the limit when ε to zero, we obtain that
+lim
+εÑ0 E r|pMγ
+8 ´ Mγ
+ε qpfq|ps “ lim
+εÑ0 lim
+tÑ8 E
+“
+|pMγ
+t ´ Mγ
+t,εqpfq|p‰
+.
+(4.16)
+and we conclude using (4.13).
+To prove (4.13), we assume that (4.1) holds. Then using the BDG inequality (we omit the
+dependence in f for ease of reading). We have for every t ě 0
+Er|Mγ
+t ´ Mγ
+t,ε|ps ď CpErxMγ ´ Mγ
+¨,εyp{2
+8 ` |Mγ
+0 ´ Mγ
+0,ε|ps.
+(4.17)
+The reader can then check by an explicit calculation of the second moment that
+lim
+εÑ0 E
+”
+|Mγ
+0 pfq ´ Mγ
+0,εpfq|pı
+ď lim
+εÑ0 E
+”
+|Mγ
+0 pfq ´ Mγ
+0,εpfq|2ıp{2
+“ 0
+(4.18)
+Hence in view of (4.17)-(4.18), to prove (4.13) we need to show that
+lim
+εÑ0 ErxMγ ´ Mγ
+¨,εyp{2
+8 s “ 0.
+(4.19)
+Expanding the product, using Itˆo calculus (Re denotes the real part) we obtain
+xMγ ´ Mγ
+¨,εy8 “ |γ|2
+ż 8
+0
+´
+At ´ 2Re
+´
+Ap1q
+t,ε
+¯
+` Ap2q
+t,ε
+¯
+dt.
+(4.20)
+where, At is defined in (4.8), and recalling (3.5), Ap1q
+t,ε and Ap2q
+t,ε are defined by
+Ap1q
+t,ε :“
+ż
+R2d fpxqfpyqQt,ε,0px, yqeγXtpxq`γXt,εpyq´ γ2
+2 Ktpxq´ γ2
+2 Kt,εpyqdxdy,
+Ap2q
+t,ε :“
+ż
+R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyqdxdy.
+(4.21)
+We are going to reduce the proof of (4.19) to that of two convergence statements concerning
+Apiq
+t,ε for i P t1, 2u (the first being valid for any fixed r ą 0)
+lim
+εÑ0 sup
+tPr0,rs
+E
+”
+|At ´ Apiq
+t,ε|
+ı
+“ 0
+for i P t1, 2u.
+(4.22)
+lim
+rÑ8 sup
+εPp0,1s
+E
+«ˆż 8
+r
+|Apiq
+t,ε|dt
+˙p{2ff
+“ 0
+for i P t1, 2u.
+(4.23)
+Before proving (4.22)-(4.23) let us explain how (4.19) is deduced from it. Note that (4.23)
+is also valid for At (this can be extracted from the proof in Section 4.1). Given δ ą 0,
+
+16
+HUBERT LACOIN
+using subadditivity (4.9), and (4.23) we can find rδ such that for every ε ą 0
+E
+«ˆż 8
+rδ
+´
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+¯
+dt
+˙p{2ff
+ď E
+«ˆż 8
+rδ
+Atdt
+˙p{2
+`
+ˆż 8
+rδ
+2|Ap1q
+t,ε |dt
+˙p{2
+`
+ˆż 8
+rδ
+Ap2q
+t,ε dt
+˙p{2ff
+ď δ{2.
+(4.24)
+Now using first Jensen’s inequality and then (4.22) (recall that At is real valued) we can
+find εδ such that for every ε P p0, εδq
+E
+«ˆż rδ
+0
+´
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+¯
+ds
+˙p{2ff
+ď
+ˆż rδ
+0
+E
+”
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+ı
+ds
+˙p{2
+ď
+ˆż rδ
+0
+E
+”
+2|At ´ Ap1q
+t,ε | ` |Ap2q
+t,ε ´ At|
+ı
+ds
+˙p{2
+ď δ{2.
+(4.25)
+Using subadditivity again we deduce from (4.24)-(4.25) that if ε P p0, εδq we have
+E
+«ˆż rδ
+0
+´
+At ´ 2Re
+`
+Ap1q
+t,ε
+˘
+` Ap2q
+t,ε
+¯
+dt
+˙p{2ff
+ď δ,
+(4.26)
+which (recalling (4.20)) concludes the proof of (4.19). Let us now prove (4.22)-(4.23).
+The proof (4.22) follows from a rather pedestrian but rather cumbersome computation
+of the L2 norm of pAt ´ Apiq
+t,εq. The following lemma summarizes the key points of this
+computation.
+Lemma 4.3. Consider the following:
+‚ Let pX, µq be a measured space and T be a set of indices.
+‚ Let Zt,εp¨q, t P T , ε P p0, 1s be a collection of complex valued Gaussian processes
+defined on X. We set
+Gt,εpx, yq :“ ErZt,εpxqZt,εpyqs
+and
+Ht,εpx, yq :“ ErZt,εpxqZt,εpyqs.
+(4.27)
+‚ Let Zt be defined on the same probability space in such a way that pZt, Zt,εq is
+jointly Gaussian. We let Gt and Ht be defined as in (4.27) and set
+Ht,ε,0px, yq :“ ErZt,εpxqZtpyqs.
+‚ Let gt,ε and gt be deterministic functions X Ñ R.
+We assume that:
+(i) The covariance functions are uniformly bounded, that is
+sup
+tPT
+εPp0,1s
+sup
+x,yPX
+max pHt,εpx, yq, Htpx, yq, Ht,ε,0px, yqq ă 8.
+(ii) There exists a µ-integrable function h such that for every t P T and ε P p0, 1s
+@x P X,
+maxp|gt,εpxq|, |gtpxq|q ď hpxq
+(iii) That for every t P T , we have the following pointwise convergence
+lim
+εÑ0 gt,ε “ gt,
+and
+lim
+εÑ0 Ht,ε “ lim
+εÑ0 Ht,ε,0 “ Ht
+(4.28)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES17
+Then setting
+Wt,ε :“
+ż
+X
+gt,εpxqeZt,εpxq´ 1
+2 Gt,εpxqµpdxq
+and
+Wt :“
+ż
+X
+gtpxqeZtpxq´ 1
+2Gtpx,xqµpdxq.
+We have
+lim
+εÑ0 sup
+tPT
+E
+“
+|Wε ´ Wt,ε|2‰
+“ 0.
+(4.29)
+Proof of Lemma 4.3. The proof is actually much shorter than the statement. We have
+E
+“
+|Wt,ε ´ Wt|2‰
+“
+ż
+X 2
+ˆ
+gt,εpxqgt,εpyqeHt,εpx,yq ´ 2Re
+´
+gt,εpxqgtpyqeHt,ε,0px,yq¯
+` gtpxqgtpyqeHtpx,yq
+˙
+µpdxqµpdyq
+(4.30)
+and using our assumptions we can apply dominated convergence.
+□
+Proof of (4.22). We consider the case i “ 2 but the other one is identical. We set X “ R2d,
+µ is Lebesgue measure, T “ r0, rs, and
+Zt,εpx, yq “ γXt,εpxq ` γXt,εpyq,
+Ztpx, yq “ γXtpxq ` γXtpyq,
+gt,εpx, yq “ Qt,εpx, yqfpxqfpyqe|γ|2Kt,εpx,yq,
+gtpx, yq “ Qtpx, yqfpxqfpyqe|γ|2Ktpxq.
+Then the assumptions of Lemma 4.3 are immediate to check.
+□
+We now provide the details for the proof of (4.23) i “ 2 (the case i “ 1 is similar). Let
+us set n0pεq “ rlogp1{εqs and assume (without loss of generality) that r is an integer and
+is smaller than n0. Using - as in the proof of Proposition 4.2 - subadditivity (4.9) and
+Jensen’s inequality we obtain
+E
+«ˆż 8
+r
+|Ap2q
+t,ε |ds
+˙p{2ff
+ď
+n0
+ÿ
+n“r
+E
+«ˆż n`1
+n
+|Ap2q
+t,ε |dt
+˙p{2ff
+ď
+n0´1
+ÿ
+n“r
+E
+«ˆż n`1
+n
+E
+”
+|Ap2q
+t,ε | | Fn
+ı
+dt
+˙p{2ff
+` E
+«ˆż 8
+n0
+E
+”
+|Ap2q
+t,ε | | Fn0
+ı
+dt
+˙p{2ff
+.
+(4.31)
+Proceeding as in (4.11), we obtain that if t P rn, n ` 1q, n P �r, n0 ´ 1�, or t ě n0, n “ n0,
+we have (using Lemma 3.1 to replace the covariance Kn,εpxq by n)
+E
+”
+|Ap2q
+t,ε | | Fn
+ı
+ď C
+ż
+R2d |fpxq|2Qs,εpx, yqe2αpXn,εpxq´
+?
+2dnq`2dndxdy.
+(4.32)
+Using (3.13) to integrate over y and setting
+Bp2q
+n,ε :“
+ż
+Rd |fpxq|2e2αpXn,εpxq´
+?
+2dnq`dndx
+(4.33)
+we obtain that
+E
+«ˆż 8
+r
+|Ap2q
+t,ε |ds
+˙p{2ff
+ď C
+n0
+ÿ
+n“r
+E
+”
+pBp2q
+n,εqp{2ı
+.
+(4.34)
+
+18
+HUBERT LACOIN
+Using Jensen’s inequality for the probability θεpy ´ xqdy, we can replace the mollification
+acting on Xn in the exponential by one acting of |f|2, we have
+e2αXn,εpxq ď
+ż
+Rd θεpx ´ yqe2αXnpyqdy
+which after multiplying by |fpxq|2 and integrating with respect to x implies that
+Bp2q
+n,ε ď
+ż
+D
+`
+|f|2 ˚ θε
+˘
+pyqe2αpXnpyq´
+?
+2dnq`dndy.
+(4.35)
+Since |f|2˚θε ď }f}2
+81t|x|ďR`1u if f is supported in Bp0, Rq, we can conclude using Lemma
+4.1, that
+E
+”
+pBp2q
+n,εqp{2ı
+ď Cn´ 3αp
+?
+8d
+for a constant which does not depend on ε. Recalling that p ą
+?
+8d{3α (cf. (4.1)) we
+obtain combining(4.32), (4.34) and (4.35) that
+E
+«ˆż 8
+r
+|Ap2q
+t,ε |ds
+˙p{2ff
+ď Cr1´ 3αp
+2
+?
+2d .
+(4.36)
+This concludes the proof of (4.23), and thus of Proposition 4.2.
+□
+5. Proof of Proposition 3.5
+5.1. Reduction to a statement concerning the total variation. Using [16, Theorem
+2.5] (which is a simpler version of Theorem 3.6 displayed above) we can reduce the proof
+of (3.19) to the following convergence statement about the quadratic variation of the
+martingale.
+Proposition 5.1. We have the following
+lim
+tÑ8 vpt, γq´2xMγpf, ωqyt “ M1pe|γ|2L|f|2q.
+(5.1)
+Proof of Proposition 3.5 from Proposition 5.1. We simply apply [16, Theorem 2.5] to the
+martingale Mγ
+t pf, ωq.
+□
+Setting, for notational simplicity Wt :“ Mγ
+t pfq. Recall that for a complex value mar-
+tingale such as Wt we use the notation xWyt for the bracket between W and its conjugate.
+Using bilinearity of the martingale brackets we have
+xMγpf, ωqyt “ 1
+2
+`
+xWyt ` Repe´2iωxW, Wytq
+˘
+(5.2)
+Hence to prove (5.1), it is sufficient to prove that following convergences hold in probability.
+lim
+tÑ8 vpt, γq´2xWyt “ 2M1pe|γ|2L|f|2q,
+lim
+tÑ8 vpt, γq´2xW, Wyt “ 0.
+(5.3)
+The expression for the bracket of Wt can be obtained by using Itˆo calculus (recall (4.4))
+More precisely we have
+xWyt “ |γ|2
+ż t
+0
+Asds
+and
+xW, Wyt “ γ2
+ż t
+0
+Bsds,
+(5.4)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES19
+where At is defined in (4.5) and
+Bt :“
+ż
+R2d fpxqfpyqQtpx, yqeγpXtpxq`Xtpyqq´ γ2
+2 pKtpxq`Ktpyqqdxdy.
+(5.5)
+Now using (5.4) our first idea is to deduce (5.3) from a convergence statement concerning
+At and Bt. A really important point here is that while At, properly rescaled, converges
+to M1pe|γ|2Lfq in probability, this type of convergence is not sufficient to say something
+about the integral
+şt
+0 Asds. A convenient framework to work with integrals is L1 conver-
+gence, but the issue we encounter is that At certainly does not converge in L1 (we have
+E
+”
+|M1pe|γ|2Lfq|
+ı
+“ 8 when f is non trivial).
+To bypass this problem, restrict ourselves to likely family of event and prove L1 convergence
+for the restriction. Recalling the definition of X (3.4), given q ě 0 and R ą 0, t ě 0 and
+x P Rd we introduce the events
+At,qpxq :“
+"
+max
+sPr0,tspXspxq ´
+?
+2dsq ă q
+*
+,
+Aq,R :“
+#
+sup
+sě0,|x|ďR
+pXspxq ´
+?
+2dtq ă q
++
+“
+č
+xPBp0,Rq
+tě0
+At,qpxq.
+(5.6)
+A very important fact, which is a direct consequence of [4, Proposition 19] (see also [17,
+Proposition 2.4] for a concise proof).
+Lemma 5.2. We have for any fixed R ą 0
+lim
+qÑ8 P rAq,Rs “ 1
+(5.7)
+We introduce (we drop the dependence in γ in most displays to make them easier to read)
+φptq “ φpt, γq :“
+d
+2
+πpt _ 1qe|γ2|j
+ˆż
+Rd Qtp0, zqe|γ2|Ktp0,zqdz
+˙
+,
+(5.8)
+which plays the role of a rescaling function for At. Our main technical result in this section
+is the proof that At{φptq converges in L1 towards M1pe|γ|2L|f|2q after restriction to the
+event Aq,R.
+Proposition 5.3. The following convergences hold for any q ě 0 and any R such that
+Supppfq Ă Bp0, Rq
+lim
+tÑ8 E
+”
+|At{φptq ´ M1pe|γ|2L|f|2q|1Aq,R
+ı
+“ 0,
+(5.9)
+lim
+tÑ8 E
+“
+|Bt{φptq| 1Aq,R
+‰
+“ 0,
+(5.10)
+and the above quantities are finite for every t ě 0.
+To show that Proposition 5.3 implies the convergence stated in Proposition 5.1, we need
+to ensure that the rescaling by φptq matches that proposed for xWyt (which is vpt, γq2)
+after integrating with respect to time. This is the purpose of the following lemma.
+Lemma 5.4. We have for any |γ| ě d
+lim
+tÑ8
+|γ|2 şt
+0 φpsqds
+2vpt, γq2
+“ 1.
+(5.11)
+
+20
+HUBERT LACOIN
+The proof of Lemma 5.4 is presented in Appendix A.3.
+Note that the goal of the
+lemma is only to obtain a more presentable expression for vpt, γq since without it, we can
+still prove that Proposition 5.1 and hence Proposition 3.5 are valid with v replaced by
+vpt, γq :“ |γ|
+b
+p
+şt
+0 φpsqdsq{2.
+Proof of Proposition 5.1. As we have seen, it is sufficient to prove (5.3). We provide the
+details concerning the convergence of xWyt (the first line in (5.3)) but that of xW, Wyt can
+be obtained exactly in the same manner. Using (5.4) and Jensen’s inequality we have
+E
+«ˇˇˇˇˇ
+xWyt
+|γ|2 şt
+0 φpsqds
+´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+ď
+şt
+0 φpsqE
+”ˇˇˇ As
+φpsq ´ M1pe|γ|2L|f|2q
+ˇˇˇ 1Aq,R
+ı
+ds
+şt
+0 φpsqds
+.
+(5.12)
+Observing that
+ş8
+0 φpsqds “ 8, the r.h.s. of (5.12) is simply a weighted Cesaro mean and
+thus we deduce from Proposition 5.3 and more precisely from (5.9) that
+lim
+tÑ8 E
+«ˇˇˇˇˇ
+xWyt
+|γ|2 şt
+0 φpsqds
+´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0
+(5.13)
+Since this holds for every q ą 0 we obtain that the following convergence holds in proba-
+bility (the replacement of |γ|2 şt
+0 φpsqds by 2vpt, γq2 simply comes from Lemma 5.2) that
+lim
+tÑ0
+ˇˇˇˇ
+xWyt
+2vpt, γq2 ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Ť
+qě1 Aq,R “ 0
+which, since the event in the indicator has probability one (cf. Lemma 5.2) is the desired
+conclusion.
+□
+5.2. Restricted convergence in L2 for the critical GMC. Before starting the proof
+of Proposition 5.3, we recall a result which play a key role in the proof, the L2 convergence
+of M
+?
+2d
+t
+pgq towards M1pgq when considering the restriction to the event Aq,R. This also
+implies convergence in L1 which is what we require for the proof of Proposition 5.3. The
+result can be deduced from the L2 convergence of the truncated version of M
+?
+2d
+t
+pgq which
+is proved in [17].
+Lemma 5.5. We have for any g in CcpRdq such that Supppgq Ă Bp0, Rq and any q ą 0
+lim
+tÑ8 E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d
+t
+pgq ´ M1pgq
+ˇˇˇˇˇ
+2
+1Aq,R
+fi
+fl “ 0,
+(5.14)
+and E
+“
+|M1pgq|21Aq,R
+‰
+ă 8.
+Proof. The fact that E
+“
+|M1pgq|21Aq,R
+‰
+ă 8 is a simple consequence of the convergence
+since for any fixed t, Er|M
+?
+2d
+t
+pgq|2s ă 8. We set (recall (5.6))
+M
+?
+2d,pqq
+t
+pgq :“
+ż
+gpxqe
+?
+2dXtpxq´dKtpxq1At,qpxqdx,
+(5.15)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES21
+From [17, Proposition 4.1], there exists an L2 variable D
+pqq
+8 pgq such that
+lim
+tÑ8 E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d,pqq
+t
+pgq ´ D
+pqq
+8 pgq
+ˇˇˇˇˇ
+2fi
+fl “ 0.
+(5.16)
+It satisfies D
+pqq
+8 pgq “ M1pgq on the event Aq,R. More precisely [17, Proposition 4.1] is only
+stated in the special case where g is an indicator function (to keep notation light) but
+the proof for g P CcpRdq is identical. On the event Aq,R we have M
+?
+2d,pqq
+t
+pgq “ M
+?
+2d
+t
+pgq.
+Hence
+lim sup
+tÑ8
+E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d
+t
+pgq ´ M1pgq
+ˇˇˇˇˇ
+2
+1Aq,R
+fi
+fl
+“ lim sup
+tÑ8
+E
+»
+–
+ˇˇˇˇˇ
+c
+πt
+2 M
+?
+2d,pqq
+t
+pfq ´ D
+pqq
+8 pgq
+ˇˇˇˇˇ
+2
+1Aq,R
+fi
+fl “ 0.
+(5.17)
+where the last equality follows from (5.16).
+□
+5.3. Organizing the proof of Proposition 5.3. The two convergences rely on similar
+ideas, we focus on (5.9) which is the more delicate of the two. The main idea is that since
+the integrand in the definition of At
+At :“
+ż
+R2d fpxqfpyqQtpx, yqeγXtpxq`γXtpyq´ γ2
+2 Ktpxq´ γ2
+2 Ktpyqdxdy,
+vanishes when |x ´ y| ě e´t (due to the presence of the multiplicative Qtpx, yq), the value
+of the integral should not be much affected much if one changes fpyq, Xtpyq and Ktpyq by
+fpxq, Xtpxq and Ktpxq in the expression.
+The quantity obtained after this replacement is, up to a multiplicative factor, of the
+form M
+?
+2d
+t
+pgq (recall that γ ` γ “
+?
+2d) for some function g. Hence we should be able to
+conclude the proof of the convergence statement using Lemma 5.5.
+While this idea is relatively simple, it requires several steps to be implemented. We set
+K˚
+t px, yq :“ K0pxq ` Ktpx, yq
+and
+r “ rptq :“ t ´ log log t
+(5.18)
+(we are assuming that t ą e so that 0 ď r ď t). We introduce the quantity rAt which will
+appear after all our “replacement” steps have been performed, it is defined by
+rAt :“
+ż
+R2d Qtpx, yqe|γ|2K˚
+t px,yq|fpxq|2e
+?
+2dXrpxq´dKrpxqdxdy
+“
+ˆż
+Rd Qtp0, zqe|γ|2Ktp0,zqdz
+˙ ż
+Rd e|γ|2K0ptq|fpxq|2e
+?
+2dXrpxq´dKrpxqdx
+“ φptq
+c
+πt
+2
+ż
+Rd e|γ|2Lpxq|fpxq|2e
+?
+2dXrpxq´dKrpxqdx “ φptq
+c
+πt
+2 M
+?
+2d
+r
+pe|γ|2L|f|2q.
+(5.19)
+As a direct consequence of Lemma 5.5 (since r “ t ´ optq the presence of
+?
+t instead of ?r
+does not affect the convergence), we have
+lim
+tÑ8 E
+«ˇˇˇˇˇ
+rAt
+φptq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0.
+(5.20)
+
+22
+HUBERT LACOIN
+With this observation the proof of (5.9) reduces to showing that
+lim
+tÑ0
+1
+φptqE
+”
+|At ´ rAt|1Aq,R
+ı
+“ 0.
+(5.21)
+This requires some care but before going in the depth of the proof, let us explain the
+heuristic behind (5.21). Note that rAt is obtained from At with two simple modifications:
+‚ We have replaced fpxqfpyq by |fpxq|2.
+‚ In the exponential, we have replaced γXtpxq`γXtpyq by
+?
+2dXrpxq “ pγ`γqXrpxq
+and ´ γ2
+2 Ktpxq ´ γ2
+2 Ktpyq by ´dKrpxq ` |γ|2K˚
+t px, yq.
+The first modification is rather straightfoward, we are integrating close to the diagonal so
+that fpyq is close to fpxq. For the second modification, the idea is that replacing Xtpyq
+with Xtpxq (and t with r) should not yield big modifications provided that we change the
+normalization to keep the expectation of the exponential unchanged (or almost so). In
+our case we have
+E
+„
+eγXtpxq`γXtpyq´ γ2
+2 Ktpxq´ γ2
+2 Ktpyq
+
+“ e|γ|2Ktpx,yq,
+E
+”
+e
+?
+2dXrpxq´dKrpxq`|γ|2K˚
+t px,yqı
+“ e|γ|2K˚
+t px,yq.
+(5.22)
+and, on the considered domain of integration, K˚
+t px, yq and Ktpx, yq are very close since
+|x ´ y| ď e´t when Qtpx, yq ‰ 0. The proof of (5.21) requires three distinct steps which
+are detailed in the next subsection.
+5.4. The proof of (5.21).
+Step 1: Changing the deterministic prefactor in the integrand. The integrand of At and
+rAt have different expectations. Our first step aims to fix this by replacing fpyq by fpxq in
+At and doing a small modification in the exponential factor. We set
+Ap1q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq` γ2
+2 Ktpxq` γ2
+2 Ktpyq`|γ|2pK0pxq´K0px,yqqdxdy (5.23)
+We are going to prove that
+lim
+tÑ8 φptq´1E
+”
+|At ´ Ap1q
+t |1Aq,R
+ı
+“ 0
+(5.24)
+Since f and K0 are uniformly continuous on the support of f and Supppfq Ă Bp0, Rq,
+there exists a positive function δ with limtÑ8 δptq “ 0, such that for |x ´ y| ď e´t setting
+Fpx, yq :“ fpxqfpyq ´ |fpxq|2e|γ|2pK0pxq´K0px,yqq
+we have
+|Fpx, yq| ď δptq1Bp0,Rqpxq1Bp0,Rqpyq
+(5.25)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES23
+Hence we obtain (since α “
+a
+d{2, we have Repγ2q “ d ´ |γ|2)
+|At ´ Ap1q
+t | “
+ˇˇˇˇ
+ż
+R2d Qtpx, yqFpx, yqeγXtpxq`γXtpyq` γ2
+2 Ktpxq` γ2
+2 Ktpyqdxdy
+ˇˇˇˇ
+ď
+ż
+R2d Qtpx, yq|Fpx, yq|e
+b
+d
+2 pXtpxq`Xtpyqq`p|γ|2´dq Ktpxq`Ktpyq
+2
+dxdy
+ď δptq
+ż
+Bp0,Rq2 Qtpx, yqe
+b
+d
+2 pXtpxq`Xtpyqq`p|γ|2´dq Ktpxq`Ktpyq
+2
+dxdy
+ď δptq
+ż
+Bp0,Rq2 Qtpx, yqe
+?
+2dXtpxq`p|γ|2´dqKtpxqdxdy
+(5.26)
+where the first inequality is simply obtained by taking the modulus of the integrand and
+in the third one we simply used
+ZpxqZpyq ď 1
+2pZpxq2 ` Zpyq2q
+with Zpxq “ e
+b
+d
+2 Xtpxq`p|γ|2´dq Ktpxq
+2
+and then symmetry in x and y. Then we observe that
+(λ denotes the Lebesgue measure) since At,qpxq Ă Aq,R we have
+E
+”
+e
+?
+2dXtpxq´dKtpxq1Aq,R
+ı
+ď E
+”
+e
+?
+2dXpxq´dKtpxq1At,qpxq
+ı
+“ Pr@s P r0, ts, Bs ď qs ď
+c
+2
+πtq
+(5.27)
+where in the last line, we used Cameron-Martin formula (see Proposition A.1 in the ap-
+pendix) and the fact that pXtpxqqtě0 is a standard Brownian Motion. The last inequality
+is simply Lemma A.2. Combining (5.26) and (5.27) and the fact that K0 is bounded, we
+have
+E
+”
+|At ´ Ap1q
+t |1Aq,R
+ı
+ď Cδptq
+?
+t
+ż
+Bp0,Rq2 Qtpx, yqe|γ|2Ktpxqdxdy ď C1δptqφptq.
+(5.28)
+□
+Step 2: Taking conditional expectation. Recalling the definition of rptq (5.18) we set
+Ap2q
+t
+:“ ErAp1q
+t
+| Frs
+(5.29)
+For this step of the proof (and only this step), we are going to assume that K0 ” 0 (and
+hence X0 ” 0q. Treating the case where X0 is a non-trivial field does not present any
+extra difficulty besides the challenge of making the equations fit within the margins. This
+assumption allows to replace Ktpxq and Ktpyq by t, and we get the following simplification
+for the expression of Ap1q
+t .
+Ap1q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq`p|γ|2´dqtdxdy
+(5.30)
+Then we have
+Ap2q
+t
+“
+ż
+R2d |fpxq|2Qtpx, yqeγXrpxq`γXrpyq`p|γ|2´dqr`|γ|2Krr,tspx,yqdxdy,
+(5.31)
+
+24
+HUBERT LACOIN
+where Krr,ts “ Kt ´ Kr (in the remainder of the paper, we use this convention for other
+quantities indexed by t). We are going to show that
+lim
+tÑ8 φptq´1E
+”
+|Ap1q
+t
+´ Ap2q
+t |1Aq,R
+ı
+“ 0
+(5.32)
+Recalling (5.6) we define
+A
+p1q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq`p|γ|2´dqt1Ar,qpxqdxdy,
+A
+p2q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqeγXrpxq`γXrpyq`p|γ|2´dqr`|γ|2Krr,tspx,yq1Ar,qpxqdxdy.
+(5.33)
+Since on Aq,R, Apiq
+t
+and A
+piq
+t
+coincide, We have
+E
+”
+pAp2q
+t
+´ Ap1q
+t q21Aq,R
+ı
+ď E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+(5.34)
+and thus we can prove that (5.32) holds by showing that
+lim
+tÑ8 φptq´2E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+“ 0.
+(5.35)
+To bound ErpA
+p2q
+t
+´ A
+p1q
+t q2s we expand the square, making it an integral on R4d. We set
+ξpx, yq :“ |fpxq|2Qtpx, yqe´p|γ|2´dqt ´
+eγXspxq`γXspyq ´ E
+”
+eγXspxq`γXspyqq | Fr
+ı¯
+1Ar,qpxq.
+We have
+E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+“
+ż
+R4d E
+“
+ξpx1, y1qξpx2, y2q
+‰
+dx1dy1dx2dy2.
+(5.36)
+As the range of correlation of the increment field Xrr,ts :“ Xt ´ Xr is smaller that e´r
+have, whenever |x1 ´ x2| ě 3e´r
+E
+“
+ξpx1, y1qξpx2, y2q | Fr
+‰
+“ 0.
+(5.37)
+Hence we only need to integrate the r.h.s. of (5.36) on the set |x1 ´ x2| ď 3e´r. In that
+case we use
+E
+“
+ξpx1, y1qξpx2, y2q
+‰
+ď E
+“
+|ξpx1, y1q|2‰1{2 E
+“
+|ξpx2, y2q|2‰1{2 .
+(5.38)
+and
+E
+“
+|ξpx, yq|2‰
+“ |fpxq|4Qtpx, yq2e2p|γ|2´dqtE
+”
+e
+?
+2dpXtpxq`Xtpyqq1Ar,qpxq
+ı
+(5.39)
+Using Cameron-Martin formula and the fact that pXtpxqqtě0 is a standard Brownian
+motion we have
+E
+”
+e
+?
+2dpXtpxq`Xtpyqq1Ar,qpxq
+ı
+“ e2dpt`Ktpx,yqqP r@u P r0, rs, Bu ď q ´ Kupx, yqs .
+Using (3.12) (and then Lemma A.2) we obtain for a constant q1 ą q
+e´4dtE
+”
+e
+?
+2dpXtpxq`Xtpyqq1Aq,rpxq
+ı
+ď P
+”
+@u P r0, rs, Bu ď q1 ´
+?
+2du
+ı
+ď Cr´3{2e´dr.
+Altogether , setting hpx, y, tq :“ 1t|x1´x2|ď3e´ru|fpx1qfpx2q|2Qtpx1, y1qQtpx2, y2q (recall
+that r is a function of t) we obtain that for t sufficiently large
+E
+”
+pA
+p2q
+t
+´ A
+p1q
+t q2ı
+ď Cr´3{2e2p|γ|2`dqt´dr
+ż
+R4d hpx, y, tqdxdy
+ď C1t´3{2e2|γ|2t´2dr ď C2t´1{2e2dpt´rqφptq2 ď t´1{4φptq2.
+(5.40)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES25
+To get the second inequality, simply observe that h is smaller than a constant times the
+indicator of the set t|x1| ď R, |x2 ´ x1| ď 3e´r, |yi ´ xi| ď e´t, i “ 1, 2u, which has volume
+of order e´dpr`2tq. The third inequality is a consequence of (A.10) (see the computation
+in the Appendix, while the last inequality follows from the the fact that with our choice
+of parameters (5.18) we have t ´ r “ oplog tq.
+Step 3: Comparing Ap2q
+t
+and rAt. Finally, we show that
+lim
+tÑ8 φptq´1E
+”
+|Ap2q
+t
+´ rAt|1Aq,R
+ı
+“ 0
+(5.41)
+which together with (5.24)-(5.32), concludes the proof of (5.21). We introduce another
+smaller time parameter, namely r “ t{2 and define p
+Xpx, yq “ Xrpxq ` Xrr,rspyq. We want
+to replace Xrpyq by Xrpxq in the exponential with an intermediate steps, so we set
+Z1pxq :“
+?
+2dXrpxq,
+Z2px, yq :“ γXrpxq ` γ p
+Xpx, yq,
+Z3px, yq :“ γXrpxq ` γXrpyq.
+(5.42)
+The reader can check that we have
+rAt :“
+ż
+R2d |fpxq|2Qtpx, yqe|γ|2K˚
+t px,yqeZ1pxq´ 1
+2ErZ1pxqsdxdy,
+Ap2q
+t
+:“
+ż
+R2d |fpxq|2Qtpx, yqe|γ|2K˚
+t px,yqeZ3px.yq´ 1
+2ErZ3px,yqsdxdy.
+(5.43)
+In order to prove (5.41) taking absolute value inside the integrand, we have
+E
+”
+|Ap2q
+t
+´ rAt|1Aq,R
+ı
+ď
+max
+|x|ďR
+|x´y|ďe´t
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+ˆ
+ż
+R2d |fpxq|2Qtpx, yqe|γ|2K˚
+t px,yqdxdy.
+(5.44)
+Since the integral is of order ep|γ|2´dqt (cf. (3.12)), which is the same order as φptq
+?
+t (cf.
+(A.10)), the estimate (5.41) boilds down to proving
+lim
+tÑ8
+?
+t
+max
+|x|ďR
+|x´y|ďe´t
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+“ 0.
+(5.45)
+To prove (5.45) we start with the decomposition
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+ď E
+„ˇˇˇeZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+ˇˇˇ1Ar,qpxq
+
+` E
+„ˇˇˇeZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+ˇˇˇ
+
+(5.46)
+
+26
+HUBERT LACOIN
+(this is just the triangle inequality and replacing Aq,R with a larger event Ar,qpxq) and
+show that each term is opt´1{2q. We start with the second one. From Lemma A.3, we have
+E
+„
+eZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+|
+
+ď C
+a
+Er|Z3 ´ Z2|2s
+“ C|γ|
+a
+ErpXrpxq ´ Xrpyqq2s ď C1e´ct,
+(5.47)
+where we have used that
+ErpXrpxq ´ Xrpyqq2s “ 2pr ´ Krpx, yqq ` pK0pxq ` K0pyq ´ 2K0px, yqq.
+The second part of the sum is smaller than |x ´ y|c since K0 is H¨older continuous and the
+first part is smaller than |x ´ y|2e2r (from (3.14)), both are exponentially small in t. For
+the first term in (5.46) we factorize the part that is Fr measureable and use independence
+to obtain
+E
+„
+|eZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+|1Aq,rpxq
+
+“ E
+”
+e
+?
+2dXrpxq´dKrpxq1Aq,rpxq
+ı
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1q2
+2s
+2
+|
+
+,
+(5.48)
+where Z1
+i “ Zi ´
+?
+2dXrpxq. Using Cameron-Martin formula and Lemma A.2, we have
+E
+”
+e
+?
+2dXrpxq´dKrpxq1Ar,qpxq
+ı
+“ P r@s P r0, rs, Bs ď qs ď
+c
+2
+rπq.
+(5.49)
+The factor r´1{2 is sufficient to cancel the
+?
+t in (5.45) and we just have to show that the
+second factor in (5.48) is small. From Lemma A.3 we have
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1
+2q2s
+2
+|
+
+ď
+b
+E r|Z1
+1 ´ Z1
+2|2s
+“ |γ|
+b
+E
+“
+|Xrr,rspxq ´ Xrr,rspyq|2‰
+ď Cer|x ´ y| ď Cer´t,
+(5.50)
+where the penultimate inequality can be deduced from (3.14). The combination of (5.47)-
+(5.49) and (5.50) concludes the proof of (5.45).
+□
+Bonus step: the case of Bt. To conclude let us sketch rapidly the proof of (5.10). We can
+repeat the argument of step 2 to show that
+lim
+tÑ8 φptq´2Er|Bt ´ ErBt | Frs|2s “ 0.
+(5.51)
+Then it is rather direct to check that
+lim
+tÑ8 φptq´1E
+“
+|ErBt | Frs|1Aq,R
+‰
+“ 0.
+(5.52)
+More precisely we have
+ErBt | Frs “
+ż
+R2d fpxqfpyqQtpx, yqeγpXrpxq`Xrpyqq` γ2
+2 p2Krr,tspx,yq´Krpxq´Krpyqqdxdy.
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES27
+Taking the absolute value of the integrand, using (3.12) to evaluate Kt and Krr,ts, then
+the inequality ab ď pa2 ` b2q{2 and symmetry, and finally (3.13)
+|ErBt | Frs| ď Cep|γ2|´dqp2r´tq
+ż
+R2d |fpxqfpyq|Qtpx, yqe
+?
+d{2pXrpxq`Xrpyqqdxdy
+ď Cep|γ2|´dqp2r´tq
+ż
+R2d |fpxq|2Qtpx, yqe
+?
+2dXrpxqdxdy
+ď C1ep|γ2|´dqp2r´tq´dt
+ż
+R2d |fpxq|2e
+?
+2dXrpxqdx
+(5.53)
+Hence we have
+E
+“
+|ErBt | Frs|1Aq,R
+‰
+ď ep|γ2|´dqp2r´tq´dt
+ż
+R2d |fpxq|2E
+”
+e
+?
+2dXrpxq1Ar,qpxq
+ı
+dx.
+(5.54)
+Using Cameron Martin formula, (3.12) and Lemma A.2 (recall that r „ t) we obtain that
+E
+”
+e
+?
+2dXrpxq1Ar,qpxq
+ı
+ď Ct´1{2edr.
+(5.55)
+Overall using (A.10) we have φptq´1E
+“
+|ErBt | Frs|1Aq,R
+‰
+ď Ce´|γ2|pt´rq.
+□
+6. Proof of Proposition 2.8
+6.1. Organization of the proof. Like for the proof of Theorem 2.1, we assume that our
+probability space contains a martingale approximation sequence pXtqtě0 of the field X,
+with covariance given by (3.3). For the same reason as the one exposed at the beginning
+of Section 4.2 this entails no loss of generality.
+The main idea is to apply Theorem 3.6 (for the filtration corresponding to pXtq) to the
+family Mγ
+ε pf, ωq with rate vpε, θ, γq and with the variable Z being equal to M1pe|γ|2L|f|2q.
+Hence need to check that the martingale Mγ
+t,εpf, ωq :“ E rMγ
+ε pf, ωq | Fts satisfy all the
+requirements in (3.22). Setting W pεq
+t
+:“ Mγ
+t,ε (recall (3.7)), and using the bilinearity of the
+martingale bracket like in (5.2) we obtain
+xMγ
+¨,εpf, ωqyt “ 1
+2
+´
+xW pεqyt ` Repe´2iωxW pεq, W pεqytq
+¯
+.
+(6.1)
+The requirements concerning the quadratic variation of Mγ
+t,εpf, ωq can be obtained as
+consequences of the following,
+Proposition 6.1. The following convergences hold
+lim
+εÑ0 E
+«ˇˇˇˇˇ
+xW pεqy8
+2vpε, θ, γq2 ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0,
+lim
+εÑ0 E
+«ˇˇˇˇˇ
+xW pεq, W pεqy8
+vpε, θ, γq2
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0.
+(6.2)
+Furthermore we have for any fixed t we have
+sup
+εPp0,1q
+ErxW pεqyts ă 8.
+(6.3)
+Proposition 6.1 is proved in the next subsection, let us first show how our main results
+can be deduced from it.
+
+28
+HUBERT LACOIN
+Proof of Proposition 2.8. We must check that the three requirements in (3.22) are satis-
+fied since the result follows then from Theorem 3.6. Given that limεÑ0 vpε, θ, γq “ 8,
+it is sufficient for the second and third requirements to show that that the sequences
+pMγ
+0,εpf, ωqqεPp0,1q, and pxMγ
+¨,εpf, ωqytqεPp0,1q (for a fixed t) are tight. The sequences are in
+fact uniformly bounded in L1. We have
+sup
+εPp0,1q
+Er|Mγ
+0,εpf, ωq|s ď sup
+εPp0,1q
+Er|Mγ
+0,εpfq|s ă 8.
+(6.4)
+Indeed taking the absolute value of the integrand, we have
+Er|Mγ
+0,εpfq|s ď
+ż
+Rd E
+„
+fpxqe
+?
+d{2X0,εpxq` β2´pd{2q
+2
+K0,εpxq
+
+dx “
+ż
+Rd fpxqeβ2K0,εpxqdx,
+(6.5)
+and the uniform bound follows from (3.12). From (6.1) we have xMγ
+¨,εpf, ωqyt ď xW pεqyt
+and thus the uniform boundedness in L1 is consequence of (6.3). Let us now turn to
+the first and main requirement in (3.22). The convergences in (6.2) imply the following
+convergence in probability
+lim
+εÑ0
+xW pεqy8
+2vpε, θ, γq2 1Ť
+qě1 Aq,R “ M1pe|γ|2L|f|2q,
+lim
+εÑ0
+xW pεq, W pεqy8
+vpε, θ, γq2
+1Ť
+qě1 Aq,R “ 0.
+(6.6)
+Using Lemma 5.2 and (6.1), we conclude that
+lim
+εÑ0 vpε, θ, γq´2xMγ
+¨,εpf, ωqy8 “ M1pe|γ|2L|f|2q
+in probability.
+□
+As another preliminary step to our proof, we reduce the convergence statement in
+Proposition 6.1 to a convergence of the derivative of the martingale brackets. Using Itˆo
+calculus we obtain that for T P r0, 8s,
+xW pεqyT “
+ż T
+0
+At,εdt and xW pεqyT “
+ż T
+0
+Bt,εdt
+where
+At,ε :“
+ż
+R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyqdxdy,
+Bt,ε :“
+ż
+R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 pKt,εpxq`Kt,εpyqqdxdy.
+(6.7)
+Similarly to what has been done in Proposition 5.3, we are going to show that, with ap-
+propriate renormalizations and restrictions, At,ε and Bt,ε converge in L1 to M1pe|γ|2L|f|2q
+and 0 respectively. To this end we introduce a couple of parameters (recall (3.5))
+tpt, εq :“ t ^ logp1{εq
+φpt, εq :“
+d
+2
+πpt _ 1qe|γ|2j
+ˆż
+Rd e|γ|2Kt,εp0,zqQt,εp0, zqdz
+˙
+.
+(6.8)
+The quantity r will on Our aim is to prove the following
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES29
+Proposition 6.2.
+lim
+εÑ0
+tÑ8
+E
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+“ 0,
+lim
+εÑ0
+tÑ8
+E
+„ˇˇˇˇ
+Bt,ε
+φpt, εq
+ˇˇˇˇ 1Aq,R
+
+“ 0,
+(6.9)
+and for any T ă 8
+sup
+tPr0,Ts
+εPp0,1q
+E r|At,ε|s ă 8.
+(6.10)
+Remark 6.3. Let us underline that lim εÑ0
+tÑ8 Fpt, εq “ 0 means that that there exists t0pδq
+and ε0pδq such that |Fpt, εq| ď δ when t ě t0 AND ε P p0, ε0q. This is a stronger statement
+than both limεÑ0 limtÑ8 Fpt, εq “ 0 or limtÑ8 limεÑ0 Fpt, εq “ 0
+Clearly (6.10) implies (6.3). To deduce (6.2) from (6.9), we need to check that renormal-
+izing factor 2vpε, θ, γq2 corresponds to the integral of φpt, εq. This is the content of the
+following lemma whose proof is presented in Appendix A.4.
+Lemma 6.4. We have for any |γ| ě d
+lim
+εÑ0
+|γ|2 ş8
+0 φpt, εqdt
+2vpε, θ, γq2
+“ 1
+(6.11)
+We can now complete the proof of Proposition 6.1 using Proposition 6.2
+Proof of Proposition 6.1. From Lemma 6.4 it is sufficient to prove the convergence of
+E
+«ˇˇˇˇˇ
+xW pεqy8
+ş8
+0 |γ|2φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+ď
+1
+ş8
+0 φpt, εqdt
+ż 8
+0
+φpt, εqE
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+dt.
+(6.12)
+Let us fix δ ą 0, and let T and ε0 be such that for all t ą T and ε ă ε0 we have
+E
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+ď δ
+2
+(6.13)
+In the integral we can distinguish the contribution from r0, Ts from the rest. We have
+from (6.10) and the fact that φpt, εq is bounded from below
+sup
+tPr0,Ts
+εPp0,ε0q
+E
+„ˇˇˇˇ
+At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇ 1Aq,R
+
+ă 8.
+(6.14)
+As a consequence, since
+ş8
+0 φpt, εqdt diverges when ε Ñ 0, taking ε1 sufficiently small we
+have forall ε P p0, ε1q
+1
+ş8
+0 φpt, εq
+ż T
+0
+φpt, εqE
+„ At,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+
+dt ď δ
+2,
+(6.15)
+
+30
+HUBERT LACOIN
+which implies that for ε ď ε0 ^ ε1 we have
+E
+«ˇˇˇˇˇ
+xW pεqy8
+ş8
+0 φpt, εqdt ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+ď δ
+(6.16)
+and thus conclude the proof.
+□
+6.2. Proof of Proposition 6.2. Let us start with the proof of (6.10). Noting that At,ε
+is positive, we have
+E rAt,εs “
+ż
+R2d fpxqfpyqQt,εpx, yqe|γ|2Kt,εpx,yqdxdy.
+(6.17)
+We can just use (3.12) and bound Qt,εpx, yq by 1 and Kt,εpx, yq by T `C to conclude. For
+the proof of the convergence of At,ε we proceed exactly as for the proof of Proposition 5.3.
+We assume that tpt, εq ą e (recall (6.8)) and set
+r “ rpt, εq :“ t ´ log log t,
+(6.18)
+Setting K˚
+t,εpx, yq :“ K0pxq ` Kt,εpx, yq, we define
+rAt,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqe|γ|2K˚
+t,εpx,yqe
+?
+2dXrpxq´2dKrpxqdxdy
+“ φpt, εqM
+?
+2d
+r
+pe|γ|2L|f|2q
+(6.19)
+Since lim εÑ0
+tÑ8 rpt, εq “ 8, we obtain, as a direct consequence of Lemma 5.5 that
+lim
+εÑ0
+tÑ8
+E
+«ˇˇˇˇˇ
+rAt,ε
+φpt, εq ´ M1pe|γ|2L|f|2q
+ˇˇˇˇˇ 1Aq,R
+ff
+“ 0.
+(6.20)
+Our task is thus to prove that
+lim
+εÑ0
+tÑ8
+φpt, εq´1E
+”
+| rAt,ε ´ At,ε|1Aq,R
+ı
+“ 0.
+(6.21)
+Like for the proof of (5.21) in the previous section, we proceed in three steps.
+Step 1: Changing the deterministic prefactor in the integrand. Set
+Ap1q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyq`|γ|2pK0pxq´K0,εpx,yqqdxdy
+Let us prove that
+lim
+εÑ0
+tÑ8
+φpt, εq´1E
+”
+|Ap1q
+t,ε ´ At,ε|1Aq,R
+ı
+“ 0.
+(6.22)
+Let As a direct consequence of the continuity of f and of K0, if one sets
+sup
+|x|,|y|ďR
+|x´y|ďet`2ε
+ˇˇˇfpxqfpyq ´ |fpxq|2e|γ|2pK0pxq´K0,εpx,yqqˇˇˇ “: δpε, tq,
+(6.23)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES31
+we have lim εÑ0
+tÑ8 δpε, tq “ 0. Since Qt,εpx, yq “ 0 when |x ´ y| ě et ` 2ε repeating the
+computation (5.26) and using (3.13) we get
+|Ap1q
+t,ε ´ At,ε| ď δpε, tq
+ż
+Bp0,Rq2 Qt,εpx, yqe
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxqdxdy
+ď Ce´dtδpε, tq
+ż
+Bp0,Rq
+e
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxqdx
+(6.24)
+Using Cameron-Martin formula, we obtain (assuming |x|, |y| ď R and |x ´ y| ď et ` 2ε)
+for a constant q1 ą q
+E
+”
+e
+?
+2dXt,εpxq´dKt,εpxq1Aq,R
+ı
+ď E
+”
+e
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxq
+ı
+“ P
+”
+@s P r0, ts, Bs ď
+?
+2dps ´ Ks,ε,0pxqq ` q
+ı
+ď Pp@s P r0, ts, Bs ď q1q ď Cpt _ 1q´1{2et|γ|2
+(6.25)
+where in the second inequality we have used (3.12) to estimate covariances. Hence, using
+(A.20) we deduce from (6.24) that
+E
+”
+|Ap1q
+t,ε ´ At,ε|1Aq,R
+ı
+ď Cδpε, tqt´1{2et|γ|2´dt ď C1δpε, tqφpt, εq.
+This concludes the proof of (6.22).
+□
+Step 2: Taking conditional expectation. We set
+Ap2q
+t,ε :“ E
+”
+Ap1q
+t,ε | Fr
+ı
+(6.26)
+and we are going to prove
+lim
+εÑ0
+tÑ8
+φpt, εq´2E
+”
+|Ap2q
+t,ε ´ Ap1q
+t,ε |21Aq,R
+ı
+“ 0.
+(6.27)
+Like what we did in the previous section, we assume here that K0 ” 0 to simplify the
+writing (but this does not affect the proof).
+In that case note that since Kt,εpxq “
+Kt,εpyq “ Kt,εpxq, we can factorize the term. We have (recall that Krr,ts,ε “ Kt,ε ´ Kr,ε)
+Ap2q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqeγXr,εpxq`γXr,εpyq`p|γ2|´dqKr,εpxq`|γ|2Krr,ts,εpx,yqdxdy
+(6.28)
+Setting
+ζpx, yq :“ eγXt,εpxq`γXt,εpyq`p|γ2|´dqKt,εpxq,
+A
+p1q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqζpx, yq1Ar,qpxqdxdy,
+A
+p2q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqErζpx, yq |Frs1Ar,qpxqdxdy
+(6.29)
+we realize that
+E
+”
+|Ap2q
+t,ε ´ Ap1q
+t,ε |21Aq,R
+ı
+ď E
+”
+|A
+p2q
+t,ε ´ A
+p1q
+t,ε |2ı
+(6.30)
+Now we set
+ξt,εpx, yq :“ |fpxq|2Qt,εpx, yq pζpx, yq ´ Erζpx, yq |Frsq 1Ar,qpxq.
+(6.31)
+
+32
+HUBERT LACOIN
+We have
+E
+”
+|A
+p2q
+t,ε ´ A
+p1q
+t,ε |2ı
+ď
+ż
+R4d E
+“
+ξpx1, y1qξpx2, y2q
+‰
+dx1dx2dy1dy2
+(6.32)
+The range of the convariance of Xrr,ts,ε is smaller than e´r ` 2ε, and Qt,εpx, yq vanishes
+when |x ´ y| ě e´t ` 2ε. All of this implies that if if |x1 ´ x2| ě 2e´r (if ε is sufficiently
+small, then e´r is much larger than both ε and e´t cf. (6.8)) then
+E
+“
+ξpx1, y1qξpx2, y2q | Fr
+‰
+“ 0.
+(6.33)
+When |x1 ´ x2| ď 2e´r we can use Cauchy-Schwarz to bound the covariance. We have
+E
+“
+|ξpx, yq|2‰
+ď |fpxq|4 pQt,εpx, yqq2 Er|ζpx, yq|21Ar,qpxqs
+(6.34)
+and from Cameron-Martin formula, we have, for |x ´ y| ď e´t ` 2ε
+Er|ζpx, yq|21Ar,qpxqs
+“ e2|γ|2Kt,εpxq`2dKt,εpx,yqP
+´
+@s P r0, rs, Bs ď
+?
+2dps ´ Ks,ε,0pxq ´ Ks,ε,0py, xqq ` q
+¯
+ď Cep|γ|2`dqtP
+´
+@s P r0, rs, Bs ď ´
+?
+2ds ` q1¯
+ď C1etp2|γ|2`dqr´3{2.
+where in the first inequality we used (3.12) which replace the Kt,ε and Ks,ε by t and
+s respectively at the cost of an additive constant and in the second inequality we used
+Lemma A.2. Altogether we obtain that
+E
+”
+|A
+p2q
+t,ε ´ A
+p1q
+t,ε |2ı
+ď Cetp2|γ|2`dqr´3{2
+ˆ
+ż
+R4d 1t|x1´x2|ď2e´ru|fpx1q|2|fpx2q|2Qt,εpx1, y1qQt,εpx2, y2qdxdy
+ď C1e2|γ|2t`dpt´rqr´3{2
+ˆż
+Rd Qt,εp0, zqdz
+˙2
+ď C2edpt´rqr´1{2φpt, εq2.
+(6.35)
+We conclude the proof of (6.27) by observing (recall (6.8)) that
+lim
+εÑ0
+tÑ8
+edpt´rqr´1{2 “ 0.
+(6.36)
+□
+Step 3: Final comparison. Finally to conclude we need to show that
+lim
+εÑ0
+tÑ8
+φpt, εq´2E
+”
+|Ap2q
+t,ε ´ rAt,ε|21Aq,R
+ı
+“ 0.
+(6.37)
+We set (recall that Xrs,ts,ε “ Xt,ε ´ Xs,ε´)
+Z1pxq :“
+?
+2dXrpxq,
+Z2px, yq :“
+?
+2dXrpxq ` γXrr,rs,εpxq ` γXrr,rs,εpyq,
+Z3px, yq :“ γXr,εpxq ` γXr,εpyq.
+(6.38)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES33
+The reader can check (using (6.26) rather than (6.28) since the latter assumes K0 ” 0)
+that
+rAt,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqe|γ|2K˚
+t,εpx,yqeZ1pxq´ 1
+2 ErZ1pxqsdxdy,
+Ap2q
+t,ε :“
+ż
+R2d |fpxq|2Qt,εpx, yqe|γ|2K˚
+t,εpx,yqeZ3px,yq´ 1
+2 ErZ3px,yqsdxdy.
+(6.39)
+In order to prove (6.37) taking absolute value inside the integrand using Jensen’s inequality,
+we just have to obtain a uniform bound on the integrand, that is, to show that
+lim
+tÑ8
+εÑ0
+t1{2
+max
+|x|ďR
+|x´y|ďe´t`2ε
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 pxqs
+2
+´ eZ3px,yq´
+ErZ2
+3 px,yqs
+2
+ˇˇˇ1Aq,R
+
+“ 0.
+(6.40)
+The restriction for x and y comes from the support of f and Qt,ε respectively. To prove
+(5.45) we start with the decomposition
+E
+„ˇˇˇeZ1pxq´
+ErZ2
+1 s
+2
+´ eZ3´
+ErZ2
+3 s
+2
+ˇˇˇ1Aq,R
+
+ď E
+„ˇˇˇeZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+ˇˇˇ1Ar,qpxq
+
+` E
+„ˇˇˇeZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+ˇˇˇ
+
+(6.41)
+(the inequality is just the triangle inequality and replacing Aq,R with a larger event), and
+show that each term is opt´1{2q. Let us start with the second one. Using Lemma A.3, we
+have
+E
+„
+eZ3´
+ErZ2
+3 s
+2
+´ eZ2
+2´
+ErZ2
+2 s
+2
+|
+
+ď C
+a
+Er|Z3 ´ Z2|2s
+“ C
+b
+Er|
+?
+2dXrpxq ´ γXr,εpxq ´ γXr,εpyq|2s
+ď C1
+ˆb
+E r|Xrpxq ´ Xr,εpxq|2s `
+b
+E r|Xrpxq ´ Xr,εpyq|2s
+˙
+“ C1 ´
+pKrpxq ` Kr,εpxq ´ 2Kr,ε,0pxqq1{2 ` pKrpxq ` Kr,εpxq ´ 2Kr,ε,0py, xqq1{2¯
+ď C2 pε ` |x ´ y|qc ď C1e´ct
+(6.42)
+where to obtain the last line we have used (3.14) and the H¨older continuity of K0. For the
+first term in (6.41) we factorize the Fr-measureable part and use independence to obtain
+E
+„
+|eZ1´
+ErZ2
+1 s
+2
+´ eZ2´ ErZ2s
+2
+|1Ar,qpxq
+
+“ E
+”
+e
+?
+2dXrpxq´dKrpxq1Ar,qpxq
+ı
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1q2
+2s
+2
+|
+
+,
+(6.43)
+where Z1
+i “ Zi ´
+?
+2dXrpxq. We have from Cameron-Martin Formula and Lemma A.2
+E
+”
+e
+?
+2dXrpxq´dKrpxq1Ar,qpxq
+ı
+“ P r@s P r0, rs, Bs ď qs ď Cr´1{2.
+(6.44)
+
+34
+HUBERT LACOIN
+This is obviously Opt´1{2q so to conclude we only need to show that the other factor in
+(6.43) goes to zero. We also have from Lemma A.3 and (3.14)
+E
+„
+|eZ1
+1´
+ErpZ1
+1q2s
+2
+´ eZ1
+2´
+ErpZ1q2
+2s
+2
+|
+
+ď C
+b
+E r|Z1
+1 ´ Z1
+2|2s
+ď C
+`
+Krr,rspxq ` Krr,rs,εpxq ` Krr,rs,εpyq ´ 2Krr,rs,ε,0pxq ´ 2Krr,rs,ε,0py, xq
+˘1{2
+ď C1erp|x ´ y| ` εq ď Cepr´tq.
+(6.45)
+This concludes the proof.
+□
+The convergence of Bt,ε. To prove the second convergence in (6.9), it is sufficient again to
+show first that
+lim
+tÑ8
+εÑ0
+ϕpt, εq´2E
+“
+|Bt,ε ´ ErBt,ε | Frs|21Aq,R
+‰
+“ 0
+(6.46)
+repeating the computation of step two, and then prove that
+lim
+tÑ8
+εÑ0
+Er|ErBt,ε | Frs|1Aq,Rs “ 0.
+We leave this part to the reader, since this is very similar to the computation performed
+at the end of Section 5.
+□
+7. Proof of Theorem 3.6
+We need to show that for any H bounded and F8-measurable and ξ P R we have
+lim
+nÑ8 E
+„
+H
+ˆ
+eiξ Wn
+vpnq ´ e´ ξ2Z
+2
+˙
+“ 0.
+(7.1)
+We first assume that the collection of variables vpnq´2xWny8 is uniformly essentially
+bounded, that is, that there exists M such that for every n ě 1
+P
+“
+vpnq´2xWny8 ě M
+‰
+“ 0
+(7.2)
+Note that this implies also that P rZ ě Ms “ 0. We assume, to simplify notation that
+ξ “ 1 (this entails no loss of generality). We set Ht :“ E rH | Fts and Zt :“ E rZ | Fts we
+have
+E
+„
+H
+ˆ
+ei ξWn
+vpnq ´ e´ ξ2Z
+2
+˙
+“ E
+”
+Hpe´ Z
+2 ´ e´ Zt
+2 q
+ı
+` E
+”
+pH ´ Htq
+´
+ei Wn
+vpnq ´ e´ Zt
+2
+¯ı
+` E
+”
+Ht
+´
+ei Wn
+vpnq ´ e´ Zt
+2
+¯ı
+“: E1pt, nq ` E2pt, nq ` E3pt, nq.
+(7.3)
+We prove the convergence (7.1) by showing that for i “ 1, 2, 3
+lim
+tÑ8 lim sup
+nÑ8 |Eipt, nq| “ 0.
+(7.4)
+Using the fact that z ÞÑ ez is 1-Lipshitz (first line) and has modulus bounded by 1 (second
+line) in tz P C : Repzq ď 0u we have
+|E1pt, nq| ď E
+”
+|H|
+ˇˇˇe´ Z
+2 ´ e´ Zt
+2
+ˇˇˇ
+ı
+ď }H}8
+2
+E r|Z ´ Zt|s ,
+|E2pt, nq| ď E
+”
+|H ´ Ht|
+ˇˇˇei Wn
+vpnq ´ e´ Z
+2
+ˇˇˇ
+ı
+ď 2E r|H ´ Ht|s .
+(7.5)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES35
+Since Ht and Zt converge respectively to H and Z in L1, (7.4) holds for i “ 1, 2. For
+i “ 3, we observe that for fixed t the process
+Mpnq
+u
+:“ e
+iWn,t`u´Wn,t
+vpnq
+`
+xWnyt`u´xWnyt
+2vpnq2
+´ Zt
+2
+is a martingale for the filtration pGuq :“ pFt`uq, which converges in L1 when u Ñ 8. In
+particular we have
+E
+„
+e
+iWn´Wn,t
+vpnq
+` xWny8´xWnyt
+2vpnq2
+| Ft
+
+“ 1.
+(7.6)
+Multiplying by Hte´ Zt
+2 and taking expectation we obtain that
+E
+„
+Hte
+iWn´Wn,t
+vpnq
+` xWny8´xWnyt
+2vpnq2
+´ Zt
+2
+
+“ E
+”
+Hte´ Zt
+2
+ı
+(7.7)
+Hence we have (using that }Ht}8 ď }H}8)
+|E3pt, nq| ď E
+„
+|Ht|
+ˇˇˇˇei Wn
+vpnq ´ e
+ipWn´Wn,tq
+vpnq
+` xWny8´xWnyt
+2vpnq2
+´ Zt
+2
+ˇˇˇˇ
+
+ď }H}8E
+„ˇˇˇˇ1 ´ e
+´
+iWn,t
+vpnq ` xWny8´xWnyt
+2vpnq2
+´ Zt
+2
+ˇˇˇˇ
+
+,
+(7.8)
+From (3.22) we have the following convergence in probability for any fixed t
+lim
+nÑ8 ´iWn,t
+vpnq ` xWny8 ´ xWnyt
+2vpnq2
+´ Zt
+2 “ Z ´ Zt
+2
+.
+(7.9)
+Using assumption (7.2), taking the limit in the r.h.s. of (7.8) and using dominated con-
+vergence, we obtain that
+lim sup
+nÑ8 |E3pt, nq| ď }H}8E
+”ˇˇˇ1 ´ e
+Z´Zt
+2
+ˇˇˇ
+ı
+.
+(7.10)
+Since Zt converges to Z we can conclude that (7.4) also holds for i “ 3 using dominated
+convergence again (both variables are uniformly bounded).
+Let us now remove the boundedness assumption. Given A ą 0 we set
+TA,n :“ inftt : vpnq´2xWnyt “ Au
+and
+W A
+n :“ Wn,TA,n.
+Note that E
+“
+W A
+n | Ft
+‰
+“ Wt^TA,n so that (using the notation xW A
+n yt to denote the qua-
+dratic variation of this martingale) we have
+lim
+nÑ8 vpnq´2xW A
+n y8 “ Z ^ A.
+(7.11)
+Since we have proved (7.1) under the assumption (7.2) we know that for every A ą 0
+lim
+nÑ8 E
+„
+H
+ˆ
+ei ξW A
+n
+vpnq ´ e´ ξ2pZ^Aq
+2
+˙
+“ 0
+(7.12)
+From the convergence assumption, we have
+lim sup
+nÑ8 PrTA,n “ 8s ď P rZ ě As
+(7.13)
+and hence
+lim
+AÑ8 lim inf
+nÑ8 P
+“
+W A
+n “ Wn
+‰
+“ lim
+AÑ8 PrZ ^ A “ Zs “ 1.
+
+36
+HUBERT LACOIN
+As a consequence we can conclude using (7.12) that
+lim
+nÑ8 E
+„
+H
+ˆ
+eiξ Wn
+vpnq ´ e´ ξ2Z
+2
+˙
+“ lim
+AÑ8 lim
+nÑ8 E
+„
+H
+ˆ
+ei ξW A
+n
+vpnq ´ e´ ξ2pZ^Aq
+2
+˙
+“ 0.
+(7.14)
+□
+Acknowledgements: This work was supported by a productivity grant from CNPq and
+a JCNE grant from FAPERJ.
+Appendix A. Technical results and their proof
+A.1. Standard Gaussian tools. We first display two standard tools which are used
+throughout the proof. The first is the standard Cameron-Martin formula which describes
+how a Gaussian process is affected by an exponential tilt.
+Proposition A.1. Let pY pzqqzPZ be a centered Gaussian field indexed by a set Z. We
+let H denote its covariance and P denote its law. Given z0 P Z let us define rPz0 the
+probability obtained from P after a tilt by Y pz0q that is
+drPz0
+dP :“ eY pz0q´ 1
+2 Hpz0,z0q
+(A.1)
+Under rPz0, Y is a Gaussian field with covariance H, and mean rEz0rY pzqs “ Hpz, z0q.
+The second is a bound on the probability for a Brownian Motion to remain below a line.
+Both estimates can be proved directly using the reflexion principle.
+Lemma A.2. Let B be a standard Brownian Motion and let P denote its distribution,
+setting gtpaq :“
+şu`
+0
+e´ z2
+2t dz. we have
+P
+«
+sup
+sPr0,ts
+Bs ď a
+ff
+“
+c
+2π
+t gtpaq ď
+c
+2π
+t a.
+(A.2)
+Additionally for any a, b ą 0 there exists Ca,b such that f
+P
+«
+sup
+sPr0,ts
+pBs ` bsq ď a
+ff
+“
+1
+?
+2πt
+ż
+e´ u2
+2t p1 ´ e
+2apa`u´bsq`
+t
+qdu ď Ca,be´ b2t
+2 t´3{2.
+(A.3)
+A.2. Comparing exponentiated Gaussians. In or comparison of partition functions
+Lemma A.3. Consider pX1, X2, Y1, Y2q an R4 valued centered Gaussian vector and set
+X :“ X1 ` iX2 and Y “ Y1 ` iY2. Assuming that
+ErX2
+2s ď 1 and Er|X ´ Y |2s ď 1
+(A.4)
+then there exits a constant C such that
+E
+”ˇˇeX´ 1
+2ErX2s ´ eY ´ 1
+2ErY 2sˇˇ
+ı
+ď CE
+“
+|X ´ Y |2‰
+(A.5)
+Proof. We factorize eX´ 1
+2ErX2s, use the Cameron-Martin formula and rearrange the ex-
+pectation terms in the exponential, we obtain
+E
+„ˇˇeY ´ ErY 2s
+2
+´ eX´ ErX2s
+2
+ˇˇ
+
+“ E
+„
+eX1`
+ErX2
+2 s´ErX2
+1 s
+2
+ˇˇeY ´X` ErX2s´ErY 2s
+2
+´ 1
+ˇˇ
+
+“ e
+ErX2
+2 s
+2
+E
+„ˇˇeY ´X´ ErpX´Y q2s
+2
+´iErX2pY ´Xqs ´ 1
+ˇˇ
+
+.
+(A.6)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES37
+The prefactor is bounded (by assumption) by e1{2. For the rest, setting Z “ Y ´ X (and
+letting Z1 and Z2 denote the real and imaginary part) we have using the triangle inequality
+E
+„ˇˇeZ´ ErZ2s
+2
+´iErX2Zs ´ 1
+ˇˇ
+
+ď
+ˇˇeiErX2Zs ´ 1
+ˇˇE
+„ˇˇeZ´ ErZ2s
+2
+ˇˇ
+
+` E
+„ˇˇeZ´ ErZ2s
+2
+´ 1
+ˇˇ
+
+.
+(A.7)
+For the first term, using that |ErX2Zs| ď
+a
+ErX2
+2sEr|Z|2s ď
+a
+Er|Z|2s ď 1, and that
+|eu ´ 1| ď e|u| for u ď 1 and computing expectation, we obtain that
+ˇˇeiErX2Zs ´ 1
+ˇˇE
+„ˇˇeZ´ ErZ2s
+2
+ˇˇ
+
+ď e
+a
+Er|Z|2se
+ErZ2
+2 s
+2
+ď e3{2a
+Er|Z|2s.
+(A.8)
+For the second term we have (using again |eu ´ 1| ď e|u|)
+E
+„ˇˇeZ´ ErZ2s
+2
+´ 1
+ˇˇ
+
+ď
+d
+E
+„ˇˇeZ´ ErZ2s
+2
+´ 1
+ˇˇ2
+
+“
+a
+eEr|Z|2s ´ 1 ď e1{2a
+Er|Z|2s,
+(A.9)
+which yields the desired result for C “ e2 ` e.
+□
+A.3. Proof of Lemma 5.4. Let us first compute the order of magnitude of φptq. Let us
+set for practical purpose φptq :“
+ş
+Qtp0, zqe|γ|2Ktp0,zqdz. Using (3.12) (recall that |z| ď e´t
+on the integrand) and (3.13) we have
+φptq — ep|γ|2´dqt
+and
+φptq — t´1{2ep|γ|2´dqt
+(A.10)
+As a consequence when |γ|2 ą d most of the integral is carried by rt ´
+?
+t, ts and we have
+ż t
+0
+φpsqds “ p1 ` op1qq
+ż t
+t´
+?
+t
+φpsqds
+“ p1 ` op1qq
+ż t
+t´
+?
+t
+c
+s _ 1
+t
+φpsqds
+“ p1 ` op1qq
+c
+2
+πte|γ|2j
+ż t
+0
+φpsqds.
+(A.11)
+We observe that
+|γ|2Qsp0, zqe|γ|2Ksp0,zq “ Bs
+´
+e|γ|2Ksp0,zq¯
+.
+Using Fubini and integrating w.r.t. time and making a change of variable we have
+|γ|2
+ż t
+0
+φpsqds “
+ż
+Rd
+´
+e|γ|2Ktp0,zq ´ 1
+¯
+dz
+“ ep|γ|2´dqt
+ż
+Rd
+´
+e|γ|2pKtp0,e´tzq´tq ´ e´|γ|2t¯
+dz.
+(A.12)
+The integrand in the second line is bounded above by p|z| _ 1q´|γ|2. This is obvious for
+|z| ě et since the integrand vanishes, and when |z| ď et this can be obtainded from (3.11).
+Furthermore it converges to e|γ|2ℓpzq and we obtain using dominated convergence that
+lim
+tÑ8
+|γ|2 şt
+0 φpsqds
+ep|γ|2´dqt
+“
+ż
+Rd e|γ|2ℓpzqdz,
+(A.13)
+which combined with (A.11), proves the lemma in the case |γ|2 ą d. When |γ|2 “ d, we
+observe that using, as in (A.12), a change of variable and dominated convergence, we have
+lim
+sÑ8 φpsq “
+ż
+Rd κpe
+η1
+η2 zqedℓpzqdz.
+(A.14)
+
+38
+HUBERT LACOIN
+On the other hand we have from (A.12)
+ż t
+0
+φpsqds “
+ż
+Rd
+´
+edpKtp0,e´tzq´tq ´ e´dt¯
+dz
+(A.15)
+We have, for 1 ď |z| ď et,
+Ktp0, e´tzq “ Kp0, e´tzq “ Kp0, e´tzq ´ K0p0, e´tzq
+“ t ` log 1
+|z| ` pL ´ K0qp0, e´tzq.
+(A.16)
+Since pL ´ K0qp0, e´t|z|q “ ´j ` δ
+`
+e´t|z|
+˘
+where δpuq tends to zero when u Ñ 0 we can
+deduce that
+ż
+Rd
+´
+edpKtp0,e´tzq´tq ´ e´dt¯
+dz “ p1 ` op1qq
+ż
+1t1ď|z|ďetuedpKtp0,e´tzq´tqdz
+“ p1 ` op1qqe´dj
+ż
+1t1ď|z|ďetu|z|´ddz
+“ p1 ` op1qqe´djΣd´1t.
+(A.17)
+Since φpsq converges, we deduce that its limit equals its Cesaro limit and thus
+lim
+sÑ8 φpsq “ e´djΣd´1,
+(A.18)
+which implies in turn that
+ż t
+0
+d
+2
+πps ^ 1qedjφpsq “ p1 ` op1qq2
+c
+2t
+π Σd´1,
+(A.19)
+and concludes the proof of the lemma.
+□
+A.4. Proof of Lemma 6.4. Let us again start with the case |γ|2 ą d. As in the proof of
+Lemma 5.4, we can compute the asymptotic of φpt, εq (when t and ε goes to infinity and
+zero respectively)
+φpt, εq — t´1{2e|γ|2t´dt.
+(A.20)
+Since suptPr0,Ts
+εPp0,1q
+φpt, εq ă 8 for every finite T, this implies that the integral
+ş8
+0 φpt, εq is
+mostly carried by values of s around logp1{εq (say ˘
+a
+logp1{εq). For this reason, we can
+replace the term pt _ 1q´1{2 by plog 1{εq´1{2.
+|γ|2
+ż 8
+0
+φpt, εqdt “ p1 ` op1qq
+d
+2
+πplog 1{εqe|γ|2j
+ż 8
+0
+ż
+Rd |γ|2e|γ|2Kt,εp0,zqQt,εp0, zqdz (A.21)
+Using Fubini and integrating with respect to time as in (A.12) we have
+ż 8
+0
+ż
+Rd |γ|2e|γ|2Kt,εp0,zqQt,εp0, zqdz “
+ż
+Rd
+´
+e|γ|2Kεp0,zq ´ 1
+¯
+dz
+(A.22)
+We then perform a change of variable for z
+ż
+Rd
+´
+e|γ|2Kεp0,zq ´ 1
+¯
+dz “ εd´|γ|2 ż
+Rd
+´
+e|γ|2pKεp0,εzq`logpεqq ´ ε|γ|2¯
+dz
+(A.23)
+Next we observe that
+Kεp0, εzq ` logpεq “ ℓθpzq ` pLεp0, εzq ´ Kεp0, εzqq .
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES39
+Using dominated convergence as ε goes to zero (the integrand is bounded above by p|z| _
+1q´|γ|2 and recalling (3.9) we obtain that
+lim
+εÑ0
+ż
+Rd
+´
+e|γ|2pKεp0,εzq`logpεqq ´ ε|γ|2¯
+dz “ e´|γ|2j
+ż
+Rd e´|γ|2ℓθpzqdz.
+(A.24)
+The combination of (A.21)-(A.24) concludes the proof in the case |γ|2 ą d. For the case
+|γ|2 “ d, based on (A.20) we know that setting Tε “ logp1{εq ´
+a
+logp1{εq
+ż 8
+0
+φpt, εqdt “ p1 ` op1qq
+ż Tε
+0
+φpt, εqdt.
+(A.25)
+Now in this range for t it is tedious but not difficult to check that
+lim
+εÑ0 sup
+tPr0,Tεs
+ş
+Rd e|γ|2Kt,εp0,zqQt,εp0, zqdz
+ş
+Rd e|γ|2Ktp0,zqQtp0, zqdz
+“ 1.
+(A.26)
+From this we obtain that
+ż 8
+0
+φpt, εqdt “ p1 ` op1qq
+ż Tε
+0
+φpt, εqdt “ p1 ` op1qq
+ż Tε
+0
+φptqdt
+(A.27)
+and we can conclude using Lemma 5.4.
+Appendix B. The convergence of Mγ
+ε as a distribution
+We have chosen for simplicity, to present our convergence results as convergence of a
+collection of random variables Mγ
+ε pfq indexed by CcpRdq. We can go further and prove
+that Mγ
+ε p¨q converges as a distribution. For this purpose we need to recall the definition
+of local Sobolev/Bessel spaces.
+The Bessel space Hs,ppRkq, s P R and p P r1, 8s on Rk is defined by
+Hs,ppRkq :“ tϕ P D1pRkq : p1 ` |ξ|2qs{2 pϕpξq P LppRkqu
+(B.1)
+where D1pRkq is the space of distribution and pϕpξq is the Fourier transform of ϕ defined
+for ϕ P C8
+c pRkq by pϕpξq “
+ş
+Rk eiξxϕpxqdx. It is a Banach space when equiped with the
+norm
+}f}Hs,p “
+ż
+Rkp1 ` |ξ|2qps{2|pϕpξq|pdξ
+(B.2)
+For U Ă Rk open, the local Bessel space Hs,p
+locpUq denotes the set of distributions which
+belongs to Hs,ppUq after multiplication by an arbitrary smooth function with compact
+support
+Hs,p
+locpUq :“
+!
+ϕ P D1pUq | ρϕ P Hs,ppRdq for all ρ P C8
+c pUq
+)
+,
+(B.3)
+where above ρϕ is identified with its extension by zero on Rk. It is equiped with the
+topology generated by the family of seminorms rρ, ρ P C8
+c pUq defined by rρpϕq :“ }ϕρ}Hs,p.
+In the particular case where p “ 2 we write HspRkq :“ Hs,2pRkq which is a Hilbert space
+(and use the same convention for the local spaces). The convergence result for Mγ
+ε p¨q as a
+distribution for γ P PI{II is the following.
+Theorem B.1. If X is a centered Gaussian field whose covariance kernel K has an
+almost star-scale invariant part, γ P PI{II, p P r1,
+?
+2dαq and s ă ´ d
+p, then there exists
+Mγ
+8 P Hs,p
+locpRdq such that for every ρ P C8
+c pRdq
+lim
+εÑ0 E
+“
+}Mγ
+ε pρ ¨q ´ Mγ
+8pρ ¨q}p
+Hs,p
+‰
+“ 0.
+(B.4)
+
+40
+HUBERT LACOIN
+In particular Mγ
+ε converges to Mγ
+8 in probability in the Hs,p
+locpRdq topology.
+Similarly in P1
+II{III the convergence in law holds also for the distribution.
+Theorem B.2. Let X be a Gaussian random field with an almost star-scale covariance.
+Then given γ P P1
+II{III, and s ă ´ d
+2 the following joint convergence in law for the Hs
+locpRdq
+topology
+ˆ
+X,
+Mγ
+ε
+vpε, θ, γq
+˙
+εÑ0
+ñ pX, Mγq,
+(B.5)
+Remark B.3. In the proof of Theorems B.1 and B.2 presented below, we are going to
+assume that our probability space contains a martingale sequence pXtqtě0 of fields with co-
+variance (3.3) approximating X. For reasons analogous to the one exposed at the beginning
+of Section 4.2 this entails no loss of generality.
+B.1. The case of Theorem B.1. Let us fix ρ P C8
+c pRdq. We want to prove that Mγ
+ε pρ ¨q
+converges in Hs,ppRdq. We first define the limit point. We set (without underlying the
+dependence in ρ to keep the notation light)
+x
+Mγ
+ε pξq “
+ż
+Rd ρpxqeiξ.xeγXεpxq´ γ2
+2 Kεpxqdx,
+x
+Mγ
+8pξq “ lim
+εÑ0
+x
+Mγ
+ε pξq.
+(B.6)
+We let Mγ
+8pρ ¨q denote the random distribution whose Fourier transform is given by x
+Mγ
+8pξq.
+The proof of (B.4), implies that Mγ
+8pρ ¨q P Hs,ppRdq with probability one. To prove (B.4),
+note that we have
+E
+“
+}Mγ
+ε pρ ¨q ´ Mγ
+8pρ ¨q}p
+Hs,p
+‰
+“
+ż
+Rd E
+”
+|px
+Mγ
+ε ´ x
+Mγ
+8qpξq|pı
+p1 ` |ξ|2q
+ps
+2 dξ.
+(B.7)
+Hence with our assumption on s ă ´ d
+p it is sufficient to prove that
+lim sup
+εÑ0
+sup
+ξPRd E
+”
+|px
+Mγ
+ε ´ x
+Mγ
+8qpξq|pı
+ă 8,
+@ξ P Rd, lim
+εÑ0 E
+”
+|px
+Mγ
+ε ´ x
+Mγ
+8qpξq|pı
+“ 0.
+(B.8)
+and we can then conclude using dominated convergence (the first line yields the domina-
+tion). The second line is simply (2.2) with fpxq “ ρpxqeiξ.x. For the first line it sufficient
+to prove that
+lim sup
+εÑ0
+sup
+ξPRd E
+”
+|x
+Mγ
+ε pξq|pı
+ă 8,
+since the bound for x
+Mγ
+8pξq| can then be obtained by Fatou. We set V pεq
+t
+pξq “ Erx
+Mγ
+ε | Fts.
+Using the BDG inequality we have
+E
+”
+|x
+Mγ
+ε pξq|pı
+ď CE
+”
+|V pεq
+0
+pξq|p ` xV pεqpξqyp{2
+8
+ı
+.
+(B.9)
+Now we have (recall that p ă
+?
+2d{α ď 2)
+E
+”
+|V pεq
+0
+pξq|2ı
+“
+ż
+R2d ρpxqρpyqeiξ.px´yqe|γ|2K0,εpx,yqdxdy
+(B.10)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES41
+and we can conclude by replacing eiξ.px´yq by 1 and observing that since K is continuous
+K0,ε is uniformly bounded for x, y in the support of ρ and ε P p0, 1q. For the quadratic
+variation part, we have
+xV pεqy8 “ |γ|2
+ż 8
+0
+At,εpξqdt,
+(B.11)
+where,
+At,εpξq :“
+ż
+R2d ρpxqρpyqQt,εpx, yqeiξpx´yqeγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyqdxdy
+ď
+ż
+R2d ρpxqρpyqQt,εpx, yqeαpXt,εpxq`Xt,εpyqq` β2´α2
+2
+pKt,ε`Kt,εpyqqdxdy “: At,ε,
+(B.12)
+the inequality being obtain by taking the modulus of the integrand. To conclude, we just
+need to prove that
+sup
+εPp0,1q
+E
+«ˆż 8
+0
+At,εdt
+˙p{2ff
+ă 8
+(B.13)
+For this part we can just repeat the computations made to prove (4.23) in Section 4.2.
+B.2. The case of Theorem B.2. Since the convergence of finite dimensional marginal
+has been established, we only need to prove tightness of the distribution of vpε, θ, γq´1Mγ
+ε pρ ¨q
+in HspRdq for every ρ. For this, we simply replicate the strategy presented in [16], with a
+minor twist. Since in our case, the Fourier transform in not in L2, we need to consider a
+restriction to the event Aq,R where R is such that the support ρ is contained in Bp0, Rq
+(recall (5.6)). Keeping the notation introduced in (B.6) for the Fourier transform, we are
+going to prove the following analogue of [16, Lemma B.2] (we use the notation vpεq for
+vpε, θ, γq for ease of reading)
+Lemma B.4. If the support of ρ is included in Bp0, Rq then the following holds for every
+a P Rd with a constant C which depends on ρ.
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|x
+Mγ
+ε pξq|21Aq,R
+ı
+ă 8,
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|x
+Mγ
+ε pξ ` aq ´ x
+Mγ
+ε pξq|21Aq,R
+ı
+ď C|a|2,
+(B.14)
+Proof. We introduce a martingale whose limit coincides with x
+Mγ
+ε pξq on the event Aq,R.
+Given x P Rd and q ą 0 we set
+Tqpxq :“ inftt ą 0 : Xtpxq “
+?
+2dt ` qu,
+(B.15)
+and define
+N pεq
+t
+pξq :“
+ż
+Rd ρpxqeiξ.xeγXt^Tqpxq,εpxq´ γ2
+2 Kt^Tqpxq,εpxqdx.
+(B.16)
+Since Tqpxq “ 8 for all x in the support of ρ on the event Aq,R, we have
+N pεq
+8 pξq1Aq,R “ x
+Mγ
+ε pξq1Aq,R.
+(B.17)
+
+42
+HUBERT LACOIN
+Hence it is sufficient to prove
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|N pεq
+8 pξq|2ı
+ă 8,
+sup
+εPp0,1q
+ξPRd
+E
+”
+vpεq´2|N pεq
+8 pξ ` aq ´ N pεq
+8 pξq|2ı
+ď C|a|2.
+(B.18)
+Let us prove the only second inequality, since the first one is only easier.
+We set for
+simplicity Wt :“ N pεq
+t
+pξ ` aq ´ N pεq
+t
+pξq. We have
+E
+”
+|N pεq
+8 pξ ` aq ´ N pεq
+8 pξq|2ı
+“ Er|W8|2s “ Er|W0|2s ` E rxWy8s .
+We are going to prove a bound for each of the term in the r.h.s. . We have
+Er|W0|2s “
+ż
+R2d ρpxqρpyq
+´
+eipξ`aq.x ´ eiξ.x¯ ´
+e´ipξ`aq.y ´ eiξ.y¯
+e|γ|2K0,εpx,yq
+ď C|a|2
+ż
+ρpxqρpyq|x||y|dxdy ď C1|a|2.
+(B.19)
+where in the second line have taken the modulus of the integrand, and used the fact
+that the complex exponential is Lipshitz. To bound the expected value of the quadratic
+variation, using Itˆo calculus, and observing that tTqpxq ă tu “ At,qpxq (recall (5.6)) we
+obtain that
+xWy8 “ |γ|2
+ż 8
+0
+Utdt.
+(B.20)
+where
+Ut :“
+ż
+R2d ρpxqρpyqQt,εpx, yq
+´
+eipξ`aq.x ´ eiξ.x¯ ´
+e´ipξ`aq.y ´ eiξ.y¯
+ˆ eγXt,εpxq`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyq1At,qpxqXAt,qpyqdx.
+(B.21)
+Taking the modulus in the integrand value everywhere inside the integral and using the
+fact that the complex exponential is Lipshitz fwe obtain
+Ut ď |a|2
+ż
+R2d ρpxqρpyq|x||y|Qt,εpx, yq
+ˆ e
+?
+d{2pXt,εpxq`Xt,εpyqq` |γ|2´d
+2
+pKt,εpxq`Kt,εpyqq1At,qpxqXAt,qpyqdxdy
+ď C|a|2
+ż
+R2d ρpxq2|x|2Qt,εpx, yqe
+?
+2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxqdxdy,
+(B.22)
+where the second line is obtained via the same step as (5.26) (ab ď a2`b2{2 and symmetry
+and in x and y). Now recalling (6.25) we have
+E
+”
+e
+?
+2dXt,εpxq´dKt,εpxq1At,qpxq
+ı
+ď Cpt _ 1q´1{2.
+(B.23)
+where to obtain the first inequality, we used (3.12) to show that Kspx, yq (and all similar
+terms) are well estimated by t for s P r0, ts. Now we have
+E rUts ď C|a|2e´dt
+?
+t _ 1
+ż
+R2d ρpxq2|x|2Qt,εpx, yqe|γ|2Kt,εpx,yqdx ď C1|a|2φpt, εq.
+(B.24)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES43
+After integrating with respect to t (recalling Lemma 6.4) we obtain that
+ErxWy8s ď C|a|2vpεq2,
+(B.25)
+for a constant C which is independent of ε and ξ and a, which combined with (B.19),
+concludes the proof.
+□
+Appendix C. Beyond star-scale invariance
+The assumption that the kernel can be written in the form (1.8) may be felt as unnec-
+essarily restrictive, since after all, given an open domain D Ă Rd and a positive definite
+Kernel kernel K :
+D2 Ñ p´8, 8s that admits a decomposition of the form (1.1), the
+mollified field Xε can be defined on
+Dε :“ tx P D : inf
+yPDA |x ´ y| ą 2εu.
+More precisely in that case the field X is indexed by CcpDq the set of functions with
+compact support on D (in (1.4), R2d is replaced by D2), and Xε remains defined by (1.5)
+(here θεpx ´ ¨q, which for x P Dε, has its support included in D, is identified with its
+restriction on D).
+It turns out that our results can be extended to the the general setup described above,
+only with an additional regularity assumption concerning the function L present in (1.1).
+Given U Ă D, we say that the restriction of K to U has an almost star-scale invariant
+part, if
+@x, y P U, Kpx, yq “ K0px, yq ` Kpx, yq
+(C.1)
+where K is an almost-star scale invariant Kernel, and K0 : U 2 Ñ R is positive definite
+and H¨older continuous.
+To extend the result we use the fact (proved in [12]) that if L is sufficiently regular
+then K is locally star-scale invariant in the sense defined above. We state this result as a
+proposition. It can be directly derived from [12, Theorem 4.5].
+Proposition C.1. If K is a positive definite kernel on D that can be written in the form
+(1.1) with L P Hs
+locpD2q with s ą d, then for every z P D, there exist δz ą 0 such that the
+restriction of K to Bpz, δzq has an almost star-scale invariant part.
+To extend Theorem 2.5 we require another technical result, which states that with the
+same assumption as above, and U an open set whose closure is included in D, K can be
+approximated by a kernel with an almost star-scale invariant part defined on U. This is
+the content of the following result, [16, Lemma 2.1]
+Proposition C.2. Given K a covariance kernel on D of the form (1.1) with L P Hs
+locpD2q
+for s ą d , U a bounded open set whose closure satisfies U Ă D and δ ą 0, then there
+exists a kernel Kpδq on U satisfying (C.1) such that
+(A) For all x, y P U,
+|Kpδqpx, yq ´ Kpx, yq| ď δ.
+(B) ∆pδqpx, yq “ Kpδqpx, yq ´ Kpx, yq is a positive definite kernel on U.
+Remark C.3. More precisely, [16, Lemma 2.1] states that one can chose η1 “ 0 (recall
+(1.8)) for the almost-star scale invariant part of Kpδq, but this refinement is not required
+for our purpose.
+
+44
+HUBERT LACOIN
+C.1. The case of Theorem 2.1. The extension of the result to the case of a general
+log-correlated field defined on a domain D is the following.
+Theorem C.4. If X is a centered Gaussian field defined on D whose covariance kernel
+K can be written in the form (1.1) with L P Hs
+locpD2q for s ą d, and f P CcpRdq, then
+there exists a complex valued random variable Mγ
+8pfq such that for any choice of mollifier
+θ the following convergence holds in Lp if p P
+”
+1,
+?
+2d{α
+¯
+.
+lim
+εÑ0 Mγ
+ε pfq “ Mγ
+8pfq.
+(C.2)
+Proof. This follows quite immediately via a localization argument using a partition of
+unity. Let f P CcpDq be fixed. Using Proposition C.1, we can cover the support of f (which
+is compact) by finitely many Euclidean balls Bpzi, εiq, i P I such that for every i P I the
+restriction of K to Bpzi, εiq has an almost star-scale invariant part. Using a partition of the
+unity, we can write f :“ ř
+iPI fi where fi is continuous with compact support included in
+Bpzi, εiq. Using Theorem 2.1 for K restricted to Bpzi, εiq, we obtain that Mγ
+ε pfiq converges
+in Lp for every fi and thus we obtain the convergence for Mγ
+ε pfq “ ř
+iPI Mγ
+ε pfiq.
+□
+C.2. The case of Theorem 2.5. To extend the result for γ P P1
+II{III it is sufficient to
+extend Proposition 2.8. In the statement below, we implicitely use the fact that the critical
+multiplicative chaos M1 is well defined under our assumptions (see [17, Theorem C.2] for
+a proof).
+Proposition C.5. If X is a centered Gaussian field defined on D whose covariance kernel
+K can be written in the form (1.1) with L P Hs
+locpD2q for s ą d, given ρ, f P CcpDq,
+ω P r0, 2πq, we have
+lim
+εÑ0 E
+„
+eixX,ρy`i Mγ
+ε pf,ωq
+vpε,θ,γq
+
+“ E
+„
+eixX,ρy´ 1
+2M1pe|γ|2L|f|2q
+
+.
+(C.3)
+Proof. Given a fixed f P CcpDq, and n ě 1, we chose U which contains the support of
+f and Kn : U 2 Ñ p´8, 8s satisfying the assumptions of Kpδq of Proposition C.2 with
+δ “ 1{n. We let Zn be a centered Gaussian field indexed by U, independent of X and with
+covariance ∆n “ Kn ´K and define Xn a field indexed by CcpUq by setting Xn “ X `Zn.
+Note that Xn has covariance Kn. We let Mγ,n
+ε
+and M
+1
+n denote the mollified GMC and
+critical GMC associated with Xn. For simplicity, we define all the pZnqně1 on the same
+probability space: the fields Zn form an independent sequence which is independent of
+X. We let P denote the corresponding probability. From Proposition 2.8 we have for each
+n ě 1
+lim
+εÑ0 E
+„
+eixXn,ρy`i Mγ,n
+ε
+pf,ωq
+vpε,θ,γq
+
+“ E
+„
+eixXn,ρy´ 1
+2 M
+1
+npe|γ|2Ln|f|2q
+
+,
+(C.4)
+where Ln :“ L ` ∆n. In order to conclude, we need to show that (for any choice of Kpδq)
+lim
+nÑ8 sup
+εPp0,1q
+ˇˇˇˇE
+„
+eixXn,ρy`i Mγ,n
+ε
+pf,ωq
+vpε,θ,γq
+
+´ E
+„
+eixX,ρy`i Mγ
+ε pf,ωq
+vpε,θ,γq
+ˇˇˇˇ “ 0
+(C.5)
+and that
+lim
+nÑ8 E
+„
+eixXn,ρy´ 1
+2M
+1
+npe|γ|2Ln|f|2q
+
+“ E
+„
+eixX,ρy´ 1
+2 M
+1pe|γ|2Lpδq|f|2q
+
+.
+(C.6)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES45
+Note that it is sufficient to show that the difference between the terms in the l.h.s. and
+the r.h.s. tend to zero in probability (uniformly in ε) that is
+lim
+nÑ8 E r|xXn, ρy ´ xX, ρy| _ 1s “ 0,
+lim
+nÑ8 E
+”ˇˇˇM
+1
+npe|γ|2Ln|f|2q ´ M
+1pe|γ|2L|f|2q
+ˇˇˇ _ 1
+ı
+“ 0,
+lim
+nÑ8 sup
+εPp0,1q
+E
+„ˇˇˇˇ
+Mγ,n
+ε
+pf, ωq
+vpε, θ, γq
+´ Mγ
+ε pf, ωq
+vpε, θ, γq
+ˇˇˇˇ _ 1
+
+“ 0.
+(C.7)
+The first line is immediate via the computation of the L2 norm (the convergence holds in
+L2). For the second line, we set gn “ e|γ|2Ln|f|2 and g “ e|γ|2L|f|2. Using the notational
+convention introduced in Section 3.1, we let Zn,ε denote the mollification of Zn and ∆n,ε
+its covariance. Letting e
+?
+2dZn,ε´d∆n,ε denote the function x ÞÑ e
+?
+2dZn,εpxq´d∆n,εpxq we have
+E
+„´
+M
+?
+2d,n
+ε
+pgnq ´ M
+?
+2d
+ε
+pgq
+¯2
+| X
+
+“ E
+”
+M
+?
+2d
+ε
+pe
+?
+2dZn,ε´d∆n,εgn ´ gq2 | X
+ı
+“
+ż
+U2
+´
+e2d∆n,εpx,yqgnpxqgnpyq ´ 2gnpxqgpyq ` gpxqgpyq
+¯
+M
+?
+2d
+ε
+pdxqM
+?
+2d
+ε
+pdyq.
+(C.8)
+From the assumption that |∆npx, yq| ď 1{n (and thus |Lnpxq ´ Lpxq| ď 1{n) we obtain
+that
+|e2d∆n,εpx,yqgnpxqgnpyq ´ 2gnpxqgpyq ` gpxqgpyq| ď Cgpxqgpyq
+n
+and hence
+E
+„´
+M
+?
+2d,n
+ε
+pgnq ´ M
+?
+2d
+ε
+pgq
+¯2
+| X
+
+ď C
+n M
+?
+2d
+ε
+pgq2
+(C.9)
+Using Fatou after renormalization we obtain that
+E
+”`
+M1
+npgnq ´ M1pgq
+˘2 | X
+ı
+ď C
+n M1pgq2
+(C.10)
+which implies the second line in (C.7). For the third line, we are going to Proposition
+(C.1). More precisely, we use a decomposition of f “ ř
+iPI fi where fi is continous with
+compact support included in Ui and the restriction of K to Ui has an almost star-scale
+invariant part. We are going to prove that for each i P I.
+lim
+nÑ8 sup
+εPp0,1q
+E
+„ˇˇˇˇ
+Mγ,n
+ε
+pfi, ωq
+vpε, θ, γq
+´ Mγ
+ε pfi, ωq
+vpε, θ, γq
+ˇˇˇˇ _ 1
+
+“ 0.
+(C.11)
+This operation shows that it is in fact sufficient to prove the third line of (C.7) assuming
+that K is an almost star-scale invariant Kernel. We can thus further our probability space
+contains pXtqtě0 a martingale sequence of fields with covariance Kt (we adopt the notation
+of Section 3.1) approximating X. We equip our space with the filtration
+Gt :“ σppXsqsPr0,ts, pZnqně1q.
+We recall the definition of Tqpxq in (B.15) and define
+W pn,εq
+t
+:“
+ż
+U
+peγZn,εpxq´ γ2
+2 ∆n,εpxq ´ 1qfpxqeγXt^Tqpxq,ε´ γ2
+2 Kt^Tq,εpxqdx
+(C.12)
+Setting
+Aq :“ t@x P Supppfq, @t ą 0, Xtpxq ď
+?
+2dt ` qu
+
+46
+HUBERT LACOIN
+We have from the definition
+W pn,εq
+8
+1Aq “ |Mγ,n
+ε
+pfq ´ Mγ
+ε pfq|1Aq
+Hence we have (for any ω P r0, 2πq since the projection on one axis reduces the modulus
+E
+“
+|Mγ,n
+ε
+pf, ωq ´ Mγ
+ε pf, ωq|21Aq
+‰
+ď Er|W pn,εq
+8
+|2s “ Er|W pn,εq
+0
+|2s ` ErxW pn,εqy8s.
+(C.13)
+Since from Lemma 5.2 we have limqÑ8 PrAqs “ 1, to prove the third line in (C.7), it is
+sufficient to show that for any q we have
+lim
+nÑ8 sup
+εPp0,1q
+vpε, θ, γq´2 ´
+Er|W pn,εq
+0
+|2s ` ErxW pn,εqy8s
+¯
+.
+(C.14)
+For the first term, we have
+Er|W pn,εq
+0
+|2s “
+ż
+U2 fpxqfpyqpe|γ|2∆n,εpx,yq´1qe|γ|2K0,εpx,yqdxdy ď Cn´1
+(C.15)
+where the inequality obtained taking the modulus of the integrand and using the fact that
+|∆npx, yq| ď 1{n and the other terms are uniformly bounded. The derivative of the bracket
+of W pn,εq is given by |γ|2 times (recall that by (5.6) we have At,qpxq “ tTqpxq ď tu)
+Dt :“
+ż
+R2d Qt,εpx, yqGn,εpxqGn,εpyq
+ˆ eγXt,ε`γXt,εpyq´ γ2
+2 Kt,εpxq´ γ2
+2 Kt,εpyq1At,qpxqXAt,qpyqdxdy.
+(C.16)
+with Gn,εpxq “ peγZn,εpxq´ γ2
+2 ∆n,εpxq ´ 1q. Repeating once more the computation in (5.26)
+we obtain that
+Dt ď
+ż
+R2d Qt,εpx, yq|Gn,εpxq|2e
+?
+2dXt,ε`p|γ|2´dqKt,εpxq1At,qpxqdxdy.
+(C.17)
+We define a martingale W
+pεq and W
+pεq
+t
+by setting
+W pεq
+t
+:“ E rMγ,n
+ε
+pf, ωq ´ Mγ
+ε pf, ωq | Gts
+(C.18)
+Now we have
+E
+“
+|Gn,εpxq|2‰
+“ e|γ|2∆n,εpxq ´ 1 ď Cn´1
+This the term is independent of the rest, thus using (B.23) we obtain that
+ErDts ď Cn´1
+ż
+R2d Qt,εpx, yqe|γ|2Kt,εpxqdxdy ď C1n´1φpt, εq.
+(C.19)
+Integrating against t we conclude that
+ErxW pn,εqy8s ď Cn´1vpε, θ, γq2
+and this concludes the proof of (C.14).
+□
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES47
+Appendix D. Proof of Lemma 4.1
+We use Kahane convexity inequality in order to compare Bn to the the partition function
+of a Gaussian branching random walk (or polymer on a 2d-adic tree). We assume without
+loss of generality that Supppfq Ă r0, 1sd. For x, y P r0, 1sd we let 2´kpx,yq be the sidelength
+of the smallest dyadic cube that contains x and y.
+kpx, yq :“ inf
+!
+n ě 0 : Dm P �0, 2n ´ 1�d, tx, yu Ă
+´
+2´nm ` r0, 2´nqd¯)
+.
+and set knpx, yq :“ kpx, yq ^ n. Note that kn defines a positive definite function and that
+kpx, yq ď log2
+´
+1
+|x´y|
+¯
+` C. Hence from Lemma 3.1 there exists a constant A ą 0 such
+that
+plog 2qknpx, yq ď Krn log 2spx, yq ` A.
+(D.1)
+Using Kahane’s convexity inequality (proved in [15] see also [27, Theorem 2.1]) which we
+introduce in a simplified setup
+Lemma D.1. If C1 and C2 are two bounded positive definite kernel on an arbitrary space
+X satisfying
+@x, y P X,
+C1px, yq ď C2px, yq
+µ is a finite measure on r0, 1sd and F : R` Ñ R is a concave function with at most
+polynomial growth at infinity and Y1 and Y2 are Gaussian fields with respective covariance
+C1 and C2 then we have for any θ P R
+E
+„
+F
+ˆż
+eθY1pxq´ θ2
+2 C1pxqµpdxq
+˙
+ď E
+„
+F
+ˆż
+eY2pxq´ θ2
+2 C2pxqµpdxq
+˙
+.
+(D.2)
+Hence if Zn denotes a field defined on r0, 1sd with covariance kn we can apply Lemma
+D.1 result for the fields ?log 2Zn and Xrn log 2s`
+?
+AN where N is an independent standard
+Gaussian (the fields have their resepective covariances given by the two sides of Equation
+(D.1)), µpdxq “ |fpxq|2dx and Fpuq “ up{2. Recalling (4.11) we have
+E
+”
+Bp{2
+rn log 2s
+ı
+ď CE
+»
+–
+˜
+2dn
+ż
+r0,1sd |fpxq|2e2α?log 2pZnpxq´?2d log 2nqdx
+¸p{2fi
+fl .
+(D.3)
+The constant C above takes care of the fact that the variance of ?log 2Znpxq and Xrn log 2s
+differ by a Op1q term, and also of the moment of the variable N. We can ignore the
+constant f at the cost of a prefactor }f}p
+8. To conclude we thus need a bound on the
+moment of order p{2 of the partition function of the Gaussian branching random walk
+Wn,ζ :“ 2dn
+ż
+r0,1sd eζpZnpxq´?2d log 2nqdx “
+ÿ
+mP�0,2n´1�d
+eζpZnpm2´nq´?2d log 2nq,
+(D.4)
+for ζ “ 2α?log 2. The following result is a particular case of [8, Theorem 1.6]. We present
+a shorter proof which is valid in our context for the sake of completeness.
+Lemma D.2. Given ζ ą ?2d log 2 and q ď
+?2d log 2
+ζ
+there exists positive constant C and
+b such that
+E rpWn,ζqqs ď Cn´
+3qζ
+2?2d log 2plog nq6
+
+48
+HUBERT LACOIN
+Proof. We split our integral in three parts. We set
+Bnpxq :“ tDm P �1, n�, Zmpxq ě
+a
+2d log 2 ` plog nq2u,
+Cnpxq :“ BA
+npxq X tZnpxq ď
+a
+2d log 2n ´ plog nq2u,
+Anpxq :“ BA
+npxq X CA
+npxq
+(D.5)
+We define Wn,ζpAq, Wn,ζpBq and Wn,ζpCq by setting, for I P tA, B, Cu
+Wn,ζpIq :“ 2dn
+ż
+r0,1sd eζpZnpxq´?2d log 2nq1Inpxqdx
+(D.6)
+Using subadditivity (4.9) we have
+E rpWn,ζqqs ď E rWn,ζpAqqs ` E rWn,ζpBqqs ` E rWn,ζpCqqs .
+(D.7)
+We are going to show that the two last terms in the r.h.s. decay faster than any negative
+power of n and then prove a bound of the right order of magnitude for E rpWn,ζqqs. Letting
+setting Bn :“ Ť
+xPr0,1s Bnpxq, and q1 “ ?2d log 2ζ´1 (q1 P rq, 1q) we have
+E rWn,ζpBqqs ď E rpWn,ζqq1Bns ď E
+”
+pWn,ζqq1ı q
+q1 P rBns1´ q
+q1 .
+(D.8)
+Using subadditivity (4.9) for the sum (D.4) with θ “ q1,
+E
+”
+pWn,ζqq1ı
+ď E
+“
+Wn,?2d log 2
+‰
+“ 1.
+(D.9)
+The inequality on the right comes from the fact that pWm,?2d log 2qmě1 is a martingale for
+the natural filtration associated with Zn. Using the optional stopping Theorem for this
+same martingale, we can obtain a bound for the probability of Bn,
+PrBns ď P
+”
+Dm, Wm,?2d log 2 ě e
+?2 log 2plog nq2ı
+ď e´?2 log 2plog nq2.
+(D.10)
+This yields a subpolynomial decay for ErWn,ζpBqqs. For Wn,ζpCq using the fact that Zn is a
+Gaussian of variance n, we obtain using Jensen’s inequality, the Cameron-Martin formula
+and Gaussian tail bounds
+E rpWn,ζpCqqqs1{q ď E rWn,ζpCqs “ 2dnE
+”
+eqpZn´?2d log 2nq1tZnď?2d log 2n´plog nq2u
+ı
+“ e
+ˆ
+d log 2` ζ2
+2 ´ζ?2d log 2
+˙
+n
+P
+”
+Znp0q ď p
+a
+2d log 2 ´ ζqn ´ plog nq2ı
+ď ep?2d log 2´ζqplog nq2,
+(D.11)
+also proving a subpolynomial decay. It remains to estimate the main part E rpWn,ζpAqqqs.
+Using first subaddivity (4.9) and then Jensen’s inequality
+E rpWn,ζpAqqqs ď E
+„
+Wn,?2d log 2pAq
+qζ
+?2d log 2
+
+ď E
+“
+Wn,?2d log 2pAq
+‰
+qζ
+?2d log 2 .
+(D.12)
+The Cameron-Martin formula directly expresses E
+“
+Wn,?2d log 2pAq
+‰
+as the probability con-
+cerning the Gaussian centered random walk pZmp0qqmě0,
+E
+“
+Wn,?2d log 2pAq
+‰
+“ P
+“
+@m P �1, n�, Zmp0q ď plog nq2 ; Znp0q ě ´plog nq2‰
+ď Cn´3{2plog nq6.
+(D.13)
+
+CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES49
+The bound for the probability of the event above is valid for any random walk with IID
+centered increments with finite second moment (see for instance [1, Lemma A.3]) which
+concludes our proof.
+□
+References
+[1] Elie A¨ıd´ekon and Zhan Shi. Weak convergence for the minimal position in a branching random walk:
+a simple proof. Period. Math. Hungar., 61(1-2):43–54, 2010.
+[2] Nathana¨el Berestycki. An elementary approach to Gaussian multiplicative chaos. Electron. Commun.
+Probab., 22:Paper No. 27, 12, 2017.
+[3] B. Derrida, M. R. Evans, and E. R. Speer. Mean field theory of directed polymers with random
+complex weights. Comm. Math. Phys., 156(2):221–244, 1993.
+[4] Bertrand Duplantier, R´emi Rhodes, Scott Sheffield, and Vincent Vargas. Critical Gaussian multiplica-
+tive chaos: convergence of the derivative martingale. Ann. Probab., 42(5):1769–1808, 2014.
+[5] Bertrand Duplantier, R´emi Rhodes, Scott Sheffield, and Vincent Vargas. Renormalization of critical
+Gaussian multiplicative chaos and KPZ relation. Comm. Math. Phys., 330(1):283–330, 2014.
+[6] Lisa Hartung and Anton Klimovsky. The glassy phase of the complex branching Brownian motion
+energy model. Electron. Commun. Probab., 20:no. 78, 15, 2015.
+[7] Lisa Hartung and Anton Klimovsky. The phase diagram of the complex branching Brownian motion
+energy model. Electron. J. Probab., 23:Paper No. 127, 27, 2018.
+[8] Yueyun Hu and Zhan Shi. Minimal position and critical martingale convergence in branching random
+walks, and directed polymers on disordered trees. Ann. Probab., 37(2):742–789, 2009.
+[9] Jean Jacod and Albert N. Shiryaev. Limit theorems for stochastic processes, volume 288 of Grundlehren
+der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-
+Verlag, Berlin, second edition, 2003.
+[10] Janne Junnila and Eero Saksman. Uniqueness of critical Gaussian chaos. Electron. J. Probab., 22:Paper
+No. 11, 31, 2017.
+[11] Janne Junnila, Eero Saksman, and Lauri Viitasaari. On the regularity of complex multiplicative chaos.
+arXiv e-prints, page arXiv:1905.12027, May 2019.
+[12] Janne Junnila, Eero Saksman, and Christian Webb. Decompositions of log-correlated fields with ap-
+plications. Ann. Appl. Probab., 29(6):3786–3820, 2019.
+[13] Janne Junnila, Eero Saksman, and Christian Webb. Imaginary multiplicative chaos: moments, regu-
+larity and connections to the Ising model. Ann. Appl. Probab., 30(5):2099–2164, 2020.
+[14] Zakhar Kabluchko and Anton Klimovsky. Complex random energy model: zeros and fluctuations.
+Probab. Theory Related Fields, 158(1-2):159–196, 2014.
+[15] Jean-Pierre Kahane. Sur le chaos multiplicatif. (On multiplicative chaos). Ann. Sci. Math. Qu´e.,
+9:105–150, 1985.
+[16] Hubert Lacoin. Convergence in law for complex Gaussian multiplicative chaos in phase III. Ann.
+Probab., 50(3):950–983, 2022.
+[17] Hubert
+Lacoin.
+Critical
+Gaussian
+Multiplicative
+Chaos
+revisited.
+arXiv
+e-prints,
+page
+arXiv:2209.06683, September 2022.
+[18] Hubert Lacoin. A universality result for subcritical complex Gaussian multiplicative chaos. Ann. Appl.
+Probab., 32(1):269–293, 2022.
+[19] Hubert Lacoin, R´emi Rhodes, and Vincent Vargas. A probabilistic approach of ultraviolet renormali-
+sation in the boundary Sine-Gordon model. to appear in Probab. Theory Related Fields.
+[20] Hubert Lacoin, R´emi Rhodes, and Vincent Vargas. Complex Gaussian multiplicative chaos. Comm.
+Math. Phys., 337(2):569–632, 2015.
+[21] Jean-Fran¸cois Le Gall. Brownian motion, martingales, and stochastic calculus, volume 274 of Graduate
+Texts in Mathematics. Springer, 2016.
+[22] Thomas Madaule. Convergence in law for the branching random walk seen from its tip. J. Theor.
+Probab., 30:27–63, 2017.
+[23] Thomas Madaule, R´emi Rhodes, and Vincent Vargas. The glassy phase of complex branching Brow-
+nian motion. Comm. Math. Phys., 334(3):1157–1187, 2015.
+[24] Thomas Madaule, R´emi Rhodes, and Vincent Vargas. Glassy phase and freezing of log-correlated
+gaussian potentials. Ann. Appl. Probab., 26(2):643–690, 04 2016.
+
+50
+HUBERT LACOIN
+[25] Ellen Powell. Critical Gaussian multiplicative chaos: a review. arXiv e-prints, page arXiv:2006.13767,
+June 2020.
+[26] Ellen Powell. Critical Gaussian multiplicative chaos:
+a review. Markov Process. Related Fields,
+27(4):557–506, 2021.
+[27] R´emi Rhodes and Vincent Vargas. Gaussian multiplicative chaos and applications: a review. Probab.
+Surv., 11:315–392, 2014.
+IMPA, Institudo de Matem´atica Pura e Aplicada, Estrada Dona Castorina 110 Rio de
+Janeiro, CEP-22460-320, Brasil.
+
diff --git a/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/load_file.txt b/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..dcb7a8095964ac36e5f1b44cd936cdc6ecdc1545
--- /dev/null
+++ b/1tE4T4oBgHgl3EQfzg0x/content/tmp_files/load_file.txt
@@ -0,0 +1,1913 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf,len=1912
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='05274v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='PR]  12 Jan 2023  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE  CHAOS ON PHASE BOUNDARIES  HUBERT LACOIN  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The complex Gaussian Multiplicative Chaos (or complex GMC) is infor-  mally defined as a random measure eγXdx where X is a log correlated Gaussian field on  Rd and γ “ α ` iβ is a complex parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The correlation function of X is of the form  Kpx, yq “ log  1  |x ´ y| ` Lpx, yq,  where L is a continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In the present paper, we consider the cases γ P PI{II  and γ P P1  II{III where  PI{II :“ tα ` iβ : α, β P R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |α| ą |β| ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |α| ` |β| “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2du,  and  P1  II{III :“ tα ` iβ : α, β P R ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |α| “  a  d{2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |β| ą  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2du,  We prove that if X is replaced by an approximation Xε obtained via mollification, then  eγXεdx, when properly rescaled, converges when ε Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The limit does not depend on  the mollification kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' When γ P PI{II, the convergence holds in probability and in Lp  for some value of p P r1,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{αq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' When γ P P1  II{III the convergence holds only in law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  In this latter case, the limit can be described a complex Gaussian white noise with a  random intensity given by a critical real GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The regions PI{II and P1  II{III correspond to  phase boundary between the three different regions of the complex GMC phase diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  These results complete previous results obtained for the GMC in phase I [18] and III [16]  and only leave as an open problem the question of convergence in phase II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2010 Mathematics Subject Classification: 60F99, 60G15, 82B99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Keywords: Random distributions, log-correlated fields, Gaussian Multiplicative Chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Introduction  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Main results  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The martingale approximation for GMC  8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proof of convergence results on for γ P PI{II  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5  18  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8  27  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6  34  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Technical results and their proof  36  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The convergence of Mγ  ε as a distribution  39  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Beyond star-scale invariance  43  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1  47  References  49  1  2  HUBERT LACOIN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Introduction  Let K : Rd ˆ Rd Ñ p´8, 8s be a positive definite kernel on Rd (d ě 1 is fixed) which  admits a decomposition of the form  Kpx, yq “ log  1  |x ´ y| ` Lpx, yq,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  (with the convention logp1{0q “ 8) where L is a continuous function on R2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' A kernel  K is positive definite if for ρ P CcpRdq (ρ continuous with compact support)  ż  R2d Kpx, yqρpxqρpyqdxdy ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  Given a centered Gaussian field X with covariance K and γ “ α ` iβ a complex number  (α, β P R) the complex Gaussian Multiplicative Chaos (or complex GMC) with parameter  γ is the random distribution formally defined by the expression  Mγpdxq “ eγXpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  A difficulty comes up when trying to give an interpretation to the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' A field  X with a covariance given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) can be defined only as a random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For a  fixed x P Rd it is not possible to make sense of Xpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The problem of providing a mathematical construction of Mγ that gives a meaning to  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3) was first considered by Kahane in [15] in the case where γ P R, we refer to [25, 27]  for reviews on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The case of γ P C was considered only more recently, see for  instance [11, 12, 13, 16, 18, 19, 20] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The standard procedure to define  the GMC is to use a sequence of approximation of the field X, consider the exponential  of the approximation and then pass to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Mostly two kinds of approximation of X  have been considered in the literature:  (A) A mollification of the field, Xε, via convolution with a smooth kernel on scale ε,  (B) A martingale approximation, Xt, via an integral decomposition of the kernel K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  In the present paper we present convergence results for the random distribution eγXεpxqdx  and eγXtpxqdx and in a certain range of parameter γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Before describing our results in more  details and provide some motivation, we first rigorously introduce the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The mollification of a log-correlated field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Log-correlated fields defined as distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Since K is infinite on the diagonal, it is not  possible to define a Gaussian field indexed by Rd with covariance function K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We consider  instead a process indexed by test functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We define pK, a bilinear form on CcpRdq (the  set of compactly supported continuous functions) by  pKpρ, ρ1q “  ż  R2d Kpx, yqρpxqρ1pyqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  Since pK is positive definite (in the usual sense: for any pρiqk  i“1, the matrix pKpρi, ρjqk  i,j“1 is  positive definite), it is possible to define X “ xX, ρyρPCcpRdq a centered Gaussian process  indexed by CcpRdq with covariance kernel given by pK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' There exists a modification of the process X which take value in a distri-  bution space (more specifically, such that X takes values in the Sobolev space Hs  locpRdq for  every s ă 0 (see the definition (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3) in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For this reason (and although we  will not use this fact) we refer to X as a random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 3  Approximation of X via mollification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The random distribution X can be approximated  by a sequence of functional fields - processes indexed by Rd - by the mean of mollification  by a smooth kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Consider θ a nonnegative function in C8  c pRdq (the set of infinitely  differentiable functions in CcpRdq) whose compact support is included in Bp0, 1q (for the  remainder of the paper Bpx, rq denotes the closed Euclidean ball of center x and radius r)  and which satisfies  ş  Bp0,1q θpxqdx “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We define for ε ą 0, θε :“ ε´dθpε´1¨q and consider  pXεpxqqxPRd, the mollified version of X, that is  Xεpxq :“ xX, θεpx ´ ¨qy  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  From (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4), the field Xεp¨q has covariance  Kεpx, yq :“ ErXεpxqXεpyqs “  ż  R2d θεpx ´ z1qθεpy ´ z2qKpz1, z2qdz1dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  We set Kεpxq :“ Kεpx, xq and extend this convention to other functions of two variables  in Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Since Kε is infinitely differentiable - thus in particular is H¨older continuous -  by Kolmogorov’s Continuity Theorem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' [21, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9]) there exists a continuous  modification of Xεp¨q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In the remainder of the paper, we always consider the continuous  modification of a process when it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  This ensures that integrals such as the one  appearing in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7) are well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We define the distribution Mγ  ε by setting for f P CcpRdq  Mγ  ε pfq :“  ż  Rd fpxqeγXεpxq´ γ2  2 Kεpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  The question of interest in the present paper is the convergence of Mγ  ε when ε Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that, even if we have chosen to omit this dependence in the notation,  Xε and Mγ  ε both depend on the particular convolution kernel θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' An important feature of  our results is that the limits obtained for Mγ  ε pfq do not depend on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Star-scale invariance and our assumption on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' On top of assuming that K  admits a decomposition like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1), we also assume that it has an almost star-scale invariant  part (see the definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This assumption might seem at first quite restrictive,  but it has been shown in [12] that it is locally satisfied as soon as the function L in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) is sufficiently regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In Appendix C we provides details concerning the regularity  assumption for L and explain how to extend the validity of our results to all sufficiently  regular log-correlated kernels using the ideas in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Following a terminology introduced in [12], we say that a the kernel K defined on Rd is  almost star-scale invariant if it can be written in the form  @x, y P Rd, Kpx, yq “  ż 8  0  p1 ´ η1e´η2tqκpetpx ´ yqqdt,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  where η1 P r0, 1s and η2 ą 0 are constants and the function κ P C8  c pRdq is radial, nonneg-  ative and definite positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' More precisely we assume the following:  (i) κ P C8  c pRdq and there exists rκ : R` Ñ r0, 8q such that κpxq :“ rκp|x|q,  (ii) rκp0q “ 1 and rκprq “ 0 for r ě 1,  (iii) The mapping px, yq ÞÑ κpx ´ yq defines a positive definite kernel on Rd ˆ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We say furthermore that a kernel K has an almost star-scale invariant part, if  @x, y P Rd, Kpx, yq “ K0px, yq ` Kpx, yq  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  where Kpx, yq is an almost star-scale invariant kernel and K0 is H¨older continuous on R2d  and positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  4  HUBERT LACOIN  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Phase transitions and phase diagrams for GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Our main results concerns  the asymptotic behavior of Mγ  ε in the specific range of γ given in the abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In order  to properly motivate and present these results, it is necessary to introduce some context,  and recall known facts about the phase diagram of the complex GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Phase transition at |α| “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d for the real valued GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The question of the existence and  identification of the limit  lim  εÑ0 Mα  ε p¨q,  has first been considered in the work of Kahane in the eighties [15], in the case when  α P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The obtained limit in that case crucially depends on α: when |α| ă  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d - referred  to as the subcritical case - then Mα  ε converges in probability to a non-trivial limiting  distribution (see for instance [2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1] for a short and self contained proof, we  refer to the introduction in [2] for a detailed chronological account of results obtained for  the subcritical case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  When |α| ě  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d, we have limεÑ0 Mα  ε pfq “ 0 and a rescaling procedure is needed in  order to obtain a non-trivial limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The phenomenology is however different according to  whether |α| “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d (α critical) or |α| ą  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d (α supercritical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  In the critical case (α “ ˘  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d), is has been shown, under fairly mild assumptions (see  [4, 5, 10, 26] and Theorem A below) that  a  log p1{εqMα  ε converges in probability to a  non-trivial limit called the critical GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  When |α| ą  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d, the results are less complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' So far the convergence has not been  proved for Mα  ε but only for an approximating martingale sequence Mα  t (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)) in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Besides this technical point, the most important differences with the case |α| ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  concerns the type of the convergence and the nature of limiting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence  only holds only in law, and the limit is a purely atomic measure (a measure supported by  a countable set) see [24, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Phase diagram for complex GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' When γ is allowed to assume complex value, the phase  diagram becomes more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The complex plane can be divided in three open regions  with intersecting boundaries  PI :“  ␣  α ` iβ : α2 ` β2 ă d  (  Y  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  α ` iβ : α P p  a  d{2,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |α| ` |β| ă  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  )  ,  PII :“  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  α ` iβ : |α| ` |β| ą  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |α| ą  a  d{2  )  ,  PIII :“  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  α ` iβ : α2 ` β2 ą d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |α| ă  a  d{2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  This diagram first appeared in the context of complex Gaussian multiplicative cascade  [3], and also serves to describe the behavior of other related models such as the complex  REM [14] or complex branching Brownian Motion [6, 7, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The region PI corresponds to the subcritical phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For γ P PI it has been proved  [12, 18] that Mγ  ε converges to a limit that does not depend on the mollifier θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The region PII corresponds to the supercritical phase, in which it is believed that Mγ  ε  after proper renormalization - converges only in law to a purely atomic random distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This conjecture is supported by rigorous results obtained in the case of Complex  Branching Brownian Motion [6, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 5  PSfrag replacements  β  α  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  d  a  d{2  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  PI  PII  PIII  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The phase diagram of the complex GMC in the complex plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Each region  correspond to a different limiting behavior for M γ  ε in terms of renormalization factor,  type of convergence and properties of the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In the present paper, we prove results  concerning the asymptotic behavior on frontier of PI Y PIII with PII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Results concerning  convergence in PI Y PIII where proved in [12, 18] (for PI) and [16] (for PIII Y PI{III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The  convergence in the region PII remains a challenging conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Finally the region PIII corresponds to yet another asymptotic behavior for Mγ  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Like in  PII, Mγ  ε - properly rescaled - only converges in law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The limit is given by a white noise  whose intensity is random and is given by the real valued GMC with parameter 2α (which  is subcritical, according to the definition of PIII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' A similar convergence result holds on  the boundary between PII and PIII that is  PII{III :“  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  α ` iβ : α2 ` β2 “ d ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' |α| ă  a  d{2  )  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  These convergence statements for γ P PIII Y PI{III are proved in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The present contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The aim of the present paper is to come closer to a completition  of the phase diagram by stating and proving convergence results for Mγ  ε on the phase  transition curves PI{II and PI{III as well as at the triple points PI{II{III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  In each case,  the limit obtained does not depend on the regularization kernel θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We leave as an open  problem the challenging task of proving a convergence result in the frozen phase PII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Main results  For simplicity of notation, we consider, for the remainder of the paper and without loss  of generality that γ is in the upper-right quarterplane of C, that is α, β ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The boundary between phase I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Our first result concerns the case when  γ lies on the boundary between regions I and II  PI{II :“ tα ` iβ : α ą β ą 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' α ` β “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  6  HUBERT LACOIN  Note that our definition of PI{II excludes one point of the boundary which correspond to  Critical Gaussian multiplicative chaos γ “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If X is a centered Gaussian field whose covariance kernel K has an almost  star-scale invariant part, γ P PI{II, and f P CcpRdq then there exists a complex valued  random variable Mγ  8pfq such that for any choice of mollifier θ the following convergence  holds in Lp if p P  ”  1,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{α  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  lim  εÑ0 Mγ  ε pfq “ Mγ  8pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  The above result extends [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2] which established convergence for γ P PI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The method which we use to prove it however, completely differs from the one employed in  [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In fact the method of proof that we employ in Section 4 balso provides an alternative  and much shorter proof of [18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2], with the additional benefit of establishing  convergence in Lp for an optimal range of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have chosen to denote the limit by Mγ  8 rather than Mγ  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' While the  latter may seem a more natural choice, it is already in use for the initial condition of the  martingale GMC approximation introduced in Section 3 (see for instance (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have chosen to put the emphasis on the proof of the convergence of  Mγ  ε pfq for all fixed f, but it is true also that Mγ  ε p¨q converges as a random distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  More precisely the convergence (in probability) of Mγ  ε in a local Sobolev space of negative  index can in fact be deduced from the estimates obtained in the proof of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We include  the argument in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The boundary between phase II and III, and the triple point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Our second  result concerns the case when γ P P1  II{III where  PII{III :“ t  a  d{2 ` iβ : β ą  a  d{2u,  P1  II{III :“ PII{III Y t  a  d{2p1 ` iqu  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  In that case Mγ  ε needs to be rescaled in order to obtain a non-trivial limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence  holds only in law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To describe the limit we need to introduce two notions: Critical Gaussian  Multiplicative Chaos, and Gaussian White Noise with a random intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Critical GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' As explained in the introduction critical Gaussian Multiplicative Chaos  is obtained as the limit of Mα  ε when α “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The value  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d represent a threshold  for the convergence of Mα  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence result below follows from a combination of  [5, Theorem 5] - which establishes the convergence for the martingale sequence Mα  t (see  Section 3) and [10, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4] which establish that the limit is the same for  the exponential of the mollified field Mα  ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Alternative concise proofs of these results have  been recently given in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let X be a Gaussian random field with an almost-star scale invariant kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  There exists a locally finite random measure M1 with dense support and no atoms such  that for every f P CcpRdq the following convergence holds in probability  lim  εÑ0  c  π log p1{εq  2  M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ε  pfq “ M1pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that we have set different conventions and that our M1 differs from  that in [5, Theorem 5] by a factor  b  2  π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 7  Complex white noise with random intensity given by a Real GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For γ P P1  II{III we  define Mγ to be a complex white noise with intensity measure given by M1pe|γ|2L¨q It is a  random linear form which is constructed jointly with X, on an extended probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Conditionally on X, for f P C8  c pDq, Mγpfq is a complex Gaussian random variable, with  independent real and imaginary parts, both with a variance equal to  M1pe|γ|2Lf 2q “  ż  D  e|γ|2Lpx,xqfpxq2M1pdxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Formally, letting P and P denote respectively the law of X and the joint law of pX, Mγp¨qq,  Mγp¨q is the random process indexed by CcpRdq which satisfies for any m, n ě 1, ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' , ρm, f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' , fn P  CcpRdq and any bounded measurable function F on Cn`m  E  “  F  `  pxX, ρiyqm  i“1, pMγpfjqqn  j“1  ˘‰  “ E b En  “  F  `  pxX, ρiyqm  i“1, Σrγ, X, pfjqn  j“1s ¨ Nn  ˘‰  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  where under Pn, Nn is an n dimensional vector whose coordinate are IID standard com-  plex Gaussian variables, and Σrγ, X, pfjqn  j“1s is the positive definite square root of the  Hermitian matrix  ´  M1pe|γ|2Lfif jq  ¯n  i,j“1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us define the function ℓθ on Rd, obtained by convoluting z ÞÑ log 1{|z|  twice with θ, that is  ℓθpzq :“  ż  Rd log  ˆ  1  |z ` z1 ´ z2|  ˙  θpz1qθpz2qdz1dz2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  and set (recall (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3))  vpε, θ, γq :“  $  ’  &  ’  %  p2π logp1{εqq´1{4ε  d´|γ|2  2  ´ş  Rd e|γ|2ℓθpzqdz  ¯1{2  if γ P PII{III,  a  Σd´1  ´  2 logp1{εq  π  ¯1{4  if γ “  a  d{2pi ` 1q  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  where Σd is the volume of the d ´ 1 dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that limεÑ0 vpε, θ, γq “ 8  in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let X be a Gaussian random field with an almost star-scale covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Then given γ P P1  II{III, we have the following joint convergence in law  ˆ  X,  Mγ  ε  vpε, θ, γq  ˙  εÑ0  ñ pX, Mγq,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8) implies that vpε, θ, γq´1Mγ  ε does not converge  in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' On the heuristic level, this can be explained as follows: The white noise  that appears in the limit is the product of local fluctuations of Xε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' These fluctuations are  produced by high frequencies in the Fourier spectrum of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The set of frequencies that  produce the fluctuations diverges to infinity when ε Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This means that the randomness  that produces the white noise become asymptotically independent of X in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8) means that for any collection pρiqm  i“1 and pfjqn  j“1 we  have the convergence in law of the Cm`n valued vector  lim  εÑ0  ˜  pxX, ρiyqm  i“1,  ˆ Mγ  ε pfjq  vpε, θ, γq  ˙n  j“1  ¸  “  ´  pxX, ρiyqm  i“1, pMγpfjqqn  j“1  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  8  HUBERT LACOIN  The convergence can also be shown to hold in a space of distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' More precisely, there  exists a modification of the process Mγ taking values in the local Sobolev space H´u  loc pRdq  with u ą d{2 and Mγ  ε pfjq  vpε,θ,γq converges in law in that space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' See Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Since both X and Mγ  ε are linear forms, the convergence of finite dimensional marginals  follows from that of one dimensional marginals (this can simply be checked using Fourier  transform and L´evy Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' More precisely, we only need to prove the convergence for  every f P CcpRdq and ω P r0, 2πq of the real valued variable (Re denotes the real part)  Mγ  ε pf, ωq :“ Re  `  e´iωMγ  ε pfq  ˘  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  Hence Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 can be reduced to the proof of the following statement  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Under the assumption of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5, given ρ, f P CcpRdq, ω P r0, 2πq,  we have  lim  εÑ0 E  „  eixX,ρy`i Mγ  ε pf,ωq  vpε,θ,γq  \uf6be  “ E  „  eixX,ρy´ 1  2M1pe|γ|2L|f|2q  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  The r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11) of corresponds to the Fourier transform of pX, Mγq (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5))  E  „  eixX,ρy´ 1  2 M1pe|γ|2L|f|2q  \uf6be  “ E  ”  eixX,ρy`iMγpfqı  ,  and the convergence of the Fourier transform implies that of finite dimensional marginals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  More detailed justifications are exposed in [16, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The martingale approximation for GMC  Before getting to the technical core of the paper, we need one more introductory section  to present an essential tool which is used in the proof of both Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 and Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5: the martingale decomposition of the field X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Under the almost star-scale assumption  for K, besides mollification, there is another natural way to approximate the log-correlated  field X by a smooth field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Extending the probability space, one can define a martingale  sequence of smooth fields pXtqtě0 that converges to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  This allows for another approach to the construction of GMC, considering the exponen-  tial of the martingale approximation of X (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)) which we call Mγ  t (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2  concerning the conflict of notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Convergence results for Mγ  t which are a analogous  to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 are also presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 we introduce an  important technical tool which is used to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The result (a central limit  Theorem proving convergence to a Gaussian with random variance) may find applications  in other context, so it is stated in a rather general setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The martingale decomposition of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given K with an almost star-scale invariant  part, and using the decomposition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8) for K, we set Qtpx, yq :“ κpet1px ´ yqq where t1 is  defined as the unique positive solution of  t1 ´ η1  η2  p1 ´ e´η2t1q “ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  We set  Ktpx, yq :“ K0px, yq `  ż t  0  Qspx, yqds  “ K0px, yq `  ż t1  0  p1 ´ η1e´η2sqκpespx ´ yqqds “: K0px, yq ` Ktpx, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES 9  Note that we have limtÑ8 Ktpx, yq “ Kpx, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We define pXtpxqqxPRd,tě0 to be a centered  Gaussian field with covariance given by (using the notation a ^ b :“ minpa, bq, a _ b :“  maxpa, bq)  ErXtpxqXspyqs “ Ks^tpx, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  Since ps, t, x, yq ÞÑ Ks^tpx, yq is H¨older continuous, the field admits a continuous modifi-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We let Ft :“ σ  ´  pXspxqqxPRd,sPr0,ts  ¯  denote the natural filtration associated with  X¨p¨q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The process X indexed by CcpRdq and defined by xX, ρy “ limtÑ8  ş  Rd Xtpxqρpxqdx,  is a centered Gaussian field with covariance, so that Xt is an approximation sequence for  a log-correlated field with covariance K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We also define X¨ :“ X¨ ´ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  we have  ErXtpxqXspyqs “ Ks^tpx, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  An important observation is that since Ktpxq :“ Ktpx, xq “ t, for any fixed x P Rd, the  process pXtpxqqtě0 is a standard Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We also introduce the field Xt,ε which  is the mollification of Xt, that is  Xt,εpxq :“  ż  Rd θεpx ´ zqXtpzqdz “ E rXεpxq | Fts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We let Kt,εpx, yq denote the covariance of the field Xt,ε and Kt,ε,0px, yq the cross-covariance  of Xt,ε and Xt  Kt,εpx, yq :“ ErXt,εpxqXt,εpyqs “  ż  Rd θεpx ´ z1qθεpy ´ z2qKtpz1, z2qdz1dz2,  Kt,ε,0px, yq :“ ErXt,εpxqXtpyqs “  ż  Rd θεpx ´ zqKtpz, yqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  The quantity Kt,ε is defined similarly and we use the notation Qt,ε and Qt,ε,0 the corre-  sponding mollified versions of Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The martingale approximation for the GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We define the distribution Mγ  t  by setting for f P CcpRdq  Mγ  t pfq :“  ż  Rd fpxqeγXtpxq´ γ2  2 Ktpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  Using the independence of the increments of X, it is elementary to check that Mtpfq is an  pFtq-martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We also define  Mγ  t,εpfq :“  ż  Rd fpxqeγXt,εpxq´ γ2  2 Kt,εpxqdx “ E rMγ  ε pfq | Fts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' A few properties of the covariance kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We introduce some technical nota-  tion and estimates that are going to be of use throughout the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us first not that  if a kernel K has an almost star-scale invariant part then it can be written in the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Indeed, if K satisfies (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8) then the function L defined for x ‰ y by  Lpx, yq :“ Kpx, yq ` log |x ´ y|,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  can be extended to a continuous function on R2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that we have  Lpxq “ lim  yÑx pKpx, yq ` log |x ´ y|q “ K0pxq ´ j  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  10  HUBERT LACOIN  where the difference term j does not depend on x and can be computed explicitely  j :“ lim  zÑ0  `  logp1{|z|q ´ Kp0, zq  ˘  “ η1  η2  `  ż 8  0  `  1 ´ rκpe´sq  ˘  ds ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  The above comes from the fact that  logp1{|z|q ´ Kp0, zq “  ż logp1{|z|q  0  p1 ´ Qlogp1{|z|q´up0, zqqdu  and the fact that the integrand on the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' converges to 1 ´ rκpe  η1  η2 ´sq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Lastly one can  observe that the following identity holds  ℓpzq :“ lim  tÑ8  `  Ktp0, e´tzq ´ t  ˘  “ lim  tÑ8  ż t  0  pκpes1´tzq ´ 1qds “  ż 8  0  pκpe  η1  η2 ´uzq ´ 1qdu, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  where in the integral in s, s1 is related to s via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To obtain the third equality, one  simply observe that s1 “ s ` η1{η2 ` op1q in the large s limit and make the change of  variable u “ t ´ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that ℓpzq is a continuous negative function and that for any  |z| ě e´ η1  η2 we have ℓpzq “ log 1  |z| ´ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  To conclude this subsection, we gather in a a technical lemma a couple of useful estimates  concerning Kt, Qt and their variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given R ą 0, there exists a constant CR such that for any x, y P Bp0, Rq,  t ą 0 and ε P r0, 1s  ˇˇˇˇKt,εpx, yq ´ log  ˆ  1  maxpe´t, ε, |x ´ y|q  ˙ˇˇˇˇ ď CR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  The bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) remains valid with Kt,ε replaced by Kt (with ε “ 0), Kt,ε,0, Kt,ε etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We also have ż  Rd Qtpx, yq “  ż  Rd Qt,εpx, yqdy “  ż  Rd Qt,ε,0px, yqdy ď Ce´dt,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  and  0 ď t ´ Kt,εpx, yq ď C  `  etp|x ´ y| ` εq  ˘2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  The estimates above can be proved rather directly from the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' A detailed proof  of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) is provided in [17, Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The bound (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13) follows directly from the  definition of Qt given above (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) and the fact that |t´t1| is uniformy bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The upper  bound in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14) can be obtained by integrating (in time and space) the inequality  1 ´ Qtpz1, z2q ď Cret1|z1 ´ z2|s2  which follows directly from the Taylor expansion at second order of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' There is an obvious conflict of notation between Kt introduced above and  Kε introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6) and the same can be said about Xt and Mγ  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This should not cause  any confusion since we keep using the letter ε for quantities related to the mollified field  Xε and latin letters for quantities related to the martingale approximation Xt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Convergence results for the martingale approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' An intermediate step  to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 is to show that similar results hold for the mar-  tingale approximation Mγ  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' These results present of course an interest in their own right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES11  The case of γ P PI{II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' When γ P PI{II, the martingale Mγ  t pfq is bounded Lp for p P r1,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{|α|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  As a consequence the limit  lim  tÑ8 Mγ  t pfq “: Mγ  8pfq  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15)  exists almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence holds in Lp and the limit is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The martingale limit in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15) is the same as the limit of Mγ  ε appearing in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 (this is the reason why we use the same notation), we have  lim  tÑ8 Mγ  t pfq “ lim  εÑ0 Mγ  ε pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16)  This observation is important, since it establishes that the limit in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) does not depend  on the choice of the mollifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The case γ P P1  II{III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In order to state the convergence in law result for Mγ  t , we need to  introduce a normalization factor vpt, γq (analogous to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7) for the mollified case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us  set  vpt, γq “  $  &  %  e  |γ2|j  2  ` 1  2πt  ˘1{4 e  p|γ|2´dqt  2  ´ş  Rd e|γ|2ℓpzqdz  ¯1{2  ,  if |γ|2 ą d  a  Σd´1  ` 2t  π  ˘1{4  if |γ|2 “ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)  and define  Mγ  t pf, ωq :“ Re  `  e´iωMγ  t pfq  ˘  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18)  The following analogue of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If X is an almost star-scale invariant field and γ P P1  II{III we have for  any ρ, f P CcpRdq  lim  tÑ0 E  „  eixX,ρy`i  Mγ  t pf,ωq  vpt,γq  \uf6be  “ E  „  eixX,ρy´ 1  2 M1pe|γ|2L|f|2q  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19)  As a consequence we have the following convergence in law (in the sense of finite dimen-  sional marginals)  ˆ  X,  Mγ  t  vpt, γq  ˙  tÑ8  ùñ  pX, Mγq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' CLT towards a Gaussian with random variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We conclude this section by  introducing a technical results which is essential to prove the convergence of a sequence of  variable towards a Gaussian with random intensity in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We provide the result  and its proof in a reasonably high level of generality since it may find application in other  contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Consider pFtqtě0 a filtration and pWnqně1 a sequence of real valued random variables  in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We introduce for each n ě 1 the martingale  Wn,t :“ E rWn | Fts .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21)  We assume that the martingale Wn,t admits a modification which is continuous in t for  every n ě 1 We prove that Wn converges to to a Gaussian with random variance if the  quadratic variation of pWn,tqtě0 satisfy a law of large number and a couple of additional  technical assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The result generalizes a similar CLT established for a single mar-  tingale process (see [9, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='50, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' VIII-Section 5c] or [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  12  HUBERT LACOIN  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us assume that and that there exists a non-negative valued random-  variable Z which is such that the three following convergences in probability hold  lim  nÑ8  xWny8  v2pnq “ Z,  @t ě 0 lim  nÑ8  xWnyt  v2pnq “ 0, and lim  nÑ8  Wn,0  vpnq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)  Then Xn{vpnq converges in distribution towards a random Gaussian with variance given  by Z, that is to say that for any F8 bounded measurable H we have  lim  nÑ8 E  ”  HeiξWn{vpnqı  “ lim  nÑ8 E  „  He´ ξ2Z  2  \uf6be  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23)  This is equivalent to saying that for any F8 random variable Y we have the following  convergence in law  pY, Wnq ùñ pY,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ZNq  where N is a standard Gaussian which is independent of Z and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We believe that with adequate assumption on the size of the jumps, the  result may extend to the case where pWn,tqtě0 is a c`ad-l`ag martingale, with the quadratic  variation is replaced by the predictable bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Since we have no application in that setup,  we restricted ourselves to the continuous case where the proof is technically simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In Section 6 we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6 for a sequence of variables indexed by  ε P p0, 1q (namely Mγ  ε pf, ωq) in the limit when ε Ñ 0 rather than n ě 1 and n Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  These setups are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Organization of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The remainder of the paper is organized as follows  ‚ In Section 4 we prove all the statements concerning convergence in PI{II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 is devoted to the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The more technical proof of Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1, which uses Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 as in imput is displayed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ The statements concerning γ P P1  II{III, namely Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 and Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8, while relying on relatively simple ideas, require a certain amount of technical  computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In Section 5 we prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5, in Section 6 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ In Section 7, we present the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  A significant amount of material is presented in appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ In Appendix A, we prove a couple of auxilliary results used in Section 5/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ In Appendix B, we present and prove an extension of our main results, that is,  the convergence of Mγ  ε p¨q as a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' After identifying the right topology,  the proof mostly boils down to repeating the computation made in Section 4 (for  γ P PI{II) and Section 6 (for γ P P1  II{III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ In Appendix C, we explain how our results can be extended to the case of a  (sufficiently regular) log-correlated Gaussian field defined on an arbitrary open  domain D Ă Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ In Appendix D, we present a relatively short proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 for the sake  of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' It the same as the one presented in [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15], except  that we include a short proof of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 instead of relying on the branching  random walk literature where more general results have been shown, albeit with  much longer proofs (see for instance [8, 22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES13  A comment on notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Throughout the paper, we use the letter C for a generic positive  constant when we need to compare two quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' It may depend on some parameters  (for instance on γ or on the kernel K) but never on the variable t or ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The value of C  might change from one equation to the other withing the same proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We use C1 and C2  if we need several constants in the same display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of convergence results on for γ P PI{II  In this section we prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The first one is easier, recall  that due to the martingale property of Mγ  t , it is sufficient to show that the sequence  is bounded in Lp to prove convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This is performed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We rely on  the Burkeholder-Davis-Gundy (BDG) inequality, compute the quadratic variation of the  martingale and studying its moment of order p{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2, we adapt the same method to estimate the Lp norm of Mγ  ε ´ Mγ  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  More precisely the BDG inequality for the martingale pMγ  t,ε ´ Mγ  t qtě0, and show that the  moment of order p{2 of its quadratic variation is uniformly small in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Recalling that  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{α ą 1, we are going to prove that  Mγ  t pfq is bounded in Lp for  p P  ´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  8d{p3αq _ 1,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{α  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  In the whole paper, when Mt is a complex valued continuous martingale, we use the  notation xMyt to denote the the bracket between M and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' It is the predictable process  such that |Mt|2 ´ xMyt is a local martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Burkeholder-Davis-Gundy (BDG)  inequality for Mγ  t pfq, there exists a constant Cp such that for every t ą 0  E  “  |Mγ  t pfq|p‰  ď Cp  ´  ErxMγpfqyp{2  t  s ` E r|Mγ  0 pfq|ps  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  We have  E  “  |Mγ  0 pfq|2‰  “  ż  R2d e|γ|2K0px,yqfpxqfpyqdxdy ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  Since p ă 2 by assumption, Jensen’s inequality implies that E r|Mγ  0 pfq|ps ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Itˆo  calculus, we obtain an explicit expression for the quadratic variation  xMγpfqy8 “ |γ|2  ż 8  0  Atdt  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  where  At :“  ż  R2d fpxqfpyqQtpx, yqeγXtpxq`γXtpyq´ γ2  2 Ktpxq´ γ2  2 Ktpyqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  Note that At is real and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2), we deduce that Mγ  t pfq is bounded in Lp if  E  “  p  ş8  0 Atdtqp{2‰  ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To bound At from above, we take the modulus of the integrand in  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5) and using the assumption that β “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d ´ α (γ P PI{II) we obtain that  At ď  ż  R2d |fpxqfpyq|Qtpx, yqeαpXtpxq`Xtpyqq` 2d´2  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dα  2  pKtpxq`Ktpyqqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  Then using the inequality ab ď a2  2 ` b2  2 with  a “ |fpxq|eαXtpxq` 2d´2  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dα  2  Ktpxq  and  b “ |fpyq|eαXtpyq` 2d´2  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dα  2  Ktpyq  14  HUBERT LACOIN  and symmetry in x and y, we have  At ď  ż  R2d |fpxq|2Qtpx, yqe2αXtpxq`p2d´2  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dαqKtpxqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  We use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13) to integrate over y and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) to replace replace Ktpxq by t (at the cost of  multiplicative constant) and we have  At ď Cedt  ż  Rd |fpxq|2e2αpXtpxq´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dtqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  Now, as α ą  a  d{2, we have by p{2 ă 1 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We can use thus the following  inequality (valid for an arbitrary collection of positive real numbers paiqiPI and q P p0, 1q)  ˜ÿ  iPI  ai  ¸q  ď  ÿ  iPI  aq  i ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  with q “ p{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In the remainder of the paper, we simply say “by subadditivity” when using  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) and Jensen’s inequality we have  E  «ˆż 8  0  Atdt  ˙p{2ff  ď  ÿ  ně0  E  «ˆż n`1  n  Atdt  ˙p{2ff  ď  ÿ  ně0  E  «ˆż n`1  n  ErAs | Fnsds  ˙p{2ff  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  Averaging with respect to pXs ´ Xnq we obtain from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  ż n`1  n  E rAs | Fns ď Cedn  ż  R2d |fpxq|2e2αpXnpxq´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dnqdx “: CBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  As p ą  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  8d{3α by assumption, we can conclude using the estimate in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 below  for the fractional moments of Bn (the assumption on p makes the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) summable  in n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' More precisely, we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10),(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) that E  ”`ş8  0 Atdt  ˘p{2ı  ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For α ą  a  d{2 and p ă  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{α we have  E  ”  Bp{2  n  ı  ď Cn´ 3αp  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  8d plog nq6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  This result is a weaker version of [20, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We provide, for the commodity of the  reader a self-contained of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 in the setup where our probability  space contains a martingale approximation pXtqtě0 of the field X with covariance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  More precisely we show that Mγ  ε pfq converges to the same limit as Mγ  t pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Working in  an enlarged probability space entails by no mean a loss of generality since the validity of  the statement “the sequence pMγ  ε pfqqεPp0,1s is Cauchy in Lp” is entirely determined by the  distribution of pXεpxqqxPRd,εPp0,1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given γ P PI{II and p P r1,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{αq we have  lim  εÑ0 sup  tą0  E  “  |pMγ  t ´ Mγ  t,εqpfq|p‰  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  As a consequence the following convergence holds in Lp  lim  εÑ0 Mγ  ε pfq “ Mγ  8pfq  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES15  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us first show indicate how (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14) follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We observe that  E  “  |pMγ  t,ε ´ Mγ  ε qpfq|2‰  “  ż 8  0  fpxqfpyq  ´  e|γ|2Kεpx,yq ´ e|γ|2Kt,εpx,yq¯  dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15)  Since limtÑ8 Kt,εpx, yq “ Kεpx, yq, using dominated convergence the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' tends to 0 when  t Ñ 8 and thus limtÑ8 Mγ  t,εpfq “ Mγ  ε pfq in L2, and hence also in Lp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Proposition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 we thus have the following convergence in Lp  lim  tÑ8pMγ  t ´ Mγ  t,εqpfq “ pMγ  8 ´ Mγ  ε qpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Taking the limit when ε to zero, we obtain that  lim  εÑ0 E r|pMγ  8 ´ Mγ  ε qpfq|ps “ lim  εÑ0 lim  tÑ8 E  “  |pMγ  t ´ Mγ  t,εqpfq|p‰  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16)  and we conclude using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  To prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13), we assume that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Then using the BDG inequality (we omit the  dependence in f for ease of reading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have for every t ě 0  Er|Mγ  t ´ Mγ  t,ε|ps ď CpErxMγ ´ Mγ  ¨,εyp{2  8 ` |Mγ  0 ´ Mγ  0,ε|ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)  The reader can then check by an explicit calculation of the second moment that  lim  εÑ0 E  ”  |Mγ  0 pfq ´ Mγ  0,εpfq|pı  ď lim  εÑ0 E  ”  |Mγ  0 pfq ´ Mγ  0,εpfq|2ıp{2  “ 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18)  Hence in view of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18), to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13) we need to show that  lim  εÑ0 ErxMγ ´ Mγ  ¨,εyp{2  8 s “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19)  Expanding the product, using Itˆo calculus (Re denotes the real part) we obtain  xMγ ´ Mγ  ¨,εy8 “ |γ|2  ż 8  0  ´  At ´ 2Re  ´  Ap1q  t,ε  ¯  ` Ap2q  t,ε  ¯  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20)  where, At is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8), and recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5), Ap1q  t,ε and Ap2q  t,ε are defined by  Ap1q  t,ε :“  ż  R2d fpxqfpyqQt,ε,0px, yqeγXtpxq`γXt,εpyq´ γ2  2 Ktpxq´ γ2  2 Kt,εpyqdxdy,  Ap2q  t,ε :“  ż  R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2  2 Kt,εpxq´ γ2  2 Kt,εpyqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21)  We are going to reduce the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19) to that of two convergence statements concerning  Apiq  t,ε for i P t1, 2u (the first being valid for any fixed r ą 0)  lim  εÑ0 sup  tPr0,rs  E  ”  |At ´ Apiq  t,ε|  ı  “ 0  for i P t1, 2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)  lim  rÑ8 sup  εPp0,1s  E  «ˆż 8  r  |Apiq  t,ε|dt  ˙p{2ff  “ 0  for i P t1, 2u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23)  Before proving (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23) let us explain how (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19) is deduced from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23)  is also valid for At (this can be extracted from the proof in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given δ ą 0,  16  HUBERT LACOIN  using subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23) we can find rδ such that for every ε ą 0  E  «ˆż 8  rδ  ´  At ´ 2Re  `  Ap1q  t,ε  ˘  ` Ap2q  t,ε  ¯  dt  ˙p{2ff  ď E  «ˆż 8  rδ  Atdt  ˙p{2  `  ˆż 8  rδ  2|Ap1q  t,ε |dt  ˙p{2  `  ˆż 8  rδ  Ap2q  t,ε dt  ˙p{2ff  ď δ{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24)  Now using first Jensen’s inequality and then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22) (recall that At is real valued) we can  find εδ such that for every ε P p0, εδq  E  «ˆż rδ  0  ´  At ´ 2Re  `  Ap1q  t,ε  ˘  ` Ap2q  t,ε  ¯  ds  ˙p{2ff  ď  ˆż rδ  0  E  ”  At ´ 2Re  `  Ap1q  t,ε  ˘  ` Ap2q  t,ε  ı  ds  ˙p{2  ď  ˆż rδ  0  E  ”  2|At ´ Ap1q  t,ε | ` |Ap2q  t,ε ´ At|  ı  ds  ˙p{2  ď δ{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='25)  Using subadditivity again we deduce from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='25) that if ε P p0, εδq we have  E  «ˆż rδ  0  ´  At ´ 2Re  `  Ap1q  t,ε  ˘  ` Ap2q  t,ε  ¯  dt  ˙p{2ff  ď δ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26)  which (recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20)) concludes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us now prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The proof (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22) follows from a rather pedestrian but rather cumbersome computation  of the L2 norm of pAt ´ Apiq  t,εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The following lemma summarizes the key points of this  computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Consider the following:  ‚ Let pX, µq be a measured space and T be a set of indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ Let Zt,εp¨q, t P T , ε P p0, 1s be a collection of complex valued Gaussian processes  defined on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set  Gt,εpx, yq :“ ErZt,εpxqZt,εpyqs  and  Ht,εpx, yq :“ ErZt,εpxqZt,εpyqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='27)  ‚ Let Zt be defined on the same probability space in such a way that pZt, Zt,εq is  jointly Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We let Gt and Ht be defined as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='27) and set  Ht,ε,0px, yq :“ ErZt,εpxqZtpyqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ Let gt,ε and gt be deterministic functions X Ñ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We assume that:  (i) The covariance functions are uniformly bounded, that is  sup  tPT  εPp0,1s  sup  x,yPX  max pHt,εpx, yq, Htpx, yq, Ht,ε,0px, yqq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (ii) There exists a µ-integrable function h such that for every t P T and ε P p0, 1s  @x P X,  maxp|gt,εpxq|, |gtpxq|q ď hpxq  (iii) That for every t P T , we have the following pointwise convergence  lim  εÑ0 gt,ε “ gt,  and  lim  εÑ0 Ht,ε “ lim  εÑ0 Ht,ε,0 “ Ht  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='28)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES17  Then setting  Wt,ε :“  ż  X  gt,εpxqeZt,εpxq´ 1  2 Gt,εpxqµpdxq  and  Wt :“  ż  X  gtpxqeZtpxq´ 1  2Gtpx,xqµpdxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We have  lim  εÑ0 sup  tPT  E  “  |Wε ´ Wt,ε|2‰  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='29)  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The proof is actually much shorter than the statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have  E  “  |Wt,ε ´ Wt|2‰  “  ż  X 2  ˆ  gt,εpxqgt,εpyqeHt,εpx,yq ´ 2Re  ´  gt,εpxqgtpyqeHt,ε,0px,yq¯  ` gtpxqgtpyqeHtpx,yq  ˙  µpdxqµpdyq  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='30)  and using our assumptions we can apply dominated convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  Proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We consider the case i “ 2 but the other one is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set X “ R2d,  µ is Lebesgue measure, T “ r0, rs, and  Zt,εpx, yq “ γXt,εpxq ` γXt,εpyq,  Ztpx, yq “ γXtpxq ` γXtpyq,  gt,εpx, yq “ Qt,εpx, yqfpxqfpyqe|γ|2Kt,εpx,yq,  gtpx, yq “ Qtpx, yqfpxqfpyqe|γ|2Ktpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Then the assumptions of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 are immediate to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  We now provide the details for the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23) i “ 2 (the case i “ 1 is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let  us set n0pεq “ rlogp1{εqs and assume (without loss of generality) that r is an integer and  is smaller than n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using - as in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 - subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) and  Jensen’s inequality we obtain  E  «ˆż 8  r  |Ap2q  t,ε |ds  ˙p{2ff  ď  n0  ÿ  n“r  E  «ˆż n`1  n  |Ap2q  t,ε |dt  ˙p{2ff  ď  n0´1  ÿ  n“r  E  «ˆż n`1  n  E  ”  |Ap2q  t,ε | | Fn  ı  dt  ˙p{2ff  ` E  «ˆż 8  n0  E  ”  |Ap2q  t,ε | | Fn0  ı  dt  ˙p{2ff  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='31)  Proceeding as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11), we obtain that if t P rn, n ` 1q, n P �r, n0 ´ 1�, or t ě n0, n “ n0,  we have (using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 to replace the covariance Kn,εpxq by n)  E  ”  |Ap2q  t,ε | | Fn  ı  ď C  ż  R2d |fpxq|2Qs,εpx, yqe2αpXn,εpxq´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dnq`2dndxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='32)  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13) to integrate over y and setting  Bp2q  n,ε :“  ż  Rd |fpxq|2e2αpXn,εpxq´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dnq`dndx  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='33)  we obtain that  E  «ˆż 8  r  |Ap2q  t,ε |ds  ˙p{2ff  ď C  n0  ÿ  n“r  E  ”  pBp2q  n,εqp{2ı  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='34)  18  HUBERT LACOIN  Using Jensen’s inequality for the probability θεpy ´ xqdy, we can replace the mollification  acting on Xn in the exponential by one acting of |f|2, we have  e2αXn,εpxq ď  ż  Rd θεpx ´ yqe2αXnpyqdy  which after multiplying by |fpxq|2 and integrating with respect to x implies that  Bp2q  n,ε ď  ż  D  `  |f|2 ˚ θε  ˘  pyqe2αpXnpyq´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dnq`dndy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='35)  Since |f|2˚θε ď }f}2  81t|x|ďR`1u if f is supported in Bp0, Rq, we can conclude using Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1, that  E  ”  pBp2q  n,εqp{2ı  ď Cn´ 3αp  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  8d  for a constant which does not depend on ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Recalling that p ą  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  8d{3α (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)) we  obtain combining(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='32), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='34) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='35) that  E  «ˆż 8  r  |Ap2q  t,ε |ds  ˙p{2ff  ď Cr1´ 3αp  2  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='36)  This concludes the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23), and thus of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Reduction to a statement concerning the total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using [16, Theorem  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5] (which is a simpler version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6 displayed above) we can reduce the proof  of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19) to the following convergence statement about the quadratic variation of the  martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have the following  lim  tÑ8 vpt, γq´2xMγpf, ωqyt “ M1pe|γ|2L|f|2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We simply apply [16, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5] to the  martingale Mγ  t pf, ωq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  Setting, for notational simplicity Wt :“ Mγ  t pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Recall that for a complex value mar-  tingale such as Wt we use the notation xWyt for the bracket between W and its conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Using bilinearity of the martingale brackets we have  xMγpf, ωqyt “ 1  2  `  xWyt ` Repe´2iωxW, Wytq  ˘  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  Hence to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1), it is sufficient to prove that following convergences hold in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  lim  tÑ8 vpt, γq´2xWyt “ 2M1pe|γ|2L|f|2q,  lim  tÑ8 vpt, γq´2xW, Wyt “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  The expression for the bracket of Wt can be obtained by using Itˆo calculus (recall (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4))  More precisely we have  xWyt “ |γ|2  ż t  0  Asds  and  xW, Wyt “ γ2  ż t  0  Bsds,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES19  where At is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5) and  Bt :“  ż  R2d fpxqfpyqQtpx, yqeγpXtpxq`Xtpyqq´ γ2  2 pKtpxq`Ktpyqqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  Now using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4) our first idea is to deduce (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3) from a convergence statement concerning  At and Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' A really important point here is that while At, properly rescaled, converges  to M1pe|γ|2Lfq in probability, this type of convergence is not sufficient to say something  about the integral  şt  0 Asds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' A convenient framework to work with integrals is L1 conver-  gence, but the issue we encounter is that At certainly does not converge in L1 (we have  E  ”  |M1pe|γ|2Lfq|  ı  “ 8 when f is non trivial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  To bypass this problem, restrict ourselves to likely family of event and prove L1 convergence  for the restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Recalling the definition of X (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4), given q ě 0 and R ą 0, t ě 0 and  x P Rd we introduce the events  At,qpxq :“  "  max  sPr0,tspXspxq ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dsq ă q ,  Aq,R :“  #  sup  sě0,|x|ďR  pXspxq ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dtq ă q  +  “  č  xPBp0,Rq  tě0  At,qpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  A very important fact, which is a direct consequence of [4, Proposition 19] (see also [17,  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4] for a concise proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have for any fixed R ą 0  lim  qÑ8 P rAq,Rs “ 1  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  We introduce (we drop the dependence in γ in most displays to make them easier to read)  φptq “ φpt, γq :“  d  2  πpt _ 1qe|γ2|j  ˆż  Rd Qtp0, zqe|γ2|Ktp0,zqdz  ˙  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  which plays the role of a rescaling function for At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Our main technical result in this section  is the proof that At{φptq converges in L1 towards M1pe|γ|2L|f|2q after restriction to the  event Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The following convergences hold for any q ě 0 and any R such that  Supppfq Ă Bp0, Rq  lim  tÑ8 E  ”  |At{φptq ´ M1pe|γ|2L|f|2q|1Aq,R  ı  “ 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  lim  tÑ8 E  “  |Bt{φptq| 1Aq,R  ‰  “ 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  and the above quantities are finite for every t ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  To show that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 implies the convergence stated in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1, we need  to ensure that the rescaling by φptq matches that proposed for xWyt (which is vpt, γq2)  after integrating with respect to time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This is the purpose of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have for any |γ| ě d  lim  tÑ8  |γ|2 şt  0 φpsqds  2vpt, γq2  “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  20  HUBERT LACOIN  The proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4 is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Note that the goal of the  lemma is only to obtain a more presentable expression for vpt, γq since without it, we can  still prove that Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 and hence Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 are valid with v replaced by  vpt, γq :“ |γ|  b  p  şt  0 φpsqdsq{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' As we have seen, it is sufficient to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We provide the  details concerning the convergence of xWyt (the first line in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)) but that of xW, Wyt can  be obtained exactly in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4) and Jensen’s inequality we have  E  «ˇˇˇˇˇ  xWyt  |γ|2 şt  0 φpsqds  ´ M1pe|γ|2L|f|2q  ˇˇˇˇˇ 1Aq,R  ff  ď  şt  0 φpsqE  ”ˇˇˇ As  φpsq ´ M1pe|γ|2L|f|2q  ˇˇˇ 1Aq,R  ı  ds  şt  0 φpsqds  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  Observing that  ş8  0 φpsqds “ 8, the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) is simply a weighted Cesaro mean and  thus we deduce from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 and more precisely from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) that  lim  tÑ8 E  «ˇˇˇˇˇ  xWyt  |γ|2 şt  0 φpsqds  ´ M1pe|γ|2L|f|2q  ˇˇˇˇˇ 1Aq,R  ff  “ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  Since this holds for every q ą 0 we obtain that the following convergence holds in proba-  bility (the replacement of |γ|2 şt  0 φpsqds by 2vpt, γq2 simply comes from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) that  lim  tÑ0  ˇˇˇˇ  xWyt  2vpt, γq2 ´ M1pe|γ|2L|f|2q  ˇˇˇˇ 1Ť  qě1 Aq,R “ 0  which, since the event in the indicator has probability one (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) is the desired  conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Restricted convergence in L2 for the critical GMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Before starting the proof  of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3, we recall a result which play a key role in the proof, the L2 convergence  of M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  t  pgq towards M1pgq when considering the restriction to the event Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This also  implies convergence in L1 which is what we require for the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The  result can be deduced from the L2 convergence of the truncated version of M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  t  pgq which  is proved in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have for any g in CcpRdq such that Supppgq Ă Bp0, Rq and any q ą 0  lim  tÑ8 E  »  –  ˇˇˇˇˇ  c  πt  2 M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  t  pgq ´ M1pgq  ˇˇˇˇˇ  2  1Aq,R  fi  fl “ 0,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  and E  “  |M1pgq|21Aq,R  ‰  ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The fact that E  “  |M1pgq|21Aq,R  ‰  ă 8 is a simple consequence of the convergence  since for any fixed t, Er|M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  t  pgq|2s ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set (recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6))  M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d,pqq  t  pgq :“  ż  gpxqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXtpxq´dKtpxq1At,qpxqdx,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES21  From [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1], there exists an L2 variable D  pqq  8 pgq such that  lim  tÑ8 E  »  –  ˇˇˇˇˇ  c  πt  2 M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d,pqq  t  pgq ´ D  pqq  8 pgq  ˇˇˇˇˇ  2fi  fl “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16)  It satisfies D  pqq  8 pgq “ M1pgq on the event Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' More precisely [17, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1] is only  stated in the special case where g is an indicator function (to keep notation light) but  the proof for g P CcpRdq is identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' On the event Aq,R we have M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d,pqq  t  pgq “ M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  t  pgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Hence  lim sup  tÑ8  E  »  –  ˇˇˇˇˇ  c  πt  2 M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  t  pgq ´ M1pgq  ˇˇˇˇˇ  2  1Aq,R  fi  fl  “ lim sup  tÑ8  E  »  –  ˇˇˇˇˇ  c  πt  2 M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d,pqq  t  pfq ´ D  pqq  8 pgq  ˇˇˇˇˇ  2  1Aq,R  fi  fl “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)  where the last equality follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Organizing the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The two convergences rely on similar  ideas, we focus on (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) which is the more delicate of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The main idea is that since  the integrand in the definition of At  At :“  ż  R2d fpxqfpyqQtpx, yqeγXtpxq`γXtpyq´ γ2  2 Ktpxq´ γ2  2 Ktpyqdxdy,  vanishes when |x ´ y| ě e´t (due to the presence of the multiplicative Qtpx, yq), the value  of the integral should not be much affected much if one changes fpyq, Xtpyq and Ktpyq by  fpxq, Xtpxq and Ktpxq in the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The quantity obtained after this replacement is, up to a multiplicative factor, of the  form M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  t  pgq (recall that γ ` γ “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d) for some function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Hence we should be able to  conclude the proof of the convergence statement using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  While this idea is relatively simple, it requires several steps to be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set  K˚  t px, yq :“ K0pxq ` Ktpx, yq  and  r “ rptq :“ t ´ log log t  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18)  (we are assuming that t ą e so that 0 ď r ď t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We introduce the quantity rAt which will  appear after all our “replacement” steps have been performed, it is defined by  rAt :“  ż  R2d Qtpx, yqe|γ|2K˚  t px,yq|fpxq|2e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxqdxdy  “  ˆż  Rd Qtp0, zqe|γ|2Ktp0,zqdz  ˙ ż  Rd e|γ|2K0ptq|fpxq|2e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxqdx  “ φptq  c  πt  2  ż  Rd e|γ|2Lpxq|fpxq|2e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxqdx “ φptq  c  πt  2 M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  r  pe|γ|2L|f|2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19)  As a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 (since r “ t ´ optq the presence of  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t instead of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='r  does not affect the convergence), we have  lim  tÑ8 E  «ˇˇˇˇˇ  rAt  φptq ´ M1pe|γ|2L|f|2q  ˇˇˇˇˇ 1Aq,R  ff  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20)  22  HUBERT LACOIN  With this observation the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) reduces to showing that  lim  tÑ0  1  φptqE  ”  |At ´ rAt|1Aq,R  ı  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21)  This requires some care but before going in the depth of the proof, let us explain the  heuristic behind (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that rAt is obtained from At with two simple modifications:  ‚ We have replaced fpxqfpyq by |fpxq|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ‚ In the exponential, we have replaced γXtpxq`γXtpyq by  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq “ pγ`γqXrpxq  and ´ γ2  2 Ktpxq ´ γ2  2 Ktpyq by ´dKrpxq ` |γ|2K˚  t px, yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The first modification is rather straightfoward, we are integrating close to the diagonal so  that fpyq is close to fpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the second modification, the idea is that replacing Xtpyq  with Xtpxq (and t with r) should not yield big modifications provided that we change the  normalization to keep the expectation of the exponential unchanged (or almost so).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In  our case we have  E  „  eγXtpxq`γXtpyq´ γ2  2 Ktpxq´ γ2  2 Ktpyq  \uf6be  “ e|γ|2Ktpx,yq,  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxq`|γ|2K˚  t px,yqı  “ e|γ|2K˚  t px,yq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)  and, on the considered domain of integration, K˚  t px, yq and Ktpx, yq are very close since  |x ´ y| ď e´t when Qtpx, yq ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21) requires three distinct steps which  are detailed in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Step 1: Changing the deterministic prefactor in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The integrand of At and  rAt have different expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Our first step aims to fix this by replacing fpyq by fpxq in  At and doing a small modification in the exponential factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set  Ap1q  t  :“  ż  R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq` γ2  2 Ktpxq` γ2  2 Ktpyq`|γ|2pK0pxq´K0px,yqqdxdy (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23)  We are going to prove that  lim  tÑ8 φptq´1E  ”  |At ´ Ap1q  t |1Aq,R  ı  “ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24)  Since f and K0 are uniformly continuous on the support of f and Supppfq Ă Bp0, Rq,  there exists a positive function δ with limtÑ8 δptq “ 0, such that for |x ´ y| ď e´t setting  Fpx, yq :“ fpxqfpyq ´ |fpxq|2e|γ|2pK0pxq´K0px,yqq  we have  |Fpx, yq| ď δptq1Bp0,Rqpxq1Bp0,Rqpyq  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='25)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES23  Hence we obtain (since α “  a  d{2, we have Repγ2q “ d ´ |γ|2)  |At ´ Ap1q  t | “  ˇˇˇˇ  ż  R2d Qtpx, yqFpx, yqeγXtpxq`γXtpyq` γ2  2 Ktpxq` γ2  2 Ktpyqdxdy  ˇˇˇˇ  ď  ż  R2d Qtpx, yq|Fpx, yq|e  b  d  2 pXtpxq`Xtpyqq`p|γ|2´dq Ktpxq`Ktpyq  2  dxdy  ď δptq  ż  Bp0,Rq2 Qtpx, yqe  b  d  2 pXtpxq`Xtpyqq`p|γ|2´dq Ktpxq`Ktpyq  2  dxdy  ď δptq  ż  Bp0,Rq2 Qtpx, yqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXtpxq`p|γ|2´dqKtpxqdxdy  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26)  where the first inequality is simply obtained by taking the modulus of the integrand and  in the third one we simply used  ZpxqZpyq ď 1  2pZpxq2 ` Zpyq2q  with Zpxq “ e  b  d  2 Xtpxq`p|γ|2´dq Ktpxq  2  and then symmetry in x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Then we observe that  (λ denotes the Lebesgue measure) since At,qpxq Ă Aq,R we have  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXtpxq´dKtpxq1Aq,R  ı  ď E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXpxq´dKtpxq1At,qpxq  ı  “ Pr@s P r0, ts, Bs ď qs ď  c  2  πtq  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='27)  where in the last line, we used Cameron-Martin formula (see Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 in the ap-  pendix) and the fact that pXtpxqqtě0 is a standard Brownian Motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The last inequality  is simply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='27) and the fact that K0 is bounded, we  have  E  ”  |At ´ Ap1q  t |1Aq,R  ı  ď Cδptq  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t  ż  Bp0,Rq2 Qtpx, yqe|γ|2Ktpxqdxdy ď C1δptqφptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='28)  □  Step 2: Taking conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Recalling the definition of rptq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18) we set  Ap2q  t  :“ ErAp1q  t  | Frs  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='29)  For this step of the proof (and only this step), we are going to assume that K0 ” 0 (and  hence X0 ” 0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Treating the case where X0 is a non-trivial field does not present any  extra difficulty besides the challenge of making the equations fit within the margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This  assumption allows to replace Ktpxq and Ktpyq by t, and we get the following simplification  for the expression of Ap1q  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Ap1q  t  :“  ż  R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq`p|γ|2´dqtdxdy  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='30)  Then we have  Ap2q  t  “  ż  R2d |fpxq|2Qtpx, yqeγXrpxq`γXrpyq`p|γ|2´dqr`|γ|2Krr,tspx,yqdxdy,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='31)  24  HUBERT LACOIN  where Krr,ts “ Kt ´ Kr (in the remainder of the paper, we use this convention for other  quantities indexed by t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We are going to show that  lim  tÑ8 φptq´1E  ”  |Ap1q  t  ´ Ap2q  t |1Aq,R  ı  “ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='32)  Recalling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6) we define  A  p1q  t  :“  ż  R2d |fpxq|2Qtpx, yqeγXtpxq`γXtpyq`p|γ|2´dqt1Ar,qpxqdxdy,  A  p2q  t  :“  ż  R2d |fpxq|2Qtpx, yqeγXrpxq`γXrpyq`p|γ|2´dqr`|γ|2Krr,tspx,yq1Ar,qpxqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='33)  Since on Aq,R, Apiq  t  and A  piq  t  coincide, We have  E  ”  pAp2q  t  ´ Ap1q  t q21Aq,R  ı  ď E  ”  pA  p2q  t  ´ A  p1q  t q2ı  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='34)  and thus we can prove that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='32) holds by showing that  lim  tÑ8 φptq´2E  ”  pA  p2q  t  ´ A  p1q  t q2ı  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='35)  To bound ErpA  p2q  t  ´ A  p1q  t q2s we expand the square, making it an integral on R4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set  ξpx, yq :“ |fpxq|2Qtpx, yqe´p|γ|2´dqt ´  eγXspxq`γXspyq ´ E  ”  eγXspxq`γXspyqq | Fr  ı¯  1Ar,qpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We have  E  ”  pA  p2q  t  ´ A  p1q  t q2ı  “  ż  R4d E  “  ξpx1, y1qξpx2, y2q  ‰  dx1dy1dx2dy2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='36)  As the range of correlation of the increment field Xrr,ts :“ Xt ´ Xr is smaller that e´r  have, whenever |x1 ´ x2| ě 3e´r  E  “  ξpx1, y1qξpx2, y2q | Fr  ‰  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='37)  Hence we only need to integrate the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='36) on the set |x1 ´ x2| ď 3e´r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In that  case we use  E  “  ξpx1, y1qξpx2, y2q  ‰  ď E  “  |ξpx1, y1q|2‰1{2 E  “  |ξpx2, y2q|2‰1{2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='38)  and  E  “  |ξpx, yq|2‰  “ |fpxq|4Qtpx, yq2e2p|γ|2´dqtE  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dpXtpxq`Xtpyqq1Ar,qpxq  ı  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='39)  Using Cameron-Martin formula and the fact that pXtpxqqtě0 is a standard Brownian  motion we have  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dpXtpxq`Xtpyqq1Ar,qpxq  ı  “ e2dpt`Ktpx,yqqP r@u P r0, rs, Bu ď q ´ Kupx, yqs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) (and then Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) we obtain for a constant q1 ą q  e´4dtE  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dpXtpxq`Xtpyqq1Aq,rpxq  ı  ď P  ”  @u P r0, rs, Bu ď q1 ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2du  ı  ď Cr´3{2e´dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Altogether , setting hpx, y, tq :“ 1t|x1´x2|ď3e´ru|fpx1qfpx2q|2Qtpx1, y1qQtpx2, y2q (recall  that r is a function of t) we obtain that for t sufficiently large  E  ”  pA  p2q  t  ´ A  p1q  t q2ı  ď Cr´3{2e2p|γ|2`dqt´dr  ż  R4d hpx, y, tqdxdy  ď C1t´3{2e2|γ|2t´2dr ď C2t´1{2e2dpt´rqφptq2 ď t´1{4φptq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='40)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES25  To get the second inequality, simply observe that h is smaller than a constant times the  indicator of the set t|x1| ď R, |x2 ´ x1| ď 3e´r, |yi ´ xi| ď e´t, i “ 1, 2u, which has volume  of order e´dpr`2tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The third inequality is a consequence of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10) (see the computation  in the Appendix, while the last inequality follows from the the fact that with our choice  of parameters (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18) we have t ´ r “ oplog tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Step 3: Comparing Ap2q  t  and rAt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Finally, we show that  lim  tÑ8 φptq´1E  ”  |Ap2q  t  ´ rAt|1Aq,R  ı  “ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='41)  which together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='32), concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We introduce another  smaller time parameter, namely r “ t{2 and define p  Xpx, yq “ Xrpxq ` Xrr,rspyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We want  to replace Xrpyq by Xrpxq in the exponential with an intermediate steps, so we set  Z1pxq :“  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq,  Z2px, yq :“ γXrpxq ` γ p  Xpx, yq,  Z3px, yq :“ γXrpxq ` γXrpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='42)  The reader can check that we have  rAt :“  ż  R2d |fpxq|2Qtpx, yqe|γ|2K˚  t px,yqeZ1pxq´ 1  2ErZ1pxqsdxdy,  Ap2q  t  :“  ż  R2d |fpxq|2Qtpx, yqe|γ|2K˚  t px,yqeZ3px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='yq´ 1  2ErZ3px,yqsdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='43)  In order to prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='41) taking absolute value inside the integrand, we have  E  ”  |Ap2q  t  ´ rAt|1Aq,R  ı  ď  max  |x|ďR  |x´y|ďe´t  E  „ˇˇˇeZ1pxq´  ErZ2  1 s  2  ´ eZ3´  ErZ2  3 s  2  ˇˇˇ1Aq,R  \uf6be  ˆ  ż  R2d |fpxq|2Qtpx, yqe|γ|2K˚  t px,yqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='44)  Since the integral is of order ep|γ|2´dqt (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)), which is the same order as φptq  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)), the estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='41) boilds down to proving  lim  tÑ8  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t  max  |x|ďR  |x´y|ďe´t  E  „ˇˇˇeZ1pxq´  ErZ2  1 s  2  ´ eZ3´  ErZ2  3 s  2  ˇˇˇ1Aq,R  \uf6be  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='45)  To prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='45) we start with the decomposition  E  „ˇˇˇeZ1pxq´  ErZ2  1 s  2  ´ eZ3´  ErZ2  3 s  2  ˇˇˇ1Aq,R  \uf6be  ď E  „ˇˇˇeZ1´  ErZ2  1 s  2  ´ eZ2´ ErZ2s  2  ˇˇˇ1Ar,qpxq  \uf6be  ` E  „ˇˇˇeZ3´  ErZ2  3 s  2  ´ eZ2  2´  ErZ2  2 s  2  ˇˇˇ  \uf6be  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='46)  26  HUBERT LACOIN  (this is just the triangle inequality and replacing Aq,R with a larger event Ar,qpxq) and  show that each term is opt´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We start with the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3, we have  E  „  eZ3´  ErZ2  3 s  2  ´ eZ2  2´  ErZ2  2 s  2  |  \uf6be  ď C  a  Er|Z3 ´ Z2|2s  “ C|γ|  a  ErpXrpxq ´ Xrpyqq2s ď C1e´ct,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='47)  where we have used that  ErpXrpxq ´ Xrpyqq2s “ 2pr ´ Krpx, yqq ` pK0pxq ` K0pyq ´ 2K0px, yqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The second part of the sum is smaller than |x ´ y|c since K0 is H¨older continuous and the  first part is smaller than |x ´ y|2e2r (from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)), both are exponentially small in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For  the first term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='46) we factorize the part that is Fr measureable and use independence  to obtain  E  „  |eZ1´  ErZ2  1 s  2  ´ eZ2´ ErZ2s  2  |1Aq,rpxq  \uf6be  “ E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxq1Aq,rpxq  ı  E  „  |eZ1  1´  ErpZ1  1q2s  2  ´ eZ1  2´  ErpZ1q2  2s  2  |  \uf6be  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='48)  where Z1  i “ Zi ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Cameron-Martin formula and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2, we have  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxq1Ar,qpxq  ı  “ P r@s P r0, rs, Bs ď qs ď  c  2  rπq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='49)  The factor r´1{2 is sufficient to cancel the  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='45) and we just have to show that the  second factor in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='48) is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' From Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 we have  E  „  |eZ1  1´  ErpZ1  1q2s  2  ´ eZ1  2´  ErpZ1  2q2s  2  |  \uf6be  ď  b  E r|Z1  1 ´ Z1  2|2s  “ |γ|  b  E  “  |Xrr,rspxq ´ Xrr,rspyq|2‰  ď Cer|x ´ y| ď Cer´t,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='50)  where the penultimate inequality can be deduced from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The combination of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='47)-  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='49) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='50) concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  Bonus step: the case of Bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To conclude let us sketch rapidly the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We can  repeat the argument of step 2 to show that  lim  tÑ8 φptq´2Er|Bt ´ ErBt | Frs|2s “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='51)  Then it is rather direct to check that  lim  tÑ8 φptq´1E  “  |ErBt | Frs|1Aq,R  ‰  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='52)  More precisely we have  ErBt | Frs “  ż  R2d fpxqfpyqQtpx, yqeγpXrpxq`Xrpyqq` γ2  2 p2Krr,tspx,yq´Krpxq´Krpyqqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES27  Taking the absolute value of the integrand, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) to evaluate Kt and Krr,ts, then  the inequality ab ď pa2 ` b2q{2 and symmetry, and finally (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  |ErBt | Frs| ď Cep|γ2|´dqp2r´tq  ż  R2d |fpxqfpyq|Qtpx, yqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  d{2pXrpxq`Xrpyqqdxdy  ď Cep|γ2|´dqp2r´tq  ż  R2d |fpxq|2Qtpx, yqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxqdxdy  ď C1ep|γ2|´dqp2r´tq´dt  ż  R2d |fpxq|2e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxqdx  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='53)  Hence we have  E  “  |ErBt | Frs|1Aq,R  ‰  ď ep|γ2|´dqp2r´tq´dt  ż  R2d |fpxq|2E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq1Ar,qpxq  ı  dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='54)  Using Cameron Martin formula, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 (recall that r „ t) we obtain that  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq1Ar,qpxq  ı  ď Ct´1{2edr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='55)  Overall using (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10) we have φptq´1E  “  |ErBt | Frs|1Aq,R  ‰  ď Ce´|γ2|pt´rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Organization of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Like for the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1, we assume that our  probability space contains a martingale approximation sequence pXtqtě0 of the field X,  with covariance given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the same reason as the one exposed at the beginning  of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 this entails no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The main idea is to apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6 (for the filtration corresponding to pXtq) to the  family Mγ  ε pf, ωq with rate vpε, θ, γq and with the variable Z being equal to M1pe|γ|2L|f|2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Hence need to check that the martingale Mγ  t,εpf, ωq :“ E rMγ  ε pf, ωq | Fts satisfy all the  requirements in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Setting W pεq  t  :“ Mγ  t,ε (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)), and using the bilinearity of the  martingale bracket like in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) we obtain  xMγ  ¨,εpf, ωqyt “ 1  2  ´  xW pεqyt ` Repe´2iωxW pεq, W pεqytq  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  The requirements concerning the quadratic variation of Mγ  t,εpf, ωq can be obtained as  consequences of the following,  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The following convergences hold  lim  εÑ0 E  «ˇˇˇˇˇ  xW pεqy8  2vpε, θ, γq2 ´ M1pe|γ|2L|f|2q  ˇˇˇˇˇ 1Aq,R  ff  “ 0,  lim  εÑ0 E  «ˇˇˇˇˇ  xW pεq, W pεqy8  vpε, θ, γq2  ˇˇˇˇˇ 1Aq,R  ff  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  Furthermore we have for any fixed t we have  sup  εPp0,1q  ErxW pεqyts ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 is proved in the next subsection, let us first show how our main results  can be deduced from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  28  HUBERT LACOIN  Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We must check that the three requirements in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22) are satis-  fied since the result follows then from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given that limεÑ0 vpε, θ, γq “ 8,  it is sufficient for the second and third requirements to show that that the sequences  pMγ  0,εpf, ωqqεPp0,1q, and pxMγ  ¨,εpf, ωqytqεPp0,1q (for a fixed t) are tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The sequences are in  fact uniformly bounded in L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have  sup  εPp0,1q  Er|Mγ  0,εpf, ωq|s ď sup  εPp0,1q  Er|Mγ  0,εpfq|s ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  Indeed taking the absolute value of the integrand, we have  Er|Mγ  0,εpfq|s ď  ż  Rd E  „  fpxqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  d{2X0,εpxq` β2´pd{2q  2  K0,εpxq  \uf6be  dx “  ż  Rd fpxqeβ2K0,εpxqdx,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  and the uniform bound follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) we have xMγ  ¨,εpf, ωqyt ď xW pεqyt  and thus the uniform boundedness in L1 is consequence of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us now turn to  the first and main requirement in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergences in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) imply the following  convergence in probability  lim  εÑ0  xW pεqy8  2vpε, θ, γq2 1Ť  qě1 Aq,R “ M1pe|γ|2L|f|2q,  lim  εÑ0  xW pεq, W pεqy8  vpε, θ, γq2  1Ť  qě1 Aq,R “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1), we conclude that  lim  εÑ0 vpε, θ, γq´2xMγ  ¨,εpf, ωqy8 “ M1pe|γ|2L|f|2q  in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  As another preliminary step to our proof, we reduce the convergence statement in  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 to a convergence of the derivative of the martingale brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Itˆo  calculus we obtain that for T P r0, 8s,  xW pεqyT “  ż T  0  At,εdt and xW pεqyT “  ż T  0  Bt,εdt  where  At,ε :“  ż  R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2  2 Kt,εpxq´ γ2  2 Kt,εpyqdxdy,  Bt,ε :“  ż  R2d fpxqfpyqQt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2  2 pKt,εpxq`Kt,εpyqqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  Similarly to what has been done in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3, we are going to show that, with ap-  propriate renormalizations and restrictions, At,ε and Bt,ε converge in L1 to M1pe|γ|2L|f|2q  and 0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To this end we introduce a couple of parameters (recall (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5))  tpt, εq :“ t ^ logp1{εq  φpt, εq :“  d  2  πpt _ 1qe|γ|2j  ˆż  Rd e|γ|2Kt,εp0,zqQt,εp0, zqdz  ˙  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  The quantity r will on Our aim is to prove the following  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES29  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  lim  εÑ0  tÑ8  E  „ˇˇˇˇ  At,ε  φpt, εq ´ M1pe|γ|2L|f|2q  ˇˇˇˇ 1Aq,R  \uf6be  “ 0,  lim  εÑ0  tÑ8  E  „ˇˇˇˇ  Bt,ε  φpt, εq  ˇˇˇˇ 1Aq,R  \uf6be  “ 0,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  and for any T ă 8  sup  tPr0,Ts  εPp0,1q  E r|At,ε|s ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us underline that lim εÑ0  tÑ8 Fpt, εq “ 0 means that that there exists t0pδq  and ε0pδq such that |Fpt, εq| ď δ when t ě t0 AND ε P p0, ε0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This is a stronger statement  than both limεÑ0 limtÑ8 Fpt, εq “ 0 or limtÑ8 limεÑ0 Fpt, εq “ 0  Clearly (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10) implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To deduce (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9), we need to check that renormal-  izing factor 2vpε, θ, γq2 corresponds to the integral of φpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This is the content of the  following lemma whose proof is presented in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have for any |γ| ě d  lim  εÑ0  |γ|2 ş8  0 φpt, εqdt  2vpε, θ, γq2  “ 1  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  We can now complete the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 using Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2  Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' From Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4 it is sufficient to prove the convergence of  E  «ˇˇˇˇˇ  xW pεqy8  ş8  0 |γ|2φpt, εq ´ M1pe|γ|2L|f|2q  ˇˇˇˇˇ 1Aq,R  ff  ď  1  ş8  0 φpt, εqdt  ż 8  0  φpt, εqE  „ˇˇˇˇ  At,ε  φpt, εq ´ M1pe|γ|2L|f|2q  ˇˇˇˇ 1Aq,R  \uf6be  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  Let us fix δ ą 0, and let T and ε0 be such that for all t ą T and ε ă ε0 we have  E  „ˇˇˇˇ  At,ε  φpt, εq ´ M1pe|γ|2L|f|2q  ˇˇˇˇ 1Aq,R  \uf6be  ď δ  2  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  In the integral we can distinguish the contribution from r0, Ts from the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have  from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10) and the fact that φpt, εq is bounded from below  sup  tPr0,Ts  εPp0,ε0q  E  „ˇˇˇˇ  At,ε  φpt, εq ´ M1pe|γ|2L|f|2q  ˇˇˇˇ 1Aq,R  \uf6be  ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  As a consequence, since  ş8  0 φpt, εqdt diverges when ε Ñ 0, taking ε1 sufficiently small we  have forall ε P p0, ε1q  1  ş8  0 φpt, εq  ż T  0  φpt, εqE  „ At,ε  φpt, εq ´ M1pe|γ|2L|f|2q  \uf6be  dt ď δ  2,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15)  30  HUBERT LACOIN  which implies that for ε ď ε0 ^ ε1 we have  E  «ˇˇˇˇˇ  xW pεqy8  ş8  0 φpt, εqdt ´ M1pe|γ|2L|f|2q  ˇˇˇˇˇ 1Aq,R  ff  ď δ  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16)  and thus conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us start with the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Noting that At,ε  is positive, we have  E rAt,εs “  ż  R2d fpxqfpyqQt,εpx, yqe|γ|2Kt,εpx,yqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)  We can just use (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) and bound Qt,εpx, yq by 1 and Kt,εpx, yq by T `C to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For  the proof of the convergence of At,ε we proceed exactly as for the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We assume that tpt, εq ą e (recall (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)) and set  r “ rpt, εq :“ t ´ log log t,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18)  Setting K˚  t,εpx, yq :“ K0pxq ` Kt,εpx, yq, we define  rAt,ε :“  ż  R2d |fpxq|2Qt,εpx, yqe|γ|2K˚  t,εpx,yqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´2dKrpxqdxdy  “ φpt, εqM  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  r  pe|γ|2L|f|2q  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19)  Since lim εÑ0  tÑ8 rpt, εq “ 8, we obtain, as a direct consequence of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 that  lim  εÑ0  tÑ8  E  «ˇˇˇˇˇ  rAt,ε  φpt, εq ´ M1pe|γ|2L|f|2q  ˇˇˇˇˇ 1Aq,R  ff  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20)  Our task is thus to prove that  lim  εÑ0  tÑ8  φpt, εq´1E  ”  | rAt,ε ´ At,ε|1Aq,R  ı  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21)  Like for the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21) in the previous section, we proceed in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Step 1: Changing the deterministic prefactor in the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Set  Ap1q  t,ε :“  ż  R2d |fpxq|2Qt,εpx, yqeγXt,εpxq`γXt,εpyq´ γ2  2 Kt,εpxq´ γ2  2 Kt,εpyq`|γ|2pK0pxq´K0,εpx,yqqdxdy  Let us prove that  lim  εÑ0  tÑ8  φpt, εq´1E  ”  |Ap1q  t,ε ´ At,ε|1Aq,R  ı  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)  Let As a direct consequence of the continuity of f and of K0, if one sets  sup  |x|,|y|ďR  |x´y|ďet`2ε  ˇˇˇfpxqfpyq ´ |fpxq|2e|γ|2pK0pxq´K0,εpx,yqqˇˇˇ “: δpε, tq,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES31  we have lim εÑ0  tÑ8 δpε, tq “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Since Qt,εpx, yq “ 0 when |x ´ y| ě et ` 2ε repeating the  computation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26) and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13) we get  |Ap1q  t,ε ´ At,ε| ď δpε, tq  ż  Bp0,Rq2 Qt,εpx, yqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXt,εpxq`p|γ|2´dqKt,εpxqdxdy  ď Ce´dtδpε, tq  ż  Bp0,Rq  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXt,εpxq`p|γ|2´dqKt,εpxqdx  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24)  Using Cameron-Martin formula, we obtain (assuming |x|, |y| ď R and |x ´ y| ď et ` 2ε)  for a constant q1 ą q  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXt,εpxq´dKt,εpxq1Aq,R  ı  ď E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxq  ı  “ P  ”  @s P r0, ts, Bs ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dps ´ Ks,ε,0pxqq ` q  ı  ď Pp@s P r0, ts, Bs ď q1q ď Cpt _ 1q´1{2et|γ|2  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='25)  where in the second inequality we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) to estimate covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Hence, using  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20) we deduce from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24) that  E  ”  |Ap1q  t,ε ´ At,ε|1Aq,R  ı  ď Cδpε, tqt´1{2et|γ|2´dt ď C1δpε, tqφpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  This concludes the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  Step 2: Taking conditional expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set  Ap2q  t,ε :“ E  ”  Ap1q  t,ε | Fr  ı  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26)  and we are going to prove  lim  εÑ0  tÑ8  φpt, εq´2E  ”  |Ap2q  t,ε ´ Ap1q  t,ε |21Aq,R  ı  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='27)  Like what we did in the previous section, we assume here that K0 ” 0 to simplify the  writing (but this does not affect the proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  In that case note that since Kt,εpxq “  Kt,εpyq “ Kt,εpxq, we can factorize the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have (recall that Krr,ts,ε “ Kt,ε ´ Kr,ε)  Ap2q  t,ε :“  ż  R2d |fpxq|2Qt,εpx, yqeγXr,εpxq`γXr,εpyq`p|γ2|´dqKr,εpxq`|γ|2Krr,ts,εpx,yqdxdy  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='28)  Setting  ζpx, yq :“ eγXt,εpxq`γXt,εpyq`p|γ2|´dqKt,εpxq,  A  p1q  t,ε :“  ż  R2d |fpxq|2Qt,εpx, yqζpx, yq1Ar,qpxqdxdy,  A  p2q  t,ε :“  ż  R2d |fpxq|2Qt,εpx, yqErζpx, yq |Frs1Ar,qpxqdxdy  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='29)  we realize that  E  ”  |Ap2q  t,ε ´ Ap1q  t,ε |21Aq,R  ı  ď E  ”  |A  p2q  t,ε ´ A  p1q  t,ε |2ı  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='30)  Now we set  ξt,εpx, yq :“ |fpxq|2Qt,εpx, yq pζpx, yq ´ Erζpx, yq |Frsq 1Ar,qpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='31)  32  HUBERT LACOIN  We have  E  ”  |A  p2q  t,ε ´ A  p1q  t,ε |2ı  ď  ż  R4d E  “  ξpx1, y1qξpx2, y2q  ‰  dx1dx2dy1dy2  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='32)  The range of the convariance of Xrr,ts,ε is smaller than e´r ` 2ε, and Qt,εpx, yq vanishes  when |x ´ y| ě e´t ` 2ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' All of this implies that if if |x1 ´ x2| ě 2e´r (if ε is sufficiently  small, then e´r is much larger than both ε and e´t cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)) then  E  “  ξpx1, y1qξpx2, y2q | Fr  ‰  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='33)  When |x1 ´ x2| ď 2e´r we can use Cauchy-Schwarz to bound the covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have  E  “  |ξpx, yq|2‰  ď |fpxq|4 pQt,εpx, yqq2 Er|ζpx, yq|21Ar,qpxqs  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='34)  and from Cameron-Martin formula, we have, for |x ´ y| ď e´t ` 2ε  Er|ζpx, yq|21Ar,qpxqs  “ e2|γ|2Kt,εpxq`2dKt,εpx,yqP  ´  @s P r0, rs, Bs ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dps ´ Ks,ε,0pxq ´ Ks,ε,0py, xqq ` q  ¯  ď Cep|γ|2`dqtP  ´  @s P r0, rs, Bs ď ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2ds ` q1¯  ď C1etp2|γ|2`dqr´3{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  where in the first inequality we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) which replace the Kt,ε and Ks,ε by t and  s respectively at the cost of an additive constant and in the second inequality we used  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Altogether we obtain that  E  ”  |A  p2q  t,ε ´ A  p1q  t,ε |2ı  ď Cetp2|γ|2`dqr´3{2  ˆ  ż  R4d 1t|x1´x2|ď2e´ru|fpx1q|2|fpx2q|2Qt,εpx1, y1qQt,εpx2, y2qdxdy  ď C1e2|γ|2t`dpt´rqr´3{2  ˆż  Rd Qt,εp0, zqdz  ˙2  ď C2edpt´rqr´1{2φpt, εq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='35)  We conclude the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='27) by observing (recall (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)) that  lim  εÑ0  tÑ8  edpt´rqr´1{2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='36)  □  Step 3: Final comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Finally to conclude we need to show that  lim  εÑ0  tÑ8  φpt, εq´2E  ”  |Ap2q  t,ε ´ rAt,ε|21Aq,R  ı  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='37)  We set (recall that Xrs,ts,ε “ Xt,ε ´ Xs,ε´)  Z1pxq :“  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq,  Z2px, yq :“  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq ` γXrr,rs,εpxq ` γXrr,rs,εpyq,  Z3px, yq :“ γXr,εpxq ` γXr,εpyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='38)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES33  The reader can check (using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26) rather than (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='28) since the latter assumes K0 ” 0)  that  rAt,ε :“  ż  R2d |fpxq|2Qt,εpx, yqe|γ|2K˚  t,εpx,yqeZ1pxq´ 1  2 ErZ1pxqsdxdy,  Ap2q  t,ε :“  ż  R2d |fpxq|2Qt,εpx, yqe|γ|2K˚  t,εpx,yqeZ3px,yq´ 1  2 ErZ3px,yqsdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='39)  In order to prove (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='37) taking absolute value inside the integrand using Jensen’s inequality,  we just have to obtain a uniform bound on the integrand, that is, to show that  lim  tÑ8  εÑ0  t1{2  max  |x|ďR  |x´y|ďe´t`2ε  E  „ˇˇˇeZ1pxq´  ErZ2  1 pxqs  2  ´ eZ3px,yq´  ErZ2  3 px,yqs  2  ˇˇˇ1Aq,R  \uf6be  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='40)  The restriction for x and y comes from the support of f and Qt,ε respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To prove  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='45) we start with the decomposition  E  „ˇˇˇeZ1pxq´  ErZ2  1 s  2  ´ eZ3´  ErZ2  3 s  2  ˇˇˇ1Aq,R  \uf6be  ď E  „ˇˇˇeZ1´  ErZ2  1 s  2  ´ eZ2´ ErZ2s  2  ˇˇˇ1Ar,qpxq  \uf6be  ` E  „ˇˇˇeZ3´  ErZ2  3 s  2  ´ eZ2  2´  ErZ2  2 s  2  ˇˇˇ  \uf6be  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='41)  (the inequality is just the triangle inequality and replacing Aq,R with a larger event), and  show that each term is opt´1{2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us start with the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3, we  have  E  „  eZ3´  ErZ2  3 s  2  ´ eZ2  2´  ErZ2  2 s  2  |  \uf6be  ď C  a  Er|Z3 ´ Z2|2s  “ C  b  Er|  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq ´ γXr,εpxq ´ γXr,εpyq|2s  ď C1  ˆb  E r|Xrpxq ´ Xr,εpxq|2s `  b  E r|Xrpxq ´ Xr,εpyq|2s  ˙  “ C1 ´  pKrpxq ` Kr,εpxq ´ 2Kr,ε,0pxqq1{2 ` pKrpxq ` Kr,εpxq ´ 2Kr,ε,0py, xqq1{2¯  ď C2 pε ` |x ´ y|qc ď C1e´ct  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='42)  where to obtain the last line we have used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14) and the H¨older continuity of K0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the  first term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='41) we factorize the Fr-measureable part and use independence to obtain  E  „  |eZ1´  ErZ2  1 s  2  ´ eZ2´ ErZ2s  2  |1Ar,qpxq  \uf6be  “ E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxq1Ar,qpxq  ı  E  „  |eZ1  1´  ErpZ1  1q2s  2  ´ eZ1  2´  ErpZ1q2  2s  2  |  \uf6be  ,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='43)  where Z1  i “ Zi ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have from Cameron-Martin Formula and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXrpxq´dKrpxq1Ar,qpxq  ı  “ P r@s P r0, rs, Bs ď qs ď Cr´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='44)  34  HUBERT LACOIN  This is obviously Opt´1{2q so to conclude we only need to show that the other factor in  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='43) goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We also have from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  E  „  |eZ1  1´  ErpZ1  1q2s  2  ´ eZ1  2´  ErpZ1q2  2s  2  |  \uf6be  ď C  b  E r|Z1  1 ´ Z1  2|2s  ď C  `  Krr,rspxq ` Krr,rs,εpxq ` Krr,rs,εpyq ´ 2Krr,rs,ε,0pxq ´ 2Krr,rs,ε,0py, xq  ˘1{2  ď C1erp|x ´ y| ` εq ď Cepr´tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='45)  This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  The convergence of Bt,ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To prove the second convergence in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9), it is sufficient again to  show first that  lim  tÑ8  εÑ0  ϕpt, εq´2E  “  |Bt,ε ´ ErBt,ε | Frs|21Aq,R  ‰  “ 0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='46)  repeating the computation of step two, and then prove that  lim  tÑ8  εÑ0  Er|ErBt,ε | Frs|1Aq,Rs “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We leave this part to the reader, since this is very similar to the computation performed  at the end of Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6  We need to show that for any H bounded and F8-measurable and ξ P R we have  lim  nÑ8 E  „  H  ˆ  eiξ Wn  vpnq ´ e´ ξ2Z  2  ˙\uf6be  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  We first assume that the collection of variables vpnq´2xWny8 is uniformly essentially  bounded, that is, that there exists M such that for every n ě 1  P  “  vpnq´2xWny8 ě M  ‰  “ 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  Note that this implies also that P rZ ě Ms “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We assume, to simplify notation that  ξ “ 1 (this entails no loss of generality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set Ht :“ E rH | Fts and Zt :“ E rZ | Fts we  have  E  „  H  ˆ  ei ξWn  vpnq ´ e´ ξ2Z  2  ˙\uf6be  “ E  ”  Hpe´ Z  2 ´ e´ Zt  2 q  ı  ` E  ”  pH ´ Htq  ´  ei Wn  vpnq ´ e´ Zt  2  ¯ı  ` E  ”  Ht  ´  ei Wn  vpnq ´ e´ Zt  2  ¯ı  “: E1pt, nq ` E2pt, nq ` E3pt, nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  We prove the convergence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) by showing that for i “ 1, 2, 3  lim  tÑ8 lim sup  nÑ8 |Eipt, nq| “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  Using the fact that z ÞÑ ez is 1-Lipshitz (first line) and has modulus bounded by 1 (second  line) in tz P C : Repzq ď 0u we have  |E1pt, nq| ď E  ”  |H|  ˇˇˇe´ Z  2 ´ e´ Zt  2  ˇˇˇ  ı  ď }H}8  2  E r|Z ´ Zt|s ,  |E2pt, nq| ď E  ”  |H ´ Ht|  ˇˇˇei Wn  vpnq ´ e´ Z  2  ˇˇˇ  ı  ď 2E r|H ´ Ht|s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES35  Since Ht and Zt converge respectively to H and Z in L1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4) holds for i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For  i “ 3, we observe that for fixed t the process  Mpnq  u  :“ e  iWn,t`u´Wn,t  vpnq  `  xWnyt`u´xWnyt  2vpnq2  ´ Zt  2  is a martingale for the filtration pGuq :“ pFt`uq, which converges in L1 when u Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In  particular we have  E  „  e  iWn´Wn,t  vpnq  ` xWny8´xWnyt  2vpnq2  | Ft  \uf6be  “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  Multiplying by Hte´ Zt  2 and taking expectation we obtain that  E  „  Hte  iWn´Wn,t  vpnq  ` xWny8´xWnyt  2vpnq2  ´ Zt  2  \uf6be  “ E  ”  Hte´ Zt  2  ı  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  Hence we have (using that }Ht}8 ď }H}8)  |E3pt, nq| ď E  „  |Ht|  ˇˇˇˇei Wn  vpnq ´ e  ipWn´Wn,tq  vpnq  ` xWny8´xWnyt  2vpnq2  ´ Zt  2  ˇˇˇˇ  \uf6be  ď }H}8E  „ˇˇˇˇ1 ´ e  ´  iWn,t  vpnq ` xWny8´xWnyt  2vpnq2  ´ Zt  2  ˇˇˇˇ  \uf6be  ,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22) we have the following convergence in probability for any fixed t  lim  nÑ8 ´iWn,t  vpnq ` xWny8 ´ xWnyt  2vpnq2  ´ Zt  2 “ Z ´ Zt  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  Using assumption (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2), taking the limit in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8) and using dominated con-  vergence, we obtain that  lim sup  nÑ8 |E3pt, nq| ď }H}8E  ”ˇˇˇ1 ´ e  Z´Zt  2  ˇˇˇ  ı  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  Since Zt converges to Z we can conclude that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4) also holds for i “ 3 using dominated  convergence again (both variables are uniformly bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Let us now remove the boundedness assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given A ą 0 we set  TA,n :“ inftt : vpnq´2xWnyt “ Au  and  W A  n :“ Wn,TA,n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Note that E  “  W A  n | Ft  ‰  “ Wt^TA,n so that (using the notation xW A  n yt to denote the qua-  dratic variation of this martingale) we have  lim  nÑ8 vpnq´2xW A  n y8 “ Z ^ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  Since we have proved (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) under the assumption (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) we know that for every A ą 0  lim  nÑ8 E  „  H  ˆ  ei ξW A  n  vpnq ´ e´ ξ2pZ^Aq  2  ˙\uf6be  “ 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  From the convergence assumption, we have  lim sup  nÑ8 PrTA,n “ 8s ď P rZ ě As  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  and hence  lim  AÑ8 lim inf  nÑ8 P  “  W A  n “ Wn  ‰  “ lim  AÑ8 PrZ ^ A “ Zs “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  36  HUBERT LACOIN  As a consequence we can conclude using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) that  lim  nÑ8 E  „  H  ˆ  eiξ Wn  vpnq ´ e´ ξ2Z  2  ˙\uf6be  “ lim  AÑ8 lim  nÑ8 E  „  H  ˆ  ei ξW A  n  vpnq ´ e´ ξ2pZ^Aq  2  ˙\uf6be  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  □  Acknowledgements: This work was supported by a productivity grant from CNPq and  a JCNE grant from FAPERJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Technical results and their proof  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Standard Gaussian tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We first display two standard tools which are used  throughout the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The first is the standard Cameron-Martin formula which describes  how a Gaussian process is affected by an exponential tilt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let pY pzqqzPZ be a centered Gaussian field indexed by a set Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We  let H denote its covariance and P denote its law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given z0 P Z let us define rPz0 the  probability obtained from P after a tilt by Y pz0q that is  drPz0  dP :“ eY pz0q´ 1  2 Hpz0,z0q  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  Under rPz0, Y is a Gaussian field with covariance H, and mean rEz0rY pzqs “ Hpz, z0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The second is a bound on the probability for a Brownian Motion to remain below a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Both estimates can be proved directly using the reflexion principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let B be a standard Brownian Motion and let P denote its distribution,  setting gtpaq :“  şu`  0  e´ z2  2t dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' we have  P  «  sup  sPr0,ts  Bs ď a  ff  “  c  2π  t gtpaq ď  c  2π  t a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  Additionally for any a, b ą 0 there exists Ca,b such that f  P  «  sup  sPr0,ts  pBs ` bsq ď a  ff  “  1  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2πt  ż  e´ u2  2t p1 ´ e  2apa`u´bsq`  t  qdu ď Ca,be´ b2t  2 t´3{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Comparing exponentiated Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In or comparison of partition functions  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Consider pX1, X2, Y1, Y2q an R4 valued centered Gaussian vector and set  X :“ X1 ` iX2 and Y “ Y1 ` iY2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Assuming that  ErX2  2s ď 1 and Er|X ´ Y |2s ď 1  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  then there exits a constant C such that  E  ”ˇˇeX´ 1  2ErX2s ´ eY ´ 1  2ErY 2sˇˇ  ı  ď CE  “  |X ´ Y |2‰  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We factorize eX´ 1  2ErX2s, use the Cameron-Martin formula and rearrange the ex-  pectation terms in the exponential, we obtain  E  „ˇˇeY ´ ErY 2s  2  ´ eX´ ErX2s  2  ˇˇ  \uf6be  “ E  „  eX1`  ErX2  2 s´ErX2  1 s  2  ˇˇeY ´X` ErX2s´ErY 2s  2  ´ 1  ˇˇ  \uf6be  “ e  ErX2  2 s  2  E  „ˇˇeY ´X´ ErpX´Y q2s  2  ´iErX2pY ´Xqs ´ 1  ˇˇ  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES37  The prefactor is bounded (by assumption) by e1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the rest, setting Z “ Y ´ X (and  letting Z1 and Z2 denote the real and imaginary part) we have using the triangle inequality  E  „ˇˇeZ´ ErZ2s  2  ´iErX2Zs ´ 1  ˇˇ  \uf6be  ď  ˇˇeiErX2Zs ´ 1  ˇˇE  „ˇˇeZ´ ErZ2s  2  ˇˇ  \uf6be  ` E  „ˇˇeZ´ ErZ2s  2  ´ 1  ˇˇ  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  For the first term, using that |ErX2Zs| ď  a  ErX2  2sEr|Z|2s ď  a  Er|Z|2s ď 1, and that  |eu ´ 1| ď e|u| for u ď 1 and computing expectation, we obtain that  ˇˇeiErX2Zs ´ 1  ˇˇE  „ˇˇeZ´ ErZ2s  2  ˇˇ  \uf6be  ď e  a  Er|Z|2se  ErZ2  2 s  2  ď e3{2a  Er|Z|2s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  For the second term we have (using again |eu ´ 1| ď e|u|)  E  „ˇˇeZ´ ErZ2s  2  ´ 1  ˇˇ  \uf6be  ď  d  E  „ˇˇeZ´ ErZ2s  2  ´ 1  ˇˇ2  \uf6be  “  a  eEr|Z|2s ´ 1 ď e1{2a  Er|Z|2s,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  which yields the desired result for C “ e2 ` e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us first compute the order of magnitude of φptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us  set for practical purpose φptq :“  ş  Qtp0, zqe|γ|2Ktp0,zqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) (recall that |z| ď e´t  on the integrand) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13) we have  φptq — ep|γ|2´dqt  and  φptq — t´1{2ep|γ|2´dqt  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  As a consequence when |γ|2 ą d most of the integral is carried by rt ´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t, ts and we have  ż t  0  φpsqds “ p1 ` op1qq  ż t  t´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t  φpsqds  “ p1 ` op1qq  ż t  t´  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t  c  s _ 1  t  φpsqds  “ p1 ` op1qq  c  2  πte|γ|2j  ż t  0  φpsqds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  We observe that  |γ|2Qsp0, zqe|γ|2Ksp0,zq “ Bs  ´  e|γ|2Ksp0,zq¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Using Fubini and integrating w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' time and making a change of variable we have  |γ|2  ż t  0  φpsqds “  ż  Rd  ´  e|γ|2Ktp0,zq ´ 1  ¯  dz  “ ep|γ|2´dqt  ż  Rd  ´  e|γ|2pKtp0,e´tzq´tq ´ e´|γ|2t¯  dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  The integrand in the second line is bounded above by p|z| _ 1q´|γ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This is obvious for  |z| ě et since the integrand vanishes, and when |z| ď et this can be obtainded from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Furthermore it converges to e|γ|2ℓpzq and we obtain using dominated convergence that  lim  tÑ8  |γ|2 şt  0 φpsqds  ep|γ|2´dqt  “  ż  Rd e|γ|2ℓpzqdz,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  which combined with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11), proves the lemma in the case |γ|2 ą d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' When |γ|2 “ d, we  observe that using, as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12), a change of variable and dominated convergence, we have  lim  sÑ8 φpsq “  ż  Rd κpe  η1  η2 zqedℓpzqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  38  HUBERT LACOIN  On the other hand we have from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  ż t  0  φpsqds “  ż  Rd  ´  edpKtp0,e´tzq´tq ´ e´dt¯  dz  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15)  We have, for 1 ď |z| ď et,  Ktp0, e´tzq “ Kp0, e´tzq “ Kp0, e´tzq ´ K0p0, e´tzq  “ t ` log 1  |z| ` pL ´ K0qp0, e´tzq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16)  Since pL ´ K0qp0, e´t|z|q “ ´j ` δ  `  e´t|z|  ˘  where δpuq tends to zero when u Ñ 0 we can  deduce that  ż  Rd  ´  edpKtp0,e´tzq´tq ´ e´dt¯  dz “ p1 ` op1qq  ż  1t1ď|z|ďetuedpKtp0,e´tzq´tqdz  “ p1 ` op1qqe´dj  ż  1t1ď|z|ďetu|z|´ddz  “ p1 ` op1qqe´djΣd´1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)  Since φpsq converges, we deduce that its limit equals its Cesaro limit and thus  lim  sÑ8 φpsq “ e´djΣd´1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18)  which implies in turn that  ż t  0  d  2  πps ^ 1qedjφpsq “ p1 ` op1qq2  c  2t  π Σd´1,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19)  and concludes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us again start with the case |γ|2 ą d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' As in the proof of  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4, we can compute the asymptotic of φpt, εq (when t and ε goes to infinity and  zero respectively)  φpt, εq — t´1{2e|γ|2t´dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20)  Since suptPr0,Ts  εPp0,1q  φpt, εq ă 8 for every finite T, this implies that the integral  ş8  0 φpt, εq is  mostly carried by values of s around logp1{εq (say ˘  a  logp1{εq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For this reason, we can  replace the term pt _ 1q´1{2 by plog 1{εq´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  |γ|2  ż 8  0  φpt, εqdt “ p1 ` op1qq  d  2  πplog 1{εqe|γ|2j  ż 8  0  ż  Rd |γ|2e|γ|2Kt,εp0,zqQt,εp0, zqdz (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21)  Using Fubini and integrating with respect to time as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) we have  ż 8  0  ż  Rd |γ|2e|γ|2Kt,εp0,zqQt,εp0, zqdz “  ż  Rd  ´  e|γ|2Kεp0,zq ´ 1  ¯  dz  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)  We then perform a change of variable for z  ż  Rd  ´  e|γ|2Kεp0,zq ´ 1  ¯  dz “ εd´|γ|2 ż  Rd  ´  e|γ|2pKεp0,εzq`logpεqq ´ ε|γ|2¯  dz  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23)  Next we observe that  Kεp0, εzq ` logpεq “ ℓθpzq ` pLεp0, εzq ´ Kεp0, εzqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES39  Using dominated convergence as ε goes to zero (the integrand is bounded above by p|z| _  1q´|γ|2 and recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) we obtain that  lim  εÑ0  ż  Rd  ´  e|γ|2pKεp0,εzq`logpεqq ´ ε|γ|2¯  dz “ e´|γ|2j  ż  Rd e´|γ|2ℓθpzqdz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24)  The combination of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24) concludes the proof in the case |γ|2 ą d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the case  |γ|2 “ d, based on (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20) we know that setting Tε “ logp1{εq ´  a  logp1{εq  ż 8  0  φpt, εqdt “ p1 ` op1qq  ż Tε  0  φpt, εqdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='25)  Now in this range for t it is tedious but not difficult to check that  lim  εÑ0 sup  tPr0,Tεs  ş  Rd e|γ|2Kt,εp0,zqQt,εp0, zqdz  ş  Rd e|γ|2Ktp0,zqQtp0, zqdz  “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26)  From this we obtain that  ż 8  0  φpt, εqdt “ p1 ` op1qq  ż Tε  0  φpt, εqdt “ p1 ` op1qq  ż Tε  0  φptqdt  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='27)  and we can conclude using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence of Mγ  ε as a distribution  We have chosen for simplicity, to present our convergence results as convergence of a  collection of random variables Mγ  ε pfq indexed by CcpRdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We can go further and prove  that Mγ  ε p¨q converges as a distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For this purpose we need to recall the definition  of local Sobolev/Bessel spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The Bessel space Hs,ppRkq, s P R and p P r1, 8s on Rk is defined by  Hs,ppRkq :“ tϕ P D1pRkq : p1 ` |ξ|2qs{2 pϕpξq P LppRkqu  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  where D1pRkq is the space of distribution and pϕpξq is the Fourier transform of ϕ defined  for ϕ P C8  c pRkq by pϕpξq “  ş  Rk eiξxϕpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' It is a Banach space when equiped with the  norm  }f}Hs,p “  ż  Rkp1 ` |ξ|2qps{2|pϕpξq|pdξ  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  For U Ă Rk open, the local Bessel space Hs,p  locpUq denotes the set of distributions which  belongs to Hs,ppUq after multiplication by an arbitrary smooth function with compact  support  Hs,p  locpUq :“  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  ϕ P D1pUq | ρϕ P Hs,ppRdq for all ρ P C8  c pUq  )  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  where above ρϕ is identified with its extension by zero on Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' It is equiped with the  topology generated by the family of seminorms rρ, ρ P C8  c pUq defined by rρpϕq :“ }ϕρ}Hs,p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  In the particular case where p “ 2 we write HspRkq :“ Hs,2pRkq which is a Hilbert space  (and use the same convention for the local spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The convergence result for Mγ  ε p¨q as a  distribution for γ P PI{II is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If X is a centered Gaussian field whose covariance kernel K has an  almost star-scale invariant part, γ P PI{II, p P r1,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dαq and s ă ´ d  p, then there exists  Mγ  8 P Hs,p  locpRdq such that for every ρ P C8  c pRdq  lim  εÑ0 E  “  }Mγ  ε pρ ¨q ´ Mγ  8pρ ¨q}p  Hs,p  ‰  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  40  HUBERT LACOIN  In particular Mγ  ε converges to Mγ  8 in probability in the Hs,p  locpRdq topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Similarly in P1  II{III the convergence in law holds also for the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let X be a Gaussian random field with an almost star-scale covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Then given γ P P1  II{III, and s ă ´ d  2 the following joint convergence in law for the Hs  locpRdq  topology  ˆ  X,  Mγ  ε  vpε, θ, γq  ˙  εÑ0  ñ pX, Mγq,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In the proof of Theorems B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 presented below, we are going to  assume that our probability space contains a martingale sequence pXtqtě0 of fields with co-  variance (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3) approximating X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For reasons analogous to the one exposed at the beginning  of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 this entails no loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The case of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let us fix ρ P C8  c pRdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We want to prove that Mγ  ε pρ ¨q  converges in Hs,ppRdq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We first define the limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set (without underlying the  dependence in ρ to keep the notation light)  x  Mγ  ε pξq “  ż  Rd ρpxqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='xeγXεpxq´ γ2  2 Kεpxqdx,  x  Mγ  8pξq “ lim  εÑ0  x  Mγ  ε pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  We let Mγ  8pρ ¨q denote the random distribution whose Fourier transform is given by x  Mγ  8pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  The proof of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4), implies that Mγ  8pρ ¨q P Hs,ppRdq with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To prove (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4),  note that we have  E  “  }Mγ  ε pρ ¨q ´ Mγ  8pρ ¨q}p  Hs,p  ‰  “  ż  Rd E  ”  |px  Mγ  ε ´ x  Mγ  8qpξq|pı  p1 ` |ξ|2q  ps  2 dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  Hence with our assumption on s ă ´ d  p it is sufficient to prove that  lim sup  εÑ0  sup  ξPRd E  ”  |px  Mγ  ε ´ x  Mγ  8qpξq|pı  ă 8,  @ξ P Rd, lim  εÑ0 E  ”  |px  Mγ  ε ´ x  Mγ  8qpξq|pı  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  and we can then conclude using dominated convergence (the first line yields the domina-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The second line is simply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2) with fpxq “ ρpxqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the first line it sufficient  to prove that  lim sup  εÑ0  sup  ξPRd E  ”  |x  Mγ  ε pξq|pı  ă 8,  since the bound for x  Mγ  8pξq| can then be obtained by Fatou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set V pεq  t  pξq “ Erx  Mγ  ε | Fts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Using the BDG inequality we have  E  ”  |x  Mγ  ε pξq|pı  ď CE  ”  |V pεq  0  pξq|p ` xV pεqpξqyp{2  8  ı  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  Now we have (recall that p ă  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{α ď 2)  E  ”  |V pεq  0  pξq|2ı  “  ż  R2d ρpxqρpyqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='px´yqe|γ|2K0,εpx,yqdxdy  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES41  and we can conclude by replacing eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='px´yq by 1 and observing that since K is continuous  K0,ε is uniformly bounded for x, y in the support of ρ and ε P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the quadratic  variation part, we have  xV pεqy8 “ |γ|2  ż 8  0  At,εpξqdt,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  where,  At,εpξq :“  ż  R2d ρpxqρpyqQt,εpx, yqeiξpx´yqeγXt,εpxq`γXt,εpyq´ γ2  2 Kt,εpxq´ γ2  2 Kt,εpyqdxdy  ď  ż  R2d ρpxqρpyqQt,εpx, yqeαpXt,εpxq`Xt,εpyqq` β2´α2  2  pKt,ε`Kt,εpyqqdxdy “: At,ε,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  the inequality being obtain by taking the modulus of the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To conclude, we just  need to prove that  sup  εPp0,1q  E  «ˆż 8  0  At,εdt  ˙p{2ff  ă 8  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  For this part we can just repeat the computations made to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The case of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Since the convergence of finite dimensional marginal  has been established, we only need to prove tightness of the distribution of vpε, θ, γq´1Mγ  ε pρ ¨q  in HspRdq for every ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For this, we simply replicate the strategy presented in [16], with a  minor twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Since in our case, the Fourier transform in not in L2, we need to consider a  restriction to the event Aq,R where R is such that the support ρ is contained in Bp0, Rq  (recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Keeping the notation introduced in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6) for the Fourier transform, we are  going to prove the following analogue of [16, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2] (we use the notation vpεq for  vpε, θ, γq for ease of reading)  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If the support of ρ is included in Bp0, Rq then the following holds for every  a P Rd with a constant C which depends on ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  sup  εPp0,1q  ξPRd  E  ”  vpεq´2|x  Mγ  ε pξq|21Aq,R  ı  ă 8,  sup  εPp0,1q  ξPRd  E  ”  vpεq´2|x  Mγ  ε pξ ` aq ´ x  Mγ  ε pξq|21Aq,R  ı  ď C|a|2,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We introduce a martingale whose limit coincides with x  Mγ  ε pξq on the event Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Given x P Rd and q ą 0 we set  Tqpxq :“ inftt ą 0 : Xtpxq “  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dt ` qu,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15)  and define  N pεq  t  pξq :“  ż  Rd ρpxqeiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='xeγXt^Tqpxq,εpxq´ γ2  2 Kt^Tqpxq,εpxqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16)  Since Tqpxq “ 8 for all x in the support of ρ on the event Aq,R, we have  N pεq  8 pξq1Aq,R “ x  Mγ  ε pξq1Aq,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)  42  HUBERT LACOIN  Hence it is sufficient to prove  sup  εPp0,1q  ξPRd  E  ”  vpεq´2|N pεq  8 pξq|2ı  ă 8,  sup  εPp0,1q  ξPRd  E  ”  vpεq´2|N pεq  8 pξ ` aq ´ N pεq  8 pξq|2ı  ď C|a|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18)  Let us prove the only second inequality, since the first one is only easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We set for  simplicity Wt :“ N pεq  t  pξ ` aq ´ N pεq  t  pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have  E  ”  |N pεq  8 pξ ` aq ´ N pεq  8 pξq|2ı  “ Er|W8|2s “ Er|W0|2s ` E rxWy8s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We are going to prove a bound for each of the term in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We have  Er|W0|2s “  ż  R2d ρpxqρpyq  ´  eipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='x ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='x¯ ´  e´ipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='y ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='y¯  e|γ|2K0,εpx,yq  ď C|a|2  ż  ρpxqρpyq|x||y|dxdy ď C1|a|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19)  where in the second line have taken the modulus of the integrand, and used the fact  that the complex exponential is Lipshitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To bound the expected value of the quadratic  variation, using Itˆo calculus, and observing that tTqpxq ă tu “ At,qpxq (recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)) we  obtain that  xWy8 “ |γ|2  ż 8  0  Utdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='20)  where  Ut :“  ż  R2d ρpxqρpyqQt,εpx, yq  ´  eipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='x ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='x¯ ´  e´ipξ`aq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='y ´ eiξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='y¯  ˆ eγXt,εpxq`γXt,εpyq´ γ2  2 Kt,εpxq´ γ2  2 Kt,εpyq1At,qpxqXAt,qpyqdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='21)  Taking the modulus in the integrand value everywhere inside the integral and using the  fact that the complex exponential is Lipshitz fwe obtain  Ut ď |a|2  ż  R2d ρpxqρpyq|x||y|Qt,εpx, yq  ˆ e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  d{2pXt,εpxq`Xt,εpyqq` |γ|2´d  2  pKt,εpxq`Kt,εpyqq1At,qpxqXAt,qpyqdxdy  ď C|a|2  ż  R2d ρpxq2|x|2Qt,εpx, yqe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXt,εpxq`p|γ|2´dqKt,εpxq1At,qpxqdxdy,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='22)  where the second line is obtained via the same step as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26) (ab ď a2`b2{2 and symmetry  and in x and y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Now recalling (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='25) we have  E  ”  e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXt,εpxq´dKt,εpxq1At,qpxq  ı  ď Cpt _ 1q´1{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23)  where to obtain the first inequality, we used (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12) to show that Kspx, yq (and all similar  terms) are well estimated by t for s P r0, ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Now we have  E rUts ď C|a|2e´dt  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  t _ 1  ż  R2d ρpxq2|x|2Qt,εpx, yqe|γ|2Kt,εpx,yqdx ď C1|a|2φpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='24)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES43  After integrating with respect to t (recalling Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4) we obtain that  ErxWy8s ď C|a|2vpεq2,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='25)  for a constant C which is independent of ε and ξ and a, which combined with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19),  concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Beyond star-scale invariance  The assumption that the kernel can be written in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8) may be felt as unnec-  essarily restrictive, since after all, given an open domain D Ă Rd and a positive definite  Kernel kernel K :  D2 Ñ p´8, 8s that admits a decomposition of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1), the  mollified field Xε can be defined on  Dε :“ tx P D : inf  yPDA |x ´ y| ą 2εu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  More precisely in that case the field X is indexed by CcpDq the set of functions with  compact support on D (in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4), R2d is replaced by D2), and Xε remains defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  (here θεpx ´ ¨q, which for x P Dε, has its support included in D, is identified with its  restriction on D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  It turns out that our results can be extended to the the general setup described above,  only with an additional regularity assumption concerning the function L present in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Given U Ă D, we say that the restriction of K to U has an almost star-scale invariant  part, if  @x, y P U, Kpx, yq “ K0px, yq ` Kpx, yq  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  where K is an almost-star scale invariant Kernel, and K0 : U 2 Ñ R is positive definite  and H¨older continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  To extend the result we use the fact (proved in [12]) that if L is sufficiently regular  then K is locally star-scale invariant in the sense defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We state this result as a  proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' It can be directly derived from [12, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If K is a positive definite kernel on D that can be written in the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) with L P Hs  locpD2q with s ą d, then for every z P D, there exist δz ą 0 such that the  restriction of K to Bpz, δzq has an almost star-scale invariant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  To extend Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5 we require another technical result, which states that with the  same assumption as above, and U an open set whose closure is included in D, K can be  approximated by a kernel with an almost star-scale invariant part defined on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This is  the content of the following result, [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1]  Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given K a covariance kernel on D of the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) with L P Hs  locpD2q  for s ą d , U a bounded open set whose closure satisfies U Ă D and δ ą 0, then there  exists a kernel Kpδq on U satisfying (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) such that  (A) For all x, y P U,  |Kpδqpx, yq ´ Kpx, yq| ď δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (B) ∆pδqpx, yq “ Kpδqpx, yq ´ Kpx, yq is a positive definite kernel on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Remark C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' More precisely, [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1] states that one can chose η1 “ 0 (recall  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)) for the almost-star scale invariant part of Kpδq, but this refinement is not required  for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  44  HUBERT LACOIN  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The extension of the result to the case of a general  log-correlated field defined on a domain D is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If X is a centered Gaussian field defined on D whose covariance kernel  K can be written in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) with L P Hs  locpD2q for s ą d, and f P CcpRdq, then  there exists a complex valued random variable Mγ  8pfq such that for any choice of mollifier  θ the following convergence holds in Lp if p P  ”  1,  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d{α  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  lim  εÑ0 Mγ  ε pfq “ Mγ  8pfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' This follows quite immediately via a localization argument using a partition of  unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Let f P CcpDq be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1, we can cover the support of f (which  is compact) by finitely many Euclidean balls Bpzi, εiq, i P I such that for every i P I the  restriction of K to Bpzi, εiq has an almost star-scale invariant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using a partition of the  unity, we can write f :“ ř  iPI fi where fi is continuous with compact support included in  Bpzi, εiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 for K restricted to Bpzi, εiq, we obtain that Mγ  ε pfiq converges  in Lp for every fi and thus we obtain the convergence for Mγ  ε pfq “ ř  iPI Mγ  ε pfiq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The case of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To extend the result for γ P P1  II{III it is sufficient to  extend Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In the statement below, we implicitely use the fact that the critical  multiplicative chaos M1 is well defined under our assumptions (see [17, Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2] for  a proof).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If X is a centered Gaussian field defined on D whose covariance kernel  K can be written in the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) with L P Hs  locpD2q for s ą d, given ρ, f P CcpDq,  ω P r0, 2πq, we have  lim  εÑ0 E  „  eixX,ρy`i Mγ  ε pf,ωq  vpε,θ,γq  \uf6be  “ E  „  eixX,ρy´ 1  2M1pe|γ|2L|f|2q  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given a fixed f P CcpDq, and n ě 1, we chose U which contains the support of  f and Kn : U 2 Ñ p´8, 8s satisfying the assumptions of Kpδq of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 with  δ “ 1{n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We let Zn be a centered Gaussian field indexed by U, independent of X and with  covariance ∆n “ Kn ´K and define Xn a field indexed by CcpUq by setting Xn “ X `Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Note that Xn has covariance Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We let Mγ,n  ε  and M  1  n denote the mollified GMC and  critical GMC associated with Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For simplicity, we define all the pZnqně1 on the same  probability space: the fields Zn form an independent sequence which is independent of  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We let P denote the corresponding probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' From Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8 we have for each  n ě 1  lim  εÑ0 E  „  eixXn,ρy`i Mγ,n  ε  pf,ωq  vpε,θ,γq  \uf6be  “ E  „  eixXn,ρy´ 1  2 M  1  npe|γ|2Ln|f|2q  \uf6be  ,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  where Ln :“ L ` ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' In order to conclude, we need to show that (for any choice of Kpδq)  lim  nÑ8 sup  εPp0,1q  ˇˇˇˇE  „  eixXn,ρy`i Mγ,n  ε  pf,ωq  vpε,θ,γq  \uf6be  ´ E  „  eixX,ρy`i Mγ  ε pf,ωq  vpε,θ,γq  \uf6beˇˇˇˇ “ 0  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  and that  lim  nÑ8 E  „  eixXn,ρy´ 1  2M  1  npe|γ|2Ln|f|2q  \uf6be  “ E  „  eixX,ρy´ 1  2 M  1pe|γ|2Lpδq|f|2q  \uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES45  Note that it is sufficient to show that the difference between the terms in the l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' and  the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' tend to zero in probability (uniformly in ε) that is  lim  nÑ8 E r|xXn, ρy ´ xX, ρy| _ 1s “ 0,  lim  nÑ8 E  ”ˇˇˇM  1  npe|γ|2Ln|f|2q ´ M  1pe|γ|2L|f|2q  ˇˇˇ _ 1  ı  “ 0,  lim  nÑ8 sup  εPp0,1q  E  „ˇˇˇˇ  Mγ,n  ε  pf, ωq  vpε, θ, γq  ´ Mγ  ε pf, ωq  vpε, θ, γq  ˇˇˇˇ _ 1  \uf6be  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  The first line is immediate via the computation of the L2 norm (the convergence holds in  L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the second line, we set gn “ e|γ|2Ln|f|2 and g “ e|γ|2L|f|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using the notational  convention introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1, we let Zn,ε denote the mollification of Zn and ∆n,ε  its covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Letting e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dZn,ε´d∆n,ε denote the function x ÞÑ e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dZn,εpxq´d∆n,εpxq we have  E  „´  M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d,n  ε  pgnq ´ M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ε  pgq  ¯2  | X  \uf6be  “ E  ”  M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ε  pe  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dZn,ε´d∆n,εgn ´ gq2 | X  ı  “  ż  U2  ´  e2d∆n,εpx,yqgnpxqgnpyq ´ 2gnpxqgpyq ` gpxqgpyq  ¯  M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ε  pdxqM  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ε  pdyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  From the assumption that |∆npx, yq| ď 1{n (and thus |Lnpxq ´ Lpxq| ď 1{n) we obtain  that  |e2d∆n,εpx,yqgnpxqgnpyq ´ 2gnpxqgpyq ` gpxqgpyq| ď Cgpxqgpyq  n  and hence  E  „´  M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d,n  ε  pgnq ´ M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ε  pgq  ¯2  | X  \uf6be  ď C  n M  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2d  ε  pgq2  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  Using Fatou after renormalization we obtain that  E  ”`  M1  npgnq ´ M1pgq  ˘2 | X  ı  ď C  n M1pgq2  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  which implies the second line in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For the third line, we are going to Proposition  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' More precisely, we use a decomposition of f “ ř  iPI fi where fi is continous with  compact support included in Ui and the restriction of K to Ui has an almost star-scale  invariant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We are going to prove that for each i P I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  lim  nÑ8 sup  εPp0,1q  E  „ˇˇˇˇ  Mγ,n  ε  pfi, ωq  vpε, θ, γq  ´ Mγ  ε pfi, ωq  vpε, θ, γq  ˇˇˇˇ _ 1  \uf6be  “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  This operation shows that it is in fact sufficient to prove the third line of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7) assuming  that K is an almost star-scale invariant Kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We can thus further our probability space  contains pXtqtě0 a martingale sequence of fields with covariance Kt (we adopt the notation  of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1) approximating X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We equip our space with the filtration  Gt :“ σppXsqsPr0,ts, pZnqně1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  We recall the definition of Tqpxq in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15) and define  W pn,εq  t  :“  ż  U  peγZn,εpxq´ γ2  2 ∆n,εpxq ´ 1qfpxqeγXt^Tqpxq,ε´ γ2  2 Kt^Tq,εpxqdx  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  Setting  Aq :“ t@x P Supppfq, @t ą 0, Xtpxq ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dt ` qu  46  HUBERT LACOIN  We have from the definition  W pn,εq  8  1Aq “ |Mγ,n  ε  pfq ´ Mγ  ε pfq|1Aq  Hence we have (for any ω P r0, 2πq since the projection on one axis reduces the modulus  E  “  |Mγ,n  ε  pf, ωq ´ Mγ  ε pf, ωq|21Aq  ‰  ď Er|W pn,εq  8  |2s “ Er|W pn,εq  0  |2s ` ErxW pn,εqy8s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  Since from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 we have limqÑ8 PrAqs “ 1, to prove the third line in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7), it is  sufficient to show that for any q we have  lim  nÑ8 sup  εPp0,1q  vpε, θ, γq´2 ´  Er|W pn,εq  0  |2s ` ErxW pn,εqy8s  ¯  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14)  For the first term, we have  Er|W pn,εq  0  |2s “  ż  U2 fpxqfpyqpe|γ|2∆n,εpx,yq´1qe|γ|2K0,εpx,yqdxdy ď Cn´1  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='15)  where the inequality obtained taking the modulus of the integrand and using the fact that  |∆npx, yq| ď 1{n and the other terms are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The derivative of the bracket  of W pn,εq is given by |γ|2 times (recall that by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6) we have At,qpxq “ tTqpxq ď tu)  Dt :“  ż  R2d Qt,εpx, yqGn,εpxqGn,εpyq  ˆ eγXt,ε`γXt,εpyq´ γ2  2 Kt,εpxq´ γ2  2 Kt,εpyq1At,qpxqXAt,qpyqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='16)  with Gn,εpxq “ peγZn,εpxq´ γ2  2 ∆n,εpxq ´ 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Repeating once more the computation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='26)  we obtain that  Dt ď  ż  R2d Qt,εpx, yq|Gn,εpxq|2e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  2dXt,ε`p|γ|2´dqKt,εpxq1At,qpxqdxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='17)  We define a martingale W  pεq and W  pεq  t  by setting  W pεq  t  :“ E rMγ,n  ε  pf, ωq ´ Mγ  ε pf, ωq | Gts  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='18)  Now we have  E  “  |Gn,εpxq|2‰  “ e|γ|2∆n,εpxq ´ 1 ď Cn´1  This the term is independent of the rest, thus using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='23) we obtain that  ErDts ď Cn´1  ż  R2d Qt,εpx, yqe|γ|2Kt,εpxqdxdy ď C1n´1φpt, εq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='19)  Integrating against t we conclude that  ErxW pn,εqy8s ď Cn´1vpε, θ, γq2  and this concludes the proof of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES47  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1  We use Kahane convexity inequality in order to compare Bn to the the partition function  of a Gaussian branching random walk (or polymer on a 2d-adic tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We assume without  loss of generality that Supppfq Ă r0, 1sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For x, y P r0, 1sd we let 2´kpx,yq be the sidelength  of the smallest dyadic cube that contains x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  kpx, yq :“ inf  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  n ě 0 : Dm P �0, 2n ´ 1�d, tx, yu Ă  ´  2´nm ` r0, 2´nqd¯)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  and set knpx, yq :“ kpx, yq ^ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Note that kn defines a positive definite function and that  kpx, yq ď log2  ´  1  |x´y|  ¯  ` C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Hence from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 there exists a constant A ą 0 such  that  plog 2qknpx, yq ď Krn log 2spx, yq ` A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)  Using Kahane’s convexity inequality (proved in [15] see also [27, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1]) which we  introduce in a simplified setup  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' If C1 and C2 are two bounded positive definite kernel on an arbitrary space  X satisfying  @x, y P X,  C1px, yq ď C2px, yq  µ is a finite measure on r0, 1sd and F : R` Ñ R is a concave function with at most  polynomial growth at infinity and Y1 and Y2 are Gaussian fields with respective covariance  C1 and C2 then we have for any θ P R  E  „  F  ˆż  eθY1pxq´ θ2  2 C1pxqµpdxq  ˙\uf6be  ď E  „  F  ˆż  eY2pxq´ θ2  2 C2pxqµpdxq  ˙\uf6be  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2)  Hence if Zn denotes a field defined on r0, 1sd with covariance kn we can apply Lemma  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1 result for the fields ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='log 2Zn and Xrn log 2s`  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  AN where N is an independent standard  Gaussian (the fields have their resepective covariances given by the two sides of Equation  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='1)), µpdxq “ |fpxq|2dx and Fpuq “ up{2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Recalling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11) we have  E  ”  Bp{2  rn log 2s  ı  ď CE  »  –  ˜  2dn  ż  r0,1sd |fpxq|2e2α?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='log 2pZnpxq´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2nqdx  ¸p{2fi  fl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3)  The constant C above takes care of the fact that the variance of ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='log 2Znpxq and Xrn log 2s  differ by a Op1q term, and also of the moment of the variable N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We can ignore the  constant f at the cost of a prefactor }f}p  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' To conclude we thus need a bound on the  moment of order p{2 of the partition function of the Gaussian branching random walk  Wn,ζ :“ 2dn  ż  r0,1sd eζpZnpxq´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2nqdx “  ÿ  mP�0,2n´1�d  eζpZnpm2´nq´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2nq,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4)  for ζ “ 2α?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' The following result is a particular case of [8, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We present  a shorter proof which is valid in our context for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Given ζ ą ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2 and q ď  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2  ζ  there exists positive constant C and  b such that  E rpWn,ζqqs ď Cn´  3qζ  2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2plog nq6  48  HUBERT LACOIN  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We split our integral in three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' We set  Bnpxq :“ tDm P �1, n�, Zmpxq ě  a  2d log 2 ` plog nq2u,  Cnpxq :“ BA  npxq X tZnpxq ď  a  2d log 2n ´ plog nq2u,  Anpxq :“ BA  npxq X CA  npxq  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='5)  We define Wn,ζpAq, Wn,ζpBq and Wn,ζpCq by setting, for I P tA, B, Cu  Wn,ζpIq :“ 2dn  ż  r0,1sd eζpZnpxq´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2nq1Inpxqdx  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='6)  Using subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) we have  E rpWn,ζqqs ď E rWn,ζpAqqs ` E rWn,ζpBqqs ` E rWn,ζpCqqs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='7)  We are going to show that the two last terms in the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' decay faster than any negative  power of n and then prove a bound of the right order of magnitude for E rpWn,ζqqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Letting  setting Bn :“ Ť  xPr0,1s Bnpxq, and q1 “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2ζ´1 (q1 P rq, 1q) we have  E rWn,ζpBqqs ď E rpWn,ζqq1Bns ď E  ”  pWn,ζqq1ı q  q1 P rBns1´ q  q1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='8)  Using subadditivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) for the sum (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='4) with θ “ q1,  E  ”  pWn,ζqq1ı  ď E  “  Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2  ‰  “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9)  The inequality on the right comes from the fact that pWm,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2qmě1 is a martingale for  the natural filtration associated with Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Using the optional stopping Theorem for this  same martingale, we can obtain a bound for the probability of Bn,  PrBns ď P  ”  Dm, Wm,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2 ě e  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 log 2plog nq2ı  ď e´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2 log 2plog nq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='10)  This yields a subpolynomial decay for ErWn,ζpBqqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' For Wn,ζpCq using the fact that Zn is a  Gaussian of variance n, we obtain using Jensen’s inequality, the Cameron-Martin formula  and Gaussian tail bounds  E rpWn,ζpCqqqs1{q ď E rWn,ζpCqs “ 2dnE  ”  eqpZn´?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2nq1tZnď?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2n´plog nq2u  ı  “ e  ˆ  d log 2` ζ2  2 ´ζ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2  ˙  n  P  ”  Znp0q ď p  a  2d log 2 ´ ζqn ´ plog nq2ı  ď ep?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2´ζqplog nq2,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='11)  also proving a subpolynomial decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' It remains to estimate the main part E rpWn,ζpAqqqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Using first subaddivity (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='9) and then Jensen’s inequality  E rpWn,ζpAqqqs ď E  „  Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2pAq  qζ  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2  \uf6be  ď E  “  Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2pAq  ‰  qζ  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='12)  The Cameron-Martin formula directly expresses E  “  Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2pAq  ‰  as the probability con-  cerning the Gaussian centered random walk pZmp0qqmě0,  E  “  Wn,?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='2d log 2pAq  ‰  “ P  “  @m P �1, n�, Zmp0q ď plog nq2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Znp0q ě ´plog nq2‰  ď Cn´3{2plog nq6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13)  CONVERGENCE FOR COMPLEX GAUSSIAN MULTIPLICATIVE CHAOS ON PHASE BOUNDARIES49  The bound for the probability of the event above is valid for any random walk with IID  centered increments with finite second moment (see for instance [1, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='3]) which  concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  □  References  [1] Elie A¨ıd´ekon and Zhan Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Weak convergence for the minimal position in a branching random walk:  a simple proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=', 61(1-2):43–54, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  [2] Nathana¨el Berestycki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' An elementary approach to Gaussian multiplicative chaos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=', 22:Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' 27, 12, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Derrida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Evans, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Speer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Mean field theory of directed polymers with random  complex weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
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+page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
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+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
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+page_content=', 26(2):643–690, 04 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  50  HUBERT LACOIN  [25] Ellen Powell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Critical Gaussian multiplicative chaos: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' arXiv e-prints, page arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='13767,  June 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  [26] Ellen Powell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Critical Gaussian multiplicative chaos:  a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Markov Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Related Fields,  27(4):557–506, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  [27] R´emi Rhodes and Vincent Vargas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Gaussian multiplicative chaos and applications: a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content=', 11:315–392, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
+page_content='  IMPA, Institudo de Matem´atica Pura e Aplicada, Estrada Dona Castorina 110 Rio de  Janeiro, CEP-22460-320, Brasil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tE4T4oBgHgl3EQfzg0x/content/2301.05274v1.pdf'}
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+arXiv:2301.01692v1  [gr-qc]  3 Jan 2023
+Another Friedman-type solution that eliminates the problem of the divergent
+cosmological constant, implemented in the framework of the lattice regularization of
+the theory of gravity
+S.N. Vergeles∗
+Landau Institute for Theoretical Physics,
+Russian Academy of Sciences, Chernogolovka,
+Moscow region, 142432 Russia
+and
+Moscow Institute of Physics and Technology,
+Department of Theoretical Physics,
+Dolgoprudnyj, Moskow region, 141707 Russia
+Lattice regularization of the theory of gravity provides a new possibility for solving the problem
+of the divergent cosmological constant. The solution of the Einstein equation within the framework
+of the Friedmann paradigm with a finite bare cosmological constant is mathematically correct, since
+all local physical quantities (energy density including vacuum energy, etc.) on the lattice are finite.
+As a result, a solution is obtained that demonstrates an exponential growth of the cosmological scale
+factor a(t) in the initial period of evolution (inflation phase of the Universe) and then passes into a
+power law (a(t) ∝
+√
+t).
+PACS numbers: 04.60.Bc
+1.
+Introduction.
+In the fundamental review [1]
+the following facts were stated regarding the problem
+of divergent energy (divergent cosmological constant)
+of the ground state in quantum field theory:
+(i) In
+flat Minkowski space-time, these divergences, generally
+speaking, take place, but in the case of supersymmetric
+theories, the energies of the ground states strictly vanish.
+(ii) In curved space-time, even in the case of supergravity,
+the cosmological constant diverges. (iii) The superstring
+theory does not save the situation either.
+Since then, many papers have been published on this
+problem. Here we note only a few approaches to solving
+the problem, which seem to us very promising.
+The first approach is represented by works [2–7]. In
+these papers, efforts were made to solve the problem of
+the cosmological constant in detail, that is, through mi-
+croscopic analysis. In particular, the probability of the
+following process was calculated. Let there be a massive
+particle in the de Sitter space. This particle gives rise to
+a pair of the same particles for a sufficiently long period
+of time. This problem has been studied for both free and
+interacting fields.
+A similar statement of the problem
+for massive charged particles in the case of a flat space-
+time in the presence of a constant electric field leads to
+the creation of particle-antiparticle pair that weaken the
+initial electric field. In the case of de Sitter space, pair
+production also leads to a decrease in the cosmological
+constant with time. Unfortunately, in these works there
+is no study of the reverse influence of quantized mate-
+rial fields on the space-time geometry. It is possible that
+continued efforts in this direction will lead to a solution
+∗e-mail:vergeles@itp.ac.ru
+to the problem of the cosmological constant.
+In the paper [8] the mean of the energy-momentum
+tensor of a quantized scalar field is calculated in the case
+of an anisotropic metric, which is considered to be clas-
+sical and variable in time. Regularization is carried out
+in the usual way: the vacuum expectation value of the
+energy-momentum tensor, calculated in the case of a sta-
+tionary vacuum, is subtracted from the obtained value.
+The authors of the paper [9] study such models of
+field theory which, although not supersymmetric, have
+the same number of boson and fermion degrees of free-
+dom. In this case, the divergences of the highest, fourth
+degree are eliminated in the quantum mean of the energy-
+momentum tensor. It is shown what conditions the al-
+ready renormalized field masses must satisfy in order to
+reduce all other divergences.
+The work [10] seems to us to be interesting and com-
+plementary to the present work, since a bare cosmological
+constant is also introduced in [10], and the reduction of
+the huge vacuum energy is a dynamic effect, not a fine-
+tuning effect.
+Another interesting approach to solving the problem,
+using the macroscopic thermodynamic ideology, is pre-
+sented in [11] (see there also references for the articles
+of F.R. Klinkhamer and G.E. Volovik). The main idea
+of this approach is as follows. A bare cosmological con-
+stant is introduced into the system describing gravity and
+matter. The bare cosmological constant plays the role of
+the chemical potential µ. If the system comes to a state
+of thermodynamic equilibrium, then a large thermody-
+namic potential is of interest. Let Ω be a large thermo-
+dynamic potential for the spatial volume V . It is known
+that
+Ω(β, µ, V ) = −P(β, µ)V.
+(1)
+
+2
+Here, β stands for inverse temperature.
+In our case,
+β should be understood as the (imaginary) time dur-
+ing which the transition quantum amplitude (or partition
+function) is calculated. We have an obvious limitation for
+β values: |βH| ≪ 1, where H = ˙a/a is Hubble constant
+and a(t) is the cosmic scale factor. Otherwise, there can
+be no thermodynamic equilibrium.
+The standard spa-
+tially flat Robertson-Walker metric
+d s2 = (d x0)2 − a2(t)(d xα)2,
+α = 1, 2, 3
+(2)
+is used in [11]. Since the gravitational degrees of free-
+dom are exhausted by only one global parameter a(t),
+then the potential (1) is saturated with the degrees of
+freedom of matter. The main idea of the authors of the
+paper [11] is that in the case of thermal equilibrium (if it
+exists), the pressure on the right side of the equality (1)
+tends to zero, since there is no external pressure at all.
+Further, the effective energy-momentum tensor of matter
+is formed by the potential (1). Therefore, the effective
+energy density of matter, including the vacuum energy,
+under the condition of thermal equilibrium is estimated
+as ε ∼ Ω/V −→ 0. Thus the problem of the divergent
+cosmological constant is removed.
+The fundamental defect of all the papers cited here
+is the fact that the vacuum energy (in particular, the
+energy of zero-point oscillations) is not limited.
+Figu-
+ratively speaking, the Dirac sea has no bottom.
+And
+although the divergences in physical quantities are elim-
+inated by subtracting vacuum values from them, there
+remains a feeling of unsteadiness of the ground under
+the feet of the researcher. The reason for this is that in
+this case the characteristic divergences are power-law of
+the fourth degree.
+On the other hand, if the hypothesis is accepted that on
+ultra-small scales, space-time has the property of gran-
+ularity (this property is modeled by a lattice), then the
+formulation and solution of at least some problems turn
+out to be mathematically correct (see below).
+This work is an ideological continuation of the work
+[11]. The essential difference between the present paper
+and the paper [11] is that we assume a lattice regular-
+ization of the theory of gravity (see [12] and references
+there). Lattice regularization provides a new possibility
+for solving the problem of the divergent cosmological
+constant. The solution of the Einstein equation within
+the framework of the Friedmann paradigm with a finite
+bare cosmological constant is mathematically correct,
+since all local physical quantities (energy density in-
+cluding vacuum energy, etc.)
+on the lattice are finite.
+Our approach assumes that all physical quantities are
+determined by taking into account quantum zero point
+fluctuations.
+In particular, the energy density and
+pressure are mainly determined by quantum fluctua-
+tions. Since the equations considered here describe such
+large energy densities that, on the characteristic time
+intervals, have actions exceeding the Planck constant
+by a huge number of times, we assume the considered
+physical quantities to be classical and use the classical
+equations [13]. As a result, a solution is obtained that
+demonstrates an exponential growth of the scale factor
+in the initial period of evolution and then passes into a
+power law.
+2. Einstein equation and solution. We use the energy-
+momentum tensor of matter in the form of the energy-
+momentum tensor of an ideal relativistic fluid:
+T a
+b = (ε + p)U aUb − pδa
+b .
+(3)
+We work in an orthonormal basis in which the metric ten-
+sor ηab = diag(1, −1, −1, −1). On the right side of (3),
+the symbols ε and p denote the energy density and pres-
+sure, respectively, and these quantities also include vac-
+uum energy and pressure. Since fermionic fields, in con-
+trast to bosonic ones, make a negative contribution to the
+vacuum energy, but there are significantly more fermionic
+degrees of freedom than bosonic ones, we have ε < 0.
+Moreover, lattice regularization means that |ε|, |p| < ∞.
+Note that in (3) the pressure p is different from the pres-
+sure P(β, µ) in (1). A comparison of these values is given
+below. U a is the averaged 4-velocity of the macroscopic
+regions of the lattice. In our case U a = (1, 0, 0, 0). To
+compensate for the vacuum energy, a bare finite positive
+cosmological constant Λ0 is introduced into the Einstein
+equation[14]:
+Ra
+b −1
+2δa
+b R = 8πG
+c4 T a
+b + Λ0δa
+b .
+(4)
+We assume that the cosmological constant
+Λ0 = const ∼ l−2
+P ,
+lP ∼
+�
+8πGℏ
+c3
+∼ 10−32cm.
+(5)
+For the metric, we use ansatz (2). In order not to clutter
+up the formulas, we introduce the notation
+8πG
+c4 ε = ˜ε,
+8πG
+c4 p = ˜p.
+(6)
+All components of the Einstein equation are reduced to
+two equations:
+3 ˙a2
+a2 = Λ0 + ˜ε,
+2¨a
+a + ˙a2
+a2 = Λ0 − ˜p.
+(7)
+Here ˙a ≡ d a/ d x0.
+Another equation ∇aT a
+b = 0 is a
+consequence of equations (7), and therefore it does not
+need to be considered. Let us introduce the Hubble con-
+stant ˜H(t) ≡ ˙a/a, with the help of which Eqs. (7) are
+rewritten as follows:
+2 ˙˜H + (˜ε + ˜p) = 0,
+3 ˜H2 − (Λ0 + ˜ε) = 0.
+(8)
+So we have 3 unknown functions {˜ε(t), ˜p(t), ˜H(t)} and 2
+equations (8). The missing equation is the equation of
+state relating energy density and pressure. Regarding the
+equation of state, the following facts are reliably known:
+(i) in the case of real dusty matter, we have ˜p = 0; (ii) in
+
+3
+the case of real ultrarelativistic matter we have ˜p = ˜ε/3;
+in the case of vacuum energy and pressure, we have ˜p =
+−˜ε. In all three cases, the energy density and pressure
+are linearly related. Therefore, we propose to accept the
+following hypothesis:
+˜p = κΛ0 + (κ − 1)˜ε ←→ ˜ε + ˜p = κ(˜ε + Λ0).
+(9)
+This equation is linear and inhomogeneous with an un-
+known function κ(t), the asymptotics of which are fur-
+ther determined based on the known dynamics. The set
+of equations (8) and (9) has a solution:
+˙˜H = −3
+2κ ˜H2 → ˜H(t) = ˜H0
+�
+1 + 3
+2H0
+� t
+0
+κ(t′) d t′
+�−1
+,
+(10)
+˜ε(t) = −Λ0 + 3 ˜H2
+0
+�
+1 + 3
+2H0
+� t
+0
+κ(t′) d t′
+�−2
+,
+(11)
+˜p(t) = Λ0 + 3
+�
+κ(t) − 1
+� ˜H2
+0
+�
+1 + 3
+2H0
+� t
+0
+κ(t′) d t′
+�−2
+.
+(12)
+Here ˜H0 ≡ H0/c is the integration constant, ˜H(t) ≡
+H(t)/c, and H0 is the Hubble constant at the beginning
+of the inflation phase, [H(t)] = [H0] = s−1.
+We indicate some of the most obvious properties of the
+solution (10), (11), (12). The estimates given below are
+very rough. Let us accept the following estimates for the
+duration of the inflation time tinf, and for the constant
+Λ0:
+tinf ∼= 10−37s,
+H0 ∼= 1039s−1,
+˜H0 ∼= 1029cm−1. (13)
+Then H0tinf ∼= 100. Let’s take κ0 ≡ κ(t = 0) ∼= 1/150.
+Assume that during the time tinf the function κ changes
+insignificantly. Then for t < tinf the solutions (10), (11),
+(12) take the form
+˜H(t) ∼= ˜H0,
+˜ε(t) ∼= −˜p ∼= −Λ0 + 3 ˜H2
+0.
+(14)
+Thus, during inflation, the scale factor a(t) increased by
+(exp H0tinf) ≈ (exp 100) ≈ 1043 times.
+Assume that when t > tinf, the function κ(t) becomes
+equal to κ = 4/3. In this case, the solutions (10), (11),
+(12) give a power-law expansion:
+H(t) ∼= 1
+2t,
+˜ε(t) ∼= −Λ0 + 3
+4t2 ,
+˜p ∼= Λ0 + 1
+4t2 .
+(15)
+Solution (15) shows that the scale factor and the density
+of real matter change according to the well-known law,
+as well as the correct equation of state in the case of
+ultrarelativistic matter:
+a(t) ∝
+√
+t,
+ρreal =
+3
+32πGt2 ,
+preal = 1
+3εreal.
+(16)
+3. Thermodynamic considerations. Here, the possibil-
+ity of using a thermodynamic approach to this problem
+is briefly discussed, and some thermodynamic relations
+are also given. The purpose of this consideration is to
+(at least superficially) explain the state equation (9).
+The estimation (13) means that
+˜H2
+0 ≪ Λ0.
+(17)
+It can be seen from Eq. (11) that the maximum frequen-
+cies of the degrees of freedom of matter in the modern
+era are of the order of
+|ωmax| ∼ c
+�
+Λ0.
+(18)
+We are interested in small times when H ∼ H0 (see Eq.
+(10).
+Since at these times the space was many orders
+of magnitude more compact, then for small times the
+estimate |ωmax| ≫ c√Λ0 was valid. Consider the time
+interval ∆t ≲ H−1, for which we have
+∆a/a ∼ H∆t ≲ 1,
+∆t|ωmax| ≫ 1.
+(19)
+Taking into account Eq. (19), we can assume that for a
+time interval ∆t the thermodynamic equilibrium of the
+vacuum degrees of freedom is realized. This assumption
+cannot be extended to those degrees of freedom whose
+frequencies ∆t|ω| ≲ 1. But such degrees of freedom make
+a small contribution to the total energy-momentum ten-
+sor.
+When passing to the Euclidean signature by Wick’s
+rotation ∆t = −i∆τ [15], the parameter
+T ≡ β−1 = ℏ(∆τ)−1 ∼ ℏH
+(20)
+acquires the meaning of temperature. Let us determine
+the temperature value in Kelvin degrees at the begin-
+ning of the inflation process, when, according to some
+estimates H0 ∼ 1039s−1. Then
+T0 ∼ ℏH0
+k
+∼
+�
+1028�◦ K.
+(21)
+Here k is the Boltzmann constant. The temperature es-
+timate (21) is within the known temperature estimates
+in the initial phase of inflation.
+Once again, we note that thermodynamic considera-
+tions do not apply to low-frequency degrees of freedom.
+In particular, ordinary real matter may, generally speak-
+ing, not be in a state of thermal equilibrium.
+According to Eqs. (10) and (20) we have:
+ℏ d β ∼ d(1/H) = 3/2κ d t.
+But in the inflation phase a(t) = a0eHt, and so d t =
+H−1 d a/a. Thus we have:
+d β/β ∼ (3/2)κ d a/a.
+(22)
+Since the temperature decreases in the inflation phase, it
+can be seen from (22) that κ(t = 0) > 0.
+
+4
+It can be seen from the first Eq. (7) that the constant
+Λ0 cancels out the huge negative energy of the vacuum,
+so that in the era of power-law expansion only the rel-
+atively extremely small positive energy density of real
+matter affects the dynamics. From the given solution of
+Einstein’s equations, it can be seen that the huge value
+of pressure is also mainly reduced by the constant Λ0.
+In the presented solution we have ˜p ∼ −˜ε ∼ Λ0. Such a
+ratio of pressure and energy density of matter is dictated
+by the relativistic invariance of quantum states.
+We will show that the contraction of the enormous vac-
+uum pressure ˜p can be interpreted as a thermodynamic
+effect. Indeed, the contribution of the cosmological con-
+stant to the action for volume V =
+� �
+|g| d3 x and time
+interval ∆t is equal to
+i AΛ0 /ℏ = − ic4
+8πGℏΛ0V ∆t.
+(23)
+As a result of the Wick rotation according to the formula
+∆t = −i∆τ and due to (20) the action (23) is trans-
+formed to the form
+i AΛ0 /ℏ −→ −
+c4
+8πGℏΛ0V ∆τ = − c4
+8πGΛ0V β.
+(24)
+Adding (24) to the Euclidean action has the same effect
+as adding µNβ.
+Here N is the number of degrees of
+freedom on the part of the lattice contained in the volume
+V , and µ is the total chemical potential of the lattice.
+Equating the value (µNβ) to the value on the right side
+of the Eq. (24), we find:
+µ = − c4
+8πGΛ0
+V
+N .
+(25)
+Usually the chemical potential is the independent vari-
+able. But here it is a function of volume. Therefore, the
+total pressure is determined by a more complex formula:
+P(β, µ) = −
+� ∂Ω
+∂V
+�
+β,µ
+−
+�∂Ω
+∂µ
+�
+β,V
+∂µ
+∂V = p −
+c4
+8πGΛ0.
+(26)
+Here we have taken into account the equality N
+=
+−(∂Ω/∂µ)β,V and Eq. (25). On the left hand side of
+Eq. (26) the pressure P(β, µ) is the same as the pressure
+in Eq. (1). The above solution of the Einstein equations
+shows that P(β, µ) is negligible compared to the total
+pressure p of matter. This fact was pointed out and used
+in the work [11]
+The estimate ˜p ∼= Λ0 together with the vacuum energy
+hypothesis ˜ε ∼= −Λ0 justifies the equation of state (9). In
+both parts of equality (˜ε + ˜p) = κ(˜ε + Λ0), the diverging
+values of the quantities cancel each other out. This fact
+is the result of solving dynamic equations.
+A more accurate equation of state should be obtained
+by microscopic analysis in the spirit of the works [2–7].
+Acknowledgments
+I thank Prof. G.E. Volovik for awakening my interest
+in the thermodynamic study of the problem. I am grate-
+ful to Prof.
+E.T. Akhmedov for numerous discussions
+and advice in the course of work. This work was carried
+out as a part of the State Program 0033-2019-0005.
+[1] S. Weinberg, Reviews of modern physics 61, 1 (1989).
+[2] D. Krotov and A. M. Polyakov, Nuclear Physics B 849,
+410 (2011).
+[3] A. Polyakov, arXiv preprint arXiv:1209.4135 (2012).
+[4] E. Akhmedov, International Journal of Modern Physics
+D 23, 1430001 (2014).
+[5] E. Akhmedov, U. Moschella, and F. Popov, Physical Re-
+view D 99, 086009 (2019).
+[6] E. Akhmedov, Modern Physics Letters A 36, 2130020
+(2021).
+[7] A. Y. Kamenshchik, A. A. Starobinsky, and T. Var-
+danyan, The European Physical Journal C 82, 1 (2022).
+[8] Y. B. Zel’Dovich and A. Starobinskiˇı, Soviet Journal of
+Experimental and Theoretical Physics 34, 1159 (1972).
+[9] A. Y. Kamenshchik, A. A. Starobinsky, A. Tronconi,
+T. Vardanyan, and G. Venturi, The European Physical
+Journal C 78, 1 (2018).
+[10] S. Appleby and E. V. Linder, Journal of Cosmology and
+Astroparticle Physics 2020, 037 (2020).
+[11] F. Klinkhamer and G. Volovik, Physical Review D 105,
+084066 (2022).
+[12] S. Vergeles, Classical and Quantum Gravity 38, 085022
+(2021).
+[13] We mean the fact that according to (5), (6) and (11)
+we have the estimate (l3
+P tP ε)/ℏ ∼ 1. Here tP ∼ lP /c ∼
+10−43s is the Planck time. However, the inflation time
+tinf is several orders of magnitude longer than the Planck
+time (see (13)), and therefore (l3
+P tinfε)/ℏ ≫ 1. This
+means that in the Planck volume, on a time interval much
+greater than the Planckian but much less than the infla-
+tion time, the action of the system is much greater than
+the Planck constant, and therefore a classical description
+is possible.
+[14] In lattice theory [12], the cosmological constant is intro-
+duced in a natural way.
+[15] The correctness of the sign during Wick rotation is es-
+tablished by the example of the action of a scalar field.
+
diff --git a/3dAzT4oBgHgl3EQfuf2L/content/tmp_files/load_file.txt b/3dAzT4oBgHgl3EQfuf2L/content/tmp_files/load_file.txt
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@@ -0,0 +1,222 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf,len=221
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='01692v1  [gr-qc]  3 Jan 2023  Another Friedman-type solution that eliminates the problem of the divergent  cosmological constant, implemented in the framework of the lattice regularization of  the theory of gravity  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Vergeles∗  Landau Institute for Theoretical Physics,  Russian Academy of Sciences, Chernogolovka,  Moscow region, 142432 Russia  and  Moscow Institute of Physics and Technology,  Department of Theoretical Physics,  Dolgoprudnyj, Moskow region, 141707 Russia  Lattice regularization of the theory of gravity provides a new possibility for solving the problem  of the divergent cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The solution of the Einstein equation within the framework  of the Friedmann paradigm with a finite bare cosmological constant is mathematically correct, since  all local physical quantities (energy density including vacuum energy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=') on the lattice are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  As a result, a solution is obtained that demonstrates an exponential growth of the cosmological scale  factor a(t) in the initial period of evolution (inflation phase of the Universe) and then passes into a  power law (a(t) ∝  √  t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  PACS numbers: 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='Bc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  In the fundamental review [1]  the following facts were stated regarding the problem  of divergent energy (divergent cosmological constant)  of the ground state in quantum field theory:  (i) In  flat Minkowski space-time, these divergences, generally  speaking, take place, but in the case of supersymmetric  theories, the energies of the ground states strictly vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (ii) In curved space-time, even in the case of supergravity,  the cosmological constant diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (iii) The superstring  theory does not save the situation either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Since then, many papers have been published on this  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Here we note only a few approaches to solving  the problem, which seem to us very promising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  The first approach is represented by works [2–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In  these papers, efforts were made to solve the problem of  the cosmological constant in detail, that is, through mi-  croscopic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In particular, the probability of the  following process was calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Let there be a massive  particle in the de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' This particle gives rise to  a pair of the same particles for a sufficiently long period  of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' This problem has been studied for both free and  interacting fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  A similar statement of the problem  for massive charged particles in the case of a flat space-  time in the presence of a constant electric field leads to  the creation of particle-antiparticle pair that weaken the  initial electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In the case of de Sitter space, pair  production also leads to a decrease in the cosmological  constant with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Unfortunately, in these works there  is no study of the reverse influence of quantized mate-  rial fields on the space-time geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' It is possible that  continued efforts in this direction will lead to a solution  ∗e-mail:vergeles@itp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='ru  to the problem of the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  In the paper [8] the mean of the energy-momentum  tensor of a quantized scalar field is calculated in the case  of an anisotropic metric, which is considered to be clas-  sical and variable in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Regularization is carried out  in the usual way: the vacuum expectation value of the  energy-momentum tensor, calculated in the case of a sta-  tionary vacuum, is subtracted from the obtained value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  The authors of the paper [9] study such models of  field theory which, although not supersymmetric, have  the same number of boson and fermion degrees of free-  dom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In this case, the divergences of the highest, fourth  degree are eliminated in the quantum mean of the energy-  momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' It is shown what conditions the al-  ready renormalized field masses must satisfy in order to  reduce all other divergences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  The work [10] seems to us to be interesting and com-  plementary to the present work, since a bare cosmological  constant is also introduced in [10], and the reduction of  the huge vacuum energy is a dynamic effect, not a fine-  tuning effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Another interesting approach to solving the problem,  using the macroscopic thermodynamic ideology, is pre-  sented in [11] (see there also references for the articles  of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Klinkhamer and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Volovik).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The main idea  of this approach is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' A bare cosmological con-  stant is introduced into the system describing gravity and  matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The bare cosmological constant plays the role of  the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' If the system comes to a state  of thermodynamic equilibrium, then a large thermody-  namic potential is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Let Ω be a large thermo-  dynamic potential for the spatial volume V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' It is known  that  Ω(β, µ, V ) = −P(β, µ)V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (1)  2  Here, β stands for inverse temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  In our case,  β should be understood as the (imaginary) time dur-  ing which the transition quantum amplitude (or partition  function) is calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' We have an obvious limitation for  β values: |βH| ≪ 1, where H = ˙a/a is Hubble constant  and a(t) is the cosmic scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Otherwise, there can  be no thermodynamic equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  The standard spa-  tially flat Robertson-Walker metric  d s2 = (d x0)2 − a2(t)(d xα)2,  α = 1, 2, 3  (2)  is used in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Since the gravitational degrees of free-  dom are exhausted by only one global parameter a(t),  then the potential (1) is saturated with the degrees of  freedom of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The main idea of the authors of the  paper [11] is that in the case of thermal equilibrium (if it  exists), the pressure on the right side of the equality (1)  tends to zero, since there is no external pressure at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Further, the effective energy-momentum tensor of matter  is formed by the potential (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Therefore, the effective  energy density of matter, including the vacuum energy,  under the condition of thermal equilibrium is estimated  as ε ∼ Ω/V −→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Thus the problem of the divergent  cosmological constant is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  The fundamental defect of all the papers cited here  is the fact that the vacuum energy (in particular, the  energy of zero-point oscillations) is not limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Figu-  ratively speaking, the Dirac sea has no bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  And  although the divergences in physical quantities are elim-  inated by subtracting vacuum values from them, there  remains a feeling of unsteadiness of the ground under  the feet of the researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The reason for this is that in  this case the characteristic divergences are power-law of  the fourth degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  On the other hand, if the hypothesis is accepted that on  ultra-small scales, space-time has the property of gran-  ularity (this property is modeled by a lattice), then the  formulation and solution of at least some problems turn  out to be mathematically correct (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  This work is an ideological continuation of the work  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The essential difference between the present paper  and the paper [11] is that we assume a lattice regular-  ization of the theory of gravity (see [12] and references  there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Lattice regularization provides a new possibility  for solving the problem of the divergent cosmological  constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The solution of the Einstein equation within  the framework of the Friedmann paradigm with a finite  bare cosmological constant is mathematically correct,  since all local physical quantities (energy density in-  cluding vacuum energy, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=')  on the lattice are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Our approach assumes that all physical quantities are  determined by taking into account quantum zero point  fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  In particular, the energy density and  pressure are mainly determined by quantum fluctua-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Since the equations considered here describe such  large energy densities that, on the characteristic time  intervals, have actions exceeding the Planck constant  by a huge number of times, we assume the considered  physical quantities to be classical and use the classical  equations [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' As a result, a solution is obtained that  demonstrates an exponential growth of the scale factor  in the initial period of evolution and then passes into a  power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Einstein equation and solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' We use the energy-  momentum tensor of matter in the form of the energy-  momentum tensor of an ideal relativistic fluid:  T a  b = (ε + p)U aUb − pδa  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (3)  We work in an orthonormal basis in which the metric ten-  sor ηab = diag(1, −1, −1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' On the right side of (3),  the symbols ε and p denote the energy density and pres-  sure, respectively, and these quantities also include vac-  uum energy and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Since fermionic fields, in con-  trast to bosonic ones, make a negative contribution to the  vacuum energy, but there are significantly more fermionic  degrees of freedom than bosonic ones, we have ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Moreover, lattice regularization means that |ε|, |p| < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Note that in (3) the pressure p is different from the pres-  sure P(β, µ) in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' A comparison of these values is given  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' U a is the averaged 4-velocity of the macroscopic  regions of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In our case U a = (1, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' To  compensate for the vacuum energy, a bare finite positive  cosmological constant Λ0 is introduced into the Einstein  equation[14]:  Ra  b −1  2δa  b R = 8πG  c4 T a  b + Λ0δa  b .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (4)  We assume that the cosmological constant  Λ0 = const ∼ l−2  P ,  lP ∼  �  8πGℏ  c3  ∼ 10−32cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (5)  For the metric, we use ansatz (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In order not to clutter  up the formulas, we introduce the notation  8πG  c4 ε = ˜ε,  8πG  c4 p = ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (6)  All components of the Einstein equation are reduced to  two equations:  3 ˙a2  a2 = Λ0 + ˜ε,  2¨a  a + ˙a2  a2 = Λ0 − ˜p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (7)  Here ˙a ≡ d a/ d x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Another equation ∇aT a  b = 0 is a  consequence of equations (7), and therefore it does not  need to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Let us introduce the Hubble con-  stant ˜H(t) ≡ ˙a/a, with the help of which Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (7) are  rewritten as follows:  2 ˙˜H + (˜ε + ˜p) = 0,  3 ˜H2 − (Λ0 + ˜ε) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (8)  So we have 3 unknown functions {˜ε(t), ˜p(t), ˜H(t)} and 2  equations (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The missing equation is the equation of  state relating energy density and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Regarding the  equation of state, the following facts are reliably known:  (i) in the case of real dusty matter, we have ˜p = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (ii) in  3  the case of real ultrarelativistic matter we have ˜p = ˜ε/3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  in the case of vacuum energy and pressure, we have ˜p =  −˜ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In all three cases, the energy density and pressure  are linearly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Therefore, we propose to accept the  following hypothesis:  ˜p = κΛ0 + (κ − 1)˜ε ←→ ˜ε + ˜p = κ(˜ε + Λ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (9)  This equation is linear and inhomogeneous with an un-  known function κ(t), the asymptotics of which are fur-  ther determined based on the known dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The set  of equations (8) and (9) has a solution:  ˙˜H = −3  2κ ˜H2 → ˜H(t) = ˜H0  �  1 + 3  2H0  � t  0  κ(t′) d t′  �−1  ,  (10)  ˜ε(t) = −Λ0 + 3 ˜H2  0  �  1 + 3  2H0  � t  0  κ(t′) d t′  �−2  ,  (11)  ˜p(t) = Λ0 + 3  �  κ(t) − 1  � ˜H2  0  �  1 + 3  2H0  � t  0  κ(t′) d t′  �−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (12)  Here ˜H0 ≡ H0/c is the integration constant, ˜H(t) ≡  H(t)/c, and H0 is the Hubble constant at the beginning  of the inflation phase, [H(t)] = [H0] = s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  We indicate some of the most obvious properties of the  solution (10), (11), (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The estimates given below are  very rough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Let us accept the following estimates for the  duration of the inflation time tinf, and for the constant  Λ0:  tinf ∼= 10−37s,  H0 ∼= 1039s−1,  ˜H0 ∼= 1029cm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (13)  Then H0tinf ∼= 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Let’s take κ0 ≡ κ(t = 0) ∼= 1/150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Assume that during the time tinf the function κ changes  insignificantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Then for t < tinf the solutions (10), (11),  (12) take the form  ˜H(t) ∼= ˜H0,  ˜ε(t) ∼= −˜p ∼= −Λ0 + 3 ˜H2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (14)  Thus, during inflation, the scale factor a(t) increased by  (exp H0tinf) ≈ (exp 100) ≈ 1043 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Assume that when t > tinf, the function κ(t) becomes  equal to κ = 4/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In this case, the solutions (10), (11),  (12) give a power-law expansion:  H(t) ∼= 1  2t,  ˜ε(t) ∼= −Λ0 + 3  4t2 ,  ˜p ∼= Λ0 + 1  4t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (15)  Solution (15) shows that the scale factor and the density  of real matter change according to the well-known law,  as well as the correct equation of state in the case of  ultrarelativistic matter:  a(t) ∝  √  t,  ρreal =  3  32πGt2 ,  preal = 1  3εreal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (16)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Thermodynamic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Here, the possibil-  ity of using a thermodynamic approach to this problem  is briefly discussed, and some thermodynamic relations  are also given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The purpose of this consideration is to  (at least superficially) explain the state equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  The estimation (13) means that  ˜H2  0 ≪ Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (17)  It can be seen from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (11) that the maximum frequen-  cies of the degrees of freedom of matter in the modern  era are of the order of  |ωmax| ∼ c  �  Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (18)  We are interested in small times when H ∼ H0 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Since at these times the space was many orders  of magnitude more compact, then for small times the  estimate |ωmax| ≫ c√Λ0 was valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Consider the time  interval ∆t ≲ H−1, for which we have  ∆a/a ∼ H∆t ≲ 1,  ∆t|ωmax| ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (19)  Taking into account Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (19), we can assume that for a  time interval ∆t the thermodynamic equilibrium of the  vacuum degrees of freedom is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' This assumption  cannot be extended to those degrees of freedom whose  frequencies ∆t|ω| ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' But such degrees of freedom make  a small contribution to the total energy-momentum ten-  sor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  When passing to the Euclidean signature by Wick’s  rotation ∆t = −i∆τ [15], the parameter  T ≡ β−1 = ℏ(∆τ)−1 ∼ ℏH  (20)  acquires the meaning of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Let us determine  the temperature value in Kelvin degrees at the begin-  ning of the inflation process, when, according to some  estimates H0 ∼ 1039s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Then  T0 ∼ ℏH0  k  ∼  �  1028�◦ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (21)  Here k is the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The temperature es-  timate (21) is within the known temperature estimates  in the initial phase of inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Once again, we note that thermodynamic considera-  tions do not apply to low-frequency degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  In particular, ordinary real matter may, generally speak-  ing, not be in a state of thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  According to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (10) and (20) we have:  ℏ d β ∼ d(1/H) = 3/2κ d t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  But in the inflation phase a(t) = a0eHt, and so d t =  H−1 d a/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Thus we have:  d β/β ∼ (3/2)κ d a/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (22)  Since the temperature decreases in the inflation phase, it  can be seen from (22) that κ(t = 0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  4  It can be seen from the first Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (7) that the constant  Λ0 cancels out the huge negative energy of the vacuum,  so that in the era of power-law expansion only the rel-  atively extremely small positive energy density of real  matter affects the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' From the given solution of  Einstein’s equations, it can be seen that the huge value  of pressure is also mainly reduced by the constant Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  In the presented solution we have ˜p ∼ −˜ε ∼ Λ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Such a  ratio of pressure and energy density of matter is dictated  by the relativistic invariance of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  We will show that the contraction of the enormous vac-  uum pressure ˜p can be interpreted as a thermodynamic  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Indeed, the contribution of the cosmological con-  stant to the action for volume V =  � �  |g| d3 x and time  interval ∆t is equal to  i AΛ0 /ℏ = − ic4  8πGℏΛ0V ∆t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (23)  As a result of the Wick rotation according to the formula  ∆t = −i∆τ and due to (20) the action (23) is trans-  formed to the form  i AΛ0 /ℏ −→ −  c4  8πGℏΛ0V ∆τ = − c4  8πGΛ0V β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (24)  Adding (24) to the Euclidean action has the same effect  as adding µNβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Here N is the number of degrees of  freedom on the part of the lattice contained in the volume  V , and µ is the total chemical potential of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Equating the value (µNβ) to the value on the right side  of the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (24), we find:  µ = − c4  8πGΛ0  V  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (25)  Usually the chemical potential is the independent vari-  able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' But here it is a function of volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Therefore, the  total pressure is determined by a more complex formula:  P(β, µ) = −  � ∂Ω  ∂V  �  β,µ  −  �∂Ω  ∂µ  �  β,V  ∂µ  ∂V = p −  c4  8πGΛ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  (26)  Here we have taken into account the equality N  =  −(∂Ω/∂µ)β,V and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' On the left hand side of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (26) the pressure P(β, µ) is the same as the pressure  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' The above solution of the Einstein equations  shows that P(β, µ) is negligible compared to the total  pressure p of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' This fact was pointed out and used  in the work [11]  The estimate ˜p ∼= Λ0 together with the vacuum energy  hypothesis ˜ε ∼= −Λ0 justifies the equation of state (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' In  both parts of equality (˜ε + ˜p) = κ(˜ε + Λ0), the diverging  values of the quantities cancel each other out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' This fact  is the result of solving dynamic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  A more accurate equation of state should be obtained  by microscopic analysis in the spirit of the works [2–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  Acknowledgments  I thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Volovik for awakening my interest  in the thermodynamic study of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' I am grate-  ful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Akhmedov for numerous discussions  and advice in the course of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' This work was carried  out as a part of the State Program 0033-2019-0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Weinberg, Reviews of modern physics 61, 1 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Krotov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Polyakov, Nuclear Physics B 849,  410 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Polyakov, arXiv preprint arXiv:1209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='4135 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Akhmedov, International Journal of Modern Physics  D 23, 1430001 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Akhmedov, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Moschella, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Popov, Physical Re-  view D 99, 086009 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Akhmedov, Modern Physics Letters A 36, 2130020  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [7] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Kamenshchik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Starobinsky, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Var-  danyan, The European Physical Journal C 82, 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Zel’Dovich and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Starobinskiˇı, Soviet Journal of  Experimental and Theoretical Physics 34, 1159 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Kamenshchik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Starobinsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Tronconi,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Vardanyan, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Venturi, The European Physical  Journal C 78, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Appleby and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Linder, Journal of Cosmology and  Astroparticle Physics 2020, 037 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [11] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Klinkhamer and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Volovik, Physical Review D 105,  084066 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [12] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Vergeles, Classical and Quantum Gravity 38, 085022  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [13] We mean the fact that according to (5), (6) and (11)  we have the estimate (l3  P tP ε)/ℏ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' Here tP ∼ lP /c ∼  10−43s is the Planck time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' However, the inflation time  tinf is several orders of magnitude longer than the Planck  time (see (13)), and therefore (l3  P tinfε)/ℏ ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content=' This  means that in the Planck volume, on a time interval much  greater than the Planckian but much less than the infla-  tion time, the action of the system is much greater than  the Planck constant, and therefore a classical description  is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [14] In lattice theory [12], the cosmological constant is intro-  duced in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
+page_content='  [15] The correctness of the sign during Wick rotation is es-  tablished by the example of the action of a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dAzT4oBgHgl3EQfuf2L/content/2301.01692v1.pdf'}
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+Mon. Not. R. Astron. Soc. 000, 1–?? (0000)
+Printed 9 January 2023
+(MN LaTEX style file v2.2)
+Photometric variable stars in the young open cluster
+NGC 6823
+Sneh Lata1⋆, W. P. Chen2,3, J. C. Pandey1, Athul Dileep1, Zhong-Han Ai3,
+Alisher S. Hojaev4, Neelam Panwar1, Santosh Joshi1, Soumen Mondal5,
+Siddhartha Biswas5, B. C. Bhatt6
+1Aryabhatta Research Institute of Observational Sciences, Manora Peak, Nainital 263002, Uttarakhand, India
+2Graduate Institute of Astronomy, National Central University, 300 Zhongda Road, Zhongli 32001 Taoyuan, Taiwan
+3Department of Physics, National Central University, 300 Zhongda Road, Zhongli 32001 Taoyuan, Taiwan
+4Ulugh Beg Astronomical Institute, Uzbekistan Academy of Sciences, Tashkent, Republic of Uzbekistan
+5S. N. Bose National Centre for Basic Sciences, Kolkata 700106, India
+6Indian Institute of Astrophysics, Koramangala, Bangalore-560034, India
+Accepted ———. Received ———;
+ABSTRACT
+We present stellar variability towards the young open cluster NGC 6823. Time series V -
+and I-band CCD photometry led to identification and characterization of 88 variable
+stars, of which only 14 have been previously recognized. We ascertain the membership
+of each variable with optical UBV I and infrared photometry, and with Gaia EDR3
+parallax and proper motion data. Seventy two variable stars are found to be cluster
+members, of which 25 are main sequence stars and 48 are pre-main-sequence stars.
+The probable cluster members collectively suggest an isochrone age of the cluster to
+be about 2 Myrs based on the GAIA photometry. With the color and magnitude, as
+well as the shape of the light curve, we have classified the main sequence variables into
+β Cep, δ Scuti, slowly pulsating B type, and new class variables. Among the pre-main-
+sequence variables, eight are classical T Tauri variables, and four are Herbig Ae/Be
+objects, whereas the remaining belong to the weak-lined T Tauri population. The
+variable nature of 32 stars is validated with TESS light curves. Our work provides
+refined classification of variability of pre-main-sequence and main-sequence cluster
+members of the active star-forming complex, Sharpless 86. Despite no strong evidence
+of the disk-locking mechanism in the present sample of TTSs, one TTS with larger
+∆(I − K) is found to be slow rotator.
+Key words:
+open clusters and associations: individual NGC 6823, Hertzsprung-
+Russell and color-magnitude diagram, stars: pre-main-sequence, stars: variables: T
+Tauri, Herbig Ae/Be
+1
+INTRODUCTION
+Young open clusters serve as useful tools for the studies
+of the star formation mechanism and early stellar evolu-
+tion. For example, young star clusters are used to trace the
+Galactic spiral structure. In particular, variability of young
+stellar members provides diagnostics on the sporadic (accre-
+tion or occultation) or periodical (rotation) properties of the
+stars, and of their relation to the circumstellar environments
+(Morales-Calderon et al. 2011).
+Pre-main-sequence (PMS) objects are categorized on
+⋆ E-mail: sneh@aries.res.in
+the basis of the spectral energy distribution in the infrared
+wavelengths: Class 0 , Class I, Class II, and Class III (Lada
+1987; Andre et al. 1993) with the classification sequence
+roughly corresponding to the evolutionary status. Namely,
+a Class 0 object signifies a clump of dust and gas heavily en-
+shrouded in the molecular envelope, and is detected only in
+far-infrared wavelengths or longer. A Class I object is more
+evolved, now emerging from the cloud to become visible in
+near- and mid-infrared. A Class I object is in the protostellar
+stage and derives the luminosity from mass accretion.
+A Class II object, corresponding to a classical T Tauri
+star (TTS), has dispersed much of the envelope of gas and
+dust but retains a circumstellar disk within which plan-
+© 0000 RAS
+arXiv:2301.02399v1  [astro-ph.SR]  6 Jan 2023
+
+2
+Sneh Lata et al.
+ets may condense or are being formed. Inside the optically
+thick but geometrically thin disk, the dust grains absorb the
+starlight and re-emit in the infrared, manifest as infrared
+excess seen typically in a classical TTS. Accretion from the
+disk onto the star, while matter is partly lost as bipolar
+jets/outflows, leads to strong emission lines in the spectrum.
+As the inner disk is dissipated (or going into planet forma-
+tion), the PMS object then evolves to Class III, now with
+negligible infrared excess and with weak emission lines, if
+any, due to surface chromospheric activity. A Class III object
+hence is called a weak-lined TTS (Joy 1945, Appenzeller &
+Mundt 1989). Variability of PMS objects hence serves as an
+important diagnosis to understand the earliest PMS stellar
+evolution, e.g., the accretion (Johnstone et al. 2018), rota-
+tion (Herbst et al. 1994), or dust properties (Huang et al.
+2019).
+Here we report the variability study of the Galactic
+young open cluster NGC 6823. At a distance of about 2 kpc,
+the cluster is associated with the prominent H II region,
+Sharpless 86. This cluster has been investigated by several
+authors (Turner 1979, Stone 1988, Bica et al. 2008, Sagar
+and Joshi 1981; Guetter 1992; Massey et al. 1995; Pigulski et
+al. 2000; Hojaev et al. 2003, Zahajkiewicz 2012). Using op-
+tical and JHK photometric observations Riaz et al. (2012)
+found a large population of young stellar sources in the re-
+gion, including two δ Scuti variables of PMS nature, and 13
+other variables such as eclipsing binaries, slowly pulsating B
+candidates and UX Ori type variables. In the line of sight to
+the cluster the reddening has been found to be from 0.7 to
+1.1 mag following a normal reddening law (Rangwal et al.
+2017). The aim of the present work is to identify variables
+in a relatively large field of ∼ 14′ ×14′ of the member versus
+nonmember variable stellar populations in the region. Par-
+ticularly, photometric rotation periods of PMS members are
+derived to add to the data inventory for the study of the
+angular momentum evolution of low-mass stars.
+We describe in Section 2 observations, data reduction
+procedure, detection of variables, and period determina-
+tions. In Section 3, membership of the identified variable
+candidates is discussed using Gaia proper motion data, pho-
+tometric two-color diagrams (TCDs) and color-magnitude
+diagrams (CMDs). Section 4 then presents the nature of
+known and newly identified variable stars, while Section 5
+deals with TESS light curves. We discuss correlation be-
+tween amplitude and rotation periods of TTSs along with
+their color excess in Section 6. The results are summarized
+in Section 7.
+2
+OBSERVATIONS AND DATA REDUCTION
+We have observed NGC 6823 with the 0.81-m f/7 Ritchey-
+Chretien Tenagra automated telescope in southern Arizona,
+equipped with a 1024 × 1024 pixel SITe camera. Each pixel
+corresponds to 0.87′′, which yields a field of view of ∼ 14.8′×
+14.8′. The observations were carried out from 2012 early
+October to 2012 December. In total, data were acquired on
+54 nights in two passbands, with 232 frames in V band and
+243 frames in I band, with typical seeing of 2–3′′. Bias and
+twilight flats were taken every observing night. The observed
+region of the cluster in I band is shown in Fig. 1. The log of
+the observations is given in Table 1.
+Figure 1. The observed region of open cluster NGC 6823. Each
+variable star identified in this work is encircled.
+Figure 2. Photometric errors as a function of instrumental mag-
+nitude in I band. Open circles represent the variables stars iden-
+tified in the present work.
+The observed images were processed using standard
+IRAF tasks: zerocombine, flatcombine and CCDPROC. We
+have performed aperture as well as point spread function
+(PSF) photometry to derive the magnitude of stars. The
+PSF photometry was obtained using program ALLSTAR
+(Stetson 1987). To match the stars between different pho-
+tometric files we used the daomatch routine of DAOPHOT
+(Stetson 1992) whereas daomaster was used to match the
+point sources, and to obtain a file having corrected mag-
+nitude of stars from all the files. The daomaster program
+also removes the flux variation of stars in different frames
+© 0000 RAS, MNRAS 000, 1–??
+
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+295.850
+295.800
+295.750
+295.700
+RA0.04
+0.02
+10
+12
+8
+14
+16
+18
+instVariable stars in NGC 6823
+3
+Figure 3. The V and I band sample light curves of a few variables
+identified in the present work where ∆m represents the differential
+magnitude.
+due to exposure time and airmass. This program makes the
+magnitudes of stars in each photometry file equal to that of
+reference file by applying a constant value.
+We have used the V and I observations of Massey et
+al. (1995) for conversion of the present instrumental magni-
+tudes to the standard ones. For this, the mean instrumental
+magnitudes in V and I bands given by DAOMASTER (Stet-
+son 1992) have been converted into standard ones with the
+following transformation equations.
+V = v + (−0.042 ± 0.001) × (V − I) + 0.818 ± 0.014
+V − I = (0.982 ± 0.004) × (v − i) + 1.185 ± 0.002,
+where v and i are the instrumental magnitudes, and V and I
+refer to the standard magnitudes of stars in V and I filters.
+The estimated photometric error as a function of the mean
+instrumental magnitude is shown in Fig. 2.
+2.1
+Variables identification
+To identify variable stars, we first produced the light curves
+of all the stars cross-matched in different CCD frames. The
+light curves were obtained by plotting the differential mag-
+nitudes (∆m) of stars (variable minus the comparison star)
+against the given Julian date (JD). We used the Lomb-
+Scargle periodogram (Lomb 1976; Scargle 1982) to derive
+the periods and produced phased light curves accordingly to
+Table 1. Log of the observations of NGC 6823. N and Exp. rep-
+resent number of frames obtained and exposure time in seconds,
+respectively.
+S. No.
+Date of
+I
+V
+Observations
+(N×Exp.)
+(N×Exp.)
+1
+13 oct 2012
+6× 30
+6 × 50
+2
+15 oct 2012
+3× 30
+3 × 50
+3
+16 oct 2012
+6× 30
+5 × 50
+4
+17 oct 2012
+6× 30
+6 × 50
+5
+19 oct 2012
+3× 30
+5 × 50
+6
+20 oct 2012
+6× 30
+6 × 50
+7
+21 oct 2012
+6× 30
+4 × 50
+8
+22 oct 2012
+6× 30
+6 × 50
+9
+23 oct 2012
+3× 30
+2 × 50
+10
+24 oct 2012
+6× 30
+6 × 50
+11
+25 oct 2012
+6× 30
+5 × 50
+12
+26 oct 2012
+5× 30
+5 × 50
+13
+27 oct 2012
+6× 30
+6 × 50
+14
+28 oct 2012
+6× 30
+6 × 50
+15
+29 oct 2012
+6× 30
+6 × 50
+16
+30 oct 2012
+6× 30
+6 × 50
+17
+31 oct 2012
+6× 30
+6 × 50
+18
+01 nov 2012
+6× 30
+4 × 50
+19
+02 nov 2012
+-
+2 × 50
+20
+03 nov 2012
+6× 30
+5 × 50
+21
+04 nov 2012
+6× 30
+5 × 50
+22
+05 nov 2012
+5× 30
+5 × 50
+23
+06 nov 2012
+6× 30
+6 × 50
+24
+08 nov 2012
+6× 30
+6 × 50
+25
+11 nov 2012
+5× 30
+5 × 50
+26
+12 nov 2012
+2× 30
+3 × 50
+27
+14 nov 2012
+6× 30
+6 × 50
+28
+17 nov 2012
+3× 30
+3 × 50
+29
+19 nov 2012
+3× 30
+3 × 50
+30
+20 nov 2012
+6× 30
+6 × 50
+31
+22 nov 2012
+6× 30
+6 × 50
+32
+25 nov 2012
+6× 30
+6 × 50
+33
+27 nov 2012
+6× 30
+6 × 50
+34
+28 nov 2012
+6× 30
+6 × 50
+35
+29 nov 2012
+5× 30
+3 × 50
+36
+30 nov 2012
+6× 30
+6 × 50
+37
+01 dec 2012
+3× 30
+3 × 50
+38
+02 dec 2012
+3× 30
+3 × 50
+39
+03 dec 2012
+3× 30
+3 × 50
+40
+04 dec 2012
+3× 30
+3 × 50
+41
+05 dec 2012
+3× 30
+3 × 50
+42
+06 dec 2012
+3× 30
+3 × 50
+43
+07 dec 2012
+3× 30
+3 × 50
+44
+08 dec 2012
+3× 30
+3 × 50
+45
+09 dec 2012
+3× 30
+3 × 50
+46
+10 dec 2012
+3× 30
+1 × 50
+47
+11 dec 2012
+2× 30
+2 × 50
+48
+12 dec 2012
+3× 30
+3 × 50
+49
+13 dec 2012
+3× 30
+3 × 50
+50
+17 dec 2012
+3× 30
+3 × 50
+51
+18 dec 2012
+3× 30
+3 × 50
+52
+20 dec 2012
+3× 30
+3 × 50
+53
+21 dec 2012
+3× 30
+3 × 50
+54
+26 dec 2012
+6× 30
+-
+ascertain their most probable periods. A few variables seem
+to show periodic variability but their periodic nature was
+not obvious in their observed light curves. The phased light
+curves of all stars were inspected, and we adopted the pe-
+riod value which produces the most consistent phased light
+curve. The light curves of a few variables are shown in Fig. 3
+as examples. The phased light curves of variables identified
+in both V and I bands are presented in Fig. 4 and Fig. 5,
+whereas Fig. 6 shows variables identified in the I band only.
+By eye inspection and periodogram analysis, we have
+detected 88 variables. We have listed optical and near-
+infrared (NIR) data of the variable stars in Table 2, including
+an identification number, coordinates, and optical as well as
+NIR photometric data. These are the star ID numbers la-
+belled in Fig. 1. These 88 variable stars include 14 known
+variables, with periods varying from ∼0.03 days to more
+than 60 days. We have plotted in Fig. 7 the root mean square
+(RMS) scatter of each star to confirm their variability. The
+observed RMS scatter includes both the intrinsic variability
+and the mean photometric error. The larger circles in Fig. 7
+show the variables identified in the present work, indicat-
+ing large RMS values for variables. Some stars have large
+RMS values but do not show noticeable brightness varia-
+tion. Some of these objects are found to be close to the edge
+© 0000 RAS, MNRAS 000, 1–??
+
+154v
+154i
+385v
+385i
+-0.6
+-0.4
+-1.0
+-0.6
+-0.5
+-0.4
+-0.4
+-0.2
+-0.2
+0.0
+-0.2
+0.0
+m
+m
+m
+m
+0.0
+0.5
+0.0
+0.2
+0.2
+1.0
+0.2
+1
+0.4
+0.4
+1.5
+0.4
+0.6
+0.6
+2.0
+0.6
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+JD-JDmin
+JD-JDmin
+JD-JDmin
+JD-JDmin
+531v
+531i
+561v
+561i
+-0.6
+-0.4
+-0.6
+-0.4
+-0.4
+-0.4
+-0.2
+-0.2
+-0.2
+-0.2
+7
+m
+m
+m
+m
+0.0
+0.0
+0.0
+0.0
+0.2
+0.2
+0.2
+0.2
+0.4
+0.4
+0.6
+0.4
+0.6
+0.4
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+JD-JDmin
+JD-JDmin
+JD-JDmin
+JD-JDmin
+655v
+655i
+752v
+752i
+-0.4
+-0.4 
+-0.4
+-0.4
+-0.2
+-0.2
+-0.2
+-0.2
+0.0
+0.0
+m
+0.2
+m
+m
+m
+0.2
+0.0
+0.0
+0.4
+0.4
+0.6
+0.2
+0.2
+0.8
+0.6
+1.0
+0.8
+0.4
+0.4
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+JD-JDmin
+JD-JDmin
+JD-JDmin
+JD-JDmin
+757v
+1235v
+757i
+1235i
+-0.4
+-0.3
+-0.4
+-0.4
+-0.2
+-0.2
+-0.2
+0.2
+0.1
+0.0
+0.0
+m
+m
+m
+m
+0.0
+0.0
+0.2
+0.2
+0.1
+0.4
+0.4
+0.2
+0.2
+0.6
+0.6±
+0.4
+0.3
+0.8
+0.8
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+0 20 40 60 80
+JD-JDmin
+JD-JDmin
+JD-JDmin
+JD-JDmin1548v
+1548i
+1268i
+-1.0
+-0.3
+-0.4
+-0.2
+-0.5
+-0.2
+-0.1
+F
+0.0
+m
+m
+0.0
+0.0
+0.5
+0.1
+0.2
+1.0
+0.2
+1.5
+0.3
+0.4
+0
+20 40 60 80
+0 20 40 60 80
+0
+20 40 60 80
+JD-JDmin
+JD-JDmin
+JD-JDmin4
+Sneh Lata et al.
+Table 2. The V I and JHK 2mass data, amplitude and period of variables identified towards NGC 6823. The 2MASS data were obtained
+from the 2mass catalog (Cutri et al. 2003). The first column with an asterisk symbol represents a known variable.
+ID
+RA
+Dec
+V
+V − I
+J
+H
+K
+(mag)
+(mag)
+(mag)
+(mag)
+(mag)
+103
+295.668444
+23.412417
+18.115±0.101
+2.190±0.048
+14.231±0.041
+13.140±0.036
+12.508±0.034
+135
+295.730111
+23.406833
+18.454±0.124
+2.428±0.054
+14.324±0.056
+12.758±0.081
+11.586±0.044
+142
+295.921528
+23.403194
+-
+-
+9.997±0.022
+8.072±0.034
+7.137±0.020
+147
+295.717833
+23.404972
+18.251±0.102
+2.442±0.043
+13.925±0.029
+12.915±0.037
+12.452±0.031
+154
+295.729444
+23.404194
+15.305±0.021
+2.514±0.015
+10.509±0.021
+9.561±0.030
+8.621±0.024
+177
+295.798972
+23.398806
+-
+-
+14.623±0.040
+13.622±0.040
+13.154±0.036
+201
+295.806361
+23.392806
+18.651±0.129
+1.686± -
+-
+-
+-
+213
+295.678611
+23.391889
+18.583±0.133
+1.531±0.099
+15.949±0.093
+15.302±0.128
+14.913±0.129
+238
+295.856583
+23.385806
+-
+-
+9.096±0.022
+7.248±0.033
+6.386±0.027
+239
+295.793083
+23.386500
+17.564±0.063
+2.161±0.036
+13.328± -
+12.411±0.044
+11.853±0.034
+240
+295.913278
+23.384861
+16.320±0.033
+1.113±0.034
+14.480±0.035
+14.178±0.051
+14.074±0.052
+264
+295.759250
+23.382944
+-
+-
+14.953±0.040
+14.097±0.043
+13.771±0.045
+298
+295.707806
+23.376667
+17.367±0.056
+1.772±0.040
+14.216±0.034
+13.319±0.040
+12.605±0.030
+369
+295.889056
+23.363222
+-
+-
+13.520±0.026
+12.672±0.033
+12.243±0.031
+377
+295.880000
+23.361417
+18.610±0.127
+2.027± -
+-
+-
+-
+385
+295.847556
+23.360861
+16.904±0.040
+1.763±0.029
+13.554±0.028
+12.719±0.033
+12.086±0.027
+402
+295.716611
+23.358944
+-
+-
+15.213±0.058
+14.159±0.063
+13.558±0.056
+449
+295.874806
+23.347806
+14.985±0.017
+1.571±0.017
+12.262±0.023
+11.665±0.028
+11.521±0.024
+452
+295.918972
+23.345917
+-
+-
+14.529±0.056
+13.475±0.055
+13.008±0.043
+478
+295.847167
+23.343222
+-
+-
+14.474±0.042
+13.418±0.046
+13.002±0.043
+502
+295.883889
+23.339000
+18.593±0.122
+2.086±0.067
+15.050±0.051
+14.019±0.053
+13.695±0.055
+510
+295.795694
+23.339139
+14.154±0.014
+1.061±0.020
+12.211±0.021
+11.957±0.031
+11.770±0.027
+527
+295.868056
+23.335778
+17.461±0.055
+2.126±0.032
+13.688±0.037
+12.840±0.040
+12.605±0.038
+529
+295.910194
+23.335083
+-
+-
+14.243±0.039
+12.844±0.040
+11.984±0.033
+531
+295.850028
+23.335583
+17.400±0.055
+2.102±0.031
+13.562±0.032
+12.740±0.036
+12.460±0.031
+546
+295.746583
+23.334750
+17.928±0.082
+1.702±0.060
+14.461± -
+13.676± -
+14.144±0.102
+561
+295.718194
+23.332694
+18.193±0.100
+2.100±0.051
+14.652± -
+13.775± -
+13.376±0.041
+576
+295.862722
+23.329111
+-
+-
+14.668±0.042
+13.423±0.039
+12.865±0.034
+614
+295.687444
+23.325333
+15.851±0.023
+1.277±0.023
+13.709±0.028
+13.328±0.041
+13.109±0.036
+619
+295.856806
+23.323000
+15.920±0.022
+1.721±0.018
+12.883±0.024
+12.309±0.031
+12.076±0.026
+623*
+295.834472
+23.322306
+13.173±0.009
+1.202±0.012
+11.141±0.019
+10.662±0.030
+10.519±0.024
+655*
+295.837528
+23.317306
+17.282±0.052
+2.287±0.028
+12.735±0.025
+11.397±0.031
+10.313±0.023
+679
+295.768611
+23.313583
+13.725±0.014
+0.967±0.018
+11.556±0.021
+10.530±0.030
+9.546±0.024
+706
+295.752944
+23.310389
+17.376±0.056
+1.963±0.036
+13.936±0.037
+13.149±0.046
+12.880±0.041
+731
+295.789306
+23.307667
+18.493±0.128
+2.239±0.060
+14.424±0.043
+13.458±0.051
+13.086±0.044
+733*
+295.798444
+23.307361
+15.211±0.018
+1.224±0.023
+12.935±0.029
+12.515±0.045
+12.256±0.044
+752
+295.737667
+23.306472
+16.172±0.028
+1.398±0.028
+13.743±0.028
+13.207±0.040
+12.978±0.037
+753*
+295.798472
+23.305694
+-
+-
+14.578±0.048
+13.491±0.062
+12.736±0.058
+757*
+295.803833
+23.305278
+16.899±0.042
+2.077±0.029
+13.186±0.028
+12.420±0.035
+12.138±0.031
+765
+295.785417
+23.304806
+17.592±0.124
+1.533±0.075
+14.421± -
+14.039±0.103
+13.848±0.076
+822*
+295.787917
+23.297000
+14.529±0.015
+1.225±0.020
+12.396±0.026
+12.024±0.030
+11.823±0.032
+826
+295.825139
+23.295944
+-
+-
+14.661±0.044
+13.740±0.051
+13.406±0.045
+831*
+295.746556
+23.296667
+-
+-
+11.266±0.033
+8.761±0.030
+7.326±0.020
+860
+295.798389
+23.292583
+17.563±0.121
+1.880±0.072
+14.754±0.055
+13.748±0.045
+13.055±0.041
+886*
+295.794056
+23.290250
+14.448±0.015
+1.065±0.018
+12.637±0.022
+12.379±0.031
+12.207±0.029
+903*
+295.801167
+23.288028
+14.382±0.014
+1.036±0.017
+12.568± -
+12.141± -
+11.928±0.033
+924*
+295.787583
+23.285444
+18.146±0.097
+2.210±0.048
+14.154± -
+13.363± -
+13.035±0.046
+945
+295.855139
+23.282167
+14.524±0.014
+1.206±0.015
+12.395±0.033
+12.051±0.040
+11.913±0.033
+950
+295.707250
+23.283611
+-
+-
+12.424±0.022
+11.060±0.026
+10.310±0.021
+951
+295.797167
+23.282278
+17.150±0.051
+2.019±0.032
+13.664±0.030
+12.913±0.043
+12.626±0.038
+965
+295.891500
+23.279944
+14.600±0.017
+1.125±0.018
+12.661±0.023
+12.150±0.031
+12.050±0.030
+979*
+295.775389
+23.280333
+-
+-
+15.201±0.083
+14.179±0.079
+13.432±0.057
+1000
+295.814639
+23.277861
+13.574±0.011
+0.579 ±0.014
+12.544±0.021
+12.472±0.033
+12.342±0.032
+1007*
+295.778417
+23.277111
+14.624±0.016
+1.181 ±0.020
+12.508±0.026
+12.220±0.039
+12.003±0.033
+1025
+295.690889
+23.276194
+-
+-
+15.839±0.088
+14.503±0.058
+13.751±0.052
+1061*
+295.788972
+23.270083
+-
+-
+14.267±0.033
+12.456± -
+12.081± -
+1063
+295.778528
+23.270139
+9.702±0.018
+0.631±0.018
+8.785±0.019
+8.712±0.029
+8.652±0.024
+1064
+295.758861
+23.270389
+14.451±0.014
+1.047±0.018
+12.514±0.022
+12.088±0.035
+11.874±0.029
+1066
+295.843528
+23.269278
+16.715±0.037
+3.602±0.014
+10.303±0.022
+9.170±0.028
+8.693±0.025
+1072
+295.820667
+23.269139
+14.714±0.014
+1.098±0.017
+12.751±0.022
+12.523±0.033
+12.317±0.030
+1087*
+295.798806
+23.266806
+-
+-
+9.945±0.019
+7.952±0.042
+7.059±0.040
+1094
+295.817722
+23.265111
+16.634±0.034
+1.949±0.023
+13.251±0.022
+12.572±0.031
+12.283±0.029
+1122
+295.709611
+23.261028
+13.788±0.011
+0.824±0.014
+12.308±0.022
+12.085±0.032
+11.956±0.028
+1151
+295.806028
+23.255889
+-
+-
+15.007±0.059
+13.956±0.041
+13.596±0.046
+1155
+295.768556
+23.255472
+18.407±0.117
+2.004±0.062
+14.928±0.039
+14.169±0.049
+13.918±0.055
+1168
+295.883528
+23.252556
+-
+-
+15.273±0.057
+14.244±0.061
+13.974±0.059
+1191
+295.875000
+23.248667
+-
+-
+15.444±0.065
+13.630± -
+12.955± -
+1228
+295.851056
+23.243861
+-
+-
+15.146±0.055
+13.810±0.063
+13.109±0.038
+1230
+295.756194
+23.244750
+14.580±0.014
+0.897±0.017
+12.948±0.026
+12.700±0.036
+12.604±0.033
+1235
+295.672972
+23.244722
+12.983±0.012
+0.937±0.012
+11.555± -
+11.322± -
+11.205±0.037
+1262
+295.775250
+23.238000
+17.173±0.050
+2.100±0.030
+13.351±0.034
+12.474±0.041
+11.998±0.032
+1266
+295.802833
+23.236806
+18.656±0.134
+2.123±0.069
+14.808±0.038
+14.078±0.049
+13.694±0.045
+1268
+295.699250
+23.237972
+-
+-
+9.148±0.019
+7.211±0.034
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+of the detector whereas a few stars contains spurious data
+points. The derived periods of stars are given in Table 2.
+3
+CLUSTER MEMBERSHIP OF VARIABLE
+STARS
+For each variable star, its UBV I plus 2MASS photometry
+along with Gaia EDR3 proper motion and parallax have
+been used to assess the likelihood of cluster membership.
+The UBV , JHK, and mid infrared (MIR) data at wave-
+© 0000 RAS, MNRAS 000, 1–??
+
+Variable stars in NGC 6823
+5
+lengths 3.6, 4.5, 5.8 and 8 micron, are taken from Massey
+et al. (1995), Cutri et al. (2003), and GLIMPSE survey, re-
+spectively.
+3.1
+Gaia Characterization of the Variable Stars
+The 88 variable stars reported in this work have been char-
+acterized with the Gaia EDR3 parallax and proper motion
+measurements. Fig. 8 plots the sky positions of all the Gaia
+sources (in gray) within 30′ toward NGC 6823. This covers
+the field of the Tenagra images (variable stars marked in
+black crosses) and is much wider than the cluster’s angular
+size of ∼ 3.′6 (red circle) (Morales et al. 2013). The stellar
+density is clearly enhanced toward the center.
+Each variable star was matched with Gaia counterparts
+within a radius of 2.′′5 as the compromise of the seeing of the
+Tenagra images, leading to 91 Gaia sources. Fig. 9 presents
+the proper motion vector plot of all the stars (gray) and
+those within 4′ nominal cluster region (black small circles,
+1294 stars) for which the members should be concentrated,
+serving as the sample of cluster members. This 4′ (posi-
+tional) sample has a mean of µα ≈ −1.7 mas yr−1 and
+µδ ≈ −5.3 mas yr−1, which agrees well with the litera-
+ture values (Cantat-Gaudin & Anders 2020). Shown in the
+bottom panel are the proper motions for variable stars (in
+black with error bars). One sees that the majority of our
+variable stars share the same proper motion ranges.
+Gaia measures repeatedly the astrometry of a source
+from which the parallax and proper motion are solved simul-
+taneously. Parallax, however, does not serve as a constraint
+for membership as stringently as the proper motion, because
+given the uncertainties, negative average values may result,
+rendering the reciprocal to estimate the distance possible
+only if a statistical inference is exercised (Bailer-Jones et al.
+2021). For our work the parallax value was used directly. The
+parallax of the 4′ sample exhibits a peak around 0.45 mas,
+indeed consistent with the literature value (Cantat-Gaudin
+& Anders 2020), and so does the variable star sample, as
+demonstrated in Fig. 10. If the 4′ sample is further di-
+vided by proper motion ranges, one finds no star within 1–
+2 mas yr−1 from the cluster’s mean having parallax between
+0.4 mas and 0.6 mas. This signifies the sufficiency as mem-
+bership criteria of (1) a radius of 1 mas yr−1 in the proper
+motion from the cluster average proper motion, and (2) a
+parallax value 0.35–0.55 mas. A variable satisfying both (1)
+and (2) is therefore considered a “highly probable” mem-
+ber, whereas one that fulfills only (1) or (2) is classified as
+a “possible” member. Table 4 lists information about the
+proper motions, parallax, and magnitudes for the 88 vari-
+ables identified in the present work.
+Fig. 11 shows the Gaia G versus BP − RP CMD for
+the highly probably members (in red) and possible mem-
+bers (in blue). Overlapped in the diagram is the PARSEC
+isochrones of 1, 2, and 4 Myr, respectively, each shifted by
+a distance modulus of 11.753 (parallax of 0.446 mas) and
+reddening of E(B −V ) = 0.8 (Sagar & Joshi 1981) adopting
+the reddening law of AV = 3.1 E(B −V ), AG = 0.83627 AV ,
+ABP = 1.08337 AV , and ARP = 0.63439 AV . The highly
+probable members indicate an age of roughly 2 Myr.
+Three variables have ambiguous Gaia counterparts
+within the matching radius. Star No. 478 has two possible
+matches, equally faint thereby with relatively large uncer-
+tainties in Gaia data but either one is consistent with being
+a member. The star was not detected in our V band im-
+age and appears progressively brighter from I = 14.47 mag,
+to 2MASS J = 13.42 mag, H = 13.42 mag, and Ks =
+13.00 mag.
+Star No. 1063 is the second brightest star in our variable
+list, with the brightest one (No. 1526) being clearly not a
+member. The star also has two Gaia matches, with contrast-
+ing brightness (G = 9.72 mag versus 12.87 mag). Given its
+V = 9.70 mag, the fainter one, having an outlying parallax
+of 0.282 mas, is eliminated. The other counterpart, however,
+has a negative parallax value with a large uncertainty. This
+compromises its membership determination. Its optical and
+NIR colors both suggest an early-type star and its position
+in the CMD suggests a main-sequence member.
+Star No. 1262 has V
+= 17.173 mag, 2MASS J =
+13.351 mag, H = 12.474 mag, and Ks = 11.998 mag. The
+brighter Gaia match has G = 16.379 mag but has a neg-
+ative parallax value and inconsistent proper motion. The
+other Gaia star is faint, with G = 19.291 and no measure-
+ments in the other two Gaia bands, has parallax and proper
+motion values well consistent with being a member.
+3.2
+Colors and Magnitudes
+3.2.1
+U − B vs B − V TCD
+To identify probable MS variables, we have plotted in Fig. 12
+the U − B versus B − V for variable stars identified in the
+cluster region with the photometric data of 22 stars found
+in Massey et al. (1995). Reddening in terms of color excess
+E(B−V ) ranges from 0.7 to 1.1 mag (Erickson 1971; Guetter
+1992; Massey et al. 1995; Pigulski et al. 2000). Pigulski et
+al. (2000) recognized the highest extinction in the eastern
+part of the cluster where a trapezium system, i.e., of O, B
+spectral types, is located. The eastern part of their observed
+field is the direction to the reflection nebula NGC 6820, and
+their study suggested more than half of the total absorption
+to arise from nearby interstellar matter toward NGC 6823.
+It is inferred that there is significant differential reddening
+within the cluster, manifest that the cluster is located behind
+at least AV = ∼ 3 mag (Riaz et al. 2012). Rangwal et al.
+(2017) studied the interstellar extinction of open clusters
+and found that NGC 6823 follows a normal extinction law in
+optical as well as in the NIR wavelengths. The U −B versus
+B − V TCD shows that the stars exhibiting within E(B −
+V ) = 0.7–1.1 mag could be MS members of the cluster,
+indicating a nonuniform reddening across the cluster. The
+reddened theoretical ZAMS of Girardi et al. (2002) is fitted
+to the TCD. The value of color excess E(V − I) was taken
+as 0.88 mag which has been calculated using the minimum
+reddening value of E(B − V )=0.70 mag.
+3.2.2
+J − H vs H − K TCD
+Fig. 13 shows the J − H versus H − K TCD for NGC 6823.
+Only 86 stars were cross-matched between the present sam-
+ple of variable stars and the 2MASS catalog, with the JHK
+counterparts of two stars No. 201 and No. 377 not found
+during the match. In the 2MASS TCD, the “F” and “T”
+regions are locations of probable Class III/field stars and
+Class II sources, respectively. The filled squares plotted in
+© 0000 RAS, MNRAS 000, 1–??
+
+6
+Sneh Lata et al.
+Figure 4. The I and V band phased light curves of variable stars identified in the region of NGC 6823.
+blue and green colors represent, respectively, probable PMS
+and MS members. Circles in the diagram represents field
+stars. Riaz et al. (2012) from their NIR CCD found early-
+type MS dwarfs concentrating close to (H −Ks) ∼ 0.7 mag,
+and (J − H) ∼ 1.5 mag, having extinction AV
+⩾ 10.
+These authors have noticed another population near the
+classical TTS locus close to (H − Ks) ∼ 0.6 mag and
+(J − H) ∼ 1.0 mag, presumably being young disk-bearing
+members. In the 2MASS TCD, about half the detected vari-
+ables are in the “T” or “F” regions hence could be T Tauri
+variables. A few PMS stars located below the TTS locus are
+probably Herbig Ae/Be stars. We note that star No. 679
+occupies the position where Herbig Ae/Be stars are placed,
+while in U − B versus B − V TCD it lies close to the MS
+locus; it could thus be either a reddened MS star or a Herbig
+Ae/Be member.
+Following Gutermuth et al. (2008) to classify young stel-
+lar sources, Riaz et al. (2012) used MIR IRAC data and
+found 2 Class I, 94 Class II, and 394 Class III or field stars
+in the region. The figure 4(a) of Riaz et al. (2012) is plot-
+ted with the (H − Ks) and [4.5] − [8] TCD for the 490
+sources. This shows both YSOs and the diskless sources
+© 0000 RAS, MNRAS 000, 1–??
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+7
+Figure 5. Continued.
+to have similar NIR colors, but from their IRAC TCD
+the photospheric and the disk population at IRAC color
+[4.5] − [8] ∼ 0.4 mag are readily distinguishable. They have
+noted that a few Class III/field stars are mixed with Class II
+sources. Their IRAC TCD in figure. 4(b) shows different lo-
+cales of Class I and Class II sources. The protostars (Class I)
+are located in the top-right corner and exhibit the reddest
+in the [3.6] − [5.8] color, whereas the Class II sources are
+placed at [4.5] − [8] ⩾ 0.5 mag, [3.6] − [5.8] ⩾ 0.4 mag, and
+the Class III/field stars are found to be near [4.5] − [8] and
+[3.6] − [5.8] ∼ 0.2 to 0.3 mag.
+3.2.3
+NIR and MIR TCDs
+To see the distribution of young variable sources we have
+plotted them in the NIR and MIR TCD (left panel) and MIR
+TCD (right panel) in Fig. 14. To obtain these plots we have
+cross-matched the coordinates of variable stars with those
+from the Spitzer Galactic Legacy Infrared Mid-Plane Survey
+Extraordinaire (GLIMPSE), yielding MIR counterparts of
+all 88 variable stars. A few stars, namely Nos. 154, 238, 313,
+1405, and 1406 do not have magnitudes at [4.5] and other
+wavelengths. The H − K versus [4.5] − [8] TCD shows most
+young stellar sources to have H − K ≳ 0.3 mag whereas the
+© 0000 RAS, MNRAS 000, 1–??
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+Phase8
+Sneh Lata et al.
+Figure 6. The I band phased light curves of probable variable candidates around the cluster.
+[4.5] − [8] and [3.6] − [5.8] TCD shows a few young stellar
+objects to be positioned as field stars or other populations.
+3.2.4
+IPHAS data
+To identify the young stellar sources with Hα emission,
+i.e., the indicator of accretion disks, we have compared the
+present data with Table 3 for IPHAS photometry of Riaz
+et al. (2012). We got 29 common stars after the match that
+have IPHAS photometry, with stars No. 502, 576, 655, 679,
+and 979 having Hα emission with equivalent width greater
+than 10 ˚A, judged by the (r′ − i′) versus (r′ − Halpha) TCD
+for NGC 6823 (Riaz et al. 2012). The location of these five
+stars is shown with the red square in the present (J − H)
+versus (H − K) TCD. The Hα emission is found to be vari-
+able in nature, therefore, it is necessary to check the location
+of the objects in spectral type/color versus magnitude dia-
+gram to know their membership and nature (Mart´ın et al.
+2000). Two stars Nos. 655 and 979 could be considered as
+PMS objects which may possess circumstellar accreting disk.
+Barrado y Navascu´es et al. (2001) showed that Hα emission
+depends on the spectral type or color in the sense that Hα
+emission is found to be larger for cooler objects in a plot
+between the Hα emission and (I − J) color. The (I − J)
+and (I − K) colors for star no. 655 is about 2.26 mag and
+4.682 mag, respectively. In the case of star no. 979, we have
+taken I magnitude from Pigulski et al. (2000) to determine
+its (I − J) and (I − K) colors due to lack of (V − I) color in
+present observations, yielding (I − J) and (I − K) colors as
+2.020 mag and 3.789 mag, respectively. Stars Nos. 655 and
+979 in particular satisfy both colors and Hα emission be-
+ing greater than 10 ˚A, hence are young stars with accretion
+disks.
+3.2.5
+V vs V − I CMD
+Sixty one variable stars were detected in both V and I
+bands. Their V magnitudes and V −I color are given in Ta-
+ble 2, and Fig. 15 shows their V versus (V − I) CMD. The
+PMS isochrones and evolutionary tracks for different masses
+are taken from Siess et al. (2000). The solid curve represents
+ZAMS by Girardi et al. (2002). We determined the distance
+modulus of the cluster to be (V −MV ) = 14.31 mag by com-
+paring the ZAMS of Girardi et al. (2002) for solar metallicity
+to the V versus V − I CMD, which corresponds to a dis-
+tance of 2.59 kpc. The present estimate of distance matches
+well with those derived in earlier works of NGC 6823. The
+isochrone of age 4 Myr also fits the data well. The CMD is
+known to be contaminated by the foreground field stars (e.g.
+Guetter 1992; Pigulski et al. 2000; Bica et al. 2008). After
+analysis of CMD, Pigulski et al. (2000) and Riaz et al. (2012)
+noted two different populations in the cluster, one consist-
+ing of older, massive stars which are located near or on the
+ZAMS, while the other one being younger objects with ages
+less than 10 Myr and are of lower masses (∼ 0.1–0.4 M⊙),
+that is, of PMS stars. Pigulski et al. (2000) also concluded
+that stars lying in B region of their figure 11(a) are cluster
+stars of PMS nature, evolving towards the MS. The present
+CMD containing variable stars also shows MS of the cluster
+to go up to around V = 16 mag; location of variables in the
+CMD suggests the majority of these stars to be probable
+members. Most the fainter and redder stars lying between
+(V − I) =∼ 2 mag and ∼ 3 mag could be PMS objects.
+In this CMD, to maintain clarity we have not plotted star
+No. 1548 despite it being detected in the I band, because it
+has V − I color more than 6 mag. Star No. 449 may be a
+possible PMS star but its placement in the U −B/B−V and
+J −H/H −K TCDs suggests a field star, even though it has
+proper motions in the range of probable cluster members.
+© 0000 RAS, MNRAS 000, 1–??
+
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+PhaseVariable stars in NGC 6823
+9
+3.2.6
+NIR CMDs
+The J versus (J − K) and J versus J − H CMDs for the
+present sample of variable stars are shown in Fig. 16. it
+is seen that the MS is almost vertical and clearly sepa-
+rated from the PMS objects/field stars as in the case of
+the V versus (V − I) CMD. Bica et al. (2008) described the
+same and from statistical cleaned CMD, they found that
+two populations are distributed separately where majority
+of PMS objects are faint and redder. They found the age
+of the cluster in the range from 2 to 7 Myr, a color excess
+E(B −V )=∼0.86 mag, and AV =2.7±0.2. In their work af-
+ter fitting theoretical models, the absolute distance modulus
+of the cluster was found to be (m − M)O = 11.5 mag.
+Although stars numbered Nos. 142, 402 826, 950, 924,
+1025, 1066, 1087, 1298, and 1548 are nonmembers from
+the analysis of the Gaia data, we have considered them as
+possible PMS objects based on their positions in different
+CMD and TCDs. We note that there are 10 stars, namely,
+Nos. 240, 614, 623, 752, 965, 1235, 1352, 1500, 1525, and
+1526, that are designated as nonmembers from proper mo-
+tion and parallax and, these could be MS stars based on
+location in TCDs and CMDs. Star 1506 is located well away
+from the MS in U − B/B − V TCD while it is lying on te
+MS in V/V −I, J/H −K and J/J −H. The analysis of Gaia
+data also found it to be nonmember. This is a doubtful case
+for being an MS member. From Gaia data analysis we found
+stars Nos. 201, 239, 449, 619, 765, 861, 1151, 1168, 1191 and
+1508 as possible or probable members but their locations in
+various TCDs and CMDs do not seem to be consistent with
+membership. Thus, these are considered field stars.
+We looked for the matches of our 8 previously identified
+PMS stars based on their 2MASS colors and their spectra
+taken at the 2.16 m telescope of Beijing Observatory (Ho-
+jaev et al. 2003). The spectra showed the strong H-alpha
+in emission, the SED in continuum and other features typi-
+cal either for TTS or Herbig Ae/Be stars. The cross-match
+yields 3 common stars namely 154, 655 and 679 for two of
+them we have already determined their features (their loca-
+tion on the diagrams, their proper motions and parallaxes).
+In the present work, we have classified 655 as classical TTS
+and 679 as MS. There was doubt to consider 679 as PMS
+because in U − B versus B − V TCD it is lying on MS but
+in J − H versus H − K diagram it is placed in the location
+of classical TTS. Now, we determined the membership of
+star No 154. It might be probable member of the cluster as
+Herbig Ae/Be type star while earlier (Hojaev et al. 2003)
+it has been classified as classical TTS, though it has very
+different position in V versus V − I CMD to be PMS star
+but GAIA suggests it to be highly probable member, and
+in J − H versus H − K it is located where Herbig Ae/Be
+stars are found. Therefore, it could be considered as Herbig
+Ae/Be star.
+We have considered members those stars which fulfill
+criteria like location in TCDs, CMDs, and have consistent
+kinematics. Therefore, using proper motion data together
+with location in various TCDs and CMDs obtained from
+present and available photometric UBV I, NIR, and MIR
+data we classify 25, 48, and 15, respectively, as MS, PMS
+members of the cluster, and field stars. The classification of
+variables is given in Table 3.
+Figure 7. Magnitude as a function of rms value of each star
+detected in I band. Open circles represent variable stars identified
+in this work.
+Figure 8. The sky positions of all the Gaia sources (in gray)
+within 30′ toward NGC 6823.
+4
+CHARACTERISTICS OF VARIABLE STARS
+The log(L/L⊙) vs log Teff diagram (H − R diagram) for 21
+members (MS variables) is shown as Fig. 17. We are not
+able to locate four MS stars Nos. 240, 752, 1107 and 1500
+in this plot due to lack of their U and B band data. Here,
+the effective temperature and bolometric correction (BC)
+have been determined from Toress relation (2010) using the
+intrinsic (B − V ) color. The Mbol values of stars are ob-
+tained from the relation Mbol = MV +BC, where MV is the
+absolute V -band magnitude. The luminosity was obtained
+from the relation log(L/L⊙) = −0.4(Mbol − Mbol⊙), where
+© 0000 RAS, MNRAS 000, 1–??
+
+0.8
+0.6
+0
+S
+0
+M
+R
+0.4
+0.2
+00
+0
+0
+0
+0
+8
+10
+12
+14
+16
+18
+instNGC 6823
+23.8
+23.6
+Decl. [deg]
+23.4
+23.2
+23.0
+22.8
+296.2
+296.0
+295.8
+295.6
+295.4
+R.A. [deg]10
+Sneh Lata et al.
+Figure 9. The upper panel represents proper motion of all the
+stars (gray) and those within 4′ cluster region (black small circles,
+1294 stars). The lower panel shows proper motions for variable
+stars (in black with error bars).
+Mbol⊙ is the bolometric magnitude for the Sun. The MS
+variable stars have been classified according to their periods
+of variability, the shape of light curves, and their positions
+in the H-R diagram. We detected one star as β Cep-type.
+Four stars Nos. 679, 886, 1122 and 1352 are located in the
+instability strip of slowly pulsating B type (SPB) stars. The
+positions in the cluster H-R diagram as well as the observed
+variability characteristics of nine stars allow us to conclude
+that these variables belong to the new class variables. One
+star based on its location in the H − R diagram should be
+δ Scuti-type variable.
+In the present study, we have detected 48 PMS stars
+as most likely cluster members in the PMS stage of evo-
+lution. Of these, 4, 8, and 36 stars are classified as Herbig
+Figure 10. Histogram of parallaxes for stars within 4 arcmin,
+where histogram shaded with black is for variable samples iden-
+tified in the present work.
+Figure 11. The G vs BP − RP CMD for the present sample
+of variable stars. The filled and open squares denote probable
+and possible cluster members, and dotted points are considered
+nonmembers.
+Ae/Be stars, classical TTSs, and weak-lined TTSs, respec-
+tively. The amplitudes of weak-lined TTSs range from ∼0.05
+to ∼0.2 mag, and most weak-lined TTSs vary with shorter
+periods of less than 1.0 days. The periods and amplitudes of
+classical TTSs are found to range from ∼ 0.05 to ∼ 30 days
+and ∼ 0.2 to ∼ 0.7 mag, respectively. The above results
+suggest that stars with disks, i.e., classical TTSs, exhibit
+relatively larger amplitudes than the weak-lined TTSs do,
+with the stellar variability in classical TTSs arising from the
+presence of the spots, hot and cold, on the stellar surfaces
+© 0000 RAS, MNRAS 000, 1–??
+
+0
+pmDE[mas/yr
+4
+-6
+-8
+-10
+-12
+6
+4
+2
+0
+-2
+-4
+9-
+-8
+PmRA[mas/yr]2
+0
+-2
+pmDE[mas/yr
+-6
+-8
+-10
++
+-12
+6
+4
+2
+0
+-2
+-4
+-6
+-8
+PmRA[mas/yr]260
+200
+150
+100
+50
+0.0
+0.5
+1.0
+1.5
+2.0
+plx [mas]10
+12
+14
+[eu]
+16
+G
+18
+20
+2
+3
+1
+4
+5
+BP - RP[mag]Variable stars in NGC 6823
+11
+Figure 12. (U − B)/(B − V ) TCD for variable stars identified in
+the present study. All the UBV data are taken from Massey et
+al. (1995). The continuous and dotted line represent the ZAMS
+(Girardi et al. 2002) which are shifted along the reddening vector
+for reddening E(B − V ) = 0.32 mag and 0.45 mag. Triangles are
+those stars that are identified as MS variables.
+as found in the previous studies (e.g. Bouvier et al. 1993;
+Pandey et al. 2019).
+4.1
+Known variables
+In the CCD search for variable stars in NGC 6823, Pigulski
+et al. (2000) demonstrated that all stars with spectral types
+later than A0 are PMS objects. They detected two variable
+stars of δ Scuti type and these stars could be at the PMS
+stage of evolution and suggested that these objects can fur-
+ther be used to test the evolutionary changes in this class of
+variable stars. The CMD was used to compare and discuss
+the position of the two discovered δ Scuti stars with refer-
+ence to the theoretical instability strip for PMS stars of this
+type. They have also 13 other variables including one bright
+cluster eclipsing binary and an SPB candidate.
+Of 15 variables identified by Pigulski et al (2000), 14
+were found to be variable in the present work. We could not
+detect variability in the star H8 (E88 or BL 4) (B0 V:pe by
+Turner 1979), B1.5 V by Massey et al. 1995), and B1 V by
+Shi & Hu (1999). Pigulski et al (2000) noted this stars as
+the brightest variable member in the observed cluster field
+and found to be a binary star where only one eclipse was
+detected in the I band.
+Now we will describe the nature of all the known 14
+variable stars individually.
+Stars BL 50 (822) and HP 57 (1007) with periods
+0.0718530 days, 0.10114 days for BL 50 and 0.0785819 days,
+0.0644149 days for HP 57 were found to be most likely clus-
+ter PMS members by Pigulski et al. (2000). With their po-
+sitions in the cluster CMD as well as the observed periods
+Figure 13. (J − H)/(H − K) TCD for variable stars detected
+in the field of NGC 6823. The JHK data have been taken from
+the 2MASS catalog (Cutri et al. 2003). The continuous and long
+dashed lines show sequences for dwarfs and giants (Bessell & Brett
+1988), respectively. The TTS locus (Meyer et al. 1997) is shown
+by a dotted line. The small dashed lines are reddening vectors
+(Cohen et al. 1981) and an increment of visual extinction of AV
+= 5 mag is denoted by crosses on the reddening vectors. Filled
+squares with blue colors represents PMS. The MS population are
+shown by green squares whereas open circles may be either MS
+members of the cluster or field stars. Triangles (black) represent
+two MS members BL 50 and HP 57.
+Pigulski et al. (2000) concluded that both objects could be δ
+Scuti variables. The present membership analysis, i.e., from
+kinematics and positions in various CMDs and TCDs, sug-
+gests both stars to be MS members. In the H-R diagram star
+No. 822 is positioned where new class variables are found
+(between the red edge of SPB and the blue edge of δ Scuti
+instability strip). Star No. 1007 could not be placed in the H-
+R diagram due to unavailability of UBV data. The present
+period of stars 822 is derived as 0.143 days and 0.084 days
+while the periodogram analysis gives period of 0.064 days
+for star 1007. The period derived for star 1007 is in good
+agreement with that derived by Pigulski et al. (2000). The
+location of these two stars were shown with red and black
+triangles in V versus (V −I) CMD and J −H versus H −K
+TCD, respectively.
+Star No. 903, a probable cluster MS member was dis-
+covered as the third pulsator (G 51) by Pigulski et al. (2000).
+Its brightness varies with a period of 0.848 days with an am-
+plitude about 0.03 mag. The star was classified by Pigulski
+et al. (2000) as an SPB variable according to their variability
+characteristics.
+The brightness of star No. 886 (G52) found to be binary
+by Pigulski et al. (2000) varies with period of 0.61 days
+with an amplitude 0.03 mag. It is diagnosed as a member of
+the cluster from proper motion and its location in various
+© 0000 RAS, MNRAS 000, 1–??
+
+1063
+-0.5
+881
+0
+1000 679
+1502
+449
+9658
+0.5
+2斤
+B
+U
+614
+1.5
+2
+1506
+0
+0.5
+1
+1.5
+2
+B-VX
+X
+2.5
+X
+X
+2
+0
+F
+T
+X
+X
+1.5
+H
+1
+0
+0.5
+0
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+H-K12
+Sneh Lata et al.
+Figure 14. (H − K) vs [4.5] − [8.0] and [3.6] − [5.8] vs [4.5] −
+[8.0] TCDs for variable stars detected in the field of NGC 6823.
+Blue circles are PMS young stellar sources while black circles are
+MS/field stars.
+Figure 15. V/(V −I) CMD for variable stars in the region of the
+cluster NGC 6823. The open circles (blue) are MS variables, and
+probable PMS variable stars are shown by filled circles (magenta).
+The open circles in black color are considered field stars. The
+continuous curve is ZAMS by Girardi et al. (2002) while dashed
+lines are PMS isochrones taken for 0.1, 1, 2, 5, 10 Myrs (Siess
+et al. 2000). The PMS evolutionary tracks for different masses
+ranging from 0.7 to 5.0 M⊙ from Siess et al. (2000) are plotted
+with dotted curves. BL 50 and HP 57 are shown by red triangles.
+Figure 16. J/(H − K) and J/(J − H) CMD for variable stars
+detected in the field of NGC 6823. The JHK data have been taken
+from the 2MASS catalogue (Cutri et al. 2003). Circles (blue) and
+circles (green) represent MS and PMS, respectively. The Open
+circles in black color demonstrate the field stars. The locations
+of stars No. 822 (BL 50) and 1007 (HP 57) are shown with open
+circle in red color.
+photometric diagrams. The present estimates for period and
+amplitude are consistent with those reported in Pigulski et
+al. (2000).
+Star No. 733 has proper motion values of µα
+=
+−1.671 mas/yr and µδ = −5.453 mas/yr, hence is a prob-
+able member of the cluster. Our analysis suggests possibly
+more than one period, with 0.143 d and 0.512 d. In Pigulski
+et al. (2000) it is H30, and they found its period of more
+than 3 days.
+The brightness of star No. 757 was found to be chang-
+ing with one single period of 0.553 days. The present work
+classified this star to be a probable PMS cluster member.
+Pigulski et al. (2000) named it V2 and derived its period of
+about 1.24 days, commenting that the true period for this
+star corresponds to an alias frequency, and they found this
+star to be of PMS type source. The present observations
+confirm its variability and its PMS nature. The light curves
+and periodogram analysis manifest that it could be an eclips-
+ing binary with primary and secondary depths being nearly
+equal. Morales-Calderon et al. (2012) found six new can-
+didate sources as PMS eclipsing binaries with multi-epoch
+data of about 2400 stars associated with the Orion Nebula
+Cluster, and it is stated that the PMS eclipsing binaries are
+valuable as they are in the stage of PMS evolution which
+is highly dynamic, therefore their detection is rare at this
+stage.
+Star No. 924 may be a PMS variable with light curve
+varying with more than one period. The proper motion sug-
+gests a nonmember of the cluster but its position in TCDs
+© 0000 RAS, MNRAS 000, 1–??
+
+2
+2
+0
+0
+00
+0
+1
+0
+0
+0
+[5.8]
+Q
+0
+K-
+二
+1
+0
+0
+0
+H
+[3.6]
+0
+00
+0
+0
+8
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+8
+0
+0
+00
+0
+0
+00
+0
+Q
+0
+8
+0
+60
+0
+0
+0
+0
+1
+2
+-1
+0
+1
+2
+[4.5]-[8.0]
+[4.5]-[8.0]8
+0
+0
+10
+0.1 Myr
+12
+0
+1 Myr
+5.0
+14
+2 Myr
+4.0
+3.5
+5 Myr
+3.0
+/2.5
+10 Myr
+1 2.0
+16
+11.7
+Q
+>1.4
+11.1
+Q.9
+0.7
+18
+20
+0
+2
+3
+4
+V-I6
+6
+8
+8
+0
+0
+0
+00
+8
+10
+10
+0
+0
+0
+0
+0
+&
+&
+0
+J12
+0
+0
+0
+0
+0
+000
+Q
+00
+14
+0
+0
+14
+00
+0000
+0o0
+8
+00
+8
+0@
+0
+0
+0
+0
+0
+0
+0
+00
+16
+0
+16
+0
+18
+18
+-1
+0
+1
+2
+3
+4
+-1
+0
+1
+2
+3
+J-K
+H-Variable stars in NGC 6823
+13
+and CMDs indicates a possible weak-lined TTSs. It is des-
+ignated as V4 by Pigulski et al. (2000).
+Star V5 of Pigulski et al. (2000) is numbered as No. 1061
+in this work. We considered it a field star based on its loca-
+tion in CMDs and TCDs. Its brightness in V and I bands
+varies with a period of 0.438 days. The variability of this
+star is confirmed in the present work.
+Star V8 (979) which could be a classical TTS based on
+its location in the J − H versus H − K diagram. It has a
+period of about 0.038 days. The kinematic data indicate it to
+be a possible member of the cluster of PMS nature. Pigulski
+et al. (2000) also found it to be suspected PMS variable.
+Star No. 753 was designated as V7 by Pigulski et al.
+(2000). The period of V7 could not be found by Pigulski et
+al. (2000) due to either irregular brightness or long-period
+variations. In our analysis, this star is considered a probable
+member of the cluster according to the proper motion study.
+Its position in the CMD and TCDs suggests a PMS Class II
+object. This star shows periodic brightness variation with
+its period and amplitude being 0.036 days and 0.225 mag,
+respectively.
+Star No. 655 (V3 in Pigulski et al. 2000) is a periodic
+variable with two possible periods, of about 17 days and of
+0.059 days. The location of this star in the J − H versus
+H − K TCD suggests a Class II source while proper motion
+data also suggest cluster membership.
+Star No. 1087 is found to be nonmember based on its
+proper motion values. Its location in TCDs suggests a PMS
+source. Its brightness changes periodically with a period of
+∼ 0.125 days. In Pigulski et al. (2000) this star (V1) was the
+reddest object among their variable sample.
+Star No. 831 is referred to as V6 by Pigulski et al. (2000)
+and they found this star too red as a member of the cluster.
+We confirm this star, with a period of about 1 day, to be a
+field star.
+Star No. 623, E 100, is a PMS object, though it was
+considered as a nonmember of the cluster in Pigulski et al.
+(2000) because its proper motion values were different from
+those of cluster members (Erickson 1971). They mentioned
+that this star might belong to the foreground population
+and it is of a late type object. The present estimation of its
+membership using Gaia data also finds it to be nonmember.
+4.2
+Newly detected variables
+Now we present newly identified variables in this work. Star
+No. 1235 is classified, on the basis of the shape of its light
+curve, to be an eclipsing binary, bearing similarity to that
+of an EA (Algol) type. In EA type eclipsing binaries, both
+stars are nearly spherical in shape, with an extremely wide
+range of periods from 0.2 days to 10000 days, and with a
+wide range of amplitude of variability. Star No. 1235 has a
+period of 1.622 days and a variation amplitude of 0.211 mag.
+In the H −R diagram, this star is found to be located in the
+region of new class variables. More observations of this star
+are required to confirm its nature. This star is a member of
+the cluster based on locations in TCDs. However, Gaia data
+suggest a nonmember of the cluster.
+The light curves of star No. 449 in both V and I bands
+reveal it to be a short-period variable. Its periodogram in
+both V and I exhibits peaks around 3.2 days and 0.789 days.
+Gaia data suggest it to be a probable member.
+[h]
+Table 3. Period and amplitude of variable stars. Last column rep-
+resents membership classification of stars along with their classifi-
+cation based on variability characteristics. The stars with asterisk
+are previously known variables. The cTTS, wTTS and HAe/Be
+are classical, weak-lined TTS and Herbig Ae/Be star, respectively.
+ID
+Period
+Period (TESS)
+Amp.
+class.
+(days)
+(days)
+(mag)
+103
+1.919
+1.913
+0.087
+PMS, wTTS
+135
+0.041, 0.010
+-
+0.336
+PMS, cTTS
+142
+0.504
+-
+0.156
+PMS, wTTS
+147
+0.940, 0.071
+-
+0.080
+PMS, wTTS
+154
+1.008
+-
+0.580
+PMS, HAe/Be
+177
+0.784
+-
+0.129
+PMS, wTTS
+201
+0.850, 0.845
+-
+0.115
+Field
+213
+0.253, 0.509
+-
+0.167
+Field
+238
+0.497
+-
+0.166
+Field
+239
+0.506, 0.969
+0.889, 2.429
+1.028
+PMS, wTTS
+240
+1.109, 0.067
+2.253
+0.032
+MS
+264
+0.546
+-
+0.202
+PMS, wTTS
+298
+0.357, 0.082
+-
+0.107
+PMS, HAe/Be
+369
+5.699
+2.400, 0.600, 7.041
+0.288
+PMS
+377
+0.0332
+-
+0.103
+Field
+385
+30.100, 0.958
+5.243, 3.960, 0.939
+0.425
+PMS, HAe/Be
+402
+0.804
+-
+0.394
+PMS, wTTS
+449
+3.852, 0.789
+5.297, 0.713, 4.084
+0.037
+Field
+452
+3.572
+-
+0.203
+PMS, wTTS
+478
+9.199, 0.898
+3.066, 2.622, 0.902
+0.366
+PMS, wTTS
+502
+1.138, 0.887
+-
+0.238
+PMS, wTTS
+510
+0.143, 0.030
+-
+0.025
+MS, New
+527
+0.671, 2.057
+2.045, 2.879
+0.168
+PMS, wTTS
+529
+0.047
+-
+0.290
+PMS, wTTS
+531
+0.755, 3.064
+3.084, 6.232, 9.713
+0.219
+PMS, wTTS
+546
+0.806, 0.057
+-
+0.044
+Field
+561
+10.300, 0.909
+3.24
+0.149
+PMS, wTTS
+576
+0.882
+0.850, 3.715
+0.237
+PMS, wTTS
+614
+0.707
+0.705, 1.775
+0.051
+MS
+619
+0.852
+3.825, 0.850
+0.030
+Field
+623*
+0.044, 0.053
+0.153, 1.595
+0.022
+MS
+655*
+17.716, 0.030
+-
+0.236
+PMS, cTTS
+679
+1.1242, 0.059, 0.564
+0.565, 3.376
+0.046
+MS
+706
+4.336, 0.561
+-
+0.049
+PMS, wTTS
+731
+0.359, 0.099
+-
+0.065
+PMS, wTTS
+733*
+0.512, 0.143
+-
+0.029
+MS
+752
+0.153
+0.153, 10.770
+0.257
+MS
+753*
+0.036
+-
+0.225
+PMS, cTTS
+757*
+0.553
+-
+0.125
+PMS, wTTS
+765
+0.112, 0.124
+-
+0.133
+Field
+822*
+0.143, 0.084
+-
+0.034
+MS, New
+826
+0.072
+-
+0.139
+PMS, wTTS
+831*
+1.009
+1.147, 1.545, 1.095
+1.463
+Field
+860
+8.517, 0.523
+-
+0.197
+PMS, cTTS
+886*
+0.446, 0.618
+-
+0.032
+MS
+903*
+0.848
+-
+0.033
+MS
+924*
+3.206, 0.59, 0.759
+-
+0.123
+PMS, wTTS
+945
+0.662, 0.663, 1.965
+1.966, 6.000
+0.026
+MS
+950
+0.504
+3.179, 2.442
+0.265
+PMS, wTTS
+951
+1.392, 0.775
+-
+0.064
+PMS, wTTS
+965
+2.661, 0.726
+2.65, 4.805
+0.053
+MS, New
+979*
+0.036
+0.064, 5.002
+0.185
+PMS, cTTS
+1000
+0.042, 0.486
+-
+0.016
+MS, New
+1007*
+0.064
+0.527, 0.064, 4.818
+0.037
+MS
+1025
+0.059
+-
+0.600
+PMS, wTTS
+1061*
+0.438
+1.581, 6.269
+0.099
+Field
+1063
+1.049, 0.028
+-
+0.131
+MS, β Cep
+1064
+0.653, 0.059, 0.395
+0.804, 5.291
+0.025
+MS
+1066
+0.0518, 0.082
+2.166, 1.203
+0.013
+PMS, wTTS
+1072
+0.484, 0.032
+0.819, 11.649
+0.025
+MS New
+1087*
+0.125
+-
+0.103
+PMS, wTTS
+1094
+0.815
+-
+0.081
+PMS, wTTS
+1122
+2.402, 0.705
+0.705, 11.649
+0.039
+MS, SPB
+1151
+0.769
+0.153, 4.834
+0.359
+Field
+1155
+10.955, 0.100
+-
+0.092
+PMS, wTTS
+1168
+0.526
+-
+0.240
+Field
+1191
+0.924
+-
+0.481
+Field
+1228
+0.902
+-
+0.448
+PMS, wTTS
+1230
+0.386, 0.629
+0.385, 1.755
+0.019
+MS, New
+1235
+1.622, 3.267
+3.243
+0.211
+MS, New
+1262
+1.215, 0.446
+-
+0.058
+PMS, wTTS
+1266
+3.382
+-
+0.148
+PMS, wTTS
+1268
+63.269
+5.240, 0.996
+0.276
+PMS, wTTS
+1295
+0.028
+-
+0.495
+PMS, cTTS
+1298
+0.983, 0.496
+-
+0.438
+PMS, cTTS
+1317
+0.485
+-
+0.671
+PMS, cTTS
+1352
+0.027, 0.336
+-
+0.014
+MS, SBP
+1389
+0.111, 0.166
+-
+0.086
+PMS, HAe/Be
+1405
+0.077, 0.064
+-
+0.128
+Field
+1406
+0.140, 0.082
+-
+0.123
+Field
+1459
+0.865
+-
+0.172
+PMS, wTTS
+1500
+0.072
+-
+0.046
+MS
+1506
+0.259, 0.491
+-
+0.031
+MS
+1508
+0.027
+-
+0.264
+Field
+1511
+0.059, 0.110
+-
+0.032
+Field
+1525
+1.132, 0.063
+-
+0.034
+MS, New
+1526
+0.058
+-
+0.075
+MS
+1548
+0.986, 0.329
+-
+0.174
+PMS, wTTS
+© 0000 RAS, MNRAS 000, 1–??
+
+14
+Sneh Lata et al.
+Figure 17.
+log(L/L⊙)/ log Teff diagram for the probable MS
+variable stars identified in the present study. The continuous curve
+represents the instability strip of SPB stars whereas dotted curve
+shows the instability region of δ Scuti stars. The dashed curve
+shows the location of β Cep stars (cf. Balona et al. 2011).
+The variability of the star No. 527 suggest that it is a pe-
+riodic variable whose light varies with a period of 0.671 days.
+It is found to be a probable member of the cluster from
+its proper motion measurements. Its location in CMDs and
+TCDs indicates a PMS object.
+The brightness of star No. 531, a probable PMS member
+of the cluster, is found to vary with a period of 0.755 days
+or 3.065 days, with the variability characteristics consistent
+with TTSs.
+The proper motion values are not in favor of star
+No. 752 to be a cluster member, though in the V versus
+V − I CMD, it is located along the MS. The period is de-
+rived as 0.153 days and the amplitude is about 0.2 mag. The
+variability characteristics of this star is similar to a pulsat-
+ing type star or eclipsing binary. After doubling its period
+its light curves show two minima which have almost equal
+depth. It could be an EW type eclipsing (W Ursae Majoris
+eclipsing system). The EW type variables have periods of
+less than one day, with almost equal depths of primary and
+secondary minima.
+Star No. 1508 has a period of ∼ 0.027 day with an am-
+plitude of about 0.2 mag. This variable resembles that of the
+SX Phe type (Cohen & Sarajedini 2012), which are similar
+to δ Scuti stars but pulsate with amplitudes up to 0.7 mag
+according to the variability types listed in the General Cat-
+alog of Variable Stars (GCVS).
+5
+TESS LIGHT CURVES
+A few variables like No. 1235 and No. 752 have times se-
+ries data from the Transiting Exoplanet Survey Satellite
+(TESS; Ricker et al. 2015). The high-quality light curves
+from the TESS can be used to understand stellar and plan-
+etary evolution and this data provide us opportunity to
+study the rotation of stars (Canto Martins et al. 2020).
+Here, we present folded light curves, exhibited as Fig. 18 for
+32 stars which do not have flux contribution from nearby
+brighter sources to account for the low spatial resolution
+of TESS. TESS observes the sky in sectors with each sec-
+tor observed for about 27 days. The eleanor pipeline to ex-
+tract times series data of objects from TESS images has
+been used, which is an open-source tool (Feinstein et al.
+2019, https://archive.stsci.edu/hlsp/eleanor). We can use
+the eleanor package to create light curves for fainter ob-
+jects for a more detailed or optimized analysis of individ-
+ual objects (Feinstein et al. 2019). The eleanor uses TESS
+Full Frame Images (FFIs) to extract systematics-corrected
+flux for any given star observed by TESS. It takes TIC ID,
+coordinates (RA and DEC) of a star. First, the raw flux
+is calculated by aperture photometry as RAW FLUX that
+is background subtracted. This raw flux is then corrected
+for possible systematic effects, which creates a flux called
+CORR FLUX. For isolated stars, to obtain the corrected
+flux we have taken default apertures. The eleanor software
+also provides the option to define one’s own aperture. We
+have extracted light curves of all the detected in the present
+photometry.
+Out of 32 variables, there are 7 stars which are diag-
+nosed as PMS and 14 as MS. The periods of all the 32 stars
+have been determined using the method described in the
+Section 2.1. The periods for 14 stars (103, 527, 531, 576,
+614, 619, 679, 752, 945, 965, 1007, 1122, 1230 and 1235)
+are found to be in good agreement with that obtained from
+the present ground based optical data. The nature of star
+No. 752 and No. 1235 as mentioned earlier is confirmed from
+their TESS light curves; that is, star No. 752 shows unequal
+maxima, likely due to the O’Connell effect (O’Connell 1951),
+for which the maxima between eclipses in some eclipsing bi-
+naries are not found equal in brightness (Knote et al. 2022).
+The phased light curves of stars Nos. 369, 561, and 619 show
+brightness variations similar to Algol type eclipsing binaries.
+The light curves of stars 527, 531, 576, 965, 1064, 1072 and
+1122 were folded by doubling the value of their derived pe-
+riod. Three stars 527, 531 and 576 of them are probable
+PMS stars while the remaining 4 stars are cluster members
+of MS type. The folded light curves and periods of 1064,
+1072 and 1122 are similar to the variability characteristic of
+EW type variables. The light curve of star 576 seems to have
+properties of EA type variable. The stars 527 and 965 could
+be weak-lined TTSs based on their variability characteris-
+tics as these sources are of PMS nature and show periodic
+variability. The period of star 531 was derived as 3.084 days
+using TESS data while its period comes out to be 0.755 days
+from V and I band light curves. The variability character-
+istics for those stars whose periods determined from present
+V and I data do not match with that derived from TESS
+observations could be revisited in the future observations
+of the cluster NGC 6823. The conflicts between periods for
+some cases may arise due to the contribution of flux from
+nearby stars in TESS data despite being selected isolated
+stars. The present work identified 5 MS, 4 PMS stars and 1
+field variable of eclipsing nature, two of which are confirmed
+eclipsing binaries and remaining are suspected ones. Their
+© 0000 RAS, MNRAS 000, 1–??
+
+5
+1063
+1526
+4
+1
+111
+1
+3
+log
+1352
+1122
+17
+606
+30
+10
+Kbr
+2
+1506
+614
+4.5
+4
+log 
+ T
+leffVariable stars in NGC 6823
+15
+Table 4. The proper motion, parallax and photometry by Gaia. The last column refers to likely or possible membership for each variable
+star.
+ID
+RA
+DEC
+µra
+µDec
+plx
+gmag
+bpmag
+rpmag
+mem
+degree
+degree
+mas/yr
+mas/yr
+mas
+(mag)
+mag
+103
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+2
+135
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+1262
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+-0.719±0.097
+-4.431±0.131
+-1.092±0.153
+16.379±0.003
+17.455±0.006
+15.228±0.007
+0
+1266
+295.802542
+23.236774
+-1.624±0.068
+-5.081±0.086
+0.554±0.093
+17.741±0.003
+18.960±0.022
+16.686±0.006
+1
+1268
+295.698738
+23.237776
+-2.864±0.083
+-5.120±0.113
+0.230±0.119
+15.750±0.004
+21.208±0.109
+13.997±0.007
+0
+1295
+295.812280
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+-1.564±0.182
+-4.156±0.212
+0.354±0.225
+18.726±0.015
+20.052±0.046
+17.061±0.037
+1
+1298
+295.671363
+23.233613
+-1.763±0.227
+-10.821±0.301
+-1.850±0.289
+18.089±0.021
+18.880±0.071
+16.384±0.040
+0
+1317
+295.899959
+23.227779
+-1.562±0.101
+-5.431±0.123
+0.589±0.137
+18.039±0.010
+19.288±0.036
+16.868±0.028
+1
+1352
+295.816693
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+-0.419±0.008
+-4.078±0.011
+1.831±0.012
+12.449±0.003
+12.760±0.003
+11.975±0.004
+0
+1389
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+-5.544±0.032
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+2
+1405
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+0
+1406
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+-3.267±0.064
+1.679±0.081
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+0
+1459
+295.758607
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+-1.710±0.090
+-5.469±0.124
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+18.147±0.003
+19.354±0.022
+17.032±0.008
+1
+1500
+295.892106
+23.195945
+3.974±0.049
+-3.001±0.071
+0.924±0.071
+15.646±0.003
+16.328±0.003
+14.820±0.004
+0
+1506
+295.737755
+23.197158
+-1.684±0.022
+-4.291±0.026
+0.674±0.028
+15.219±0.003
+15.857±0.004
+14.435±0.004
+0
+1508
+295.846040
+23.195095
+-2.071±0.089
+-4.858±0.124
+0.392±0.142
+18.061±0.005
+19.437±0.035
+16.831±0.013
+2
+1511
+295.813526
+23.194842
+4.169±0.025
+20.500±0.032
+3.135±0.035
+15.890±0.003
+16.829±0.004
+14.933±0.004
+0
+1525
+295.817542
+23.191915
+-0.781±0.009
+-7.011±0.012
+0.655±0.013
+13.206±0.003
+13.785±0.003
+12.468±0.004
+0
+1526
+295.740382
+23.192673
+59.603±0.010
+-58.093±0.014
+9.066±0.015
+08.663±0.003
+08.977±0.003
+08.173±0.004
+0
+1548
+295.840568
+23.186899
+-1.083±0.043
+-3.305±0.056
+0.248±0.060
+13.737±0.003
+18.375±0.013
+12.116±0.006
+0
+derived parameters are listed in Table 5. The masses and
+ages of two suspected PMS binaries could not be obtained
+due to unavailability of their V −I color. Since UBV data of
+MS star no. 752 are not available, the temperature for this
+star has been obtained using theoretical models of Girardi
+et al. (2002) and present V magnitude.
+© 0000 RAS, MNRAS 000, 1–??
+
+16
+Sneh Lata et al.
+Figure 18. The phased light curves of variable stars using TESS data.
+Figure 19. Amplitude of variability and rotation period of TTSs with ∆(I − K) is shown.
+6
+CORRELATION BETWEEN
+CIRCUMSTELLAR DISKS AND
+VARIABILITY
+Accretion onto the stellar surface creates hotspots that
+brighten the light curve up to 3 mag, whereas the magnetic
+field is responsible for cool and therefore dark spots. Herbst
+et al. (1994) studied photometric variability of PMS stars in
+the Orion Nebula Cluster, and showed that slower rotators
+have larger IR excess than fast rotators, indicating disk lock-
+ing, for which the angular momentum is transported through
+magnetic field lines from the central star to the circumstellar
+disk. This supported the results by Edwards et al. (1993) for
+which low-mass young stars with accretion disks have peri-
+ods more than 4 days, whereas stars without have periods
+ranging from 1.5 to 16 days. Rotation seems to be regulated
+after the disk is dissipated, as the star spins up while con-
+© 0000 RAS, MNRAS 000, 1–??
+
+103
+239
+240
+264
+1230
+160
+230
+100
+1225
+155
+90
+220
+1220
+xni
+150
+xni
+xnl
+80
+Xr
+210
+145
+1215
+70
+200
+1210
+140
+60
+1205
+135
+190
+50
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase
+369
+385
+449
+478
+3000
+340
+1270E
+275
+335
+2990
+270
+1260
+330
+2980
+265
+xnl
+Xr
+xni
+325
+1250
+2970
+260
+320
+1240
+2960
+255
+315
+2950
+310
+1230
+250
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase
+527
+531
+561
+576
+160
+550
+620
+205 F
+540
+615
+200
+155
+610
+530
+195
+xnl
+xnl
+xnl
+xn
+150
+605
+520
+190
+600
+145
+510
+185
+595
+140
+500
+590
+180
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase
+614
+619
+623
+679
+06
+345
+2160E
+2230
+85
+340
+2220
+2150
+80
+335
+2210
+xn
+xn
+xn
+75
+x
+330
+2140
+L
+2200
+70
+325
+2130
+2190
+65
+320
+60
+315
+2120
+2180
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase752
+831
+945
+950
+300
+815
+430
+205
+810
+425
+290
+200
+805
+420
+280
+xnl
+xnl
+195
+xni
+xni
+800
+415
+270
+190
+795
+410
+260
+185
+790
+405
+250
+785
+400
+180
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase
+965
+979
+1007
+1061
+355
+6920
+1.784×104
+6940
+350
+1.782×104
+6920
+6900
+345
+xnl
+1.780×104
+6900
+xn
+xni
+340
+6880
+F
+1.778×104
+6880
+335
+6860
+1.776×104
+6860
+330
+325
+6840
+1.774×104
+6840
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.01.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase
+1064
+1066
+1072
+1122
+1110
+415
+1540
+380 
+410
+1100
+1530
+370
+405
+xn
+xni
+1090
+400
+1520
+360
+395
+1080
+1510
+350
+390
+1070
+385
+1500
+340
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase
+1151
+1230
+1235
+1268
+540
+2330
+1000
+165
+530
+2325
+900
+160
+520
+2320
+800
+155
+xn
+xn
+xni
+xn
+510
+2315
+700
+150
+500
+2310
+600
+490
+145
+2305
+500
+480
+2300
+400
+140
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+0.00.5 1.0 1.52.0
+Phase
+Phase
+Phase
+Phase0.4
+0.3
+(mag
+Amp.
+0.2
+0.1
+0
+0
+2
+△(I-K)30
+28
+26
+24
+22
+20
+18
+(days)
+16
+Period
+14
+12
+10
+8
+6
+4
+2
+0
+0
+2
+△(I-K)Variable stars in NGC 6823
+17
+[h]
+Table 5. The derived parameters of the confirmed/suspected
+eclipsing binaries. The last column refers to binary classification.
+ID
+Age
+Mass
+Mbol
+log(L/L⊙)
+log Teff
+class.
+Binary
+Myrs
+M⊙
+mag
+class.
+369
+-
+-
+-
+-
+-
+PMS
+EA?
+561
+1.3
+1.28
+-
+-
+-
+PMS
+EA?
+576
+-
+-
+-
+-
+-
+PMS
+EA?
+619
+-
+-
+-
+-
+-
+Field
+EA?
+752
+4.0
+1.95
+-
+3.915
+MS
+EW
+757
+0.4
+1.50
+-
+-
+-
+PMS
+EW?
+1064
+4.0
+3.40
+-0.794
+2.211
+4.148
+MS
+EW?
+1072
+4.0
+3.20
+0.1394
+1.837
+4.047
+MS
+EW?
+1122
+4.0
+4.59
+-1.286
+2.407
+4.141
+MS
+EW?
+1235
+4.0
+6.75
+-1.272
+2.402
+3.979
+MS
+EA
+tracting towards the MS (Bouvier et al. 2007). These results
+support the magnetic-disk model which controls PMS winds
+and angular momentum of young stellar objects during the
+PMS evolution.
+Models for a disk-star interaction (Ostriker & Shu 1995,
+Shu et al. 1994, Ghosh & Lamb 1979) are supported by
+the rotation periods of PMS objects in young open star
+clusters (Attridge & Herbst 1992, Herbst et al. 2001, 2002,
+2004). Kearns & Herbst (1998) and Nordhagen et al. (2006)
+determined the rotation periods in two clusters. James et
+al. (2010) derived light-curve periods of sun-like sources in
+the young cluster NGC 1039. Lamm et al. (2004) presented
+the rotation period of PMS objects, which supports the
+disk-locking mechanism in young stars. Broeg et al. (2006)
+measured rotational periods of young objects to under-
+stand the star formation scenario that the off-cloud young
+sources should rotate faster if these objects were ejected from
+the cloud. They did not find significant period distribution
+off-cloud weak-lined TTS south of Taurus-Auriga with re-
+spect to weak-lined TTS inside the Taurus-Auriga molecu-
+lar cloud. Godoy-Rivera (2021) studied stellar rotation and
+found that the distribution of period with mass in the case
+of open clusters gives important constraints to study angu-
+lar momentum evolution and it is evident that spin down
+process depends on the mass. The rotation periods of the
+members of cluster have been presented, which are found
+to be in range from 0.5 days up to 11.5 days (Meibom et
+al. 2009, 2011). Gondoin (2018) concluded that the stellar
+rotation evolution in open star clusters could be from loss
+of angular momentum, which occurs due to strong winds
+during the early evolution of young solar type stars.
+A disk-bearing YSO spins down due to magnetic brak-
+ing (Koenigl 1991; Ostriker & Shu 1995). The increasing disk
+fraction with rotation period in open clusters was reported
+by Cieza & Baliber (2007). As discussed above, to explore
+a correlation between the variability of classical TTSs with
+color excess, mass, and age, we have plotted amplitude of
+variability with ∆(I −K) excess in Fig. 19 (left panel), while
+the right panel of Fig. 19 shows the rotation period with
+∆(I − K) excess. Here the ∆(I − K) excess of PMS sources
+is determined using the following relation,
+∆(I − K) = (I − K)obs − (AI − AK) − (I − K)0,
+where (I −K)obs and (I −K)0 are the observed and intrinsic
+colors of stars, whereas AI and AK denote the interstellar
+extinction in the I and K bands, respectively. To estimate
+the value of (I − K)0 of YSOs, it was necessary to estimate
+their masses and ages. These values are available for only 22
+PMS objects from the V versus V −I CMD after comparing
+with the theoretical models of Siess et al. (2000). Of the
+22 sources, there are 4 classical TTSs for which we could
+estimate the age and mass. The AI and AK are estimated
+using the relations given by Cohen et al. (1981) by adopting
+AV = 2.24 mag. The (I − K)0 value is obtained from the
+PMS evolutionary models of Siess et al. (2000) of a given
+mass and age. Fig. 19 (left panel) shows that a larger ∆(I −
+K) value for classical TTSs corresponds to a relatively larger
+amplitude of variability, consistent with those found in the
+literature, though we find no clear correlation between ∆(I−
+K) and rotation period. However one classical TTS No. 655
+with larger ∆(I − K) excess is found to be rotating with a
+longer period.
+7
+SUMMARY
+This work presents 88 variable stars in the young star cluster
+NGC 6823. The association of detected variables to the clus-
+ter has been discussed with the Gaia kinematic data, and
+the optical and NIR TCDs and CMDs. The membership
+of previously known variables has also been discussed. We
+have detected 48 stars as PMS stars, of which eight are clas-
+sified as classical TTSs while 36 and 4 as weak-lined TTSs
+and Herbig Ae/Be stars, respectively. Three known variables
+H30, V2 and V8 are found to PMS variables as suggested
+by Pigulski et al. (2000) while two stars BL 50 and HP 57
+previously detected as PMS δ Scuti pulsators are turned out
+to be MS members of the cluster from their proper motion,
+parallax values and positions on the TCDs and CMDs. TTSs
+have periods ranging from 0.01 days to 30 days, and ampli-
+tudes of brightness variation from 0.05 mag to 0.7 mag, with
+the classical TTSs varying generally with larger amplitudes
+than weak-lined TTSs do. It is noted that 3 of the 4 classi-
+cal TTSs with larger values of the disk indicator (∆(I −K))
+are found to have relatively larger amplitude variation. The
+present results do not support the disk-locking mechanism,
+however one classical TTS having large ∆(I − K) is found
+to be rotating slowly. In addition, we have identified 25 stars
+to be MS variables (SPB stars, δ Scuti, β Cephei and new
+class variable stars). Their variability has been characterized
+based on the period, amplitude, shape of the light curves,
+and location on the H − R diagram. Fifteen variable stars
+may belong to the field star population.
+8
+ACKNOWLEDGMENTS
+We are thankful to Prof. E. L. Mart´ın for the valuable sug-
+gestions that improved scientific content of the present work.
+Late Dr. A. K. Pandey facilitated this collaboration project
+as Director of ARIES during WPC’s visit. He will be for-
+ever remembered. SL will always be grateful to him for all
+the support and encouragement. We acknowledge the assis-
+tance of Michael Schwartz who managed the Tenagra Ob-
+servatory in acquisition of the images of this study. ASH
+and JCP thanks Ministry of Innovation Development of
+Uzbekistan and Department of Science and Technology of
+India for financing the joint project (Project References:
+UZB-Ind-2021-99 & INT/UZBEK/P-19). This publication
+makes use of data products from the 2MASS, which is a
+joint project of the University of Massachusetts and the
+© 0000 RAS, MNRAS 000, 1–??
+
+18
+Sneh Lata et al.
+Infrared Processing and Analysis Center/California Insti-
+tute of Technology, funded by the National Aeronautics
+and Space Administration and the National Science Foun-
+dation. This paper includes data collected by the TESS
+mission. Funding for the TESS mission is provided by the
+NASA’s Science Mission Directorate. We also acknowledge
+”Galactic Legacy Infrared Midplane Survey Extraordinaire”
+(GLIMPSE) Legacy Program for Spitzer IRAC data. This
+work also used data from the European Space Agency (ESA)
+space mission Gaia. Gaia data are being processed by the
+Gaia Data Processing and Analysis Consortium (DPAC).
+Funding for the DPAC is provided by national institutions,
+in particular the institutions participating in the Gaia Mul-
+tiLateral Agreement (MLA). The Gaia mission website is
+https://www.cosmos.esa.int/gaia. The Gaia archive website
+is https://archives.esac.esa.int/gaia.
+AVAILABILITY OF DATA
+The data underlying this article will be shared upon
+request
+to
+the
+corresponding
+author.
+The
+2MASS
+data
+are
+available
+at
+https://vizier.u-strasbg.fr/viz-
+bin/VizieR?-source=II/246. The Gaia and Spitzer IRAC
+data are obtained from https://gea.esac.esa.int/archive/
+and
+https://irsa.ipac.caltech.edu/cgi-bin/Gator/nph-
+scan?submit=Select&projshort=SPITZER,
+re-
+spectively.
+We
+used
+the
+following
+links
+https://archive.stsci.edu/hlsp/eleanor
+and
+https://adina.feinste.in/eleanor/ to obtain TESS data.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf,len=5974
+page_content='Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (0000)  Printed 9 January 2023  (MN LaTEX style file v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2)  Photometric variable stars in the young open cluster  NGC 6823  Sneh Lata1⋆, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Chen2,3, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pandey1, Athul Dileep1, Zhong-Han Ai3,  Alisher S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Hojaev4, Neelam Panwar1, Santosh Joshi1, Soumen Mondal5,  Siddhartha Biswas5, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Bhatt6  1Aryabhatta Research Institute of Observational Sciences, Manora Peak, Nainital 263002, Uttarakhand, India  2Graduate Institute of Astronomy, National Central University, 300 Zhongda Road, Zhongli 32001 Taoyuan, Taiwan  3Department of Physics, National Central University, 300 Zhongda Road, Zhongli 32001 Taoyuan, Taiwan  4Ulugh Beg Astronomical Institute, Uzbekistan Academy of Sciences, Tashkent, Republic of Uzbekistan  5S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Bose National Centre for Basic Sciences, Kolkata 700106, India  6Indian Institute of Astrophysics, Koramangala, Bangalore-560034, India  Accepted ———.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Received ———;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  ABSTRACT  We present stellar variability towards the young open cluster NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Time series V -  and I-band CCD photometry led to identification and characterization of 88 variable  stars, of which only 14 have been previously recognized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We ascertain the membership  of each variable with optical UBV I and infrared photometry, and with Gaia EDR3  parallax and proper motion data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Seventy two variable stars are found to be cluster  members, of which 25 are main sequence stars and 48 are pre-main-sequence stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The probable cluster members collectively suggest an isochrone age of the cluster to  be about 2 Myrs based on the GAIA photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' With the color and magnitude, as  well as the shape of the light curve, we have classified the main sequence variables into  β Cep, δ Scuti, slowly pulsating B type, and new class variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Among the pre-main-  sequence variables, eight are classical T Tauri variables, and four are Herbig Ae/Be  objects, whereas the remaining belong to the weak-lined T Tauri population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  variable nature of 32 stars is validated with TESS light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Our work provides  refined classification of variability of pre-main-sequence and main-sequence cluster  members of the active star-forming complex, Sharpless 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Despite no strong evidence  of the disk-locking mechanism in the present sample of TTSs, one TTS with larger  ∆(I − K) is found to be slow rotator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Key words:  open clusters and associations: individual NGC 6823, Hertzsprung-  Russell and color-magnitude diagram, stars: pre-main-sequence, stars: variables: T  Tauri, Herbig Ae/Be  1  INTRODUCTION  Young open clusters serve as useful tools for the studies  of the star formation mechanism and early stellar evolu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' For example, young star clusters are used to trace the  Galactic spiral structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In particular, variability of young  stellar members provides diagnostics on the sporadic (accre-  tion or occultation) or periodical (rotation) properties of the  stars, and of their relation to the circumstellar environments  (Morales-Calderon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Pre-main-sequence (PMS) objects are categorized on  ⋆ E-mail: sneh@aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='in  the basis of the spectral energy distribution in the infrared  wavelengths: Class 0 , Class I, Class II, and Class III (Lada  1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Andre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1993) with the classification sequence  roughly corresponding to the evolutionary status.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Namely,  a Class 0 object signifies a clump of dust and gas heavily en-  shrouded in the molecular envelope, and is detected only in  far-infrared wavelengths or longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' A Class I object is more  evolved, now emerging from the cloud to become visible in  near- and mid-infrared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' A Class I object is in the protostellar  stage and derives the luminosity from mass accretion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  A Class II object, corresponding to a classical T Tauri  star (TTS), has dispersed much of the envelope of gas and  dust but retains a circumstellar disk within which plan-  © 0000 RAS  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='02399v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='SR]  6 Jan 2023  2  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  ets may condense or are being formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Inside the optically  thick but geometrically thin disk, the dust grains absorb the  starlight and re-emit in the infrared, manifest as infrared  excess seen typically in a classical TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Accretion from the  disk onto the star, while matter is partly lost as bipolar  jets/outflows, leads to strong emission lines in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  As the inner disk is dissipated (or going into planet forma-  tion), the PMS object then evolves to Class III, now with  negligible infrared excess and with weak emission lines, if  any, due to surface chromospheric activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' A Class III object  hence is called a weak-lined TTS (Joy 1945, Appenzeller &  Mundt 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Variability of PMS objects hence serves as an  important diagnosis to understand the earliest PMS stellar  evolution, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=', the accretion (Johnstone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2018), rota-  tion (Herbst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1994), or dust properties (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Here we report the variability study of the Galactic  young open cluster NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' At a distance of about 2 kpc,  the cluster is associated with the prominent H II region,  Sharpless 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This cluster has been investigated by several  authors (Turner 1979, Stone 1988, Bica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2008, Sagar  and Joshi 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Guetter 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pigulski et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Hojaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2003, Zahajkiewicz 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Using op-  tical and JHK photometric observations Riaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2012)  found a large population of young stellar sources in the re-  gion, including two δ Scuti variables of PMS nature, and 13  other variables such as eclipsing binaries, slowly pulsating B  candidates and UX Ori type variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In the line of sight to  the cluster the reddening has been found to be from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7 to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1 mag following a normal reddening law (Rangwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The aim of the present work is to identify variables  in a relatively large field of ∼ 14′ ×14′ of the member versus  nonmember variable stellar populations in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Par-  ticularly, photometric rotation periods of PMS members are  derived to add to the data inventory for the study of the  angular momentum evolution of low-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  We describe in Section 2 observations, data reduction  procedure, detection of variables, and period determina-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In Section 3, membership of the identified variable  candidates is discussed using Gaia proper motion data, pho-  tometric two-color diagrams (TCDs) and color-magnitude  diagrams (CMDs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Section 4 then presents the nature of  known and newly identified variable stars, while Section 5  deals with TESS light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We discuss correlation be-  tween amplitude and rotation periods of TTSs along with  their color excess in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The results are summarized  in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2  OBSERVATIONS AND DATA REDUCTION  We have observed NGC 6823 with the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='81-m f/7 Ritchey-  Chretien Tenagra automated telescope in southern Arizona,  equipped with a 1024 × 1024 pixel SITe camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Each pixel  corresponds to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='87′′, which yields a field of view of ∼ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8′×  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The observations were carried out from 2012 early  October to 2012 December.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In total, data were acquired on  54 nights in two passbands, with 232 frames in V band and  243 frames in I band, with typical seeing of 2–3′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Bias and  twilight flats were taken every observing night.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The observed  region of the cluster in I band is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The log of  the observations is given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The observed region of open cluster NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Each  variable star identified in this work is encircled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Photometric errors as a function of instrumental mag-  nitude in I band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Open circles represent the variables stars iden-  tified in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The observed images were processed using standard  IRAF tasks: zerocombine, flatcombine and CCDPROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We  have performed aperture as well as point spread function  (PSF) photometry to derive the magnitude of stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  PSF photometry was obtained using program ALLSTAR  (Stetson 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' To match the stars between different pho-  tometric files we used the daomatch routine of DAOPHOT  (Stetson 1992) whereas daomaster was used to match the  point sources, and to obtain a file having corrected mag-  nitude of stars from all the files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The daomaster program  also removes the flux variation of stars in different frames  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='400  42  213  240  264  298  4498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content="  0SE'E2  402  6FF  478  546  521  I  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='614  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='3  DEC  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='300  7月  826  831  965  950  10002  1025  1122  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='250  1151  1156  1226  1235  1268  1298  1317  1352  682  1406  1405  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='200  1459  1500  1155  15506  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='91010  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='850  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='800  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='750  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='700  RA0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='02  10  12  8  14  16  18  instVariable stars in NGC 6823  3  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The V and I band sample light curves of a few variables  identified in the present work where ∆m represents the differential  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  due to exposure time and airmass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This program makes the  magnitudes of stars in each photometry file equal to that of  reference file by applying a constant value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  We have used the V and I observations of Massey et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (1995) for conversion of the present instrumental magni-  tudes to the standard ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' For this, the mean instrumental  magnitudes in V and I bands given by DAOMASTER (Stet-  son 1992) have been converted into standard ones with the  following transformation equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  V = v + (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='042 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='001) × (V − I) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='818 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='014  V − I = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='982 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='004) × (v − i) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='185 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='002,  where v and i are the instrumental magnitudes, and V and I  refer to the standard magnitudes of stars in V and I filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The estimated photometric error as a function of the mean  instrumental magnitude is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1  Variables identification  To identify variable stars, we first produced the light curves  of all the stars cross-matched in different CCD frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  light curves were obtained by plotting the differential mag-  nitudes (∆m) of stars (variable minus the comparison star)  against the given Julian date (JD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We used the Lomb-  Scargle periodogram (Lomb 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Scargle 1982) to derive  the periods and produced phased light curves accordingly to  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Log of the observations of NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' N and Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' rep-  resent number of frames obtained and exposure time in seconds,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Date of  I  V  Observations  (N×Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=')  (N×Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=')  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='13 oct 2012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='6× 30 ascertain their most probable periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' A few variables seem  to show periodic variability but their periodic nature was  not obvious in their observed light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The phased light  curves of all stars were inspected, and we adopted the pe-  riod value which produces the most consistent phased light  curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The light curves of a few variables are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 3  as examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The phased light curves of variables identified  in both V and I bands are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 4 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 5,  whereas Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 6 shows variables identified in the I band only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  By eye inspection and periodogram analysis, we have  detected 88 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We have listed optical and near-  infrared (NIR) data of the variable stars in Table 2, including  an identification number, coordinates, and optical as well as  NIR photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' These are the star ID numbers la-  belled in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' These 88 variable stars include 14 known  variables, with periods varying from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='03 days to more  than 60 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We have plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 7 the root mean square  (RMS) scatter of each star to confirm their variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  observed RMS scatter includes both the intrinsic variability  and the mean photometric error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content=' Some stars have large  RMS values but do not show noticeable brightness varia-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='020  of the detector whereas a few stars contains spurious data  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The derived periods of stars are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  3  CLUSTER MEMBERSHIP OF VARIABLE  STARS  For each variable star, its UBV I plus 2MASS photometry  along with Gaia EDR3 proper motion and parallax have  been used to assess the likelihood of cluster membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The UBV , JHK, and mid infrared (MIR) data at wave-  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Variable stars in NGC 6823  5  lengths 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8 and 8 micron, are taken from Massey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (1995), Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2003), and GLIMPSE survey, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1  Gaia Characterization of the Variable Stars  The 88 variable stars reported in this work have been char-  acterized with the Gaia EDR3 parallax and proper motion  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 8 plots the sky positions of all the Gaia  sources (in gray) within 30′ toward NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This covers  the field of the Tenagra images (variable stars marked in  black crosses) and is much wider than the cluster’s angular  size of ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='′6 (red circle) (Morales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The stellar  density is clearly enhanced toward the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Each variable star was matched with Gaia counterparts  within a radius of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='′′5 as the compromise of the seeing of the  Tenagra images, leading to 91 Gaia sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 9 presents  the proper motion vector plot of all the stars (gray) and  those within 4′ nominal cluster region (black small circles,  1294 stars) for which the members should be concentrated,  serving as the sample of cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This 4′ (posi-  tional) sample has a mean of µα ≈ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7 mas yr−1 and  µδ ≈ −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='3 mas yr−1, which agrees well with the litera-  ture values (Cantat-Gaudin & Anders 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Shown in the  bottom panel are the proper motions for variable stars (in  black with error bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' One sees that the majority of our  variable stars share the same proper motion ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Gaia measures repeatedly the astrometry of a source  from which the parallax and proper motion are solved simul-  taneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Parallax, however, does not serve as a constraint  for membership as stringently as the proper motion, because  given the uncertainties, negative average values may result,  rendering the reciprocal to estimate the distance possible  only if a statistical inference is exercised (Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' For our work the parallax value was used directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  parallax of the 4′ sample exhibits a peak around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='45 mas,  indeed consistent with the literature value (Cantat-Gaudin  & Anders 2020), and so does the variable star sample, as  demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' If the 4′ sample is further di-  vided by proper motion ranges, one finds no star within 1–  2 mas yr−1 from the cluster’s mean having parallax between  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4 mas and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This signifies the sufficiency as mem-  bership criteria of (1) a radius of 1 mas yr−1 in the proper  motion from the cluster average proper motion, and (2) a  parallax value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='35–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='55 mas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' A variable satisfying both (1)  and (2) is therefore considered a “highly probable” mem-  ber, whereas one that fulfills only (1) or (2) is classified as  a “possible” member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Table 4 lists information about the  proper motions, parallax, and magnitudes for the 88 vari-  ables identified in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 11 shows the Gaia G versus BP − RP CMD for  the highly probably members (in red) and possible mem-  bers (in blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Overlapped in the diagram is the PARSEC  isochrones of 1, 2, and 4 Myr, respectively, each shifted by  a distance modulus of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='753 (parallax of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='446 mas) and  reddening of E(B −V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8 (Sagar & Joshi 1981) adopting  the reddening law of AV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1 E(B −V ), AG = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='83627 AV ,  ABP = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='08337 AV , and ARP = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='63439 AV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The highly  probable members indicate an age of roughly 2 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Three variables have ambiguous Gaia counterparts  within the matching radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 478 has two possible  matches, equally faint thereby with relatively large uncer-  tainties in Gaia data but either one is consistent with being  a member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The star was not detected in our V band im-  age and appears progressively brighter from I = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='47 mag,  to 2MASS J = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='42 mag, H = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='42 mag, and Ks =  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='00 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1063 is the second brightest star in our variable  list, with the brightest one (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1526) being clearly not a  member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The star also has two Gaia matches, with contrast-  ing brightness (G = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='72 mag versus 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='87 mag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Given its  V = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='70 mag, the fainter one, having an outlying parallax  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='282 mas, is eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The other counterpart, however,  has a negative parallax value with a large uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This  compromises its membership determination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Its optical and  NIR colors both suggest an early-type star and its position  in the CMD suggests a main-sequence member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1262 has V  = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='173 mag, 2MASS J =  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='351 mag, H = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='474 mag, and Ks = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='998 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  brighter Gaia match has G = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='379 mag but has a neg-  ative parallax value and inconsistent proper motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  other Gaia star is faint, with G = 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='291 and no measure-  ments in the other two Gaia bands, has parallax and proper  motion values well consistent with being a member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  Colors and Magnitudes  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1  U − B vs B − V TCD  To identify probable MS variables, we have plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 12  the U − B versus B − V for variable stars identified in the  cluster region with the photometric data of 22 stars found  in Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Reddening in terms of color excess  E(B−V ) ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1 mag (Erickson 1971;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Guetter  1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pigulski et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) recognized the highest extinction in the eastern  part of the cluster where a trapezium system, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=', of O, B  spectral types, is located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The eastern part of their observed  field is the direction to the reflection nebula NGC 6820, and  their study suggested more than half of the total absorption  to arise from nearby interstellar matter toward NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  It is inferred that there is significant differential reddening  within the cluster, manifest that the cluster is located behind  at least AV = ∼ 3 mag (Riaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Rangwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  (2017) studied the interstellar extinction of open clusters  and found that NGC 6823 follows a normal extinction law in  optical as well as in the NIR wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The U −B versus  B − V TCD shows that the stars exhibiting within E(B −  V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1 mag could be MS members of the cluster,  indicating a nonuniform reddening across the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  reddened theoretical ZAMS of Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2002) is fitted  to the TCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The value of color excess E(V − I) was taken  as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='88 mag which has been calculated using the minimum  reddening value of E(B − V )=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='70 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  J − H vs H − K TCD  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 13 shows the J − H versus H − K TCD for NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Only 86 stars were cross-matched between the present sam-  ple of variable stars and the 2MASS catalog, with the JHK  counterparts of two stars No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 201 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 377 not found  during the match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In the 2MASS TCD, the “F” and “T”  regions are locations of probable Class III/field stars and  Class II sources, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The filled squares plotted in  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  6  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The I and V band phased light curves of variable stars identified in the region of NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  blue and green colors represent, respectively, probable PMS  and MS members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Circles in the diagram represents field  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Riaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2012) from their NIR CCD found early-  type MS dwarfs concentrating close to (H −Ks) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='  These authors have noticed another population near the  classical TTS locus close to (H − Ks) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content=' We note that star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 679  occupies the position where Herbig Ae/Be stars are placed,  while in U − B versus B − V TCD it lies close to the MS  locus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='  Following Gutermuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content=' (2012) used MIR IRAC data and  found 2 Class I, 94 Class II, and 394 Class III or field stars  in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content=' Continued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='8] ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='0  Phase  Phase  Phase  Phase8  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The I band phased light curves of probable variable candidates around the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5] − [8] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6] − [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8] TCD shows a few young stellar  objects to be positioned as field stars or other populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  IPHAS data  To identify the young stellar sources with Hα emission,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=', the indicator of accretion disks, we have compared the  present data with Table 3 for IPHAS photometry of Riaz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We got 29 common stars after the match that  have IPHAS photometry, with stars No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 502, 576, 655, 679,  and 979 having Hα emission with equivalent width greater  than 10 ˚A, judged by the (r′ − i′) versus (r′ − Halpha) TCD  for NGC 6823 (Riaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The location of these five  stars is shown with the red square in the present (J − H)  versus (H − K) TCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The Hα emission is found to be vari-  able in nature, therefore, it is necessary to check the location  of the objects in spectral type/color versus magnitude dia-  gram to know their membership and nature (Mart´ın et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Two stars Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 655 and 979 could be considered as  PMS objects which may possess circumstellar accreting disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Barrado y Navascu´es et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2001) showed that Hα emission  depends on the spectral type or color in the sense that Hα  emission is found to be larger for cooler objects in a plot  between the Hα emission and (I − J) color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The (I − J)  and (I − K) colors for star no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 655 is about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='26 mag and  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='682 mag, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In the case of star no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 979, we have  taken I magnitude from Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) to determine  its (I − J) and (I − K) colors due to lack of (V − I) color in  present observations, yielding (I − J) and (I − K) colors as  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='020 mag and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='789 mag, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Stars Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 655 and  979 in particular satisfy both colors and Hα emission be-  ing greater than 10 ˚A, hence are young stars with accretion  disks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  V vs V − I CMD  Sixty one variable stars were detected in both V and I  bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Their V magnitudes and V −I color are given in Ta-  ble 2, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 15 shows their V versus (V − I) CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  PMS isochrones and evolutionary tracks for different masses  are taken from Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The solid curve represents  ZAMS by Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We determined the distance  modulus of the cluster to be (V −MV ) = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='31 mag by com-  paring the ZAMS of Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2002) for solar metallicity  to the V versus V − I CMD, which corresponds to a dis-  tance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='59 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present estimate of distance matches  well with those derived in earlier works of NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  isochrone of age 4 Myr also fits the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The CMD is  known to be contaminated by the foreground field stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Guetter 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Bica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' After  analysis of CMD, Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) and Riaz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2012)  noted two different populations in the cluster, one consist-  ing of older, massive stars which are located near or on the  ZAMS, while the other one being younger objects with ages  less than 10 Myr and are of lower masses (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4 M⊙),  that is, of PMS stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) also concluded  that stars lying in B region of their figure 11(a) are cluster  stars of PMS nature, evolving towards the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present  CMD containing variable stars also shows MS of the cluster  to go up to around V = 16 mag;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' location of variables in the  CMD suggests the majority of these stars to be probable  members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Most the fainter and redder stars lying between  (V − I) =∼ 2 mag and ∼ 3 mag could be PMS objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  In this CMD, to maintain clarity we have not plotted star  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1548 despite it being detected in the I band, because it  has V − I color more than 6 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 449 may be a  possible PMS star but its placement in the U −B/B−V and  J −H/H −K TCDs suggests a field star, even though it has  proper motions in the range of probable cluster members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  142i  177i  238i  264i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='2  m  m  m  m  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  Phase  Phase  PhaseVariable stars in NGC 6823  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6  NIR CMDs  The J versus (J − K) and J versus J − H CMDs for the  present sample of variable stars are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' it  is seen that the MS is almost vertical and clearly sepa-  rated from the PMS objects/field stars as in the case of  the V versus (V − I) CMD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Bica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2008) described the  same and from statistical cleaned CMD, they found that  two populations are distributed separately where majority  of PMS objects are faint and redder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' They found the age  of the cluster in the range from 2 to 7 Myr, a color excess  E(B −V )=∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='86 mag, and AV =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In their work af-  ter fitting theoretical models, the absolute distance modulus  of the cluster was found to be (m − M)O = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Although stars numbered Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 142, 402 826, 950, 924,  1025, 1066, 1087, 1298, and 1548 are nonmembers from  the analysis of the Gaia data, we have considered them as  possible PMS objects based on their positions in different  CMD and TCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We note that there are 10 stars, namely,  Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 240, 614, 623, 752, 965, 1235, 1352, 1500, 1525, and  1526, that are designated as nonmembers from proper mo-  tion and parallax and, these could be MS stars based on  location in TCDs and CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Star 1506 is located well away  from the MS in U − B/B − V TCD while it is lying on te  MS in V/V −I, J/H −K and J/J −H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The analysis of Gaia  data also found it to be nonmember.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This is a doubtful case  for being an MS member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' From Gaia data analysis we found  stars Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 201, 239, 449, 619, 765, 861, 1151, 1168, 1191 and  1508 as possible or probable members but their locations in  various TCDs and CMDs do not seem to be consistent with  membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Thus, these are considered field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  We looked for the matches of our 8 previously identified  PMS stars based on their 2MASS colors and their spectra  taken at the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='16 m telescope of Beijing Observatory (Ho-  jaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The spectra showed the strong H-alpha  in emission, the SED in continuum and other features typi-  cal either for TTS or Herbig Ae/Be stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The cross-match  yields 3 common stars namely 154, 655 and 679 for two of  them we have already determined their features (their loca-  tion on the diagrams, their proper motions and parallaxes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  In the present work, we have classified 655 as classical TTS  and 679 as MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' There was doubt to consider 679 as PMS  because in U − B versus B − V TCD it is lying on MS but  in J − H versus H − K diagram it is placed in the location  of classical TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Now, we determined the membership of  star No 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' It might be probable member of the cluster as  Herbig Ae/Be type star while earlier (Hojaev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2003)  it has been classified as classical TTS, though it has very  different position in V versus V − I CMD to be PMS star  but GAIA suggests it to be highly probable member, and  in J − H versus H − K it is located where Herbig Ae/Be  stars are found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Therefore, it could be considered as Herbig  Ae/Be star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  We have considered members those stars which fulfill  criteria like location in TCDs, CMDs, and have consistent  kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Therefore, using proper motion data together  with location in various TCDs and CMDs obtained from  present and available photometric UBV I, NIR, and MIR  data we classify 25, 48, and 15, respectively, as MS, PMS  members of the cluster, and field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The classification of  variables is given in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Magnitude as a function of rms value of each star  detected in I band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Open circles represent variable stars identified  in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The sky positions of all the Gaia sources (in gray)  within 30′ toward NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  4  CHARACTERISTICS OF VARIABLE STARS  The log(L/L⊙) vs log Teff diagram (H − R diagram) for 21  members (MS variables) is shown as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We are not  able to locate four MS stars Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 240, 752, 1107 and 1500  in this plot due to lack of their U and B band data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Here,  the effective temperature and bolometric correction (BC)  have been determined from Toress relation (2010) using the  intrinsic (B − V ) color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The Mbol values of stars are ob-  tained from the relation Mbol = MV +BC, where MV is the  absolute V -band magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The luminosity was obtained  from the relation log(L/L⊙) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4(Mbol − Mbol⊙), where  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6  0  S  0  M  R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  00  0  0  0  0  8  10  12  14  16  18  instNGC 6823  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6  Decl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' [deg]  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' [deg]10  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The upper panel represents proper motion of all the  stars (gray) and those within 4′ cluster region (black small circles,  1294 stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The lower panel shows proper motions for variable  stars (in black with error bars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Mbol⊙ is the bolometric magnitude for the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The MS  variable stars have been classified according to their periods  of variability, the shape of light curves, and their positions  in the H-R diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We detected one star as β Cep-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Four stars Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 679, 886, 1122 and 1352 are located in the  instability strip of slowly pulsating B type (SPB) stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  positions in the cluster H-R diagram as well as the observed  variability characteristics of nine stars allow us to conclude  that these variables belong to the new class variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' One  star based on its location in the H − R diagram should be  δ Scuti-type variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  In the present study, we have detected 48 PMS stars  as most likely cluster members in the PMS stage of evo-  lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Of these, 4, 8, and 36 stars are classified as Herbig  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Histogram of parallaxes for stars within 4 arcmin,  where histogram shaded with black is for variable samples iden-  tified in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The G vs BP − RP CMD for the present sample  of variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The filled and open squares denote probable  and possible cluster members, and dotted points are considered  nonmembers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Ae/Be stars, classical TTSs, and weak-lined TTSs, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The amplitudes of weak-lined TTSs range from ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='05  to ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2 mag, and most weak-lined TTSs vary with shorter  periods of less than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The periods and amplitudes of  classical TTSs are found to range from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='05 to ∼ 30 days  and ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2 to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7 mag, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The above results  suggest that stars with disks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=', classical TTSs, exhibit  relatively larger amplitudes than the weak-lined TTSs do,  with the stellar variability in classical TTSs arising from the  presence of the spots, hot and cold, on the stellar surfaces  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  0  pmDE[mas/yr  4  6  8  10  12  6  4  2  0  2  4  9-  8  PmRA[mas/yr]2  0  2  pmDE[mas/yr  6  8  10  +  12  6  4  2  0  2  4  6  8  PmRA[mas/yr]260  200  150  100  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  plx [mas]10  12  14  [eu]  16  G  18  20  2  3  1  4  5  BP - RP[mag]Variable stars in NGC 6823  11  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (U − B)/(B − V ) TCD for variable stars identified in  the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' All the UBV data are taken from Massey et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The continuous and dotted line represent the ZAMS  (Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2002) which are shifted along the reddening vector  for reddening E(B − V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='32 mag and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='45 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Triangles are  those stars that are identified as MS variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  as found in the previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Pandey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1  Known variables  In the CCD search for variable stars in NGC 6823, Pigulski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) demonstrated that all stars with spectral types  later than A0 are PMS objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' They detected two variable  stars of δ Scuti type and these stars could be at the PMS  stage of evolution and suggested that these objects can fur-  ther be used to test the evolutionary changes in this class of  variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The CMD was used to compare and discuss  the position of the two discovered δ Scuti stars with refer-  ence to the theoretical instability strip for PMS stars of this  type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' They have also 13 other variables including one bright  cluster eclipsing binary and an SPB candidate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Of 15 variables identified by Pigulski et al (2000), 14  were found to be variable in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We could not  detect variability in the star H8 (E88 or BL 4) (B0 V:pe by  Turner 1979), B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5 V by Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1995), and B1 V by  Shi & Hu (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pigulski et al (2000) noted this stars as  the brightest variable member in the observed cluster field  and found to be a binary star where only one eclipse was  detected in the I band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Now we will describe the nature of all the known 14  variable stars individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Stars BL 50 (822) and HP 57 (1007) with periods  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0718530 days, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='10114 days for BL 50 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0785819 days,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0644149 days for HP 57 were found to be most likely clus-  ter PMS members by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' With their po-  sitions in the cluster CMD as well as the observed periods  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (J − H)/(H − K) TCD for variable stars detected  in the field of NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The JHK data have been taken from  the 2MASS catalog (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The continuous and long  dashed lines show sequences for dwarfs and giants (Bessell & Brett  1988), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The TTS locus (Meyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1997) is shown  by a dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The small dashed lines are reddening vectors  (Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1981) and an increment of visual extinction of AV  = 5 mag is denoted by crosses on the reddening vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Filled  squares with blue colors represents PMS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The MS population are  shown by green squares whereas open circles may be either MS  members of the cluster or field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Triangles (black) represent  two MS members BL 50 and HP 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) concluded that both objects could be δ  Scuti variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present membership analysis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=', from  kinematics and positions in various CMDs and TCDs, sug-  gests both stars to be MS members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In the H-R diagram star  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 822 is positioned where new class variables are found  (between the red edge of SPB and the blue edge of δ Scuti  instability strip).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1007 could not be placed in the H-  R diagram due to unavailability of UBV data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present  period of stars 822 is derived as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='143 days and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='084 days  while the periodogram analysis gives period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='064 days  for star 1007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The period derived for star 1007 is in good  agreement with that derived by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  location of these two stars were shown with red and black  triangles in V versus (V −I) CMD and J −H versus H −K  TCD, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 903, a probable cluster MS member was dis-  covered as the third pulsator (G 51) by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Its brightness varies with a period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='848 days with an am-  plitude about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='03 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The star was classified by Pigulski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) as an SPB variable according to their variability  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The brightness of star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 886 (G52) found to be binary  by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) varies with period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='61 days  with an amplitude 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='03 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' It is diagnosed as a member of  the cluster from proper motion and its location in various  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  1063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  881  0  1000 679  1502  449  9658  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  2斤  B  U  614  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  2  1506  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  2  B-VX  X  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  X  X  2  0  F  T  X  X  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  H  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8  2  H-K12  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (H − K) vs [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5] − [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0] and [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6] − [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8] vs [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5] −  [8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0] TCDs for variable stars detected in the field of NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Blue circles are PMS young stellar sources while black circles are  MS/field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' V/(V −I) CMD for variable stars in the region of the  cluster NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The open circles (blue) are MS variables, and  probable PMS variable stars are shown by filled circles (magenta).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The open circles in black color are considered field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  continuous curve is ZAMS by Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2002) while dashed  lines are PMS isochrones taken for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1, 1, 2, 5, 10 Myrs (Siess  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The PMS evolutionary tracks for different masses  ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7 to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0 M⊙ from Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) are plotted  with dotted curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' BL 50 and HP 57 are shown by red triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' J/(H − K) and J/(J − H) CMD for variable stars  detected in the field of NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The JHK data have been taken  from the 2MASS catalogue (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Circles (blue) and  circles (green) represent MS and PMS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The Open  circles in black color demonstrate the field stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The locations  of stars No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 822 (BL 50) and 1007 (HP 57) are shown with open  circle in red color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  photometric diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present estimates for period and  amplitude are consistent with those reported in Pigulski et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 733 has proper motion values of µα  =  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='671 mas/yr and µδ = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='453 mas/yr, hence is a prob-  able member of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Our analysis suggests possibly  more than one period, with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='143 d and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='512 d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In Pigulski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) it is H30, and they found its period of more  than 3 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The brightness of star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 757 was found to be chang-  ing with one single period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='553 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present work  classified this star to be a probable PMS cluster member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) named it V2 and derived its period of  about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='24 days, commenting that the true period for this  star corresponds to an alias frequency, and they found this  star to be of PMS type source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present observations  confirm its variability and its PMS nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The light curves  and periodogram analysis manifest that it could be an eclips-  ing binary with primary and secondary depths being nearly  equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Morales-Calderon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2012) found six new can-  didate sources as PMS eclipsing binaries with multi-epoch  data of about 2400 stars associated with the Orion Nebula  Cluster, and it is stated that the PMS eclipsing binaries are  valuable as they are in the stage of PMS evolution which  is highly dynamic, therefore their detection is rare at this  stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 924 may be a PMS variable with light curve  varying with more than one period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The proper motion sug-  gests a nonmember of the cluster but its position in TCDs  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2  2  0  0  00  0  1  0  0  0  [5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='8]  Q  0  K-  二  1  0  0  0  H  [3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='6]  0  00  0  0  8  0  0  0  0  0  0  0  0  0  0  0  0  0  8  0  0  00  0  0  00  0  Q  0  8  0  60  0  0  0  0  1  2  1  0  1  2  [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5]-[8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0]  [4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5]-[8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0]8  0  0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1 Myr  12  0  1 Myr  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  14  2 Myr  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  5 Myr  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  10 Myr  1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  16  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7  Q  >1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='4  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7  18  20  0  2  3  4  V-I6  6  8  8  0  0  0  00  8  10  10  0  0  0  0  0  &  &  0  J12  0  0  0  0  0  000  Q  00  14  0  0  14  00  0000  0o0  8  00  8  0@  0  0  0  0  0  0  0  00  16  0  16  0  18  18  1  0  1  2  3  4  1  0  1  2  3  J-K  H-Variable stars in NGC 6823  13  and CMDs indicates a possible weak-lined TTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' It is des-  ignated as V4 by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star V5 of Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) is numbered as No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1061  in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We considered it a field star based on its loca-  tion in CMDs and TCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Its brightness in V and I bands  varies with a period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='438 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The variability of this  star is confirmed in the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star V8 (979) which could be a classical TTS based on  its location in the J − H versus H − K diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' It has a  period of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='038 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The kinematic data indicate it to  be a possible member of the cluster of PMS nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pigulski  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) also found it to be suspected PMS variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 753 was designated as V7 by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The period of V7 could not be found by Pigulski et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) due to either irregular brightness or long-period  variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In our analysis, this star is considered a probable  member of the cluster according to the proper motion study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Its position in the CMD and TCDs suggests a PMS Class II  object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This star shows periodic brightness variation with  its period and amplitude being 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='036 days and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='225 mag,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 655 (V3 in Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2000) is a periodic  variable with two possible periods, of about 17 days and of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='059 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The location of this star in the J − H versus  H − K TCD suggests a Class II source while proper motion  data also suggest cluster membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1087 is found to be nonmember based on its  proper motion values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Its location in TCDs suggests a PMS  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Its brightness changes periodically with a period of  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='125 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) this star (V1) was the  reddest object among their variable sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 831 is referred to as V6 by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000)  and they found this star too red as a member of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  We confirm this star, with a period of about 1 day, to be a  field star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 623, E 100, is a PMS object, though it was  considered as a nonmember of the cluster in Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  (2000) because its proper motion values were different from  those of cluster members (Erickson 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' They mentioned  that this star might belong to the foreground population  and it is of a late type object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present estimation of its  membership using Gaia data also finds it to be nonmember.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2  Newly detected variables  Now we present newly identified variables in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Star  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1235 is classified, on the basis of the shape of its light  curve, to be an eclipsing binary, bearing similarity to that  of an EA (Algol) type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In EA type eclipsing binaries, both  stars are nearly spherical in shape, with an extremely wide  range of periods from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2 days to 10000 days, and with a  wide range of amplitude of variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1235 has a  period of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='622 days and a variation amplitude of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='211 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  In the H −R diagram, this star is found to be located in the  region of new class variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' More observations of this star  are required to confirm its nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This star is a member of  the cluster based on locations in TCDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' However, Gaia data  suggest a nonmember of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The light curves of star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 449 in both V and I bands  reveal it to be a short-period variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Its periodogram in  both V and I exhibits peaks around 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2 days and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='789 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Gaia data suggest it to be a probable member.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  [h]  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Period and amplitude of variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Last column rep-  resents membership classification of stars along with their classifi-  cation based on variability characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The stars with asterisk  are previously known variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The cTTS, wTTS and HAe/Be  are classical, weak-lined TTS and Herbig Ae/Be star, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  ID  Period  Period (TESS)  Amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  (days)  (days)  (mag)  103  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='919  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='913  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='087  PMS, wTTS  135  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='041, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='336  PMS, cTTS  142  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='156  PMS, wTTS  147  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='940, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='080  PMS, wTTS  154  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='580  PMS, HAe/Be  177  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='129  PMS, wTTS  201  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='850, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='115  Field  213  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='253, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='167  Field  238  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='166  Field  239  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='506, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='969  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='889, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='429  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='028  PMS, wTTS  240  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='109, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='067  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='253  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='032  MS  264  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='202  PMS, wTTS  298  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='357, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='107  PMS, HAe/Be  369  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='699  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='400, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='600, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='041  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='288  PMS  377  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='103  Field  385  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='100, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='958  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='243, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='960, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='939  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='425  PMS, HAe/Be  402  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='394  PMS, wTTS  449  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='852, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='789  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='297, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='713, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='084  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='037  Field  452  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  14  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  log(L/L⊙)/ log Teff diagram for the probable MS  variable stars identified in the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The continuous curve  represents the instability strip of SPB stars whereas dotted curve  shows the instability region of δ Scuti stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The dashed curve  shows the location of β Cep stars (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Balona et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The variability of the star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 527 suggest that it is a pe-  riodic variable whose light varies with a period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='671 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  It is found to be a probable member of the cluster from  its proper motion measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Its location in CMDs and  TCDs indicates a PMS object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The brightness of star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 531, a probable PMS member  of the cluster, is found to vary with a period of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='755 days  or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='065 days, with the variability characteristics consistent  with TTSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The proper motion values are not in favor of star  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 752 to be a cluster member, though in the V versus  V − I CMD, it is located along the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The period is de-  rived as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='153 days and the amplitude is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  variability characteristics of this star is similar to a pulsat-  ing type star or eclipsing binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' After doubling its period  its light curves show two minima which have almost equal  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' It could be an EW type eclipsing (W Ursae Majoris  eclipsing system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The EW type variables have periods of  less than one day, with almost equal depths of primary and  secondary minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1508 has a period of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='027 day with an am-  plitude of about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='2 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This variable resembles that of the  SX Phe type (Cohen & Sarajedini 2012), which are similar  to δ Scuti stars but pulsate with amplitudes up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7 mag  according to the variability types listed in the General Cat-  alog of Variable Stars (GCVS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  5  TESS LIGHT CURVES  A few variables like No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1235 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 752 have times se-  ries data from the Transiting Exoplanet Survey Satellite  (TESS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The high-quality light curves  from the TESS can be used to understand stellar and plan-  etary evolution and this data provide us opportunity to  study the rotation of stars (Canto Martins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Here, we present folded light curves, exhibited as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 18 for  32 stars which do not have flux contribution from nearby  brighter sources to account for the low spatial resolution  of TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' TESS observes the sky in sectors with each sec-  tor observed for about 27 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The eleanor pipeline to ex-  tract times series data of objects from TESS images has  been used, which is an open-source tool (Feinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  2019, https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='edu/hlsp/eleanor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We can use  the eleanor package to create light curves for fainter ob-  jects for a more detailed or optimized analysis of individ-  ual objects (Feinstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The eleanor uses TESS  Full Frame Images (FFIs) to extract systematics-corrected  flux for any given star observed by TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' It takes TIC ID,  coordinates (RA and DEC) of a star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' First, the raw flux  is calculated by aperture photometry as RAW FLUX that  is background subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This raw flux is then corrected  for possible systematic effects, which creates a flux called  CORR FLUX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' For isolated stars, to obtain the corrected  flux we have taken default apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The eleanor software  also provides the option to define one’s own aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We  have extracted light curves of all the detected in the present  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Out of 32 variables, there are 7 stars which are diag-  nosed as PMS and 14 as MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The periods of all the 32 stars  have been determined using the method described in the  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The periods for 14 stars (103, 527, 531, 576,  614, 619, 679, 752, 945, 965, 1007, 1122, 1230 and 1235)  are found to be in good agreement with that obtained from  the present ground based optical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The nature of star  No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 752 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1235 as mentioned earlier is confirmed from  their TESS light curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' that is, star No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 752 shows unequal  maxima, likely due to the O’Connell effect (O’Connell 1951),  for which the maxima between eclipses in some eclipsing bi-  naries are not found equal in brightness (Knote et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The phased light curves of stars Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 369, 561, and 619 show  brightness variations similar to Algol type eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The light curves of stars 527, 531, 576, 965, 1064, 1072 and  1122 were folded by doubling the value of their derived pe-  riod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Three stars 527, 531 and 576 of them are probable  PMS stars while the remaining 4 stars are cluster members  of MS type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The folded light curves and periods of 1064,  1072 and 1122 are similar to the variability characteristic of  EW type variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The light curve of star 576 seems to have  properties of EA type variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The stars 527 and 965 could  be weak-lined TTSs based on their variability characteris-  tics as these sources are of PMS nature and show periodic  variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The period of star 531 was derived as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='084 days  using TESS data while its period comes out to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='755 days  from V and I band light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The variability character-  istics for those stars whose periods determined from present  V and I data do not match with that derived from TESS  observations could be revisited in the future observations  of the cluster NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The conflicts between periods for  some cases may arise due to the contribution of flux from  nearby stars in TESS data despite being selected isolated  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The present work identified 5 MS, 4 PMS stars and 1  field variable of eclipsing nature, two of which are confirmed  eclipsing binaries and remaining are suspected ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Their  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  5  1063  1526  4  1  111  1  3  log  1352  1122  17  606  30  10  Kbr  2  1506  614  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5  4  log  T  leffVariable stars in NGC 6823  15  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The proper motion, parallax and photometry by Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The last column refers to likely or possible membership for each variable  star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  ID  RA  DEC  µra  µDec  plx  gmag  bpmag  rpmag  mem  degree  degree  mas/yr  mas/yr  mas  (mag)  mag  103  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='667936  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='412217  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='392±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='046  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='264±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='077  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='500±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='074  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='288±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='003  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='560±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='015  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='172±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='006  2  135  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='729818  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='406842  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='335±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='050  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='330±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='079  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='605±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='085  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='340±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='007  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='463±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='025  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='006±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='017  1  142  295.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='921757  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='403326  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='224±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='059  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='330±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='096  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='006  0  derived parameters are listed in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The masses and  ages of two suspected PMS binaries could not be obtained  due to unavailability of their V −I color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Since UBV data of  MS star no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 752 are not available, the temperature for this  star has been obtained using theoretical models of Girardi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2002) and present V magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  16  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The phased light curves of variable stars using TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Figure 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Amplitude of variability and rotation period of TTSs with ∆(I − K) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  6  CORRELATION BETWEEN  CIRCUMSTELLAR DISKS AND  VARIABILITY  Accretion onto the stellar surface creates hotspots that  brighten the light curve up to 3 mag, whereas the magnetic  field is responsible for cool and therefore dark spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Herbst  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (1994) studied photometric variability of PMS stars in  the Orion Nebula Cluster, and showed that slower rotators  have larger IR excess than fast rotators, indicating disk lock-  ing, for which the angular momentum is transported through  magnetic field lines from the central star to the circumstellar  disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This supported the results by Edwards et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (1993) for  which low-mass young stars with accretion disks have peri-  ods more than 4 days, whereas stars without have periods  ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content=' Rotation seems to be regulated  after the disk is dissipated, as the star spins up while con-  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='3  (mag  Amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content=' The derived parameters of the confirmed/suspected  eclipsing binaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The last column refers to binary classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  ID  Age  Mass  Mbol  log(L/L⊙)  log Teff  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Binary  Myrs  M⊙  mag  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  369 PMS  EA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='28 PMS  EA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  576 PMS  EA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  619 Field  EA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  752  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='  1235  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='0  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+page_content='979  MS  EA  tracting towards the MS (Bouvier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' These results  support the magnetic-disk model which controls PMS winds  and angular momentum of young stellar objects during the  PMS evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Models for a disk-star interaction (Ostriker & Shu 1995,  Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 1994, Ghosh & Lamb 1979) are supported by  the rotation periods of PMS objects in young open star  clusters (Attridge & Herbst 1992, Herbst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2001, 2002,  2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Kearns & Herbst (1998) and Nordhagen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2006)  determined the rotation periods in two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' James et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2010) derived light-curve periods of sun-like sources in  the young cluster NGC 1039.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Lamm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2004) presented  the rotation period of PMS objects, which supports the  disk-locking mechanism in young stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Broeg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2006)  measured rotational periods of young objects to under-  stand the star formation scenario that the off-cloud young  sources should rotate faster if these objects were ejected from  the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' They did not find significant period distribution  off-cloud weak-lined TTS south of Taurus-Auriga with re-  spect to weak-lined TTS inside the Taurus-Auriga molecu-  lar cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Godoy-Rivera (2021) studied stellar rotation and  found that the distribution of period with mass in the case  of open clusters gives important constraints to study angu-  lar momentum evolution and it is evident that spin down  process depends on the mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The rotation periods of the  members of cluster have been presented, which are found  to be in range from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5 days up to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='5 days (Meibom et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 2009, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Gondoin (2018) concluded that the stellar  rotation evolution in open star clusters could be from loss  of angular momentum, which occurs due to strong winds  during the early evolution of young solar type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  A disk-bearing YSO spins down due to magnetic brak-  ing (Koenigl 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Ostriker & Shu 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The increasing disk  fraction with rotation period in open clusters was reported  by Cieza & Baliber (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' As discussed above, to explore  a correlation between the variability of classical TTSs with  color excess, mass, and age, we have plotted amplitude of  variability with ∆(I −K) excess in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 19 (left panel), while  the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 19 shows the rotation period with  ∆(I − K) excess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Here the ∆(I − K) excess of PMS sources  is determined using the following relation,  ∆(I − K) = (I − K)obs − (AI − AK) − (I − K)0,  where (I −K)obs and (I −K)0 are the observed and intrinsic  colors of stars, whereas AI and AK denote the interstellar  extinction in the I and K bands, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' To estimate  the value of (I − K)0 of YSOs, it was necessary to estimate  their masses and ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' These values are available for only 22  PMS objects from the V versus V −I CMD after comparing  with the theoretical models of Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Of the  22 sources, there are 4 classical TTSs for which we could  estimate the age and mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The AI and AK are estimated  using the relations given by Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (1981) by adopting  AV = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='24 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The (I − K)0 value is obtained from the  PMS evolutionary models of Siess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) of a given  mass and age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 19 (left panel) shows that a larger ∆(I −  K) value for classical TTSs corresponds to a relatively larger  amplitude of variability, consistent with those found in the  literature, though we find no clear correlation between ∆(I−  K) and rotation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' However one classical TTS No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' 655  with larger ∆(I − K) excess is found to be rotating with a  longer period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  7  SUMMARY  This work presents 88 variable stars in the young star cluster  NGC 6823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The association of detected variables to the clus-  ter has been discussed with the Gaia kinematic data, and  the optical and NIR TCDs and CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The membership  of previously known variables has also been discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We  have detected 48 stars as PMS stars, of which eight are clas-  sified as classical TTSs while 36 and 4 as weak-lined TTSs  and Herbig Ae/Be stars, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Three known variables  H30, V2 and V8 are found to PMS variables as suggested  by Pigulski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' (2000) while two stars BL 50 and HP 57  previously detected as PMS δ Scuti pulsators are turned out  to be MS members of the cluster from their proper motion,  parallax values and positions on the TCDs and CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' TTSs  have periods ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='01 days to 30 days, and ampli-  tudes of brightness variation from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='05 mag to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='7 mag, with  the classical TTSs varying generally with larger amplitudes  than weak-lined TTSs do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' It is noted that 3 of the 4 classi-  cal TTSs with larger values of the disk indicator (∆(I −K))  are found to have relatively larger amplitude variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The  present results do not support the disk-locking mechanism,  however one classical TTS having large ∆(I − K) is found  to be rotating slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' In addition, we have identified 25 stars  to be MS variables (SPB stars, δ Scuti, β Cephei and new  class variable stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Their variability has been characterized  based on the period, amplitude, shape of the light curves,  and location on the H − R diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Fifteen variable stars  may belong to the field star population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  8  ACKNOWLEDGMENTS  We are thankful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Mart´ın for the valuable sug-  gestions that improved scientific content of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Late Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Pandey facilitated this collaboration project  as Director of ARIES during WPC’s visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' He will be for-  ever remembered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' SL will always be grateful to him for all  the support and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We acknowledge the assis-  tance of Michael Schwartz who managed the Tenagra Ob-  servatory in acquisition of the images of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ASH  and JCP thanks Ministry of Innovation Development of  Uzbekistan and Department of Science and Technology of  India for financing the joint project (Project References:  UZB-Ind-2021-99 & INT/UZBEK/P-19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This publication  makes use of data products from the 2MASS, which is a  joint project of the University of Massachusetts and the  © 0000 RAS, MNRAS 000, 1–?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  18  Sneh Lata et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Infrared Processing and Analysis Center/California Insti-  tute of Technology, funded by the National Aeronautics  and Space Administration and the National Science Foun-  dation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This paper includes data collected by the TESS  mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Funding for the TESS mission is provided by the  NASA’s Science Mission Directorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' We also acknowledge  ”Galactic Legacy Infrared Midplane Survey Extraordinaire”  (GLIMPSE) Legacy Program for Spitzer IRAC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' This  work also used data from the European Space Agency (ESA)  space mission Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' Gaia data are being processed by the  Gaia Data Processing and Analysis Consortium (DPAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  Funding for the DPAC is provided by national institutions,  in particular the institutions participating in the Gaia Mul-  tiLateral Agreement (MLA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The Gaia mission website is  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='int/gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The Gaia archive website  is https://archives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='int/gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  AVAILABILITY OF DATA  The data underlying this article will be shared upon  request  to  the  corresponding  author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  The  2MASS  data  are  available  at  https://vizier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='u-strasbg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='fr/viz-  bin/VizieR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='-source=II/246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content=' The Gaia and Spitzer IRAC  data are obtained from https://gea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='esac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='int/archive/  and  https://irsa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='ipac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='edu/cgi-bin/Gator/nph-  scan?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='submit=Select&projshort=SPITZER,  re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='  We  used  the  following  links  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='edu/hlsp/eleanor  and  https://adina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='feinste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
+page_content='in/eleanor/ to obtain TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69E0T4oBgHgl3EQffAC0/content/2301.02399v1.pdf'}
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+arXiv:2301.04862v1  [cs.PL]  12 Jan 2023
+Naturalistic Static Program Analysis
+Mohammad Mehdi Pourhashem Kallehbasti
+Department of Electrical and Computer Engineering
+University of Science and Technology of Mazandaran
+P.O. Box 48518-78195, Behshahr, Iran
+pourhashem@mazust.ac.ir
+Mohammad Ghafari
+TU Clausthal, Germany
+mohammad.ghafari@tu-clausthal.de
+Abstract—Static program analysis development is a non-trivial
+and time-consuming task. We present a framework through
+which developers can define static program analyses in natural
+language. We show the application of this framework to identify
+cryptography misuses in Java programs, and we discuss how it
+facilitates static program analysis development for developers.
+Index Terms—Static program analysis, cryptography, natural
+language programming
+I. INTRODUCTION
+Static program analysis is the art of examining programs
+without requiring to execute the code. However, static analysis
+tools generate false positives and tuning them requires exper-
+tise. Likewise, program analysis development requires a deep
+knowledge of compiler or mastering an analysis framework.
+End-user programming is a set of techniques that enable
+end users to write programs at a level of complexity that is
+adequate to their practices, background, and skills. For in-
+stance, it includes visual languages to program robots through
+visual blocks [1], and simplified programming languages to
+translate English sentences to Bash commands [2]. We believe
+that end-user programming techniques can also help to hide the
+complexity of writing a static program analysis task for non-
+professional programmers and empower them in this domain.
+We introduce NASRA (NAturalistic Static pRogram Analy-
+sis), a framework that enables developers to define a program
+analysis task in natural language (NL), and it generates the
+corresponding Query Language (QL) query that underlies
+CodeQL program analysis engine.1 We illustrate the appli-
+cation of this framework to find cryptography misuses in Java
+programs. NASRA is open source and publicly available.2
+The ultimate goal of NASRA is to enable “naturalistic”
+static program analysis development in a way that developers
+can specify what they need without deep knowledge of static
+program analysis and how a specific framework works. Its
+higher level of abstraction than existing static analysis frame-
+works may facilitate a more intuitive formulation of program
+analysis tasks. Similarly, its agnostic nature to programming
+languages can provide a cross-language interface for program
+analysis, which obviates the need to learn the specifics of a
+program analysis framework. This paper presents a prelimi-
+nary step to realize the above goal.
+1https://codeql.github.com
+2https://doi.org/10.5281/zenodo.7495044
+II. THE NASRA FRAMEWORK
+Cryptography is an essential component to security, but it is
+one of the notorious topics where developers struggle a lot [3],
+[4]. Locating the init method invoked on a Cipher object
+is often deemed to be the first step to analyze cryptography
+code in Java programs. For instance, in CodeQL, one should
+write the following query to implement this task.
+from
+MethodAccess init
+where init.getMethod().getName() = "init" and
+init.getReceiverType().getName() = "Cipher"
+select init
+We have developed a framework, called NASRA, that
+enables a more intuitive formulation of the above task in the
+form below:
+An object of Cipher invokes init.
+NASRA is a rule-driven synthesizer. We rely on predefined
+rules due to a lack of trustworthy labeled examples required
+for a data-driven approach in this domain. NASRA receives a
+program analysis inquiry in natural language, applies semantic
+parsing, and generates CodeQL commands. The input inquiry
+should comply with a subset of the syntax of Attempto
+Controlled English (ACE) controlled natural language. We
+use Attempto Parsing Engine (APE), a tool that receives a
+series of ACE statements and produces the corresponding
+Discourse Representation Structures (DRS) that is a semantic
+representation of the input text. NASRA applies the translation
+rules, explained later in this section, on the given DRS and
+produces the corresponding CodeQL statements. Thanks to
+APE, the way one can formulate NASRA statements is very
+flexible and there is no need for absolute correspondence with
+the NASRA syntax. We chose CodeQL as our code analysis
+engine because it is an industry-leading and community-
+powered tool, and its publicly available to all GitHub users
+without any installation hassle. To employ NASRA for a new
+static analysis framework, only the transformation rules have
+to be adapted. To support a new application domain, we should
+identify the types of queries that the current syntax does not
+support, add the corresponding production rules to the syntax,
+and develop translations for them. NASRA is open source,
+and currently, supports program analysis tasks that concern
+cryptography misuses in Java programs.
+
+A. Syntax and Semantics
+Each NASRA query comprises one or more Statement. The
+syntax is shown below (terminals have different color).
+Query ::= Statement Query | Statement
+Statement ::= BasicStatement | LogicalStatement
+| Extension
+BasicStatement ::= Exp is (Exp | in List)
+Exp ::= Prefix Exp | type | ID | Literal
+Prefix ::= ((adjective|ε) attribute of)
+LogicalStatement ::= Statement and Statement |
+Statement or Statement | It is false that Statement
+| If Statement then Statement
+a) Expression: The smallest building block is Exp. It
+includes a Literal (String or int) or an ID (user defined
+identifier) that are directly mapped to CodeQL expressions.
+An Exp can also be a CodeQL type such as class, variable,
+and method access that are mapped to Class, Variable,
+and MethodAccess, respectively.
+b) Prefix: Each Exp can have an optional Prefix in
+the form of “attribute of” that indicates an attribute of
+the expression. For instance, name, type, argument, and
+method are attributes of an entity (i.e., Exp), and they cor-
+respond to getName(), getType(), getArgument(),
+and getMethod() methods in CodeQL, respectively.
+For example, “name of method1” is an Exp, where “name”
+is an attribute and method1 is an ID, and the whole expres-
+sion is translated to method1.getName() in CodeQL.
+Additionally, the attribute itself can have an optional
+ordinal
+number
+as
+an
+adjective,
+like
+second
+in
+the
+Exp “second argument of init” that is translated to
+“init.getArgument(1)”, where second is translated to
+1 as an argument according to zero-based numbering.
+Note that “attribute of” can be repeated several times, where
+each attribute may have an adjective. For example, the Exp
+“The type of the second argument of init” has one ID
+(i.e.,init) and two attributes (i.e., type and argument).
+c) Basic Statement: Each BasicStatement is a statement
+that can serve as a Boolean condition as well as an assumption.
+As a Boolean condition, BasicStatement produces equiva-
+lence of two Exps, as well as membership of an Exp in a list.
+In “Exp is Exp” structure, both sides of equivalence are Exps
+and they need to be equal, while in “Exp is in List” structure,
+the Exp needs to be equal to an item in a list. Accordingly, a
+statement like “arg1 is in ["RSA", "AES"].” is a disjunctive
+expression and can be rephrased to “arg1 is "RSA" or arg1
+is "AES".”, that is ultimately translated to “arg1 = "RSA"
+or arg1 = "AES"”.
+The syntax structure Exp is Exp can also produce as-
+sumptions when the second Exp is a CodeQL type. The
+assumptions are mapped to the from part of a CodeQL query.
+For instance, the statement “var1 is a variable.” translates to
+“Variable var1” and belongs to the from part.
+d) Logical Statement: A LogicalStatement can be a
+negation, conjunction, disjunction, or implication. For exam-
+ple, “If arg1 is "RSA" then arg2 is "AES".” is translated to
+“not (arg1 = "RSA") or arg2 = "AES"” in Cod-
+eQL, since p ⇒ q is equivalent to ¬p ∨ q.
+B. Extensibility
+One can extend NASRA to cover auxiliary statements and
+statement patterns. Their corresponding production rules are
+as follows.
+Extension ::= Pattern | AuxiliaryStatement
+We introduce these features through three statement pat-
+terns and one auxiliary statement that are helpful to cover
+constraints on using Java cryptography objects.
+1) Patterns: We present three patterns that extend Pattern
+nonterminal in the syntax. We discuss each in the following.
+a) Invocation: We use this pattern to state that a method
+is invoked by an instance of a specific class. It can also be
+used to make sure that there is no invocation of a method by
+any instance of a specific class.
+Pattern1 ::= An object of ID (invokes|does not invoke) ID.
+The NASRA query shown in Section II is an example of this
+pattern. The transformation follows a number of steps. First, a
+MethodAccess is declared with the same name used in the
+NASRA statement (i.e.,init). Then the conditions need to be
+added to the where part. Specifically, the name of the method
+of the MethodAccess init should be "init" that is
+stated in the second line. Finally, a MethodAccess has a
+receiver, that is the object invoking its method. In this case,
+the name of the type of the receiver should be "Cipher",
+that is expressed in CodeQL in the third line.
+If one needs to make sure that no invocation occurs, an
+existential quantifier must be used, as shown in the following.
+from
+where not
+(exists
+(MethodAccess
+init
+|
+init.getMethod().getName() = "init" and
+init.getReceiverType().getName() = "Cipher"))
+It means that there is no such MethodAccess init that
+has these conditions. We can state this in NASRA in the form
+below.
+An object of Cipher doesn’t invoke init.
+b) Partial order constraints: This pattern enables one to
+put partial order constraints on method invocations. In other
+words, one can enforce a method invocation to be preceded
+(or followed) by another method invocation.
+Pattern2::=MethodName (precedes|follows) MethodName.
+For example, there are two steps in CodeQL for stating that
+“invocation of getInstance is earlier than invocation of
+init”. First, one should specify that both methods are in the
+same scope. Next, the line number of the preceding method
+invocation has to be smaller than the line number of the other
+method invocation. This is shown below.
+
+getInstance.getEnclosingCallable()
+=
+init.getEnclosingCallable() and
+getInstance.getLocation().getEndLine()
+<
+init.getLocation().getEndLine()
+We can express this query in NASRA as follows.
+getInstance precedes init.
+c) Method signature constraint: It is possible to express
+signature of a method using Pattern3.
+Pattern3 ::= MethodName’s signature is List.
+A method signature can be seen as an ordered list of data
+types. This list contains names of data types as strings, such
+that the first string is the name of the first argument’s data
+type and so on. For example, the following NASRA query
+states that getInstance method has two arguments and
+the names of their types are "int" and "Certificate",
+respectively.
+getInstance’s signature is ["int", "Certificate"].
+This query is translated to the following CodeQL query.
+(count (getInstance.getAnArgument()) = 2) and
+getInstance.getArgument(0).getType().
+toString()="int" and getInstance.
+getArgument(1).getType().toString()=
+"Certificate"
+First, the number of arguments is set to the size of the user
+defined list, then the type of arguments are constrained one
+by one. count (method.getAnArgument()) returns
+the number of arguments of the method. getArgument(i)
+returns
+the
+argument number i
+in
+the
+given
+method,
+getType() returns the type of the given argument, and
+finally toString() converts the given data type to a String.
+2) AuxiliaryStatement: We aim to find misuses in code
+that violate one or more mandatory constraints. For instance,
+suppose that if the second argument of init method is
+"private key" then it is mandatory that the encryption al-
+gorithm, i.e., the second argument of getInstance method
+is "RSA", and also if the encryption algorithm is "AES"
+then it is mandatory that the mode of encryption, i.e., the
+first argument of the getInstance method, is "CBC". The
+following NASRA query will find such violations.
+It is false that if the type of the second argument of init
+is "PrivateKey", then the algorithm of getInstance’s
+first argument is "RSA" or it is false that if the algorithm of
+getInstance’s first argument is "AES" then the mode of
+getInstance’s first argument is "CBC".
+In order to find any violation of these constraints, dis-
+junction of their negation has to be stated in the query.3
+3For example, in “X is driving in an urban area(Cond1). It is necessary that
+X is driving slower than 60 km/h (Cons1). It is necessary that X fastens the
+seat belt (Cons2).”, the query needs to find an X that is driving in an urban
+area and is driving faster than 60 km/h or is not using the seat belt. If we
+assign a Boolean variable to each statement as mentioned in the statements, it
+should aim Cond1 ∧(¬Cons1 ∨¬Cons2) whose necessity part is translated
+to the disjunction of negation of two constraints.
+Nevertheless, the above statement becomes much longer and
+harder to comprehend as the number of constraints increases.
+We define auxiliary statements to ease the formulation as
+well as the comprehension of complex queries for developers.
+Particularly, NecessityStatements are auxiliary statements that
+enable developers to enforce mandatory constraints in short
+and independent statements. It starts with “It is necessary that”
+and follows the syntax below.
+NecessityStatement
+::=
+It is necessary that Statement.
+Therefore, instead of writing disjunction of negation of all
+constraints in one single statement, developers can benefit
+this construct (i.e., NecessiyStatement) to define all such
+constraints in several statements within a query. Accordingly,
+the single but long previous statement can be stated as two
+separate statements shown below.
+It is necessary that if the type of the second argument of init is
+"PrivateKey", then the algorithm of getInstance’s first
+argument is "RSA".
+It is necessary that if the algorithm of getInstance’s first
+argument is "AES" then the mode of getInstance’s first
+argument is "CBC".
+Necessity statements are treated differently from other state-
+ments. If there is only one NecessityStatement, its enclosing
+statement is negated and added to the where part of the
+CodeQL query. If there are more than one, e.g., n constraints
+Cons1, Cons2, ..., Consn, then the “(not T Cons1 or
+not T Cons2 or ... or not T Consn)” will be added
+to the where part, where T Consi is the translation of Consi.
+III. WORKING EXAMPLES
+Cipher is one of the most misused APIs in Java cryp-
+tography [4]. Listing 1 shows how to create a Cipher
+object in Java. We should call the Cipher’s getInstance
+method. This method receives a number of arguments. The
+first one is transformation that is a string containing
+three parts separated by “/”. These parts are algorithm,
+mode, and padding, respectively. Next, we should call the
+init method on the cipher object with two arguments to
+indicate the operation mode of the cipher, and to initialize
+this object with a Key or Certificate.
+Cipher cipher = Cipher.getInstance("AES/ECB/
+PKCS5Padding");
+cipher.init(Cipher.ENCRYPT_MODE,new SecretKeySpec(
+keyBytes, "AES"));
+Listing 1. Setting up the Cipher object in Java
+In the rest of this section, we present three different program
+analysis tasks to ensure secure use of Java Cipher.
+A. Key vs. Algorithm
+Task 1: If the key has a type of PublicKey, PrivateKey,
+or Certificate, or encryption mode is WRAP MODE or UN-
+WRAP MODE, then algorithm of transformation must be
+“RSA”.
+
+Listing 2 shows how to check this constraint in CodeQL.
+from MethodAccess getInstance, MethodAccess init
+where init.getMethod().getName() = "init" and init.
+getReceiverType().getName() = "Cipher" and
+getInstance.getMethod().getName() = "getInstance
+" and getInstance.getReceiverType().getName() =
+"Cipher" and (((init.getArgument(0).toString() =
+"Cipher.WRAP MODE" or init.getArgument(0).
+toString() = "Cipher.UNWRAP MODE") or (init.
+getArgument(1).getType().toString() = "java.
+security.PublicKey" or init.getArgument(1).
+getType().toString() = "java.security.PrivateKey
+" or init.getArgument(1).toString() = "java.
+security.cert.Certificate")) and not(getInstance
+.getArgument(0).toString().replaceAll("\","").
+splitAt("/",0) = "RSA"))
+select getInstance, init
+Listing 2. Key vs. Algorithm constraint in CodeQL
+This constraint can be expressed in NASRA as follows.
+An
+object
+of
+Cipher
+invokes
+init.
+An
+object
+of
+Cipher invokes getInstance. It is necessary that if
+init’s
+first
+argument
+is
+in
+["Cipher.WRAP_MODE",
+"Cipher.UNWRAP_MODE"]
+or
+the
+type
+of
+the
+second
+argument of init is in ["PublicKey", "PrivateKey",
+"Certificate"] then the algorithm of getInstance’s
+first argument is "RSA".
+B. Algorithm vs. Transformation Mode
+Task 2: If the algorithm of transformation is “RSA” then
+the mode of transformation must be either “” or “ECB”.
+Listing 3 shows the corresponding query to check this
+constraint in CodeQL. We should look for code in which the
+algorithm is “RSA”, but neither “ECB” nor “” is set for the
+mode.
+from MethodAccess getInstance
+where getInstance.getMethod().getName() = "
+getInstance" and getInstance.getReceiverType().
+getName() = "Cipher" and (getInstance.
+getArgument(0).toString().replaceAll("\"","").
+splitAt("/", 0) = "RSA") and not (getInstance.
+getArgument(0).toString().replaceAll("\"","").
+splitAt("/", 1) = "" or getInstance.getArgument
+(0).toString().replaceAll("\"","").splitAt("/",
+1) = "ECB")
+select getInstance
+Listing 3. Algorithm vs. Transformation Mode constraint in CodeQL
+This constraint can be expressed in NASRA as follows.
+An object of Cipher invokes getInstance. It is necessary
+that if the algorithm of getInstance’s first argument is
+"RSA" then the mode of getInstance’s first argument is in
+["", "ECB"].
+Thanks to Attempto Parsing Engine (APE), NASRA state-
+ments do not need to exactly follow the syntax rules meaning
+that a degree of freedom in paraphrasing is possible. For
+instance, the part “the algorithm of getInstance’s first
+argument is "RSA"” can also be written in two other forms:
+(i) the algorithm of the first argument of getInstance is
+"RSA".
+(ii) "RSA" is the algorithm of getInstance’s first argument.
+C. Transformation and Encryption Mode vs. Signature
+Task 3: If the transformation mode is either of “CBC”,
+“PCBC”, “CTR”, “CTS”, “CFB”, or “OFB”, and the en-
+cryption mode is not “Cipher.ENCRYPT MODE”, then the
+invoked init method should not have any of the following
+signature: init(encmode, cert), init(encmode, cert, ranGen),
+init(encmode, key), init(encmode, key, ranGen).
+Listing 4 presents how to enforce this constraint in CodeQL.
+from MethodAccess getInstance, MethodAccess init
+where init.getMethod().getName() = "init" and init.
+getReceiverType().getName() = "Cipher" and
+getInstance.getMethod().getName() = "getInstance
+" and getInstance.getReceiverType().getName() =
+"Cipher" and ((getInstance.getArgument(0).
+toString().replaceAll("\"","").splitAt("/", 1) =
+"CBC" or getInstance.getArgument(0).toString().
+replaceAll("\"","").splitAt("/", 1) = "PCBC" or
+getInstance.getArgument(0).toString().replaceAll
+("\"","").splitAt("/", 1) = "CTR" and
+getInstance.getArgument(0).toString().replaceAll
+("\"","").splitAt("/", 1) = "CTS" or getInstance
+.getArgument(0).toString().replaceAll("\"","").
+splitAt("/", 1) = "CFB" or getInstance.
+getArgument(0).toString().replaceAll("\"","").
+splitAt("/", 1) = "OFB") and not (init.
+getArgument(0).toString() = "Cipher.ENCRYPT_MODE
+")) and ((count (getInstance.getAnArgument()) =
+2 and getInstance.getArgument(0).getType().
+toString() = "int" and getInstance.getArgument
+(1).getType().toString() = "Certificate") or (
+count (getInstance.getAnArgument()) = 3 and
+getInstance.getArgument(0).getType().toString()
+= "int" and getInstance.getArgument(1).getType()
+.toString() = "Certificate" and getInstance.
+getArgument(2).getType().toString() = "
+SecureRandom") or (count (getInstance.
+getAnArgument()) = 2 and getInstance.getArgument
+(0).getType().toString() = "int" and getInstance
+.getArgument(1).getType().toString() = "Key") or
+(count (getInstance.getAnArgument()) = 3 and
+getInstance.getArgument(0).getType().toString()
+= "int" and getInstance.getArgument(1).getType()
+.toString() = "Key" and getInstance.getArgument
+(2).getType().toString() = "SecureRandom"))
+select init, getInstance
+Listing 4. Transformation and Encryption mode vs. Signature constraint in
+CodeQL
+The implementation of this task in NASRA is shown below.
+An
+object
+of
+Cipher
+invokes
+getInstance.
+An
+object
+of
+Cipher
+invokes
+init.
+It
+is
+necessary
+that
+if
+the
+mode
+of
+getInstance’s
+first
+argument
+is
+in
+["CBC","PCBC","CTR","CTS","CFB","OFB"]
+and
+init’s first argument is not "Cipher.ENCRYPT_MODE"
+then
+getInstance’s
+signature
+is
+not
+["int","Certificate"]
+and
+is
+not
+["int","Certificate","SecureRandom"]
+and
+is
+not
+["int","Key"]
+and
+is
+not
+["int","Key","SecureRandom"].
+D. Discussion
+Table I presents the number of distinct operators and
+operands (i.e., vocabulary), and the total number of operators
+and operands (i.e., length) needed for each analysis task.4
+4In NASRA, we consider user defined terminals such as init, “RSA”, and
+getInstance as operands and count the rest of language constructs as operators.
+
+TABLE I
+CODEQL VS. NASRA
+Analysis Task
+Vocabulary
+Length
+CodeQL
+NASRA
+CodeQL
+NASRA
+Key vs. Algorithm (III-A)
+32
+19
+179
+39
+Algorithm vs. Mode (III-B)
+27
+18
+107
+24
+Mode vs. Signature (III-C)
+42
+26
+434
+56
+Evidently, queries in NASRA are significantly shorter than
+queries in CodeQL (i.e., up to 87% reduction in length), and
+they consume a lot fewer programming constructs (i.e., up to
+38% fewer vocabularies). We computed Halstead complexity
+measures to estimate the coding time and the difficulty to
+write or understand these queries [5]. The results showed that
+developers require a lot less effort and time to develop these
+tasks in NASRA than in CodeQL.
+We also asked ten developers to share their opinion about
+queries in NASRA. They unanimously stated that they are
+succinct and easy to understand, and one commented that
+“these queries read like API documentation”.
+It is noteworthy that NASRA’s performance, i.e., how well
+it can detect API misuses, depends on its underlying analy-
+sis framework which is currently CodeQL. In other words,
+NASRA obviates the low-level details needed to define static
+program analyses, but the issues with false positives remain
+to be relevant. Moreover, despite being natural, the use of
+NASRA still requires knowledge of its syntax.
+IV. RELATED WORK
+Mapping a natural language statement into a formal repre-
+sentation has received great attention in the community but
+not much in the program analysis development domain.
+Schlegel
+et
+al.
+developed
+an
+end-user
+programming
+paradigm for Python, that maps natural language commands
+into Python code [6]. Landhauber et al. proposed a domain
+agnostic command interpreter that receives natural language
+commands in English and uses ontology to produce relevant
+API calls [7]. Yaghmazadeh et al. developed SQLIZER, a
+system to automatically synthesize SQL queries from a natural
+language [8]. Luo et al. investigated the translation from
+a natural language query to visualization with the goal of
+simplifying the creation of data visualizations [9].
+Heyman et al. developed a Python code completion tool
+that enriches developers’ code with the natural language de-
+scription of the intended data science task [10]. Nguyen et al.
+presented an approach that takes as input an English descrip-
+tion of a programming task and synthesizes the corresponding
+API code template for the task [11]. Desai et al. built a general
+framework for constructing program synthesizers that take
+natural language inputs and produce expressions in a target
+Domain Specific Language [12]. Zhai et al. proposed a search-
+based technique to automatically translate NL comments to
+formal program specifications that specify the expected pre
+and post conditions [13].
+The work presented in this paper is also related to cryp-
+tography domain. There exist tools that find cryptography
+misuses [14] and libraries that facilitate the adoption of
+cryptography for developers [15]. Nevertheless, none of them
+employed a natural language approach.
+V. CONCLUSION
+We introduced NASRA, an open-source framework to de-
+fine static program analyses in natural language. We demon-
+strated the application of this framework to find misuses in
+Java cryptography. The ultimate goal of NASRA is to enable
+a naturalistic way to develop static program analyses, which is
+usable for mainstream developers. To realize this goal, further
+studies are needed to determine NASRA’s effectiveness in
+real-world settings. The expressiveness of its queries and the
+effort required to extend it to other problem domains have to
+be investigated as well. Finally, automatic translation without
+pre-defined rules is also an exciting future research direction.
+REFERENCES
+[1] E. Barakova, J. Gillesen, B. Huskens, and T. Lourens, “End-user
+programming architecture facilitates the uptake of robots in social
+therapies,” Robotics and Autonomous Systems, vol. 61, no. 7, 2013.
+[2] X. V. Lin, C. Wang, L. Zettlemoyer, and M. D. Ernst, “NL2Bash:
+A corpus and semantic parser for natural language interface to the
+linux operating system,” in Proceedings of the Eleventh International
+Conference on Language Resources and Evaluation (LREC 2018), 2018.
+[3] M. Hazhirpasand, O. Nierstrasz, M. Shabani, and M. Ghafari, “Hurdles
+for developers in cryptography,” in 2021 IEEE International Conference
+on Software Maintenance and Evolution (ICSME), 2021, pp. 659–663.
+[4] M. Hazhirpasand, M. Ghafari, and O. Nierstrasz, “Java cryptography
+uses in the wild,” in Proceedings of the 14th ACM / IEEE International
+Symposium on Empirical Software Engineering and Measurement, 2020.
+[5] M. H. Halstead, Elements of Software Science (Operating and program-
+ming systems series).
+Elsevier Science Inc., 1977.
+[6] V. Schlegel, B. Lang, S. Handschuh, and A. Freitas, “Vajra: Step-by-
+step programming with natural language,” in Proceedings of the 24th
+International Conference on Intelligent User Interfaces, 2019.
+[7] M. Landh¨auber, S. Weigelt, and W. F. Tichy, “Nlci: A natural language
+command interpreter,” Automated Software Engg., vol. 24, no. 4, p.
+839–861, dec 2017.
+[8] N. Yaghmazadeh, Y. Wang, I. Dillig, and T. Dillig, “Sqlizer: Query
+synthesis from natural language,” Proc. ACM Program. Lang., vol. 1,
+no. OOPSLA, oct 2017.
+[9] Y. Luo, N. Tang, G. Li, J. Tang, C. Chai, and X. Qin, “Natural language
+to visualization by neural machine translation,” IEEE Transactions on
+Visualization and Computer Graphics, vol. 28, no. 1, pp. 217–226, 2022.
+[10] G. Heyman, R. Huysegems, P. Justen, and T. Van Cutsem, “Natural
+language-guided programming,” ser. Onward!, 2021, p. 39–55.
+[11] A. T. Nguyen, P. C. Rigby, T. Nguyen, D. Palani, M. Karanfil, and T. N.
+Nguyen, “Statistical translation of english texts to api code templates,”
+in 2018 IEEE International Conference on Software Maintenance and
+Evolution (ICSME), 2018, pp. 194–205.
+[12] A. Desai, S. Gulwani, V. Hingorani, N. Jain, A. Karkare, M. Marron,
+S. R, and S. Roy, “Program synthesis using natural language,” in Pro-
+ceedings of the 38th International Conference on Software Engineering,
+ser. ICSE ’16, 2016.
+[13] J. Zhai, Y. Shi, M. Pan, G. Zhou, Y. Liu, C. Fang, S. Ma, L. Tan,
+and X. Zhang, “C2s: Translating natural language comments to formal
+program specifications,” in Proceedings of the 28th ACM Joint Meeting
+on European Software Engineering Conference and Symposium on the
+Foundations of Software Engineering, ser. ESEC/FSE 2020, 2020.
+[14] Y. Zhang, M. M. A. Kabir, Y. Xiao, D. D. Yao, and N. Meng, “Automatic
+detection of java cryptographic api misuses: Are we there yet,” IEEE
+Transactions on Software Engineering, 2022.
+[15] S. Kafader and M. Ghafari, “Fluentcrypto: Cryptography in easy mode,”
+in 2021 IEEE International Conference on Software Maintenance and
+Evolution (ICSME), 2021, pp. 402–412.
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf,len=510
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='04862v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='PL]  12 Jan 2023  Naturalistic Static Program Analysis  Mohammad Mehdi Pourhashem Kallehbasti  Department of Electrical and Computer Engineering  University of Science and Technology of Mazandaran  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Box 48518-78195, Behshahr, Iran  pourhashem@mazust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='ir  Mohammad Ghafari  TU Clausthal, Germany  mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='ghafari@tu-clausthal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='de  Abstract—Static program analysis development is a non-trivial  and time-consuming task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We present a framework through  which developers can define static program analyses in natural  language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We show the application of this framework to identify  cryptography misuses in Java programs, and we discuss how it  facilitates static program analysis development for developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Index Terms—Static program analysis, cryptography, natural  language programming  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' INTRODUCTION  Static program analysis is the art of examining programs  without requiring to execute the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' However, static analysis  tools generate false positives and tuning them requires exper-  tise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Likewise, program analysis development requires a deep  knowledge of compiler or mastering an analysis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  End-user programming is a set of techniques that enable  end users to write programs at a level of complexity that is  adequate to their practices, background, and skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For in-  stance, it includes visual languages to program robots through  visual blocks [1], and simplified programming languages to  translate English sentences to Bash commands [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We believe  that end-user programming techniques can also help to hide the  complexity of writing a static program analysis task for non-  professional programmers and empower them in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  We introduce NASRA (NAturalistic Static pRogram Analy-  sis), a framework that enables developers to define a program  analysis task in natural language (NL), and it generates the  corresponding Query Language (QL) query that underlies  CodeQL program analysis engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='1 We illustrate the appli-  cation of this framework to find cryptography misuses in Java  programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' NASRA is open source and publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='2  The ultimate goal of NASRA is to enable “naturalistic”  static program analysis development in a way that developers  can specify what they need without deep knowledge of static  program analysis and how a specific framework works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Its  higher level of abstraction than existing static analysis frame-  works may facilitate a more intuitive formulation of program  analysis tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Similarly, its agnostic nature to programming  languages can provide a cross-language interface for program  analysis, which obviates the need to learn the specifics of a  program analysis framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' This paper presents a prelimi-  nary step to realize the above goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  1https://codeql.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='com  2https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='7495044  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' THE NASRA FRAMEWORK  Cryptography is an essential component to security, but it is  one of the notorious topics where developers struggle a lot [3],  [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Locating the init method invoked on a Cipher object  is often deemed to be the first step to analyze cryptography  code in Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For instance, in CodeQL, one should  write the following query to implement this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  from  MethodAccess init  where init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getMethod().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "init" and  init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getReceiverType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "Cipher"  select init  We have developed a framework, called NASRA, that  enables a more intuitive formulation of the above task in the  form below:  An object of Cipher invokes init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  NASRA is a rule-driven synthesizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We rely on predefined  rules due to a lack of trustworthy labeled examples required  for a data-driven approach in this domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' NASRA receives a  program analysis inquiry in natural language, applies semantic  parsing, and generates CodeQL commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The input inquiry  should comply with a subset of the syntax of Attempto  Controlled English (ACE) controlled natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We  use Attempto Parsing Engine (APE), a tool that receives a  series of ACE statements and produces the corresponding  Discourse Representation Structures (DRS) that is a semantic  representation of the input text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' NASRA applies the translation  rules, explained later in this section, on the given DRS and  produces the corresponding CodeQL statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Thanks to  APE, the way one can formulate NASRA statements is very  flexible and there is no need for absolute correspondence with  the NASRA syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We chose CodeQL as our code analysis  engine because it is an industry-leading and community-  powered tool, and its publicly available to all GitHub users  without any installation hassle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' To employ NASRA for a new  static analysis framework, only the transformation rules have  to be adapted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' To support a new application domain, we should  identify the types of queries that the current syntax does not  support, add the corresponding production rules to the syntax,  and develop translations for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' NASRA is open source,  and currently, supports program analysis tasks that concern  cryptography misuses in Java programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Syntax and Semantics  Each NASRA query comprises one or more Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The  syntax is shown below (terminals have different color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Query ::= Statement Query | Statement  Statement ::= BasicStatement | LogicalStatement  | Extension  BasicStatement ::= Exp is (Exp | in List)  Exp ::= Prefix Exp | type | ID | Literal  Prefix ::= ((adjective|ε) attribute of)  LogicalStatement ::= Statement and Statement |  Statement or Statement | It is false that Statement  | If Statement then Statement  a) Expression: The smallest building block is Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' It  includes a Literal (String or int) or an ID (user defined  identifier) that are directly mapped to CodeQL expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  An Exp can also be a CodeQL type such as class, variable,  and method access that are mapped to Class, Variable,  and MethodAccess, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  b) Prefix: Each Exp can have an optional Prefix in  the form of “attribute of” that indicates an attribute of  the expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For instance, name, type, argument, and  method are attributes of an entity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', Exp), and they cor-  respond to getName(), getType(), getArgument(),  and getMethod() methods in CodeQL, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  For example, “name of method1” is an Exp, where “name”  is an attribute and method1 is an ID, and the whole expres-  sion is translated to method1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() in CodeQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Additionally, the attribute itself can have an optional  ordinal  number  as  an  adjective,  like  second  in  the  Exp “second argument of init” that is translated to  “init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(1)”, where second is translated to  1 as an argument according to zero-based numbering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Note that “attribute of” can be repeated several times, where  each attribute may have an adjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For example, the Exp  “The type of the second argument of init” has one ID  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=',init) and two attributes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', type and argument).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  c) Basic Statement: Each BasicStatement is a statement  that can serve as a Boolean condition as well as an assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  As a Boolean condition, BasicStatement produces equiva-  lence of two Exps, as well as membership of an Exp in a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  In “Exp is Exp” structure, both sides of equivalence are Exps  and they need to be equal, while in “Exp is in List” structure,  the Exp needs to be equal to an item in a list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Accordingly, a  statement like “arg1 is in ["RSA", "AES"].” is a disjunctive  expression and can be rephrased to “arg1 is "RSA" or arg1  is "AES".”, that is ultimately translated to “arg1 = "RSA"  or arg1 = "AES"”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  The syntax structure Exp is Exp can also produce as-  sumptions when the second Exp is a CodeQL type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The  assumptions are mapped to the from part of a CodeQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  For instance, the statement “var1 is a variable.” translates to  “Variable var1” and belongs to the from part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  d) Logical Statement: A LogicalStatement can be a  negation, conjunction, disjunction, or implication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For exam-  ple, “If arg1 is "RSA" then arg2 is "AES".” is translated to  “not (arg1 = "RSA") or arg2 = "AES"” in Cod-  eQL, since p ⇒ q is equivalent to ¬p ∨ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Extensibility  One can extend NASRA to cover auxiliary statements and  statement patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Their corresponding production rules are  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Extension ::= Pattern | AuxiliaryStatement  We introduce these features through three statement pat-  terns and one auxiliary statement that are helpful to cover  constraints on using Java cryptography objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  1) Patterns: We present three patterns that extend Pattern  nonterminal in the syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We discuss each in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  a) Invocation: We use this pattern to state that a method  is invoked by an instance of a specific class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' It can also be  used to make sure that there is no invocation of a method by  any instance of a specific class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Pattern1 ::= An object of ID (invokes|does not invoke) ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  The NASRA query shown in Section II is an example of this  pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The transformation follows a number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' First, a  MethodAccess is declared with the same name used in the  NASRA statement (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=',init).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Then the conditions need to be  added to the where part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Specifically, the name of the method  of the MethodAccess init should be "init" that is  stated in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Finally, a MethodAccess has a  receiver, that is the object invoking its method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' In this case,  the name of the type of the receiver should be "Cipher",  that is expressed in CodeQL in the third line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  If one needs to make sure that no invocation occurs, an  existential quantifier must be used, as shown in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  from  where not  (exists  (MethodAccess  init  |  init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getMethod().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "init" and  init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getReceiverType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "Cipher"))  It means that there is no such MethodAccess init that  has these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We can state this in NASRA in the form  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  An object of Cipher doesn’t invoke init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  b) Partial order constraints: This pattern enables one to  put partial order constraints on method invocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' In other  words, one can enforce a method invocation to be preceded  (or followed) by another method invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Pattern2::=MethodName (precedes|follows) MethodName.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  For example, there are two steps in CodeQL for stating that  “invocation of getInstance is earlier than invocation of  init”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' First, one should specify that both methods are in the  same scope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Next, the line number of the preceding method  invocation has to be smaller than the line number of the other  method invocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' This is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getEnclosingCallable()  =  init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getEnclosingCallable() and  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getLocation().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getEndLine()  <  init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getLocation().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getEndLine()  We can express this query in NASRA as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getInstance precedes init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  c) Method signature constraint: It is possible to express  signature of a method using Pattern3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Pattern3 ::= MethodName’s signature is List.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  A method signature can be seen as an ordered list of data  types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' This list contains names of data types as strings, such  that the first string is the name of the first argument’s data  type and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For example, the following NASRA query  states that getInstance method has two arguments and  the names of their types are "int" and "Certificate",  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getInstance’s signature is ["int", "Certificate"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  This query is translated to the following CodeQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  (count (getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getAnArgument()) = 2) and  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  toString()="int" and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getArgument(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString()=  "Certificate"  First, the number of arguments is set to the size of the user  defined list, then the type of arguments are constrained one  by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' count (method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getAnArgument()) returns  the number of arguments of the method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' getArgument(i)  returns  the  argument number i  in  the  given  method,  getType() returns the type of the given argument, and  finally toString() converts the given data type to a String.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  2) AuxiliaryStatement: We aim to find misuses in code  that violate one or more mandatory constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For instance,  suppose that if the second argument of init method is  "private key" then it is mandatory that the encryption al-  gorithm, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', the second argument of getInstance method  is "RSA", and also if the encryption algorithm is "AES"  then it is mandatory that the mode of encryption, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', the  first argument of the getInstance method, is "CBC".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The  following NASRA query will find such violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  It is false that if the type of the second argument of init  is "PrivateKey", then the algorithm of getInstance’s  first argument is "RSA" or it is false that if the algorithm of  getInstance’s first argument is "AES" then the mode of  getInstance’s first argument is "CBC".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  In order to find any violation of these constraints, dis-  junction of their negation has to be stated in the query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='3  3For example, in “X is driving in an urban area(Cond1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' It is necessary that  X is driving slower than 60 km/h (Cons1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' It is necessary that X fastens the  seat belt (Cons2).”, the query needs to find an X that is driving in an urban  area and is driving faster than 60 km/h or is not using the seat belt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' If we  assign a Boolean variable to each statement as mentioned in the statements, it  should aim Cond1 ∧(¬Cons1 ∨¬Cons2) whose necessity part is translated  to the disjunction of negation of two constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Nevertheless, the above statement becomes much longer and  harder to comprehend as the number of constraints increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  We define auxiliary statements to ease the formulation as  well as the comprehension of complex queries for developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Particularly, NecessityStatements are auxiliary statements that  enable developers to enforce mandatory constraints in short  and independent statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' It starts with “It is necessary that”  and follows the syntax below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  NecessityStatement  ::=  It is necessary that Statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Therefore, instead of writing disjunction of negation of all  constraints in one single statement, developers can benefit  this construct (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', NecessiyStatement) to define all such  constraints in several statements within a query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Accordingly,  the single but long previous statement can be stated as two  separate statements shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  It is necessary that if the type of the second argument of init is  "PrivateKey", then the algorithm of getInstance’s first  argument is "RSA".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  It is necessary that if the algorithm of getInstance’s first  argument is "AES" then the mode of getInstance’s first  argument is "CBC".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Necessity statements are treated differently from other state-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' If there is only one NecessityStatement, its enclosing  statement is negated and added to the where part of the  CodeQL query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' If there are more than one, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', n constraints  Cons1, Cons2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', Consn, then the “(not T Cons1 or  not T Cons2 or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' or not T Consn)” will be added  to the where part, where T Consi is the translation of Consi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' WORKING EXAMPLES  Cipher is one of the most misused APIs in Java cryp-  tography [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Listing 1 shows how to create a Cipher  object in Java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We should call the Cipher’s getInstance  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' This method receives a number of arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The  first one is transformation that is a string containing  three parts separated by “/”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' These parts are algorithm,  mode, and padding, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Next, we should call the  init method on the cipher object with two arguments to  indicate the operation mode of the cipher, and to initialize  this object with a Key or Certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Cipher cipher = Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getInstance("AES/ECB/  PKCS5Padding");' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='init(Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='ENCRYPT_MODE,new SecretKeySpec(  keyBytes, "AES"));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Listing 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Setting up the Cipher object in Java  In the rest of this section, we present three different program  analysis tasks to ensure secure use of Java Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Key vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Algorithm  Task 1: If the key has a type of PublicKey, PrivateKey,  or Certificate, or encryption mode is WRAP MODE or UN-  WRAP MODE, then algorithm of transformation must be  “RSA”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Listing 2 shows how to check this constraint in CodeQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  from MethodAccess getInstance, MethodAccess init  where init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getMethod().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "init" and init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getReceiverType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "Cipher" and  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getMethod().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "getInstance  " and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getReceiverType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() =  "Cipher" and (((init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() =  "Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='WRAP MODE" or init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  toString() = "Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='UNWRAP MODE") or (init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getArgument(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='PublicKey" or init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='PrivateKey  " or init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "java.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='cert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='Certificate")) and not(getInstance  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll("\\","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  splitAt("/",0) = "RSA"))  select getInstance, init  Listing 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Key vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Algorithm constraint in CodeQL  This constraint can be expressed in NASRA as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  An  object  of  Cipher  invokes  init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  An  object  of  Cipher invokes getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' It is necessary that if  init’s  first  argument  is  in  ["Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='WRAP_MODE",  "Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='UNWRAP_MODE"]  or  the  type  of  the  second  argument of init is in ["PublicKey", "PrivateKey",  "Certificate"] then the algorithm of getInstance’s  first argument is "RSA".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Algorithm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Transformation Mode  Task 2: If the algorithm of transformation is “RSA” then  the mode of transformation must be either “” or “ECB”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Listing 3 shows the corresponding query to check this  constraint in CodeQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We should look for code in which the  algorithm is “RSA”, but neither “ECB” nor “” is set for the  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  from MethodAccess getInstance  where getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getMethod().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "  getInstance" and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getReceiverType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getName() = "Cipher" and (getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  splitAt("/", 0) = "RSA") and not (getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  splitAt("/", 1) = "" or getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument  (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='splitAt("/",  1) = "ECB")  select getInstance  Listing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Algorithm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Transformation Mode constraint in CodeQL  This constraint can be expressed in NASRA as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  An object of Cipher invokes getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' It is necessary  that if the algorithm of getInstance’s first argument is  "RSA" then the mode of getInstance’s first argument is in  ["", "ECB"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Thanks to Attempto Parsing Engine (APE), NASRA state-  ments do not need to exactly follow the syntax rules meaning  that a degree of freedom in paraphrasing is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' For  instance, the part “the algorithm of getInstance’s first  argument is "RSA"” can also be written in two other forms:  (i) the algorithm of the first argument of getInstance is  "RSA".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  (ii) "RSA" is the algorithm of getInstance’s first argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Transformation and Encryption Mode vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Signature  Task 3: If the transformation mode is either of “CBC”,  “PCBC”, “CTR”, “CTS”, “CFB”, or “OFB”, and the en-  cryption mode is not “Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='ENCRYPT MODE”, then the  invoked init method should not have any of the following  signature: init(encmode, cert), init(encmode, cert, ranGen),  init(encmode, key), init(encmode, key, ranGen).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Listing 4 presents how to enforce this constraint in CodeQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  from MethodAccess getInstance, MethodAccess init  where init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getMethod().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "init" and init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getReceiverType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "Cipher" and  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getMethod().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() = "getInstance  " and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getReceiverType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getName() =  "Cipher" and ((getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='splitAt("/", 1) =  "CBC" or getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  replaceAll("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='splitAt("/", 1) = "PCBC" or  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll  ("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='splitAt("/", 1) = "CTR" and  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll  ("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='splitAt("/", 1) = "CTS" or getInstance  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  splitAt("/", 1) = "CFB" or getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='replaceAll("\\"","").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  splitAt("/", 1) = "OFB") and not (init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='ENCRYPT_MODE  ")) and ((count (getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getAnArgument()) =  2 and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  toString() = "int" and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument  (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "Certificate") or (  count (getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getAnArgument()) = 3 and  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString()  = "int" and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType()  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "Certificate" and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getArgument(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "  SecureRandom") or (count (getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  getAnArgument()) = 2 and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument  (0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "int" and getInstance  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "Key") or  (count (getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getAnArgument()) = 3 and  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString()  = "int" and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType()  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "Key" and getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getArgument  (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='getType().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='toString() = "SecureRandom"))  select init, getInstance  Listing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Transformation and Encryption mode vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Signature constraint in  CodeQL  The implementation of this task in NASRA is shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  An  object  of  Cipher  invokes  getInstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  An  object  of  Cipher  invokes  init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  It  is  necessary  that  if  the  mode  of  getInstance’s  first  argument  is  in  ["CBC","PCBC","CTR","CTS","CFB","OFB"]  and  init’s first argument is not "Cipher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='ENCRYPT_MODE"  then  getInstance’s  signature  is  not  ["int","Certificate"]  and  is  not  ["int","Certificate","SecureRandom"]  and  is  not  ["int","Key"]  and  is  not  ["int","Key","SecureRandom"].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Discussion  Table I presents the number of distinct operators and  operands (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', vocabulary), and the total number of operators  and operands (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', length) needed for each analysis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='4  4In NASRA, we consider user defined terminals such as init, “RSA”, and  getInstance as operands and count the rest of language constructs as operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  TABLE I  CODEQL VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' NASRA  Analysis Task  Vocabulary  Length  CodeQL  NASRA  CodeQL  NASRA  Key vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Algorithm (III-A)  32  19  179  39  Algorithm vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Mode (III-B)  27  18  107  24  Mode vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Signature (III-C)  42  26  434  56  Evidently, queries in NASRA are significantly shorter than  queries in CodeQL (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', up to 87% reduction in length), and  they consume a lot fewer programming constructs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', up to  38% fewer vocabularies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We computed Halstead complexity  measures to estimate the coding time and the difficulty to  write or understand these queries [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The results showed that  developers require a lot less effort and time to develop these  tasks in NASRA than in CodeQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  We also asked ten developers to share their opinion about  queries in NASRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' They unanimously stated that they are  succinct and easy to understand, and one commented that  “these queries read like API documentation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  It is noteworthy that NASRA’s performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=', how well  it can detect API misuses, depends on its underlying analy-  sis framework which is currently CodeQL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' In other words,  NASRA obviates the low-level details needed to define static  program analyses, but the issues with false positives remain  to be relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Moreover, despite being natural, the use of  NASRA still requires knowledge of its syntax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' RELATED WORK  Mapping a natural language statement into a formal repre-  sentation has received great attention in the community but  not much in the program analysis development domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Schlegel  et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  developed  an  end-user  programming  paradigm for Python, that maps natural language commands  into Python code [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Landhauber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' proposed a domain  agnostic command interpreter that receives natural language  commands in English and uses ontology to produce relevant  API calls [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Yaghmazadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' developed SQLIZER, a  system to automatically synthesize SQL queries from a natural  language [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' investigated the translation from  a natural language query to visualization with the goal of  simplifying the creation of data visualizations [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  Heyman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' developed a Python code completion tool  that enriches developers’ code with the natural language de-  scription of the intended data science task [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  presented an approach that takes as input an English descrip-  tion of a programming task and synthesizes the corresponding  API code template for the task [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Desai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' built a general  framework for constructing program synthesizers that take  natural language inputs and produce expressions in a target  Domain Specific Language [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Zhai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' proposed a search-  based technique to automatically translate NL comments to  formal program specifications that specify the expected pre  and post conditions [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  The work presented in this paper is also related to cryp-  tography domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' There exist tools that find cryptography  misuses [14] and libraries that facilitate the adoption of  cryptography for developers [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Nevertheless, none of them  employed a natural language approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' CONCLUSION  We introduced NASRA, an open-source framework to de-  fine static program analyses in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' We demon-  strated the application of this framework to find misuses in  Java cryptography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The ultimate goal of NASRA is to enable  a naturalistic way to develop static program analyses, which is  usable for mainstream developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' To realize this goal, further  studies are needed to determine NASRA’s effectiveness in  real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' The expressiveness of its queries and the  effort required to extend it to other problem domains have to  be investigated as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
+page_content=' Finally, automatic translation without  pre-defined rules is also an exciting future research direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE4T4oBgHgl3EQfCwsd/content/2301.04862v1.pdf'}
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+arXiv:2301.13075v1  [quant-ph]  30 Jan 2023
+Threshold theorem in quantum annealing with deterministic analog control errors
+Manaka Okuyama1 and Masayuki Ohzeki1,2,3
+1Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan
+2Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo,152-8551, Japan and
+3Sigma-i Co., Ltd., Tokyo 108-0075, Japan
+(Dated: January 31, 2023)
+We investigate the effect of deterministic analog control errors in the time-dependent Hamiltonian on iso-
+lated quantum dynamics. Deterministic analog control errors are formulated as time-dependent operators in the
+Schr¨odinger equation. We give an upper bound on the distance between two states in time evolution with and
+without deterministic analog control errors. As a result, we prove that, if the strength of deterministic analog
+control errors is less than the inverse of computational time, the final state in quantum dynamics without deter-
+ministic analog control errors can be obtained through a constant-order number of measurements in quantum
+dynamics with deterministic analog control errors.
+I.
+INTRODUCTION
+Quantum annealing [1–8] is an analog quantum computa-
+tion that utilizes continuous time evolution of quantum sys-
+tems, and, thereby, analog control errors of the parameters
+are inevitable in experimental systems. Because the theory
+of quantum error correction and suppression is incomplete
+in quantum annealing [9–13], estimating the effect of analog
+control errors is one of the most critical problems.
+There are two main types of analog control errors in quan-
+tum annealing.
+One is a stochastic control error [14–17],
+which represents an instantaneous parameter fluctuation. For
+this type of control error, recent studies [18, 19] proved that,
+if the strength of the stochastic control errors is less than the
+inverse of the computation time, information about the final
+state in quantum dynamics without analog control errors can
+be recovered from quantum dynamics with stochastic control
+errors. The other is deterministic control error, which is, for
+example, a bias acting on the magnetic field or a deviation in
+the value of the interaction. Deterministic control errors have
+been discussed so far in many literatures [20–26], but they are
+limited to specific problems.
+The present study investigates in general whether it is pos-
+sible to recover information about the target state, which is
+the final state in ideal time evolution, from quantum dynam-
+ics with deterministic analog control errors. We give an upper
+bound on the distance between two states in quantum dynam-
+ics with and without deterministic control errors using only in-
+formation about the deterministic control errors. Furthermore,
+using this bound, we prove that, if the strength of the deter-
+ministic control errors is less than the inverse of the computa-
+tion time, information about the target state can be recovered
+through a constant-order number of measurements in quan-
+tum dynamics with deterministic analog control errors. This
+result is intuitively obvious but it is important from the per-
+spective of experimental systems to give mathematical proof.
+The proof is based on the method proposed by Kieu to derive
+a quantum speed limit [27, 28].
+The organization of this paper is as follows. In Sec. II,
+we define the model and obtain the main result. Finally, our
+conclusion is presented in Sec. III.
+II.
+RESULT
+We consider the following isolated quantum dynamics:
+i d
+dt|ψ(t)⟩ = ˆH(t)|ψ(t)⟩,
+(1)
+where 0 ≤ t ≤ T and ℏ = 1. In general, it is difficult to com-
+pletely control the time-dependent Hamiltonian ˆH(t) without
+control errors in experimental systems. Deterministic ana-
+log control errors can take any form physically permissible
+but should also be described as a Hermitian operator since
+we consider isolated quantum dynamics. Thus, we incorpo-
+rate the deterministic analog control errors of ˆH(t) into the
+Schr¨odinger equation as a Hermitian operator ˆV(t). We ex-
+press the Schr¨odinger equation with deterministic analog con-
+trol errors as follows:
+i d
+dt|φ(t)⟩ = ( ˆH(t) + ˆV(t))|φ(t)⟩.
+(2)
+Then, we obtain the following result.
+Theorem 1. The distance between two final states |ψ(T)⟩ and
+|φ(T)⟩ is bounded from above by
+∥ |ψ(T)⟩ − |φ(T)⟩ ∥ ≤ v,
+(3)
+where ∥ |a⟩ ∥ ≡ √⟨a|a⟩, v ≡
+� T
+0 dt
+��� ˆV(t)
+���, and
+��� ˆA
+��� is the eigen-
+value of ˆA with the largest absolute value.
+Proof of Theorem 1. From Eqs. (1) and (2), we obtain
+d
+dt(|ψ(t)⟩ − |φ(t)⟩) = −i ˆH(t)(|ψ(t)⟩ − |φ(t)⟩) + i ˆV(t) |φ(t)⟩ ,(4)
+and
+d
+dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥2 = 2 Re
+�
+(⟨ψ(t)| − ⟨φ(t)|) d
+dt(|ψ(t)⟩ − |φ(t)⟩)
+�
+= 2 Re
+�
+(⟨ψ(t)| − ⟨φ(t)|)i ˆV(t) |φ(t)⟩
+�
+≤ 2∥ |ψ(t)⟩ − |φ(t)⟩ ∥ · ∥ ˆV(t) |φ(t)⟩ ∥,
+(5)
+where we used the Cauchy-Schwartz inequality. On the other
+hand, we find
+d
+dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥2 = 2∥ |ψ(t)⟩ − |φ(t)⟩ ∥ · d
+dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥.
+(6)
+
+2
+Thus, we obtain
+d
+dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥ ≤ ∥ ˆV(t) |φ(t)⟩ ∥ ≤ ∥ ˆV(t)∥.
+(7)
+Finally, by integrating both sides from 0 to T, we arrive at Eq.
+(3).
+□
+It is worth mentioning that the right hand side of Eq. (3)
+contains only information about the control errors ˆV and not
+about ˆH(t).
+The inequality (3) makes sense only if v < 2 is satisfied
+because
+∥ |ψ(T)⟩ − |φ(T)⟩ ∥ =
+�
+2 − 2 Re ⟨ψ(t)|φ(t)⟩ ≤ 2.
+(8)
+In particular, when the strength of deterministic control errors
+is less than the inverse of the computation time,
+��� ˆV(t)
+��� <
+√
+2
+T ,
+(9)
+we have
+∥ |ψ(T)⟩ − |φ(T)⟩ ∥ ≤ v <
+√
+2.
+(10)
+This means that the two final states have non-zero overlap
+Re ⟨ψ(T)|φ(T)⟩ ≥ 1 − v2
+2 > 0.
+(11)
+Then, it is possible to recover the information about |ψ(T)⟩
+from |φ(T)⟩.
+For example, we expand the two final states |ψ(T)⟩ and
+|φ(T)⟩ as
+|ψ(T)⟩ =
+�
+n
+Cn |n⟩ ,
+(12)
+|φ(T)⟩ =
+�
+n
+Dn |n⟩ ,
+(13)
+where |n⟩ is the measurement basis. We are interested in the
+mth eigenstate of the measurement basis and its probability
+amplitude Cm is given by
+|Cm|2 = 1 − ǫ2,
+(14)
+with 0 ≤ ǫ < 1. Then, we arrive at:
+Corollary 2. If
+1 − v2/2 > ǫ ≥ 0,
+(15)
+then the probability amplitude of the mth eigenstate in the
+Schr¨odinger equation with deterministic analog control errors
+(2) takes a non-zero value,
+|Dm| ≥ 1 − v2
+2 − ǫ
+√
+1 − ǫ2 > 0.
+(16)
+Corollary 2 states that the number of measurements re-
+quired to obtain |m⟩ is independent of the computation time
+T in quantum dynamics with deterministic analog control er-
+rors (2). Thus, under the condition (15), deterministic control
+errors do not seriously affect the efficiency of quantum anneal-
+ing.
+The condition (15) can be rewritten as
+� T
+0
+dt
+��� ˆV(t)
+��� <
+�
+2(1 − ǫ).
+(17)
+It may seem difficult to satisfy this condition for large T.
+However, when T is large, the parameters should change
+slowly and the strength of the analog control errors is expected
+to be smaller. Thus, the condition (15) is not far from experi-
+mental systems and may be acceptable.
+Proof of Corollary 2. From Eq. (11), we obtain
+0 < 1 − v2
+2 ≤ Re ⟨ψ(T)|φ(T)⟩ ≤ | ⟨ψ(T)|φ(T)⟩ |
+≤
+�
+n
+|CnDn| =
+√
+1 − ǫ2|Dm| +
+�
+n(�m)
+|CnDn|
+≤
+√
+1 − ǫ2|Dm| + ǫ
+�
+1 − |Dm|2|
+≤
+√
+1 − ǫ2|Dm| + ǫ,
+(18)
+where we used the Cauchy-Schwartz inequality. Thus, using
+Eq. (15), we obtain
+|Dm| ≥
+1 − v2
+2 − ǫ
+√
+1 − ǫ2 > 0.
+(19)
+□
+III.
+CONCLUSIONS
+We have established a threshold theorem that provides a
+sufficient condition for obtaining the target state in isolated
+quantum dynamics with any deterministic analog control er-
+ror.
+We have considered only deterministic analog control er-
+rors. A similar threshold theorem for stochastic analog control
+errors has already been obtained in Ref. [18]. For both types
+of analog control error, the same point is that, if the strength
+of the control errors is less than the inverse of the computation
+time, the target state can be obtained through a constant-order
+number of measurements in quantum dynamics with analog
+control errors. It is an interesting future problem to combine
+these results.
+Finally, we emphasize that we do not impose any assump-
+tions on time evolution. Considering a specific schedule for
+each problem, such as adiabatic time evolution, might im-
+prove the present results.
+The present work was financially supported by JSPS KAK-
+ENHI Grant No. 19H01095, 20H02168 and 21K13848.
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diff --git a/9tFPT4oBgHgl3EQfYzRy/content/tmp_files/load_file.txt b/9tFPT4oBgHgl3EQfYzRy/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf,len=266
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='13075v1  [quant-ph]  30 Jan 2023  Threshold theorem in quantum annealing with deterministic analog control errors  Manaka Okuyama1 and Masayuki Ohzeki1,2,3  1Graduate School of Information Sciences, Tohoku University, Sendai 980-8579, Japan  2Department of Physics, Tokyo Institute of Technology, Oh-okayama, Meguro-ku, Tokyo,152-8551, Japan and  3Sigma-i Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=', Ltd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=', Tokyo 108-0075, Japan  (Dated: January 31, 2023)  We investigate the effect of deterministic analog control errors in the time-dependent Hamiltonian on iso-  lated quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Deterministic analog control errors are formulated as time-dependent operators in the  Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' We give an upper bound on the distance between two states in time evolution with and  without deterministic analog control errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' As a result, we prove that, if the strength of deterministic analog  control errors is less than the inverse of computational time, the final state in quantum dynamics without deter-  ministic analog control errors can be obtained through a constant-order number of measurements in quantum  dynamics with deterministic analog control errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  INTRODUCTION  Quantum annealing [1–8] is an analog quantum computa-  tion that utilizes continuous time evolution of quantum sys-  tems, and, thereby, analog control errors of the parameters  are inevitable in experimental systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Because the theory  of quantum error correction and suppression is incomplete  in quantum annealing [9–13], estimating the effect of analog  control errors is one of the most critical problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  There are two main types of analog control errors in quan-  tum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  One is a stochastic control error [14–17],  which represents an instantaneous parameter fluctuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' For  this type of control error, recent studies [18, 19] proved that,  if the strength of the stochastic control errors is less than the  inverse of the computation time, information about the final  state in quantum dynamics without analog control errors can  be recovered from quantum dynamics with stochastic control  errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' The other is deterministic control error, which is, for  example, a bias acting on the magnetic field or a deviation in  the value of the interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Deterministic control errors have  been discussed so far in many literatures [20–26], but they are  limited to specific problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  The present study investigates in general whether it is pos-  sible to recover information about the target state, which is  the final state in ideal time evolution, from quantum dynam-  ics with deterministic analog control errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' We give an upper  bound on the distance between two states in quantum dynam-  ics with and without deterministic control errors using only in-  formation about the deterministic control errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Furthermore,  using this bound, we prove that, if the strength of the deter-  ministic control errors is less than the inverse of the computa-  tion time, information about the target state can be recovered  through a constant-order number of measurements in quan-  tum dynamics with deterministic analog control errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' This  result is intuitively obvious but it is important from the per-  spective of experimental systems to give mathematical proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  The proof is based on the method proposed by Kieu to derive  a quantum speed limit [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' II,  we define the model and obtain the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Finally, our  conclusion is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  RESULT  We consider the following isolated quantum dynamics:  i d  dt|ψ(t)⟩ = ˆH(t)|ψ(t)⟩,  (1)  where 0 ≤ t ≤ T and ℏ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' In general, it is difficult to com-  pletely control the time-dependent Hamiltonian ˆH(t) without  control errors in experimental systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Deterministic ana-  log control errors can take any form physically permissible  but should also be described as a Hermitian operator since  we consider isolated quantum dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Thus, we incorpo-  rate the deterministic analog control errors of ˆH(t) into the  Schr¨odinger equation as a Hermitian operator ˆV(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' We ex-  press the Schr¨odinger equation with deterministic analog con-  trol errors as follows:  i d  dt|φ(t)⟩ = ( ˆH(t) + ˆV(t))|φ(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (2)  Then, we obtain the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' The distance between two final states |ψ(T)⟩ and  |φ(T)⟩ is bounded from above by  ∥ |ψ(T)⟩ − |φ(T)⟩ ∥ ≤ v,  (3)  where ∥ |a⟩ ∥ ≡ √⟨a|a⟩, v ≡  � T  0 dt  ��� ˆV(t)  ���, and  ��� ˆA  ��� is the eigen-  value of ˆA with the largest absolute value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' (1) and (2), we obtain  d  dt(|ψ(t)⟩ − |φ(t)⟩) = −i ˆH(t)(|ψ(t)⟩ − |φ(t)⟩) + i ˆV(t) |φ(t)⟩ ,(4)  and  d  dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥2 = 2 Re  �  (⟨ψ(t)| − ⟨φ(t)|) d  dt(|ψ(t)⟩ − |φ(t)⟩)  �  = 2 Re  �  (⟨ψ(t)| − ⟨φ(t)|)i ˆV(t) |φ(t)⟩  �  ≤ 2∥ |ψ(t)⟩ − |φ(t)⟩ ∥ · ∥ ˆV(t) |φ(t)⟩ ∥,  (5)  where we used the Cauchy-Schwartz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' On the other  hand, we find  d  dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥2 = 2∥ |ψ(t)⟩ − |φ(t)⟩ ∥ · d  dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (6)  2  Thus, we obtain  d  dt∥ |ψ(t)⟩ − |φ(t)⟩ ∥ ≤ ∥ ˆV(t) |φ(t)⟩ ∥ ≤ ∥ ˆV(t)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (7)  Finally, by integrating both sides from 0 to T, we arrive at Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  □  It is worth mentioning that the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' (3)  contains only information about the control errors ˆV and not  about ˆH(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  The inequality (3) makes sense only if v < 2 is satisfied  because  ∥ |ψ(T)⟩ − |φ(T)⟩ ∥ =  �  2 − 2 Re ⟨ψ(t)|φ(t)⟩ ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (8)  In particular, when the strength of deterministic control errors  is less than the inverse of the computation time,  ��� ˆV(t)  ��� <  √  2  T ,  (9)  we have  ∥ |ψ(T)⟩ − |φ(T)⟩ ∥ ≤ v <  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (10)  This means that the two final states have non-zero overlap  Re ⟨ψ(T)|φ(T)⟩ ≥ 1 − v2  2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (11)  Then, it is possible to recover the information about |ψ(T)⟩  from |φ(T)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  For example, we expand the two final states |ψ(T)⟩ and  |φ(T)⟩ as  |ψ(T)⟩ =  �  n  Cn |n⟩ ,  (12)  |φ(T)⟩ =  �  n  Dn |n⟩ ,  (13)  where |n⟩ is the measurement basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' We are interested in the  mth eigenstate of the measurement basis and its probability  amplitude Cm is given by  |Cm|2 = 1 − ǫ2,  (14)  with 0 ≤ ǫ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Then, we arrive at:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' If  1 − v2/2 > ǫ ≥ 0,  (15)  then the probability amplitude of the mth eigenstate in the  Schr¨odinger equation with deterministic analog control errors  (2) takes a non-zero value,  |Dm| ≥ 1 − v2  2 − ǫ  √  1 − ǫ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (16)  Corollary 2 states that the number of measurements re-  quired to obtain |m⟩ is independent of the computation time  T in quantum dynamics with deterministic analog control er-  rors (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Thus, under the condition (15), deterministic control  errors do not seriously affect the efficiency of quantum anneal-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  The condition (15) can be rewritten as  � T  0  dt  ��� ˆV(t)  ��� <  �  2(1 − ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (17)  It may seem difficult to satisfy this condition for large T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  However, when T is large, the parameters should change  slowly and the strength of the analog control errors is expected  to be smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Thus, the condition (15) is not far from experi-  mental systems and may be acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' (11), we obtain  0 < 1 − v2  2 ≤ Re ⟨ψ(T)|φ(T)⟩ ≤ | ⟨ψ(T)|φ(T)⟩ |  ≤  �  n  |CnDn| =  √  1 − ǫ2|Dm| +  �  n(�m)  |CnDn|  ≤  √  1 − ǫ2|Dm| + ǫ  �  1 − |Dm|2|  ≤  √  1 − ǫ2|Dm| + ǫ,  (18)  where we used the Cauchy-Schwartz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Thus, using  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' (15), we obtain  |Dm| ≥  1 − v2  2 − ǫ  √  1 − ǫ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  (19)  □  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  CONCLUSIONS  We have established a threshold theorem that provides a  sufficient condition for obtaining the target state in isolated  quantum dynamics with any deterministic analog control er-  ror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  We have considered only deterministic analog control er-  rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' A similar threshold theorem for stochastic analog control  errors has already been obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' For both types  of analog control error, the same point is that, if the strength  of the control errors is less than the inverse of the computation  time, the target state can be obtained through a constant-order  number of measurements in quantum dynamics with analog  control errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' It is an interesting future problem to combine  these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  Finally, we emphasize that we do not impose any assump-  tions on time evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content=' Considering a specific schedule for  each problem, such as adiabatic time evolution, might im-  prove the present results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
+page_content='  The present work was financially supported by JSPS KAK-  ENHI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tFPT4oBgHgl3EQfYzRy/content/2301.13075v1.pdf'}
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+Revealing Rheological Parameters of Cotton-stitch-modified Cotton Fabrics by 
+Three-Network Modeling (TNM) of Materials 
+Harmony Werth1, #, Kazi Zihan Hossain1, #,  M. Rashed Khan1,* 
+1Department of Chemical and Materials Engineering, University of Nevada, Reno 
+#contributed equally 
+*Corresponding author: mrkhan@unr.edu 
+Abstract 
+Cotton threads and fabrics are the most used textile materials and have garnered 
+widespread interest for smart textiles to capture human-centered cyber-physical and 
+human-health-related bioanalytical data. Cotton threads are sewn (manually or digitally) 
+into fabrics to achieve functional and fashion stitches that soften or stiffen the base fabric. 
+There has been limited investigation into the influence of a single stitch on the mechanical 
+properties of knitted cotton fabric. Such understanding may become critical to producing 
+optimized textile-based composites/smart materials involving sewing operations. While 
+stitching operations are investigated in numerous ways to produce a range of smart 
+wearables, herein, we demonstrate the rheological modification of base cotton fabric 
+induced by two types of singular stitches (straight and zigzag). We have sewn simple 
+straight and zigzag cotton stitches to investigate the rheological modification of the base 
+cotton fabrics. Uniaxial stress-strain experimental data, combined with constitutive 
+modeling (i.e., three-network model, TNM) obtained from the calibration software 
+(MCalibration), revealed the feasibility of a data-driven approach to investigate the 
+rheological parameters. Our experimental analyses, combined with the calibrated data, 
+suggest a 99.99% confidence in assessing the influence of a single stitch on knitted cotton 
+fabrics. We have also used distributed strain energy to analyze the mechanics and failure 
+of the base and stitched fabrics. Our study may enable the design and study of integrating 
+smart threads in cotton fabrics to produce smart wearables, e-textile, biomedical and e-
+fashion textiles.    
+ 
+ 
+ 
+ 
+ 
+ 
+
+Introduction 
+Knitted cotton fabrics have been utilized in everyday garment materials and 
+emerged as one of the popular base materials to generate smart wearables. In this article, 
+we reveal rheological parameters- also known as phenomenological parameters, of 
+cotton-stitch-modified cotton fabrics, harnessing Three Network Models (TNM).1,2 We 
+produce two types of stitches for demonstrations to modify the mechanical behavior of 
+knitted cotton fabrics. While the utility of cotton fabrics is ubiquitous, and numerous 
+demonstrations are currently published in the literature, our study focuses on 
+understanding the tuned mechanics of cotton fabrics by sewn stitches and unravels data 
+that we often overlook through stress-strain analyses. Anisotropy of knitted cotton fabric 
+and its modified structural properties exhibited deformations during mechanical 
+performance analyses.3 Several studies focused on understanding knit fabrics, fabric 
+elongation, deformation, and failures at critical applied stress.4–10 Advanced applications 
+on knitted fabrics are approached mainly by trial and error methods where in-plane 
+stitches are randomly generated, leaving a knowledge gap in understanding the impact 
+of final sewn stitches on fabrics. Sewing- one of the ancient fabric manufacturing 
+techniques, loops a thread into fabrics, leveraging an analog or computerized sewing 
+machine. Different sewing stages,11 sewing parameters,12, and sewing machines13,14 
+have also been reported to alter the properties of the sewing thread. The looping process 
+integrates different threads and produces entanglements with aesthetic colors, body 
+shapes, and on-demand geometries. Different stitch patterns have also been reported to 
+change the rheological behavior of the sewing thread.15,16 However, the rheological 
+impact on the fabric due to stitching has not been adequately investigated. While the 
+original purpose of sewing has been joining two pieces of fabric together, the most 
+advanced applications integrate smart threads so that biomedical, biochemical, and 
+biological analyses can be performed in situ.17,18 From the design of fashion to human-
+centered smart wearables, state-of-sewing leverages many stitch patterns; however, a 
+data-informed approach to dissect the role of sewn stitches in manipulating the final 
+fabrics' properties is currently lacking in the literature.  
+Using a sewing machine, sewn stitches create entanglements between two 
+threads- an upper and a bobbin thread, the bottom thread. During the entanglement, the 
+sewing needle loops the upper thread between the bottom thread through the fabric, and 
+the threads entangle. When the tension of both threads matches, the entanglement lays 
+in-plane of the fabric on both sides with minimal damage.19 The resulting stitch can be 
+varied in numerous ways to create functional and non-functional patterns based on the 
+types of sewing threads, fabric types, or choice of materials for the final composite 
+structures.20 Concurrently, stitches are used to bind two or more layers of fabric together, 
+which is known as a seam. To determine the impact of stitches on the mechanical 
+behavior of fabrics, a few research groups have tested seams in woven cotton fabric.21–
+23 A few investigations on the mechanical behavior of seams in knitted fabrics are also 
+available in the literature.24,25 However, studies involving a single stitch thread to modify 
+the mechanical behavior of cotton fabric are currently lacking for a single layer of fabric. 
+Such studies, we believe, will become significantly crucial for future applications, i.e., 
+electronic textiles- because reducing materials consumption at an optimum number of 
+
+trials and errors seems crucial to pursue robust design configurations during the 
+development of smart threads and electronic fabrics.  
+The base fabric and thread used in this work are made from cotton. We assume 
+both as an elastomeric network for modeling. Elastomers having 3,000 to 10,000 
+repeating 
+units 
+exhibit 
+structural 
+flexibility 
+and 
+experience 
+stretch-induced 
+softening/hardening (also known as Mullins damage) under applied loads.26 For data 
+calibration, we use TNM in MCalibration software which maps the entire stress-strain 
+spectra from the uniaxial test. The TNM is also knowns as a phenomenological model to 
+describe deformation-induced structural evolution (i.e., the transition between soft to stiff 
+network) and how strain-energy density becomes redistributed (i.e., hysteresis) 
+throughout the experiments. Different hyperelastic models have been utilized in the 
+literature to represent the rheological behavior of fabrics.27–31 However, according to our 
+knowledge, this work presents the constitutive modeling of stitched fabrics with TNM for 
+the first time.  The data-calibration process in MCalibration can start using default or user-
+induced settings. For this study, we have chosen to start calibration using default settings 
+in MCalibration. The kinematics of the TNM consists of three parallel molecular networks. 
+We have assumed spring-dashpot domains connected in parallel for the first two, and the 
+third one is only a spring depicting the hyperelasticity of the first two networks. The semi-
+crystalline domains are captured through the spring dashpots. While a single network can 
+be used to evaluate the property of the entire composite structure, we have chosen TNM 
+to capture effective viscoplasticity.1  
+ 
+Here, we have chosen two types of sewing stitches: straight and zigzag, to 
+establish and reveal rheological parameters of cotton-stitch modified cotton fabrics, 
+harnessing TNM. These stitches are common in sewn garments and are pre-programmed 
+into the default settings of modern sewing machines. Also, we investigate several 
+variations of the zigzag stitch that has varying stitch length and width. A commercial 
+sewing machine creates stitches with 100% cotton materials (i.e., threads and fabrics). 
+For our analyses, we investigate the surface topography of the fabric and samples with 
+sewn stitches using optical and scanning electron microscope (SEM) images. We perform 
+(a) uniaxial stress-strain and (b) repeated cyclic tests in an Instron to dissect the 
+mechanical behaviors of the (a) base fabric, (b) base threads, and (c) threads-laid-fabric 
+structures. The uniaxial stress-strain analyses have revealed three regions of interest 
+(i.e., elastic, yield, and viscoplastic). Also, we have investigated the permanent failure of 
+the entire composite (fracture) to find the extremities of experimental analyses. The cyclic 
+tests provide information about hysteresis, which we leverage to understand distributed 
+strain-energy density and the loss due to hysteresis. We outline calibration using TNM in 
+MCalibration to provide a simple route to test the impact of specific sewing patterns on 
+the mechanical behavior of the final fabric. We hypothesize that the thread, which has 
+significantly denser strain energy, shifts the fabric's macroscale stress-strain behavior 
+after stitching. We proved our hypothesis through the uniaxial test and then altered the 
+stitch length to investigate the factors that cause specific changes in the stress-strain 
+behavior. The understanding developed by investigating the changes in mechanical 
+behavior can be used to optimize the mechanical properties of a composite made with 
+cotton thread and fabric. 
+
+Materials and Methods 
+Our experiments were performed to determine a constitutive equation to represent the 
+behavior of cotton fabric with different types of stitches. This was accomplished by 
+analyzing uniaxial stress-strain curves for each component (cotton fabric and cotton 
+thread) and different variations of the overall composite (straight stitched fabric, zigzag 
+stitch, and fabric with stitching holes but no thread).  
+Materials 
+The fabric used in this experiment was 100% cotton jersey knit with a unit weight of 427 
+g/m2 obtained from Hobby Lobby Stores, Inc. The measured thickness of the fabric was 
+0.45 mm. Similarly, the thread used in this experiment was a 50-weight, 4-ply, 100% 
+cotton thread of Sew-Ology Brand from Hobby Lobby Stores, Inc., produced for machine 
+quilting. The measured outer diameter of the thread was 0.30 mm. The same thread was 
+used for the top and bobbin threads for all samples prepared and presented in this work.  
+Sample Preparation 
+We used a Brother SE600 sewing machine from Amazon to create stitches of manually 
+adjustable length and width. The tension was selected so that the tension on the bobbin 
+thread and the upper thread were equal, preventing the bottom thread from showing on 
+the top or vice-versa, as is common sewing practice. A swatch of fabric approximately 20 
+cm in length was cut with scissors and then sewn wale-wise with the appropriate type of 
+stitch for the sample. Two samples were prepared with stitching: straight stitched and 
+zigzag stitched. The straight stitch was 2 mm in length. The zigzag stitches were (listed 
+as stitch length x stitch width): 2x5 mm, 1x5 mm, 3x5 mm, 2x3 mm, and 2x4 mm. Several 
+samples without any sewn stitches were also prepared for comparison. The fabric 
+samples were then cut to the same size with a Cricut Maker fabric cutter bought from 
+Amazon, allowing for accurate and reproducible sample cutting. For every sample, the 
+fabric was cut to the dimensions of 6 cm in length by 2 cm in width. Sample cutting was 
+done carefully to keep the stitching in the center of the sample. Damaged or samples with 
+uncentered stitches were discarded without any analysis.   
+Image Acquisition 
+Olympus SZ61 Stereo-microscope loaded with an Amscope MU1000-HS camera was 
+used to capture the optical microscopic images. Secondary electron images of the 
+samples were captured using a Thermo Scientific Scios 2 SEM. For SEM imaging, small 
+representative samples were cut and loaded on the sample holder with double-sided 
+carbon tape. Attention was given to keeping the stitch undamaged while loading on the 
+holder. Since the samples were nonconductive, samples were sputter-coated with Gold 
+(Au) to create a ~10 nm layer on the surface of the sample before imaging. Further optical 
+images were captured with the camera on an iPhone 13 mini. 
+Experimental Methods 
+
+The data was collected using an Instron 5982 test machine for uniaxial tensile testing. 
+The tested area was 4 cm by 2 cm. The extra centimeter on each side allowed the grip to 
+hold the sample during testing. Each sample type was examined with tensile testing to 
+determine the stress and strain until failure at a 40 mm/min strain rate. Cyclic testing of 
+four cycles was then conducted for specific samples up to a sustainable strain level for 
+that sample type. Samples with straight stitches could only withstand slightly more than 
+10% strain. Therefore, cyclic tests with straight stitches were conducted up to 10% strain. 
+For comparison, cyclic testing up to 10% strain was also conducted for unaltered fabric 
+samples and the 2x5 mm zigzag stitch.  
+Strain-energy Density Calculation  
+Using the trapezoidal rule, we calculated strain-energy density from the time-dependent 
+force and stress data at varying strain rates. The area under the stress-strain or force-
+strain curve is divided into equal-time steps. Each small area under the curve is added 
+until we reach the last data point to get the total area under the curve. The reported energy 
+density from different observations is the total after each experimental stress-strain 
+observation.  
+Constitutive Modeling of Different Fabrics 
+An initial prediction of the strain energy density of the straight stitched sample was 
+obtained based on the data collected from the unaltered fabric and thread samples. In 
+order to obtain the prediction, the strain energy density of the straight stitch sample and 
+the unaltered fabric was obtained by finding the area under the stress-strain curve of the 
+sample with the trapezoidal rule. The strain energy density was calculated up to 4.5% 
+strain because the thread samples failed around 5% strain. The strain energy of the 
+samples was calculated by multiplying the strain energy density by the volume of the 
+sample. The volume of the fabric was calculated using the sample's length, width, and 
+thickness. The thread volume was calculated from the measured diameter and length of 
+the thread sample. The straight stitch sample can be approximated by one sample of 
+fabric and two samples of thread, so the volume of the straight stitch sample was 
+calculated by adding the volume of the unaltered fabric and two threads. Similarly, the 
+predicted strain energy of the straight stitch sample was calculated by adding the strain 
+energy of the unaltered fabric and two threads. The prediction for the strain energy density 
+of the straight stitched sample could then be obtained by dividing the predicted stored 
+energy by the calculated volume. 
+MCalibration, from PolymerFEM,32 was used to obtain parameters for a material model 
+capable of representing the mechanical behavior of the fabric samples prepared in this 
+work. MCalibration fits the experimentally collected uniaxial stress-strain data to the 
+PolyUMod Three Network model.2 An average of the stress-strain behavior of each 
+sample type was obtained first. This set of averaged data was then processed using the 
+MCalibration software tools to prepare the data for calibration. The default settings were 
+used for the calibration. The calibrated parameters were exported and analyzed after the 
+automatic convergence of the calibration process.  
+
+Results and Discussion 
+Surface Topography 
+We formed two different types of stitches on the base fabric. Figures 1a and 1b are top-
+down optical microscope images of the base and sewn fabrics for visual inspection. 
+Figure 1b is a zoomed-in visual inspection of Figure 1a to identify differences between a 
+straight stitch and a zigzag stitch on the in-laid fabric. These images show that the straight 
+and zigzag stitches went through the fabric without significant internal damage. The 
+straight stitch shown in Figures 1a(ii) and 1b(ii) do not have significant bunching due to 
+the stitch compared with the only fabric shown in Figures 1a(i) and 1b(i); However, a 
+meandering network of the zigzag stitches caused the fabric within the stitch to 
+significantly bunch together, as shown in Figures 1 a(iii) and 1b(iii). The fabric is unable 
+to maintain its shape during sewing and is pulled into the stitch instead. The structural 
+stiffness and flexibility of the fabric may have contributed to the bunching, as observed 
+within the stitch dimensions. Figure 1c is the sewn fabric's SEM images to investigate the 
+surface topography of the stitches and fabric. SEM images in Figure 1c(i) and Figure 1c(ii) 
+reveal the undamaged fabric by fibers. From these visual inspections, we assume the 
+fabric remains structurally robust during the sewing and stitches only alter the mechanical 
+behavior.  
+ 
+Figure 1:  (a) Images were taken of samples under normal lighting conditions for visual inspection. 
+(b) A stereoscope was used to examine the samples. (c) Secondary electron SEM images were 
+taken of the fabric and sewn stitches. 
+Uniaxial Tensile Behavior 
+We investigated plain thread, plain fabric, and stitched fabrics using Instron for 
+mechanical behavior analyses. The uniaxial tensile test behavior of plain thread is shown 
+
+a(i)
+b(i)
+c(i)
+Cotton
+Fabric
+a(ii)
+b(i)
+500μm
+Straight
+Fabric<
+Stich
+Stitch
+c(ii)
+b(ili)
+Zigzag
+Stitch
+5 mm
+2 mm
+500 μmin Figure 2a, and the plain fabric is shown in Figure 2b. For comparison, Figure 2b also 
+shows the behaviors of straight and zigzag (2x5mm) stitched fabrics. Four other zigzag 
+stitched fabrics' behavior is shown in Figure 2c. Figures 2d and 2e show the side view of 
+a zigzag stitched fabric loaded into Intron during tensile testing and at the end of failure 
+analyses.  
+We tested two samples of the plain threads, and both samples' behavior is shown 
+in Figure 2a. The plain thread failed at 5% strain but exhibited the highest strain energy 
+density compared to other samples tested. In contrast to the plain thread, the cotton fabric 
+in Figure 2b exhibited reproducible stretchability of up to 70% strain in two samples. The 
+inclusion of straight stitches into the plain fabric induced failure at ~12% strain, and the 
+zigzag stitched sample failed at ~32% strain.  
+The unaltered fabric had the highest stain at the point of failure, shown in Figure 
+2b, between 60-80%, with a strain energy density of around 1.0 MJ/m3 at failure. In 
+comparison, the thread samples had a strain energy density of approximately 5.0 MJ/m3 
+at failure, which occurred at around 5% strain. The strain energy density of the unaltered 
+fabric and thread at 4.5% were 4.19x10-4 MJ/m3 and 4.64 MJ/m3, respectively. Examining 
+the stress-strain data for the cotton fabric and the cotton thread individually, we conclude 
+that combining these materials would result in a sample with a strain energy density that 
+falls between the different materials at a given strain. The samples with sewn straight 
+stitches of 2mm length failed between 10-15% strain. At a strain of 4.5%, the straight 
+stitched samples exhibited an average strain energy density of 1.88x10-3 MJ/m3. A 
+prediction of the strain energy density of the straight stitched samples at 4.5% was 
+obtained using the experimental values of the thread and fabric alone. The predicted 
+value was 7.2x10-2 MJ/m3, more significant than the measured strain energy density. This 
+discrepancy is expected as the sewing process exposes the thread to dynamic loads and 
+friction known to reduce the strength of the thread.33 Overall, the sample with sewn 
+straight stitches failed at all fabric samples' lowest stress and strain. The cause of the low 
+stress and strain at failure is suspected to be the structure of the stitches, which cannot 
+withstand as much strain as the fabric. The fabric, with a higher elongation at failure than 
+the thread, can deform under the load. Therefore, the thread in the straight stitch 
+withstands the load for the entire sample until the thread breaks, equivalent to sample 
+failure. It was observed that the thread failed before the fabric in all samples with straight 
+stitches. 
+Samples with zigzag stitches of 2mm length and 5mm width also failed at stress 
+and strain lower than the unaltered fabric but higher strain than the straight stitched 
+sample. An analysis of the strain energy density of the 2x5mm zigzag sample reveals 
+aspects of the mechanical behavior. At strains below 20%, the strain energy density of 
+the 2x5mm zigzag sample is indistinguishable from the strain energy density of the fabric; 
+Therefore, the fabric's mechanical properties dominate the thread's properties in the 
+2x5mm zigzag sample at strains under 20%. At 30% strain, the strain energy density of 
+the zigzag sample is nearly double that of the fabric sample. The departure of the 2x5mm 
+zigzag sample from the mechanical behavior of the fabric indicates that at strains above 
+20%, the thread is the dominant influence on the mechanical behavior. This behavior is 
+
+investigated further in zigzag samples with varying stitch lengths and widths, as indicated 
+in Figure 2c. 
+ 
+Figure 2: (a) The graph of the stress-strain curve for the samples of the cotton thread indicates 
+maximum stress of approximately 100MPa at a strain of approximately 4.5% before failure. (b) 
+The graph shows the stress-strain curves of the unaltered fabric, fabric with straight stitches of 
+2mm length, and fabric with zigzag stitches of a length of 2mm and a width of 5mm. (c) The stress-
+strain graph shows the impact of varying the properties of zigzag stitches. (d) An image of a 
+sample with 1x5mm zigzag stitches shows the condition of the sample before uniaxial tensile 
+loading. (e) An image of a sample with 1x5mm zigzag stitches shows the condition of the sample 
+after uniaxial tensile loading. Notably, the fabric has failed while the sewn thread is intact. 
+During uniaxial tensile testing, it was revealed that stitch length and width are both critical 
+factors that influence the tensile behavior of the samples with zigzag stitches. Figure 2 
+(c) shows stress-strain curves for samples with zigzag stitches of varying length and 
+width. The fabric samples with 2x5mm, 2x4mm, and 3x5mm zigzag stitches failed at a 
+higher strain than those with straight stitches but at a similar stress. As with the 2x5mm 
+sample, the 2x4mm and 3x5mm had similar strain energy densities at low strain until the 
+thread became a dominant influence. The 1x5mm zigzag sample exhibited drastically 
+
+(a)
+25
+Thread Sample 1
+b
+FabricOnly 1
+-ThreadSample2
+★—FabricOnly2
+- Straight Stitch 1
+100
+8 -
+- Straight Stitch 2
+—2x5mm Zigzag Stitch 1
+Stress (MPa)
+Stress (MPa)
+75
+2x5mm Zigzag Stitch 2
+6.
+50-
+4
+25 
+2.
+0
+0.
+0
+2
+6
+8
+10
+0
+20
+40
+60
+80
+100
+Strain (%)
+Strain (%)
+(c)
+10
+(d)
+2x4mm
+e
+一1x5mm
+2x3mm
+8-
+★一3x5mm
+FabricStress (Mpa)
+6
+Zigzag
+Stitch
+Failure
+4 
+Point
+2-
+Sample
+Grip
+0：
+0
+20
+40
+60
+80
+100
+Strain (%)different behavior from the other samples with zigzag stitches. The strain energy density 
+of the 1x5mm zigzag samples matched that of the unaltered fabric sample up to 
+approximately 60% strain, indicating that the stitches had little impact on the tensile 
+behavior of the sample overall. The 1x5mm samples also had more extended elongation 
+at failure than the unaltered fabric sample. The structure of the zigzag stitch contributes 
+to the behavior of all the zigzag samples. Since zigzag stitches have both a stitch length 
+and a stitch width, the stitch could change shape as the fabric elongates.  
+Figures 2(d) and (e) show a 1x5mm zigzag sample before and after uniaxial tensile 
+testing. After testing, the stitches are longer in the direction parallel to loading and shorter 
+in the direction perpendicular to loading compared to before tensile testing. In other 
+words, the stitch could shrink in the direction perpendicular to loading while elongating in 
+the direction of loading. A shorter stitch length results in more threads in the sample, 
+which allows the stitches to deform enough to match the elongation of the fabric. The 
+consequence of the stitch deformation is that the fabric withstands the load while the stitch 
+can deform, but the stitch bears the load when it is no longer able to match the elongation 
+of the fabric. Eventually, the load exhausts the ability of the stitch to deform, which is 
+when the strain energy density of the zigzag sample deviates from that of the unaltered 
+fabric. It was observed that the sewn thread had snapped in all samples after the sample 
+had failed during tensile testing, except for the 1x5mm zigzag sample. The 1x5mm zigzag 
+sample in Figure 2(c) had a shorter stitch length. Another point of interest is shown in 
+Figure 2(e), which shows that the thread was intact after the fabric failed, which is the 
+opposite of all other samples that contained sewn stitches. Therefore, it is possible to 
+alter stitch properties to alter the fabric's tensile behavior, and the properties determine 
+the extent of the influence from the thread and the fabric at particular strains. 
+Repeated Cycling Behavior 
+The unaltered fabric sample, straight stitch sample, and 2x5mm zigzag stitch sample 
+were examined under cyclic loading to analyze stress softening and hysteresis. Any 
+fabrics are subjected to cyclic loading during use from body movements such as the 
+expansion of the chest during breathing or the movement of joints. An analysis of the 
+behavior of the fabric samples during cyclic loading provides information that can inform 
+design decisions. All samples were strained up to 10% because the straight stitch 
+samples failed at approximately 12% strain. Hysteresis, the change in behavior from the 
+loading to the unloading cycle, was observed in all samples, as shown in Figure 3. Across 
+all samples, the most extensive hysteresis occurred during the first cycle. Additionally, all 
+samples had the highest strain energy density during the loading of the first cycle. The 
+hysteresis between the loading and unloading cycle of the overall sample is impacted by 
+the relationship between the yarns' properties and the fabric's structure. The plastic 
+deformation of the yarns, which relates to the slippage and viscoelasticity of the fibers 
+within the yarn, influences hysteresis.10 The structure dictates the number and nature of 
+the contact points between loops of thread, which impacts the friction during loading. 
+Friction is the main factor determining the amount of hysteresis that will occur.5 In the 
+samples with stitches, the causes of tensile hysteresis are further complicated by the 
+presence the stitched threads, which impact the overall properties and structure of the 
+
+sample. In Figure 3b, the straight stitch sample showed more hysteresis than the zigzag 
+stitched sample shown in Figure 3c, indicating that the straight stitched threads 
+experienced more plastic deformation than the zigzag stitched threads. The difference in 
+the plastic deformation experienced in the threads relates to the behavior observed in the 
+uniaxial tensile testing. The straight-stitched thread sustains more of the load for the entire 
+sample than the zigzag stitch; the thread in the straight-stitched sample experiences more 
+plastic deformation. Repeated cycles allow for an investigation of the hysteresis in 
+additional cycles and an analysis of the stress-softening behavior of the samples. The 
+second cycle revealed that stress softening occurred in all samples between the first and 
+second cycles, which can be observed in whole Figure 3 as a reduction in the strain 
+energy density of the loading curve between the first and second cycles. The unaltered 
+fabric sample in Figure 3a showed a minor stress softening, which can be attributed to 
+the significant difference between the maximum strain during cyclic loading and the strain 
+required to cause failure. Since the unaltered fabric sample has minor unrecoverable 
+deformation at the 10% strain tested in this experiment, minimal stress softening 
+occurred. In additional cycles after the second cycle, hysteresis in the fabric sample and 
+the zigzag sample remained the same; however, hysteresis decreased slightly in the 
+straight stitched sample from the second to the third cycle. The decrease in hysteresis is 
+attributable to the stress softening in the straight stitch sample between the second and 
+the third cycles, which indicates that further unrecoverable deformation occurred during 
+each cycle. In comparison, the fabric and the zigzag stitch samples do not experience 
+significant unrecoverable deformation in cycles after the second cycle. 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 2: (a) The unaltered fabric sample showed less stress softening than the 2x5mm zigzag 
+stitch sample but still showed hysteresis. (b) The straight stitch sample had the most stress 
+softening and also showed hysteresis. (c) The cyclic loading of the 2x5mm zigzag stitch sample 
+showed stress softening after the first cycle and hysteresis. 
+ 
+ 
+
+(a)
+0.06
+-OnlyFabricCycle1
+-OnlyFabricCycle2
+OnlyFabric Cycle3
+★一
+Only Fabric Cycle 4
+Stress (Mpa)
+0.04 -
+0.02
+0.00+
+0
+10
+5
+Strain (%)
+(b)
+1.0
+StraightStitchCycle1
+-Straight StitchCycle2
+Straight Stitch Cycle3
+0.8 -
+★一
+Straight Stitch Cycle 4
+0.2 -
+0.0 +
+5
+10
+Strain (%)
+(c)
+0.06
+2x5mmZigzagCycle1
+—2x5mmZigzagCycle2
+2x5mmZigzagCycle3
+★一
+2x5mmZigzagCycle4
+Stress (Mpa)
+0.04
+0.02
+0.00
+0
+5
+10
+Strain (%)Revealing rheological parameters of Fabric and Composite Systems 
+The TNM is a powerful constitutive model capturing the flow and deformation (rheology) 
+behaviors of materials. Bergstrom and Bischoff explained the mathematical details of the 
+TNM in their work.1 While the stress-strain analysis directly measures the mechanical 
+behavior, the rheological parameters we often overlook in stress-strain analyses can be 
+revealed through constitutive models. Studies on such parameters also enable data-
+informed design decisions.  
+We used MCalibration software to perform rheological analyses using TNM and calibrate 
+the TNM parameters to assess unaltered and altered fabrics. MCalibration software 
+begins calibration with a set of initially estimated parameter values by observing the 
+experimental data. It tries to reduce the deviation between the predicted and the 
+experimental behavior by continuously updating the parameters. This process is also 
+known as data calibration and rheological parameter identification. When the coefficient 
+of determination or the R2 value stops changing significantly by reaching convergence, 
+the software reveals the rheological parameters in its user interface. The experimental 
+data and MCalibration predicted data with their respective R2 fitness are shown in Figure 
+4, indicating that the TNM model effectively captures the uniaxial tensile behavior of 
+unaltered, straight- and zigzag-stitched fabrics. The predicted data fits closely with the 
+experimental data for all investigated samples with this method. The prediction of the 
+2x5mm zigzag sample in Figure 4a matched with an R2 fitness of 0.999, which was a 
+closer fit than the unaltered fabric sample or the straight stitch sample. The reason for the 
+closer match indicates that the 2x5mm zigzag sample had behavior closest to that of a 
+thermoplastic polymer, which is the material on which the TNM is based. Furthermore, 
+the calibration calculates the material model parameters, revealing information about the 
+behavior of the samples that cannot be determined from an analysis of the experimental 
+data alone. Table 1 shows several such parameters. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Figure 3: The material calibration with the PolyUMod TNM resulted in a good prediction for (a) 
+the unaltered fabric sample, (b) the 2mm straight stitch sample, and (c) the 2x5mm zigzag sample. 
+ 
+
+(a)
+3
+ExperimentalFabricData
+Predicted Fabric Data
+2.5
+Stress
+1.5
+1
+0.5
+R2Fitness=0.997
+0
+0
+10
+20
+3040
+50
+6070
+Strain (%)
+(b)
+1.5
+ExperimentalStraightStitchData
+-Predicted Straight StitchData
+1.25
+(MPa)
+1
+stress
+0.75
+0.5
+0.25
+R2Fitness=0.99
+0
+0
+2.5
+5
+7.5
+10
+12.5
+Strain (%)
+(c)
+0.7
+Experimental2x5mmZigzagData
+-Predicted2x5mmZigzagData
+0.6
+0.5
+(edw) :
+0.4
+0.3
+0.2
+0.1
+R2Fitness=0.999
+0
+0
+5
+10
+15
+520
+25
+30
+Strain (%)Table 1: The Three-Network Model (TNM) parameters of unaltered fabric, straight 
+stitch, and the 2x5 mm Zigzag stitch 
+Description 
+Symbol 
+Unit 
+Unaltered 
+Fabric 
+Straight Stitch 
+2x5 mm Zigzag 
+Shear modulus of network 
+A 
+𝜇� 
+KPa 
+11.40 
+77.95 
+0.65 
+Locking stretch 
+𝜆� 
+- 
+1.08 
+1.02 
+1.04 
+Bulk modulus 
+𝜅 
+KPa 
+656.37 
+1369.34 
+1194.72 
+Flow resistance of network 
+A 
+𝜏̂� 
+KPa 
+127.80 
+901.91 
+352.31 
+Stress exponential of 
+network A 
+𝑚� 
+- 
+3.83 
+11.11 
+9.59 
+Initial shear modulus of 
+network B 
+𝜇�� 
+KPa 
+96.46 
+15.05 
+40.75 
+Final shear modulus of 
+network B 
+𝜇�� 
+KPa 
+96.46 
+9.31 
+58.99 
+Evolution rate of 𝜇� 
+𝛽 
+- 
+9.69 
+10.20 
+10.50 
+Flow resistance of network 
+B 
+𝜏̂� 
+KPa 
+348.78 
+1226.80 
+636.76 
+Stress exponential of 
+network B 
+𝑚� 
+- 
+7.89 
+9.65 
+10.85 
+Shear modulus of network 
+C 
+𝜇� 
+KPa 
+398.95 
+1180.98 
+207.47 
+Earlier investigation27  on assessing the hyperelastic material model calibrated 
+parameters, leveraging Mooney-Rivlin, Ogden, neo-Hookean, Arruda Boyce, Gent, Yeoh, 
+and Blatz-Ko constitutive models. The higher-order Mooney-Rivlin and Yeoh models fitted 
+the experimental data properly. The Arruda-Boyce model also showed good relation with 
+the experimental data. Also, we noticed a similarity in the stress-strain behavior from that 
+investigation that is close to our unaltered fabric behavior shown in Figure 2(b). We want 
+to compare the parameters we obtained with that literature27. We noted a shear modulus 
+of 3.8913 KPa, and a limiting locking stretch (𝜆�,���� of 0.65907 from that investigation. 
+The Cauchy stress acting on any networks in the TNM model is based on the Arruda-
+Boyce or eight-chain model.1 The reported shear modulus and the shear modulus of the 
+Network A of the unaltered fabric are also not significantly different here. As the shear 
+modulus of the Arruda-Boyce model gets distributed in three networks, we should only 
+compare the locking stretch directly. The locking stretch is defined as the ratio of the 
+current chain length and the initial chain length. From the literature, the relation between 
+the locking stretch (𝜆�� and limited locking stretch can be found,34,35, which is 
+𝜆� � �1
+3 �𝜆�,���
+�
+�
+2
+𝜆�,���
+� 
+
+The reported limiting locking stretch converted to 𝜆� will be 1.0753, which is very close to 
+our reported locking stretch value of the unaltered fabric, 1.08. Additionally, for all the 
+samples, the locking stretch was close to 1, indicating that the sample did not go through 
+a significant strain level. The locking stretch values of the straight stitch and the zigzag 
+stitch are also smaller than the unaltered fabric, indicating less deformation observed in 
+Figure 2(b). The final calibrated parameters depend significantly on the initially guessed 
+parameters. It would be easier to compare the parameters between three samples if an 
+identical set of initial values was used. As we are using the uniaxial tensile testing here, 
+bulk modulus should not impact the predicted behavior significantly. 36 In the TNM, 
+network A and B utilize separate energy activation mechanisms to represent the 
+amorphous and semi-crystalline domains. Network C represents the large strain response 
+controlled by entropic resistance. The shear modulus and the flow resistance of network 
+A in the straight stitch are significantly higher than the other two samples indicating higher 
+resistance by the spring represented in the network. Figure 4(b) also indicates that up to 
+10% strain straight-stitched fabric is stiffer than the other two matching the observation in 
+the parameters. Comparatively close initial and final shear modulus of network B and 
+almost similar evolution rates indicate a similar effective shear modulus for all the 
+samples. The flow resistance of network B and the shear modulus of network C of the 
+straight-stitched sample are also higher, indicating higher stiffness of the materials.  
+Conclusion 
+This work determined that altering the parameters of the stitching when sewing 
+with cotton thread into a single layer of jersey-knit cotton fabric impacts the strain-energy 
+density, hysteresis, and stress softening of the sample. When examined with optical and 
+scanning electron microscopes, the stitched samples did not show damage to the fabric 
+from the sewing process. The stitch type and parameters of a zigzag stitch were shown 
+to directly impact the sample's behavior under uniaxial tensile loading. Depending on the 
+stitch type, the fabric can be altered to have a higher or lower strain energy density at 
+certain strains. We also note that stitches capable of less elongation than the fabric will 
+increase the strain energy density at lower strains and result in failure at a lower strain. 
+Stitches that can match or exceed the elongation may have minimal impact on the strain 
+energy density of the sample at the same strains as a sample without stitches but will fail 
+at higher strains, resulting in a higher strain energy density at failure. Stitches will also 
+impact the hysteresis and stress softening of the sample. Also, stitches capable of less 
+elongation than the fabric will be subjected to higher stress during loading, resulting in 
+plastic deformation and more significant hysteresis and stress softening during cyclic 
+loading. The tensile and cyclic tests reveal that the mechanical behavior of samples 
+composed of fabric with stitches varies greatly depending on the relationships between 
+the property of the materials and their structure. When data from tensile tests were 
+calibrated with the PolyUMod TNM, the materials presented in this work matched well 
+with the calibrated model; therefore, materials calibration provides an opportunity to aid 
+the selection of materials and structure by offering insight into hidden parameters that 
+allow for a data-driven approach to design.  
+
+Limitations of this work include the number of materials and structures investigated, as 
+the behavior observed may differ from samples with different compositions and 
+structures. Furthermore, many other properties may be impacted by the presence of 
+sewing stitches that were not investigated in this paper, such as abrasive strength, 
+bursting strength, torsional properties, ability to withstand washing and drying, and many 
+other characteristics. Future works may investigate the impact of additional types of 
+stitches on fabrics of different materials and structures and analyze additional properties 
+of the samples. 
+Acknowledgments 
+MRK acknowledges the funding support from VPRI's startup account. HW 
+acknowledges the Nevada undergraduate research award (NURA) fund from the 
+Undergraduate Research Office, and KZH acknowledges funding from the College of 
+Engineering Dean's Office at the University of Nevada, Reno. HW acknowledges 
+contributions from Sydney Fields, Jake Kattelman, Thomas Kaps, and Braden Norris for 
+the MSE 470 (Polymer Engineering instructed by MRK) in-class project. KZH 
+acknowledges the opportunity to train and mentor all the groups in CHE/MSE 470 and 
+Brian Perdue in CHE 495 using the concepts from this article. MRK acknowledges the 
+support received from Dean's Office to purchase Instron 5982 with Dr. Jefferey Lacombe, 
+Dr. Bin Li, and Zachary Karmiol. 
+ 
+ 
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+25.  Admassu Y, Edae A, Getahun G, et al. Experimental analysis on the effect of fabric 
+structures and seam performance characteristics of weft knitted cotton apparels. J 
+Eng Fibers Fabr 2022; 17: 15589250221113480. 
+26.  Qi HJ, Boyce MC. Constitutive model for stretch-induced softening of the stress–
+stretch behavior of elastomeric materials. J Mech Phys Solids 2004; 52: 2187–2205. 
+27.  Julio García Ruíz M, Yarime Suárez González L. Comparison of hyperelastic 
+material models in the analysis of fabrics. Int J Cloth Sci Technol 2006; 18: 314–325. 
+28.  Khiêm VN, Krieger H, Itskov M, et al. An averaging based hyperelastic modeling and 
+experimental analysis of non-crimp fabrics. Int J Solids Struct 2018; 154: 43–54. 
+29.  Gong Y, Peng X, Yao Y, et al. An anisotropic hyperelastic constitutive model for 
+thermoplastic woven composite prepregs. Compos Sci Technol 2016; 128: 17–24. 
+30.  Peng X, Guo Z, Du T, et al. A simple anisotropic hyperelastic constitutive model for 
+textile fabrics with application to forming simulation. Compos Part B Eng 2013; 52: 
+275–281. 
+31.  Peng XQ, Guo ZY, Zia-Ur-Rehman, et al. A Simple Anisotropic Fiber Reinforced 
+Hyperelastic Constitutive Model for Woven Composite Fabrics. Int J Mater Form 
+2010; 3: 723–726. 
+
+32.  MCalibration. PolymerFEM.com, https://polymerfem.com/mcalibration/ (accessed 
+21 February 2022). 
+33.  Geršak J, Knez B. REDUCTION IN THREAD STRENGTH AS A CAUSE OF 
+LOADING IN THE SEWING PROCESS. Int J Cloth Sci Technol 1991; 3: 6–12. 
+34.  Bergstrom JS. Mechanics of Solid Polymers: Theory and Computational Modeling. 
+William Andrew, 2015. 
+35.  Nguyen H-D, Huang S-C. The Uniaxial Stress–Strain Relationship of Hyperelastic 
+Material Models of Rubber Cracks in the Platens of Papermaking Machines Based 
+on Nonlinear Strain and Stress Measurements with the Finite Element Method. 
+Materials 2021; 14: 7534. 
+36.  Jorgen. How Important is the Bulk Modulus in FEA? PolymerFEM.com, 
+https://polymerfem.com/how-important-is-the-bulk-modulus/ (2021, accessed 17 
+November 2022). 
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+ 
+
diff --git a/BNFQT4oBgHgl3EQfNDZv/content/tmp_files/load_file.txt b/BNFQT4oBgHgl3EQfNDZv/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf,len=553
+page_content='Revealing Rheological Parameters of Cotton-stitch-modified Cotton Fabrics by  Three-Network Modeling (TNM) of Materials  Harmony Werth1, #, Kazi Zihan Hossain1, #,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Rashed Khan1,*  1Department of Chemical and Materials Engineering, University of Nevada, Reno  #contributed equally  Corresponding author: mrkhan@unr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='edu  Abstract  Cotton threads and fabrics are the most used textile materials and have garnered  widespread interest for smart textiles to capture human-centered cyber-physical and  human-health-related bioanalytical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Cotton threads are sewn (manually or digitally)  into fabrics to achieve functional and fashion stitches that soften or stiffen the base fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  There has been limited investigation into the influence of a single stitch on the mechanical  properties of knitted cotton fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Such understanding may become critical to producing  optimized textile-based composites/smart materials involving sewing operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' While  stitching operations are investigated in numerous ways to produce a range of smart  wearables, herein, we demonstrate the rheological modification of base cotton fabric  induced by two types of singular stitches (straight and zigzag).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We have sewn simple  straight and zigzag cotton stitches to investigate the rheological modification of the base  cotton fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Uniaxial stress-strain experimental data, combined with constitutive  modeling (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=', three-network model, TNM) obtained from the calibration software  (MCalibration), revealed the feasibility of a data-driven approach to investigate the  rheological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Our experimental analyses, combined with the calibrated data,  suggest a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='99% confidence in assessing the influence of a single stitch on knitted cotton  fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We have also used distributed strain energy to analyze the mechanics and failure  of the base and stitched fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Our study may enable the design and study of integrating  smart threads in cotton fabrics to produce smart wearables, e-textile, biomedical and e-  fashion textiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Introduction  Knitted cotton fabrics have been utilized in everyday garment materials and  emerged as one of the popular base materials to generate smart wearables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In this article,  we reveal rheological parameters- also known as phenomenological parameters, of  cotton-stitch-modified cotton fabrics, harnessing Three Network Models (TNM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='1,2 We  produce two types of stitches for demonstrations to modify the mechanical behavior of  knitted cotton fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' While the utility of cotton fabrics is ubiquitous, and numerous  demonstrations are currently published in the literature, our study focuses on  understanding the tuned mechanics of cotton fabrics by sewn stitches and unravels data  that we often overlook through stress-strain analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Anisotropy of knitted cotton fabric  and its modified structural properties exhibited deformations during mechanical  performance analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='3 Several studies focused on understanding knit fabrics, fabric  elongation, deformation, and failures at critical applied stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='4–10 Advanced applications  on knitted fabrics are approached mainly by trial and error methods where in-plane  stitches are randomly generated, leaving a knowledge gap in understanding the impact  of final sewn stitches on fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Sewing- one of the ancient fabric manufacturing  techniques, loops a thread into fabrics, leveraging an analog or computerized sewing  machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Different sewing stages,11 sewing parameters,12, and sewing machines13,14  have also been reported to alter the properties of the sewing thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The looping process  integrates different threads and produces entanglements with aesthetic colors, body  shapes, and on-demand geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Different stitch patterns have also been reported to  change the rheological behavior of the sewing thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='15,16 However, the rheological  impact on the fabric due to stitching has not been adequately investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' While the  original purpose of sewing has been joining two pieces of fabric together, the most  advanced applications integrate smart threads so that biomedical, biochemical, and  biological analyses can be performed in situ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='17,18 From the design of fashion to human-  centered smart wearables, state-of-sewing leverages many stitch patterns;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" however, a  data-informed approach to dissect the role of sewn stitches in manipulating the final  fabrics' properties is currently lacking in the literature." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Using a sewing machine, sewn stitches create entanglements between two  threads- an upper and a bobbin thread, the bottom thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' During the entanglement, the  sewing needle loops the upper thread between the bottom thread through the fabric, and  the threads entangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' When the tension of both threads matches, the entanglement lays  in-plane of the fabric on both sides with minimal damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='19 The resulting stitch can be  varied in numerous ways to create functional and non-functional patterns based on the  types of sewing threads, fabric types, or choice of materials for the final composite  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='20 Concurrently, stitches are used to bind two or more layers of fabric together,  which is known as a seam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' To determine the impact of stitches on the mechanical  behavior of fabrics, a few research groups have tested seams in woven cotton fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='21–  23 A few investigations on the mechanical behavior of seams in knitted fabrics are also  available in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='24,25 However, studies involving a single stitch thread to modify  the mechanical behavior of cotton fabric are currently lacking for a single layer of fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Such studies, we believe, will become significantly crucial for future applications, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=',  electronic textiles- because reducing materials consumption at an optimum number of  trials and errors seems crucial to pursue robust design configurations during the  development of smart threads and electronic fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  The base fabric and thread used in this work are made from cotton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We assume  both as an elastomeric network for modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Elastomers having 3,000 to 10,000  repeating  units  exhibit  structural  flexibility  and  experience  stretch-induced  softening/hardening (also known as Mullins damage) under applied loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='26 For data  calibration, we use TNM in MCalibration software which maps the entire stress-strain  spectra from the uniaxial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The TNM is also knowns as a phenomenological model to  describe deformation-induced structural evolution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=', the transition between soft to stiff  network) and how strain-energy density becomes redistributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=', hysteresis)  throughout the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Different hyperelastic models have been utilized in the  literature to represent the rheological behavior of fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='27–31 However, according to our  knowledge, this work presents the constitutive modeling of stitched fabrics with TNM for  the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The data-calibration process in MCalibration can start using default or user-  induced settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' For this study, we have chosen to start calibration using default settings  in MCalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The kinematics of the TNM consists of three parallel molecular networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  We have assumed spring-dashpot domains connected in parallel for the first two, and the  third one is only a spring depicting the hyperelasticity of the first two networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The semi-  crystalline domains are captured through the spring dashpots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' While a single network can  be used to evaluate the property of the entire composite structure, we have chosen TNM  to capture effective viscoplasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='1  Here, we have chosen two types of sewing stitches: straight and zigzag, to  establish and reveal rheological parameters of cotton-stitch modified cotton fabrics,  harnessing TNM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' These stitches are common in sewn garments and are pre-programmed  into the default settings of modern sewing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Also, we investigate several  variations of the zigzag stitch that has varying stitch length and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' A commercial  sewing machine creates stitches with 100% cotton materials (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=', threads and fabrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  For our analyses, we investigate the surface topography of the fabric and samples with  sewn stitches using optical and scanning electron microscope (SEM) images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We perform  (a) uniaxial stress-strain and (b) repeated cyclic tests in an Instron to dissect the  mechanical behaviors of the (a) base fabric, (b) base threads, and (c) threads-laid-fabric  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The uniaxial stress-strain analyses have revealed three regions of interest  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=', elastic, yield, and viscoplastic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Also, we have investigated the permanent failure of  the entire composite (fracture) to find the extremities of experimental analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The cyclic  tests provide information about hysteresis, which we leverage to understand distributed  strain-energy density and the loss due to hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We outline calibration using TNM in  MCalibration to provide a simple route to test the impact of specific sewing patterns on  the mechanical behavior of the final fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" We hypothesize that the thread, which has  significantly denser strain energy, shifts the fabric's macroscale stress-strain behavior  after stitching." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We proved our hypothesis through the uniaxial test and then altered the  stitch length to investigate the factors that cause specific changes in the stress-strain  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The understanding developed by investigating the changes in mechanical  behavior can be used to optimize the mechanical properties of a composite made with  cotton thread and fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Materials and Methods  Our experiments were performed to determine a constitutive equation to represent the  behavior of cotton fabric with different types of stitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' This was accomplished by  analyzing uniaxial stress-strain curves for each component (cotton fabric and cotton  thread) and different variations of the overall composite (straight stitched fabric, zigzag  stitch, and fabric with stitching holes but no thread).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Materials  The fabric used in this experiment was 100% cotton jersey knit with a unit weight of 427  g/m2 obtained from Hobby Lobby Stores, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The measured thickness of the fabric was  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='45 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Similarly, the thread used in this experiment was a 50-weight, 4-ply, 100%  cotton thread of Sew-Ology Brand from Hobby Lobby Stores, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=', produced for machine  quilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The measured outer diameter of the thread was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='30 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The same thread was  used for the top and bobbin threads for all samples prepared and presented in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Sample Preparation  We used a Brother SE600 sewing machine from Amazon to create stitches of manually  adjustable length and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The tension was selected so that the tension on the bobbin  thread and the upper thread were equal, preventing the bottom thread from showing on  the top or vice-versa, as is common sewing practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' A swatch of fabric approximately 20  cm in length was cut with scissors and then sewn wale-wise with the appropriate type of  stitch for the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Two samples were prepared with stitching: straight stitched and  zigzag stitched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The straight stitch was 2 mm in length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The zigzag stitches were (listed  as stitch length x stitch width): 2x5 mm, 1x5 mm, 3x5 mm, 2x3 mm, and 2x4 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Several  samples without any sewn stitches were also prepared for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The fabric  samples were then cut to the same size with a Cricut Maker fabric cutter bought from  Amazon, allowing for accurate and reproducible sample cutting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' For every sample, the  fabric was cut to the dimensions of 6 cm in length by 2 cm in width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Sample cutting was  done carefully to keep the stitching in the center of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Damaged or samples with  uncentered stitches were discarded without any analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Image Acquisition  Olympus SZ61 Stereo-microscope loaded with an Amscope MU1000-HS camera was  used to capture the optical microscopic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Secondary electron images of the  samples were captured using a Thermo Scientific Scios 2 SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' For SEM imaging, small  representative samples were cut and loaded on the sample holder with double-sided  carbon tape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Attention was given to keeping the stitch undamaged while loading on the  holder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Since the samples were nonconductive, samples were sputter-coated with Gold  (Au) to create a ~10 nm layer on the surface of the sample before imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Further optical  images were captured with the camera on an iPhone 13 mini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Experimental Methods  The data was collected using an Instron 5982 test machine for uniaxial tensile testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  The tested area was 4 cm by 2 cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The extra centimeter on each side allowed the grip to  hold the sample during testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Each sample type was examined with tensile testing to  determine the stress and strain until failure at a 40 mm/min strain rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Cyclic testing of  four cycles was then conducted for specific samples up to a sustainable strain level for  that sample type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Samples with straight stitches could only withstand slightly more than  10% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Therefore, cyclic tests with straight stitches were conducted up to 10% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  For comparison, cyclic testing up to 10% strain was also conducted for unaltered fabric  samples and the 2x5 mm zigzag stitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Strain-energy Density Calculation  Using the trapezoidal rule, we calculated strain-energy density from the time-dependent  force and stress data at varying strain rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The area under the stress-strain or force-  strain curve is divided into equal-time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Each small area under the curve is added  until we reach the last data point to get the total area under the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The reported energy  density from different observations is the total after each experimental stress-strain  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Constitutive Modeling of Different Fabrics  An initial prediction of the strain energy density of the straight stitched sample was  obtained based on the data collected from the unaltered fabric and thread samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In  order to obtain the prediction, the strain energy density of the straight stitch sample and  the unaltered fabric was obtained by finding the area under the stress-strain curve of the  sample with the trapezoidal rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The strain energy density was calculated up to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5%  strain because the thread samples failed around 5% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The strain energy of the  samples was calculated by multiplying the strain energy density by the volume of the  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" The volume of the fabric was calculated using the sample's length, width, and  thickness." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The thread volume was calculated from the measured diameter and length of  the thread sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The straight stitch sample can be approximated by one sample of  fabric and two samples of thread, so the volume of the straight stitch sample was  calculated by adding the volume of the unaltered fabric and two threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Similarly, the  predicted strain energy of the straight stitch sample was calculated by adding the strain  energy of the unaltered fabric and two threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The prediction for the strain energy density  of the straight stitched sample could then be obtained by dividing the predicted stored  energy by the calculated volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  MCalibration, from PolymerFEM,32 was used to obtain parameters for a material model  capable of representing the mechanical behavior of the fabric samples prepared in this  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' MCalibration fits the experimentally collected uniaxial stress-strain data to the  PolyUMod Three Network model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='2 An average of the stress-strain behavior of each  sample type was obtained first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' This set of averaged data was then processed using the  MCalibration software tools to prepare the data for calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The default settings were  used for the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The calibrated parameters were exported and analyzed after the  automatic convergence of the calibration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Results and Discussion  Surface Topography  We formed two different types of stitches on the base fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Figures 1a and 1b are top-  down optical microscope images of the base and sewn fabrics for visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Figure 1b is a zoomed-in visual inspection of Figure 1a to identify differences between a  straight stitch and a zigzag stitch on the in-laid fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' These images show that the straight  and zigzag stitches went through the fabric without significant internal damage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The  straight stitch shown in Figures 1a(ii) and 1b(ii) do not have significant bunching due to  the stitch compared with the only fabric shown in Figures 1a(i) and 1b(i);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' However, a  meandering network of the zigzag stitches caused the fabric within the stitch to  significantly bunch together, as shown in Figures 1 a(iii) and 1b(iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The fabric is unable  to maintain its shape during sewing and is pulled into the stitch instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The structural  stiffness and flexibility of the fabric may have contributed to the bunching, as observed  within the stitch dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" Figure 1c is the sewn fabric's SEM images to investigate the  surface topography of the stitches and fabric." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' SEM images in Figure 1c(i) and Figure 1c(ii)  reveal the undamaged fabric by fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' From these visual inspections, we assume the  fabric remains structurally robust during the sewing and stitches only alter the mechanical  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Figure 1:  (a) Images were taken of samples under normal lighting conditions for visual inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  (b) A stereoscope was used to examine the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' (c) Secondary electron SEM images were  taken of the fabric and sewn stitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Uniaxial Tensile Behavior  We investigated plain thread, plain fabric, and stitched fabrics using Instron for  mechanical behavior analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The uniaxial tensile test behavior of plain thread is shown  a(i)  b(i)  c(i)  Cotton  Fabric  a(ii)  b(i)  500μm  Straight  Fabric<  Stich  Stitch  c(ii)  b(ili)  Zigzag  Stitch  5 mm  2 mm  500 μmin Figure 2a, and the plain fabric is shown in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' For comparison, Figure 2b also  shows the behaviors of straight and zigzag (2x5mm) stitched fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" Four other zigzag  stitched fabrics' behavior is shown in Figure 2c." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Figures 2d and 2e show the side view of  a zigzag stitched fabric loaded into Intron during tensile testing and at the end of failure  analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content="  We tested two samples of the plain threads, and both samples' behavior is shown  in Figure 2a." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The plain thread failed at 5% strain but exhibited the highest strain energy  density compared to other samples tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In contrast to the plain thread, the cotton fabric  in Figure 2b exhibited reproducible stretchability of up to 70% strain in two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The  inclusion of straight stitches into the plain fabric induced failure at ~12% strain, and the  zigzag stitched sample failed at ~32% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  The unaltered fabric had the highest stain at the point of failure, shown in Figure  2b, between 60-80%, with a strain energy density of around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='0 MJ/m3 at failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In  comparison, the thread samples had a strain energy density of approximately 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='0 MJ/m3  at failure, which occurred at around 5% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The strain energy density of the unaltered  fabric and thread at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5% were 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='19x10-4 MJ/m3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='64 MJ/m3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Examining  the stress-strain data for the cotton fabric and the cotton thread individually, we conclude  that combining these materials would result in a sample with a strain energy density that  falls between the different materials at a given strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The samples with sewn straight  stitches of 2mm length failed between 10-15% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' At a strain of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5%, the straight  stitched samples exhibited an average strain energy density of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='88x10-3 MJ/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' A  prediction of the strain energy density of the straight stitched samples at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5% was  obtained using the experimental values of the thread and fabric alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The predicted  value was 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='2x10-2 MJ/m3, more significant than the measured strain energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' This  discrepancy is expected as the sewing process exposes the thread to dynamic loads and  friction known to reduce the strength of the thread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content="33 Overall, the sample with sewn  straight stitches failed at all fabric samples' lowest stress and strain." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The cause of the low  stress and strain at failure is suspected to be the structure of the stitches, which cannot  withstand as much strain as the fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The fabric, with a higher elongation at failure than  the thread, can deform under the load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Therefore, the thread in the straight stitch  withstands the load for the entire sample until the thread breaks, equivalent to sample  failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' It was observed that the thread failed before the fabric in all samples with straight  stitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Samples with zigzag stitches of 2mm length and 5mm width also failed at stress  and strain lower than the unaltered fabric but higher strain than the straight stitched  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' An analysis of the strain energy density of the 2x5mm zigzag sample reveals  aspects of the mechanical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' At strains below 20%, the strain energy density of  the 2x5mm zigzag sample is indistinguishable from the strain energy density of the fabric;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content="  Therefore, the fabric's mechanical properties dominate the thread's properties in the  2x5mm zigzag sample at strains under 20%." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' At 30% strain, the strain energy density of  the zigzag sample is nearly double that of the fabric sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The departure of the 2x5mm  zigzag sample from the mechanical behavior of the fabric indicates that at strains above  20%, the thread is the dominant influence on the mechanical behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' This behavior is  investigated further in zigzag samples with varying stitch lengths and widths, as indicated  in Figure 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Figure 2: (a) The graph of the stress-strain curve for the samples of the cotton thread indicates  maximum stress of approximately 100MPa at a strain of approximately 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5% before failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' (b)  The graph shows the stress-strain curves of the unaltered fabric, fabric with straight stitches of  2mm length, and fabric with zigzag stitches of a length of 2mm and a width of 5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' (c) The stress-  strain graph shows the impact of varying the properties of zigzag stitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' (d) An image of a  sample with 1x5mm zigzag stitches shows the condition of the sample before uniaxial tensile  loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' (e) An image of a sample with 1x5mm zigzag stitches shows the condition of the sample  after uniaxial tensile loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Notably, the fabric has failed while the sewn thread is intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  During uniaxial tensile testing, it was revealed that stitch length and width are both critical  factors that influence the tensile behavior of the samples with zigzag stitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Figure 2  (c) shows stress-strain curves for samples with zigzag stitches of varying length and  width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The fabric samples with 2x5mm, 2x4mm, and 3x5mm zigzag stitches failed at a  higher strain than those with straight stitches but at a similar stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' As with the 2x5mm  sample, the 2x4mm and 3x5mm had similar strain energy densities at low strain until the  thread became a dominant influence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The 1x5mm zigzag sample exhibited drastically  (a)  25  Thread Sample 1  b  FabricOnly 1  ThreadSample2  ★—FabricOnly2  Straight Stitch 1  100  8 -  Straight Stitch 2  —2x5mm Zigzag Stitch 1  Stress (MPa)  Stress (MPa)  75  2x5mm Zigzag Stitch 2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  50-  4  25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  0  2  6  8  10  0  20  40  60  80  100  Strain (%)  Strain (%)  (c)  10  (d)  2x4mm  e  一1x5mm  2x3mm  8-  ★一3x5mm  FabricStress (Mpa)  6  Zigzag  Stitch  Failure  4  Point  2-  Sample  Grip  0：  0  20  40  60  80  100  Strain (%)different behavior from the other samples with zigzag stitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The strain energy density  of the 1x5mm zigzag samples matched that of the unaltered fabric sample up to  approximately 60% strain, indicating that the stitches had little impact on the tensile  behavior of the sample overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The 1x5mm samples also had more extended elongation  at failure than the unaltered fabric sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The structure of the zigzag stitch contributes  to the behavior of all the zigzag samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Since zigzag stitches have both a stitch length  and a stitch width, the stitch could change shape as the fabric elongates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Figures 2(d) and (e) show a 1x5mm zigzag sample before and after uniaxial tensile  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' After testing, the stitches are longer in the direction parallel to loading and shorter  in the direction perpendicular to loading compared to before tensile testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In other  words, the stitch could shrink in the direction perpendicular to loading while elongating in  the direction of loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' A shorter stitch length results in more threads in the sample,  which allows the stitches to deform enough to match the elongation of the fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The  consequence of the stitch deformation is that the fabric withstands the load while the stitch  can deform, but the stitch bears the load when it is no longer able to match the elongation  of the fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Eventually, the load exhausts the ability of the stitch to deform, which is  when the strain energy density of the zigzag sample deviates from that of the unaltered  fabric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' It was observed that the sewn thread had snapped in all samples after the sample  had failed during tensile testing, except for the 1x5mm zigzag sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The 1x5mm zigzag  sample in Figure 2(c) had a shorter stitch length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Another point of interest is shown in  Figure 2(e), which shows that the thread was intact after the fabric failed, which is the  opposite of all other samples that contained sewn stitches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" Therefore, it is possible to  alter stitch properties to alter the fabric's tensile behavior, and the properties determine  the extent of the influence from the thread and the fabric at particular strains." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Repeated Cycling Behavior  The unaltered fabric sample, straight stitch sample, and 2x5mm zigzag stitch sample  were examined under cyclic loading to analyze stress softening and hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Any  fabrics are subjected to cyclic loading during use from body movements such as the  expansion of the chest during breathing or the movement of joints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' An analysis of the  behavior of the fabric samples during cyclic loading provides information that can inform  design decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' All samples were strained up to 10% because the straight stitch  samples failed at approximately 12% strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Hysteresis, the change in behavior from the  loading to the unloading cycle, was observed in all samples, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Across  all samples, the most extensive hysteresis occurred during the first cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Additionally, all  samples had the highest strain energy density during the loading of the first cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" The  hysteresis between the loading and unloading cycle of the overall sample is impacted by  the relationship between the yarns' properties and the fabric's structure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The plastic  deformation of the yarns, which relates to the slippage and viscoelasticity of the fibers  within the yarn, influences hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='10 The structure dictates the number and nature of  the contact points between loops of thread, which impacts the friction during loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Friction is the main factor determining the amount of hysteresis that will occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5 In the  samples with stitches, the causes of tensile hysteresis are further complicated by the  presence the stitched threads, which impact the overall properties and structure of the  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In Figure 3b, the straight stitch sample showed more hysteresis than the zigzag  stitched sample shown in Figure 3c, indicating that the straight stitched threads  experienced more plastic deformation than the zigzag stitched threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The difference in  the plastic deformation experienced in the threads relates to the behavior observed in the  uniaxial tensile testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The straight-stitched thread sustains more of the load for the entire  sample than the zigzag stitch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' the thread in the straight-stitched sample experiences more  plastic deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Repeated cycles allow for an investigation of the hysteresis in  additional cycles and an analysis of the stress-softening behavior of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The  second cycle revealed that stress softening occurred in all samples between the first and  second cycles, which can be observed in whole Figure 3 as a reduction in the strain  energy density of the loading curve between the first and second cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The unaltered  fabric sample in Figure 3a showed a minor stress softening, which can be attributed to  the significant difference between the maximum strain during cyclic loading and the strain  required to cause failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Since the unaltered fabric sample has minor unrecoverable  deformation at the 10% strain tested in this experiment, minimal stress softening  occurred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In additional cycles after the second cycle, hysteresis in the fabric sample and  the zigzag sample remained the same;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' however, hysteresis decreased slightly in the  straight stitched sample from the second to the third cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The decrease in hysteresis is  attributable to the stress softening in the straight stitch sample between the second and  the third cycles, which indicates that further unrecoverable deformation occurred during  each cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' In comparison, the fabric and the zigzag stitch samples do not experience  significant unrecoverable deformation in cycles after the second cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Figure 2: (a) The unaltered fabric sample showed less stress softening than the 2x5mm zigzag  stitch sample but still showed hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' (b) The straight stitch sample had the most stress  softening and also showed hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' (c) The cyclic loading of the 2x5mm zigzag stitch sample  showed stress softening after the first cycle and hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='06  OnlyFabricCycle1  OnlyFabricCycle2  OnlyFabric Cycle3  ★一  Only Fabric Cycle 4  Stress (Mpa)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='04 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='00+  0  10  5  Strain (%)  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='0  StraightStitchCycle1  Straight StitchCycle2  Straight Stitch Cycle3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='8 -  ★一  Straight Stitch Cycle 4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='2 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='0 +  5  10  Strain (%)  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='06  2x5mmZigzagCycle1  —2x5mmZigzagCycle2  2x5mmZigzagCycle3  ★一  2x5mmZigzagCycle4  Stress (Mpa)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='00  0  5  10  Strain (%)Revealing rheological parameters of Fabric and Composite Systems  The TNM is a powerful constitutive model capturing the flow and deformation (rheology)  behaviors of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Bergstrom and Bischoff explained the mathematical details of the  TNM in their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='1 While the stress-strain analysis directly measures the mechanical  behavior, the rheological parameters we often overlook in stress-strain analyses can be  revealed through constitutive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Studies on such parameters also enable data-  informed design decisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  We used MCalibration software to perform rheological analyses using TNM and calibrate  the TNM parameters to assess unaltered and altered fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' MCalibration software  begins calibration with a set of initially estimated parameter values by observing the  experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' It tries to reduce the deviation between the predicted and the  experimental behavior by continuously updating the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' This process is also  known as data calibration and rheological parameter identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' When the coefficient  of determination or the R2 value stops changing significantly by reaching convergence,  the software reveals the rheological parameters in its user interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The experimental  data and MCalibration predicted data with their respective R2 fitness are shown in Figure  4, indicating that the TNM model effectively captures the uniaxial tensile behavior of  unaltered, straight- and zigzag-stitched fabrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The predicted data fits closely with the  experimental data for all investigated samples with this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The prediction of the  2x5mm zigzag sample in Figure 4a matched with an R2 fitness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='999, which was a  closer fit than the unaltered fabric sample or the straight stitch sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The reason for the  closer match indicates that the 2x5mm zigzag sample had behavior closest to that of a  thermoplastic polymer, which is the material on which the TNM is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Furthermore,  the calibration calculates the material model parameters, revealing information about the  behavior of the samples that cannot be determined from an analysis of the experimental  data alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Table 1 shows several such parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Figure 3: The material calibration with the PolyUMod TNM resulted in a good prediction for (a)  the unaltered fabric sample, (b) the 2mm straight stitch sample, and (c) the 2x5mm zigzag sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  (a)  3  ExperimentalFabricData  Predicted Fabric Data  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  Stress  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  R2Fitness=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='997  0  0  10  20  3040  50  6070  Strain (%)  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  ExperimentalStraightStitchData  Predicted Straight StitchData  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='25  (MPa)  1  stress  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='25  R2Fitness=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='99  0  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  10  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  Strain (%)  (c)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='7  Experimental2x5mmZigzagData  Predicted2x5mmZigzagData  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='5  (edw) :  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='1  R2Fitness=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='999  0  0  5  10  15  520  25  30  Strain (%)Table 1: The Three-Network Model (TNM) parameters of unaltered fabric, straight  stitch, and the 2x5 mm Zigzag stitch  Description  Symbol  Unit  Unaltered  Fabric  Straight Stitch  2x5 mm Zigzag  Shear modulus of network  A  𝜇�  KPa  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='40  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='65  Locking stretch  𝜆� 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='08  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='04  Bulk modulus  𝜅  KPa  656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='37  1369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='34  1194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='72  Flow resistance of network  A  𝜏̂�  KPa  127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='80  901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='91  352.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='31  Stress exponential of  network A  𝑚� 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='83  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='11  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='59  Initial shear modulus of  network B  𝜇��  KPa  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='46  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='05  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='75  Final shear modulus of  network B  𝜇��  KPa  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='46  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='31  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='99  Evolution rate of 𝜇�  𝛽 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='69  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='20  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='50  Flow resistance of network  B  𝜏̂�  KPa  348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='78  1226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='80  636.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='76  Stress exponential of  network B  𝑚� 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='89  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='65  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='85  Shear modulus of network  C  𝜇�  KPa  398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='95  1180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='98  207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='47  Earlier investigation27  on assessing the hyperelastic material model calibrated  parameters, leveraging Mooney-Rivlin, Ogden, neo-Hookean, Arruda Boyce, Gent, Yeoh,  and Blatz-Ko constitutive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The higher-order Mooney-Rivlin and Yeoh models fitted  the experimental data properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The Arruda-Boyce model also showed good relation with  the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Also, we noticed a similarity in the stress-strain behavior from that  investigation that is close to our unaltered fabric behavior shown in Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We want  to compare the parameters we obtained with that literature27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We noted a shear modulus  of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='8913 KPa, and a limiting locking stretch (𝜆�,���� of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='65907 from that investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  The Cauchy stress acting on any networks in the TNM model is based on the Arruda-  Boyce or eight-chain model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='1 The reported shear modulus and the shear modulus of the  Network A of the unaltered fabric are also not significantly different here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' As the shear  modulus of the Arruda-Boyce model gets distributed in three networks, we should only  compare the locking stretch directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The locking stretch is defined as the ratio of the  current chain length and the initial chain length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' From the literature, the relation between  the locking stretch (𝜆�� and limited locking stretch can be found,34,35, which is  𝜆� � �1  3 �𝜆�,���  �  �  2  𝜆�,���  �  The reported limiting locking stretch converted to 𝜆� will be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='0753, which is very close to  our reported locking stretch value of the unaltered fabric, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Additionally, for all the  samples, the locking stretch was close to 1, indicating that the sample did not go through  a significant strain level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The locking stretch values of the straight stitch and the zigzag  stitch are also smaller than the unaltered fabric, indicating less deformation observed in  Figure 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The final calibrated parameters depend significantly on the initially guessed  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' It would be easier to compare the parameters between three samples if an  identical set of initial values was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' As we are using the uniaxial tensile testing here,  bulk modulus should not impact the predicted behavior significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' 36 In the TNM,  network A and B utilize separate energy activation mechanisms to represent the  amorphous and semi-crystalline domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Network C represents the large strain response  controlled by entropic resistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The shear modulus and the flow resistance of network  A in the straight stitch are significantly higher than the other two samples indicating higher  resistance by the spring represented in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Figure 4(b) also indicates that up to  10% strain straight-stitched fabric is stiffer than the other two matching the observation in  the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Comparatively close initial and final shear modulus of network B and  almost similar evolution rates indicate a similar effective shear modulus for all the  samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The flow resistance of network B and the shear modulus of network C of the  straight-stitched sample are also higher, indicating higher stiffness of the materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Conclusion  This work determined that altering the parameters of the stitching when sewing  with cotton thread into a single layer of jersey-knit cotton fabric impacts the strain-energy  density, hysteresis, and stress softening of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' When examined with optical and  scanning electron microscopes, the stitched samples did not show damage to the fabric  from the sewing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" The stitch type and parameters of a zigzag stitch were shown  to directly impact the sample's behavior under uniaxial tensile loading." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Depending on the  stitch type, the fabric can be altered to have a higher or lower strain energy density at  certain strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' We also note that stitches capable of less elongation than the fabric will  increase the strain energy density at lower strains and result in failure at a lower strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Stitches that can match or exceed the elongation may have minimal impact on the strain  energy density of the sample at the same strains as a sample without stitches but will fail  at higher strains, resulting in a higher strain energy density at failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Stitches will also  impact the hysteresis and stress softening of the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Also, stitches capable of less  elongation than the fabric will be subjected to higher stress during loading, resulting in  plastic deformation and more significant hysteresis and stress softening during cyclic  loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' The tensile and cyclic tests reveal that the mechanical behavior of samples  composed of fabric with stitches varies greatly depending on the relationships between  the property of the materials and their structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' When data from tensile tests were  calibrated with the PolyUMod TNM, the materials presented in this work matched well  with the calibrated model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' therefore, materials calibration provides an opportunity to aid  the selection of materials and structure by offering insight into hidden parameters that  allow for a data-driven approach to design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  Limitations of this work include the number of materials and structures investigated, as  the behavior observed may differ from samples with different compositions and  structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Furthermore, many other properties may be impacted by the presence of  sewing stitches that were not investigated in this paper, such as abrasive strength,  bursting strength, torsional properties, ability to withstand washing and drying, and many  other characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Future works may investigate the impact of additional types of  stitches on fabrics of different materials and structures and analyze additional properties  of the samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content="  Acknowledgments  MRK acknowledges the funding support from VPRI's startup account." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" HW  acknowledges the Nevada undergraduate research award (NURA) fund from the  Undergraduate Research Office, and KZH acknowledges funding from the College of  Engineering Dean's Office at the University of Nevada, Reno." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' HW acknowledges  contributions from Sydney Fields, Jake Kattelman, Thomas Kaps, and Braden Norris for  the MSE 470 (Polymer Engineering instructed by MRK) in-class project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' KZH  acknowledges the opportunity to train and mentor all the groups in CHE/MSE 470 and  Brian Perdue in CHE 495 using the concepts from this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=" MRK acknowledges the  support received from Dean's Office to purchase Instron 5982 with Dr." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Jefferey Lacombe,  Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Bin Li, and Zachary Karmiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' Bergstrom JS, Bischoff JE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' An Advanced Thermomechanical Constitutive Model for  UHMWPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
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+page_content=' MCalibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' PolymerFEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
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+page_content=' Mechanics of Solid Polymers: Theory and Computational Modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
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+page_content=' Nguyen H-D, Huang S-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
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+page_content=' Jorgen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' How Important is the Bulk Modulus in FEA?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content=' PolymerFEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='com,  https://polymerfem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
+page_content='com/how-important-is-the-bulk-modulus/ (2021, accessed 17  November 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/BNFQT4oBgHgl3EQfNDZv/content/2301.13270v1.pdf'}
diff --git a/CdAyT4oBgHgl3EQfePjS/content/tmp_files/2301.00319v1.pdf.txt b/CdAyT4oBgHgl3EQfePjS/content/tmp_files/2301.00319v1.pdf.txt
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+Quantum Hairy Black Hole Formation and Horizon Quantum Mechanics
+R. T. Cavalcanti∗ and J. M. Hoff da Silva†
+Departamento de F´ısica, Universidade Estadual Paulista (Unesp), Guaratinguet´a 12516-410, Brazil
+After introducing the gravitational decoupling method and the hairy black hole recently derived
+from it, we investigate the formation of quantum hairy black holes by applying the horizon quan-
+tum mechanics formalism. It enables us to determine how external fields, characterized by hairy
+parameters, affect the probability of spherically symmetric black hole formation and the generalized
+uncertainty principle.
+I.
+INTRODUCTION
+Given their intrinsic connection with intense gravitational fields, solid theoretical basis [1–3], and several observa-
+tional results corroborating their existences, black holes play a central role in contemporary high-energy physics and
+astrophysics [4–7]. Despite the characterization of the horizon of stationary black hole solutions being well-known
+within general relativity [3, 8], the nature of the horizons of non-stationary or stationary solutions beyond general
+relativity is still a source of extensive research [9–12]. The investigation of black holes is not restricted to astrophysical
+objects; they are also expected to be formed whenever a high concentration of energy is confined to a small region of
+spacetime, producing so-called quantum black holes [7, 13–17]. However, the precise formation mechanism of classical
+and quantum black holes is still unknown. Although we do not have a theory of quantum gravity, phenomenology
+suggests that some features of quantum black holes are expected to be model-independent [7]. From a certain scale,
+candidate theories should modify the results of general relativity, giving birth to some alternatives to Einsteins’s
+theory of gravity [18, 19]. Examples could allow for the presence of non-minimal coupled fundamental fields or higher
+derivative terms during the action, which directly affects the uniqueness theorems of black holes in general relativity.
+The famous no-hair theorem is not preserved outside the general relativity realm. These solutions lead to effects that
+are potentially detectable near the horizon of astrophysical black holes [20–22], or in quantum black holes’ formation
+[23, 24], and may provide hints for the quantum path.
+One of the major challenges in general relativity is finding physically relevant solutions to Einstein’s field equations.
+On the other hand, deriving new solutions from other previously known ones is a widespread technique. This approach
+is precisely what the so-called gravitational decoupling (GD) method intends to achieve. It has recently commanded
+the community’s attention due to its simplicity and effectiveness [25–27] in generating new, exact analytical solutions
+by considering additional sources to the stress-energy tensor. The recent description of anisotropic stellar distribu-
+tions [28, 29], whose predictions might be tested in astrophysical observations [30–33], as well as the hairy black hole
+solutions by gravitational decoupling, are particularly interesting. The latter describes a black hole with hair sourced
+by generic fields, possibly of quantum nature, surrounding the vacuum Schwarzschild solution [27]. Exciting results
+have been found during investigation of this solution [34–36].
+From the quantum side, one of the key features of quantum gravity phenomenology is the generalized uncertainty
+principle (GUP), which modifies the Heisenberg uncertainty principle accordingly
+∆x∆p ≳ ℏ
+�
+1 + ϵ(∆p)2�
+.
+(1)
+This expression of the GUP, which stems from different approaches to quantum gravity [37–46], characterizes a
+minimum scale length ∆x. This feature emerges quite naturally in the horizon quantum mechanics formalism (HQM)
+[16, 47]. In addition to the GUP, HQM also provides an estimation of the probability of quantum black hole formation.
+In a scenario of extra-dimensional spacetimes, the HQM gave an explanation for the null results of quantum black
+hole formation in current colliders [23, 24]. Could it also tell us something about a mechanism for decreasing the
+fundamental scale to something near the scale of current colliders? Our aim is to investigate the quantitative and
+qualitative effects of black hole hair, regarding the probability of black hole formation and the GUP by applying the
+horizon quantum mechanics formalism.
+This paper is organized as follows: Section II is dedicated to reviewing the gravitational decoupling procedure, the
+metric for GD hairy black holes, and an approximation for the horizon radius. In Section III, we apply the horizon
+∗Electronic address: rogerio.cavalcanti@unesp.br
+†Electronic address: julio.hoff@unesp.br
+arXiv:2301.00319v1  [gr-qc]  1 Jan 2023
+
+2
+quantum mechanics formalism to the hairy black hole solution of the previous section. We compare the probability of
+quantum black hole formation and the GUPs of hairy black holes for a range of hair parameters, unveiling the effects
+of the hair fields. Finally, Section IV is dedicated to conclusions and discussion.
+II.
+HAIRY BLACK HOLES AND HORIZON RADIUS
+Starting from Einstein’s field equations
+Gµν = 8π ˇTµν,
+(2)
+where Gµν = Rµν − 1
+2Rgµν denotes the Einstein tensor, the gravitational decoupling (GD) [25] method takes the
+energy–momentum tensor decomposed as
+ˇTµν = Tµν + Θµν.
+(3)
+Here, Tµν is the source of a known solution to general relativity, while Θµν introduces a new field or extension of the
+gravitational sector. From ∇µ Gµν = 0, we also have ∇µ ˇT µν = 0. The effective density and the tangential and radial
+pressures can be determined by examining the field equations
+ˇρ = ρ + Θ 0
+0 ,
+(4a)
+ˇpt = p − Θ 2
+2 ,
+(4b)
+ˇpr = p − Θ 1
+1 .
+(4c)
+The idea is to deform a known solution to split the field equations in a sector containing the known solution with
+source Tµν and a decoupled one governing the deformation, encompassing Θµν. In fact, assuming a known spherically
+symmetric metric,
+ds2 = −eκ(r)dt2 + eζ(r)dr2 + r2dΩ2,
+(5)
+and deforming κ(r) and ζ(r) as
+κ(r) �→ κ(r) + αf2(r),
+(6a)
+e−ζ(r) �→ e−ζ(r) + αf1(r),
+(6b)
+the resulting decoupled field equations read
+8π Θ 0
+0
+= α
+�f1
+r2 + f ′
+1
+r
+�
+,
+(7a)
+8π Θ 1
+1 − α e−ζ f ′
+2
+r
+= α f1
+� 1
+r2 + κ′(r) + αf ′
+2(r)
+r
+�
+,
+(7b)
+8πΘ 2
+2 −αf1Z1(r) =αf ′
+1
+4
+�
+κ′(r) + αf ′
+2(r)+ 2
+r
+�
++αZ2(r),
+(7c)
+where [25]
+Z1(r) = α2f ′
+2 (r)2 + 2 α
+�
+f ′
+2 (r) κ′ (r) + f ′
+2 (r)
+r
++ f ′′
+2 (r)
+�
++ κ′ (r)2 + 2 κ′ (r)
+r
++ 2 κ′′ (r) ,
+(8a)
+Z2(r) = αe−ζ
+�
+2f ′′
+2 + f 2′
+2 + 2f ′
+2
+r
++ 2κ′f ′
+2 − ζ′f ′
+2
+�
+.
+(8b)
+The above equations state that if the deformation parameter α goes to zero, then Θµν must go to zero. It is worth
+mentioning that for extended geometric deformation, that is, for f2 ̸= 0, the sources are not individually conserved
+in general. However, as discussed in [26], in this case, the decoupling of the field equations without an exchange of
+energy is allowed in two scenarios: (a) when Tµν is a barotropic fluid whose equation of state is T00 = T11 or (b) for
+vacuum regions of the first system Tµν = 0. When minimal geometric deformation is applied, on the other hand, the
+sources are shown to be individually conserved [25, 26].
+Assuming the Schwarzschild solution to be the known one and requiring a well-defined horizon structure [27], from
+grr = − 1
+gtt follows
+�
+1 − 2M
+r
+� �
+eαf2(r) − 1
+�
+= αf1(r).
+(9)
+
+3
+Therefore, one is able to write
+ds2 = −
+�
+1 − 2M
+r
+�
+eαf2(r)dt2+
+�
+1 − 2M
+r
+�−1
+e−α f2(r)dr2 + r2 dΩ2.
+(10)
+Further, assuming strong energy conditions,
+ˇρ + ˇpr + 2 ˇpt ≥ 0,
+(11a)
+ˇρ + ˇpr ≥ 0,
+(11b)
+ˇρ + ˇpt ≥ 0,
+(11c)
+and managing the field equations, a new hairy black hole solution was found [27]
+ds2 = −f(r)dt2 +
+1
+f(r)dr2 + r2dΩ2,
+(12)
+where
+f(r) = 1 − 2GM + αℓ
+r
++ αe−
+r
+GM .
+(13)
+The dimensionless parameter 0 ≤ α ≤ 1 tracks the deformation of the Schwarzschild black hole, e is the Euler constant,
+and ℓ is the direct effect of the nonvanishing additional font Θµν. Notice that by taking α = 0, the Schwarzschild
+solution is restored. Further, the ℓ parameter is limited to 2GM/e2 ≤ ℓ ≤ 1 due to the assumption of a strong energy
+condition. In extreme cases, ℓ = 2GM/e2 and
+fe(r) = 1 − 2GM
+r
++ α
+�
+e−
+r
+GM − 2GM
+e2 r
+�
+.
+(14)
+The hairy black hole has a single horizon, located at r = rH, such that
+�
+1 + αe− rH
+GM
+�
+rH = 2GM + αℓ.
+(15)
+Such an equation has no analytical solution. Nevertheless, a very accurate analytical approximation is found by Taylor
+expanding it around the Schwarzschild horizon radius rS = 2GM,
+rH
+GM ≈ 4
+�
+αℓe2/GM − 3 α + e2�
+αℓe2/GM − 4 α + 2 e2 .
+(16)
+Figure 1 shows a comparison between the exact and approximated horizon radii for different values of the hairy
+parameters. In the following section, we are going to use Equation (16) for the analytical expression of the hairy black
+hole’s horizon radius.
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+ℓ
+GM
+2.0
+2.1
+2.2
+2.3
+2.4
+2.5
+2.6
+2.7
+2.8
+rH
+GM
+α = 0.00
+α = 0.20
+α = 0.40
+α = 0.60
+α = 0.80
+α = 1.0
+Exact
+FIG. 1: The radius of the hairy black hole horizon rH as a function of ℓ for different values of the parameter α. The colored
+dashed lines represent the approximated radius, and the gray lines are the exact ones. It shows how the hairy horizon deviates
+from the Schwarzschild horizon for an increasing α and ℓ. The ranges for α and ℓ were fixed due to the assumption of a strong
+energy condition [27].
+
+4
+III.
+THE HORIZON QUANTUM MECHANICS FORMALISM
+Horizon quantum mechanics (also known as horizon wave function formalism) is an effective approach capable of
+providing the signatures of black hole physics to the Planck scale [48–51] (see [47] for a comprehensive review). The
+main idea is to extend quantum mechanics and gravity further than the current experimental limits. In such an
+approach, we face the conceptual challenge of consistently describing classical and quantum mechanical objects, such
+as horizons and particles. This is achieved by assigning wave functions to the quantum black hole horizon. This
+association allows the use of quantum mechanical machinery to distinguish between particles and quantum black
+holes and to estimate the GUPs. Nevertheless, first, we must choose a model describing the particle wave function to
+derive the results. Due to the previous results’ simplicity and efficiency, we shall use the Gaussian model.
+From classical general relativity, we know that the horizons of black holes are described by trapping surfaces, whose
+locations are determined by
+gij∇ir∇jr = 0 ,
+(17)
+where ∇ir is orthogonal to the surfaces of the constant area A = 4πr2. A trapping surface then exists if there are
+values of r and t such that the gravitational radius RH satisfies
+RH(r, t) ≥ r .
+(18)
+Considering a spinless point-particle of mass m, an uncertainty in the spatial particle localization of the same order
+of the Compton scale λm ≃ ℏ/m = lp mp/m follows from the uncertainty principle, where lp and mp are the Planck
+length and mass, respectively. Arguing that quantum mechanics gives a more precise description of physics, RH makes
+sense only if it is larger than the Compton wavelength associated with the same mass, namely RH ≳ λm. Thus, for
+the Schwarzschild radius RS = 2Gm = 2 lp
+mp m,
+lp m/mp ≳ lp mp/m
+=⇒
+m ≳ mp .
+(19)
+This suggests that the Planck mass is the minimum mass such that the Schwarzchild radius can be defined.
+From quantum mechanics, the spectral decomposition of a spherically symmetric matter distribution is given by
+the expression
+|ψS⟩ =
+�
+E
+C(E) |ψE⟩ ,
+(20)
+with the usual eigenfunction equation
+ˆH |ψE⟩ = E |ψE⟩ ,
+(21)
+regardless of the specific form of the actual Hamiltonian operator ˆH. Using the energy spectrum and inverting the
+expression of the Schwarzschild radius, we have
+E = mp
+rH
+2lp
+.
+(22)
+Putting it back into the wave function, one can define the (unnormalized) horizon wave function as
+ψH(rH) = C
+�
+mp
+rH
+2lp
+�
+(23)
+whose normalization is fixed, as usual, by the inner product
+⟨ψH | φH⟩ = 4π
+� ∞
+0
+ψ∗
+H(rH)φH(rH)r2
+HdrH.
+(24)
+However, the classical radius RH is thus replaced by the expected value of the operator ˆRH. From the uncertainty
+of the expectation value, it follows that the radius will necessarily be “fuzzy”, similar to the position of the source
+itself. The next aspect one has to approach to establish a criterion for deciding if a mass distribution does or does
+not form a black hole is if it lies inside its horizon of radius r = rH. From quantum mechanics, one finds that it is
+given by the product
+P<(r < rH) = PS(r < rH)PH(rH),
+(25)
+
+5
+where the first term,
+PS(r < rH) = 4π
+� rH
+0
+|ψS(r)|2r2dr,
+(26)
+is the probability that the particle resides inside the sphere of radius r = rH, while the second term,
+PH(rH) = 4πr2
+H|ψH(rH)|2
+(27)
+is the probability density that the value of the gravitational radius is rH. Finally, the probability that the particle
+described by the wave function ψS is a BH will be given by the integral of (25) over all possible values of the horizon
+radius rH. Namely,
+PBH =
+� ∞
+0
+P<(r < rH)drH,
+(28)
+which is one of the main outcomes of the formalism.
+A.
+Gaussian Sources
+The previous construction can be made explicit by applying the Gaussian model for the wave function. To implement
+this idea, let us recall that spectral decomposition is also assumed to be valid for momentum. Therefore, from (20),
+⟨p |ψS⟩ = C(p) ≡ ψH(p). The Gaussian wave function for ψS scales as r2 in the position space and leads to a Gaussian
+wave function in the momentum space, scaling as p2, naturally. Finally, since the dispersion relation relates p2 with
+energy, we are able to have ⟨p |ψS⟩ = ψH(rH) via (22). Hence, starting with a Gaussian wave function, we can describe
+a spherically symmetric massive particle at rest, such as
+ψS(r) =
+e− r2
+2 l2
+(l √π)3/2 .
+(29)
+The corresponding function in momentum space is thus given by
+˜ψS(p) = 4π
+� ∞
+0
+sin(rp)
+√
+8π3rp
+e− r2
+2 l2
+(l √π)3/2 r2dr
+=
+e−
+p2
+2 ∆2
+(∆ √π)3/2 ,
+(30)
+where ∆ = mp lp/l is the spread of the wave packet in momentum space, whose width l the Compton length of the
+particle should diminish,
+l ≥ λm ∼ mp lp
+m
+.
+(31)
+In addition to the straightforward handling of a Gaussian wave packet, it is also relevant to recall that the Gaussian
+wave function leads to a minimal uncertainty for the expected values computed with it. Had we used another wave
+function, it would certainly imply a worsening uncertainty, eventually leading to unnecessary extra difficulties relating
+to the HQM and GUP (see next section). Back to our problem, assuming the relativistic mass-shell relation in flat
+space [48]
+p2 = E2 − m2 ,
+(32)
+the energy E of the particle is expressed in terms of the related horizon radius rH = RH(E), following from Equation
+(16),
+E = αmpℓe2 +
+�
+α − e2�
+mprH
+2 (2 α − e2)lp
+.
+(33)
+
+6
+Thus, from Equations (30) and (33), one finds the the horizon wave function of the hairy black hole
+ψH(rH) = NHΘ(rH − RH) e(C2r2
+H+C1rH+C0),
+where
+C0 = − α2l2m2
+pℓ2e4
+8 (2 α − e2)2l2p
+,
+C1 = −
+�
+α − e2�
+αl2m2
+pℓe2
+4 (2 α − e2)2l2p
+,
+C2 = −
+�
+α − e2�2l2m2
+p
+8 (2 α − e2)2l2p
+.
+(34)
+The Heaviside step function Θ appears above due to the imposition E ≥ m. The normalisation factor NH is fixed
+according to
+N −2
+H
+= 4π
+� ∞
+0
+|ψH(rH)|2 r2
+H drH.
+The normalized horizon wave function is thus given as follows
+ψH(rH) = −
+2 C
+3
+2
+2 e
+A(rH )
+2
+√π
+�
+4 C1C2eA(RH) −
+�
+2
+√
+2C2Γ
+� 3
+2 , −A(RH)
+�
++
+√
+2πC2
+1
+�
+erf
+� √
+2(2 C2RH+C1)
+2 √−C2
+�
+− 1
+��√−C2
+,
+(35)
+A(x) = 4 C2
+2x2 + 4 C1C2x + C2
+1
+2 C2
+.
+Here, Γ(s, x) denotes the upper incomplete Euler–Gamma function and erf(x) the error function. The expression above
+has two classes of parameters. Two of these, α and ℓ, are related to the hairy black hole, and two are non-fixed a priori:
+the particle mass m, encoded in RH, and the Gaussian width l. The resulting probability PBH = PBH(l, m, ℓ, α) will
+also depend on the same parameters.
+According to the previous discussion, before finding the probability distribution, we have first to find the probability
+that the particle resides inside a sphere with the radius r = rH. From Equations (26) and (29), one obtains
+PS(r < rH) = 4π
+� rH
+0
+|ψS(r)|2r2dr =
+2
+√π γ
+�3
+2, r2
+H
+l2
+�
+,
+with γ(s, x) = Γ(s) − Γ(s, x), the lower incomplete Gamma function.
+Equations (27) and (35) yield PH(rH), as
+depicted in Figure 2.
+0
+1
+2
+3
+4
+5
+mprH
+lpm
+0
+1
+PH(rH)
+l = 0.50
+l = 1.0
+l = 1.5
+l = 2.0
+FIG. 2: The probability density for the value of the gravitational radius is rH for α = ℓ/(GM) = 0.5 and different values of
+the Gaussian width.
+Combining the previous results, one finds that the probability density for the particle resides within its own
+gravitational radius
+P<(r < rH) = 8√πγ
+�3
+2, r2
+H
+l2
+�
+r2
+H|ψH(rH)|2.
+
+7
+The probability of the particle described by the Gaussian to be a black hole is finally given by
+PBH(l, m, ℓ, α) = 8√π
+� ∞
+RH
+γ
+�3
+2, r2
+H
+l2
+�
+r2
+H|ψH(rH)|2,
+(36)
+which has to be calculated numerically. Assuming the Gaussian width has the same order as the particle Compton
+length, we could set l ∼ m−1 on Equation (36) and find the probability depending on either l or m. On the other
+hand, by departing again from Equation (31), we may set values for m in terms of the Planck mass and find the
+probability in this scenario. Applying l ∼ m−1 yields
+PBH(l, ℓ, α) = 8√π
+� ∞
+RH
+γ
+�3
+2, r2
+H
+l2
+�
+r2
+H|ψH(rH)|2,
+(37)
+or
+PBH(m, ℓ, α) = 8√π
+� ∞
+RH
+γ
+�3
+2, r2
+Hm2
+�
+r2
+H|ψH(rH)|2.
+(38)
+The resulting probabilities are shown in Figure 3 below. Figure 4 displays the probability for m given as a fraction of
+the Planck mass.
+0
+1
+2
+3
+4
+5
+l
+lp
+0
+1
+PBH
+ℓmp
+lpm = α = 0.00
+ℓmp
+lpm = α = 0.30
+ℓmp
+lpm = α = 0.60
+ℓmp
+lpm = α = 0.90
+1
+2
+m
+mp
+0
+1
+PBH
+ℓmp
+lpm = α = 0.00
+ℓmp
+lpm = α = 0.30
+ℓmp
+lpm = α = 0.60
+ℓmp
+lpm = α = 0.90
+FIG. 3: The probability of a ”particle” being a black hole depending on the Gaussian width or mass, assuming l ∼ m−1.
+1
+2
+3
+4
+5
+l
+lp
+0
+1
+PBH
+ℓmp
+lpm = α = 0.00
+ℓmp
+lpm = α = 0.30
+ℓmp
+lpm = α = 0.60
+ℓmp
+lpm = α = 0.90
+FIG. 4: The probability of a ”particle” being a black hole depending on the Gaussian width and mass m given as a fraction of
+the Planck mass, with m = mp (solid), m = 3mp/4 (dashed), and m = mp/2 (dotted).
+
+8
+B.
+HQM and GUP
+Since the horizon quantum mechanics formalism applies the standard wave function description for particles, a
+natural question is whether it affects the Heisenberg uncertainty principle. As mentioned, it produces a GUP similar
+to that produced by Equation (1). In quantum mechanics, the uncertainty principle may be derived by calculating
+the uncertainty associated with the wave function. Here, we start from the same point. From the Gaussian wave
+function (29), the particle size uncertainty is given by
+∆r2
+0 = ⟨r2⟩ − ⟨r⟩2
+= 4π
+� ∞
+0
+|ψS(r)|2r4dr −
+�
+4π
+� ∞
+0
+|ψS(r)|2r3dr
+�2
+= 3π − 8
+2π
+l2.
+(39)
+One might find the uncertainty of the horizon radius in an analogous way,1
+∆r2
+H = ⟨r2
+H⟩ − ⟨rH⟩2.
+(40)
+The total radial uncertainty can now be taken as a linear combination of the quantities calculated above, ∆r =
+∆r0 + ϵ∆rH. For the uncertainty in momentum, we have
+∆p2 = ⟨p2⟩ − ⟨p⟩2 = 3π − 8
+2π
+m2
+pl2
+p
+l2
+.
+Note that the momentum uncertainty and the width l are related such that ∆p ∼ 1/l. Using this fact in ∆r =
+∆r0 + ϵ∆rH, one is able to find
+∆r
+lp
+= 3π − 8
+2π
+mp
+∆p + ϵ∆H
+�∆p
+mp
+�
+,
+(41)
+which is similar to the GUP discussed previously. The function ∆H also depends on the wave function and hairy black
+hole parameters. Figure 5 shows the behavior of the GUP as a function of the momentum uncertainty, taking ϵ = 1.
+There, we can see a minimum ∆r placed around the Planck scale. From the GUP expression, it is straightforward
+to see that a larger ϵ means significant correction to the quantum mechanics’ uncertainty. The hairy parameters,
+however, have a small qualitative effect on fixing the minimum scale. As shown in Figure 5, their effects become
+prominent for a large ∆p.
+1
+2
+∆p
+mp
+1
+2
+∆r
+lp
+ℓmp
+lpm = α = 0.00
+ℓmp
+lpm = α = 0.30
+ℓmp
+lpm = α = 0.60
+ℓmp
+lpm = α = 0.90
+FIG. 5: GUP profile emerged from the horizon wave function formalism for ϵ = 1. The dotted line represents the particle size
+uncertainty ∆r0, the dashed line represents the uncertainty of the horizon radius ∆rH, and the solid lines describe the GUP.
+1 The analytical expression of ∆r2
+H is huge and little enlightening.
+
+9
+IV.
+DISCUSSION
+A few years ago, effective theories suggested lowering the scale of quantum black hole formation to TeV. Thus, in
+principle, it became experimentally accessible. In spite of no quantum black holes being detected, solid theoretical
+results point out that such objects should exist in nature [7, 14]. They could give us valuable hints about quantum
+gravity features [7, 13, 14]. One of this paper’s motivating questions was whether a generic black hole hair could
+significantly change the scale of quantum black hole formation. However, regarding the analysis carried out here, the
+hairy black holes look qualitatively similar to the Schwarzschild one, with a probability PBH of a similar shape and a
+related GUP, leading to the existence of a minimum length scale. Nevertheless, one of the main results of the present
+paper is that the existence of hair increases the probability PBH. This is indeed a point to be stressed. Its explanation
+rests upon the fact that the hairy black hole radius is slightly larger than the one for Schwarzschild. This implies
+that, although the scale of quantum black hole formation is still beyond the current experimental scale, additional
+fields may lower such scale. Those results might impact future colliders’ estimations of quantum black holes coming
+from alternative theories of gravity and potentially stimulate investigations of specific models of quantum hairy black
+holes [17].
+Acknowledgements
+R.T.C. thanks Unesp—AGRUP for the financial support. J.M.H.d.S. thanks CNPq (grant No. 303561/2018-1) for
+the financial support.
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+https://doi.org/ 10.1088/0034-4885/78/12/126001.
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+Phys. D 2016, 25, 1630006, https://doi.org/10.1142/S0218271816300068.
+[48] Casadio, R. Localised particles and fuzzy horizons: A tool for probing Quantum Black Holes. arXiv 2013, arXiv:1305.3195.
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+https://doi.org/10.1007/s10714-017-2198-7.
+
diff --git a/CdAyT4oBgHgl3EQfePjS/content/tmp_files/load_file.txt b/CdAyT4oBgHgl3EQfePjS/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf,len=792
+page_content='Quantum Hairy Black Hole Formation and Horizon Quantum Mechanics  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Cavalcanti∗ and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Hoff da Silva†  Departamento de F´ısica, Universidade Estadual Paulista (Unesp), Guaratinguet´a 12516-410, Brazil  After introducing the gravitational decoupling method and the hairy black hole recently derived  from it, we investigate the formation of quantum hairy black holes by applying the horizon quan-  tum mechanics formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' It enables us to determine how external fields, characterized by hairy  parameters, affect the probability of spherically symmetric black hole formation and the generalized  uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  INTRODUCTION  Given their intrinsic connection with intense gravitational fields, solid theoretical basis [1–3], and several observa-  tional results corroborating their existences, black holes play a central role in contemporary high-energy physics and  astrophysics [4–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Despite the characterization of the horizon of stationary black hole solutions being well-known  within general relativity [3, 8], the nature of the horizons of non-stationary or stationary solutions beyond general  relativity is still a source of extensive research [9–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The investigation of black holes is not restricted to astrophysical  objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' they are also expected to be formed whenever a high concentration of energy is confined to a small region of  spacetime, producing so-called quantum black holes [7, 13–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' However, the precise formation mechanism of classical  and quantum black holes is still unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Although we do not have a theory of quantum gravity, phenomenology  suggests that some features of quantum black holes are expected to be model-independent [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' From a certain scale,  candidate theories should modify the results of general relativity, giving birth to some alternatives to Einsteins’s  theory of gravity [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Examples could allow for the presence of non-minimal coupled fundamental fields or higher  derivative terms during the action, which directly affects the uniqueness theorems of black holes in general relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  The famous no-hair theorem is not preserved outside the general relativity realm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' These solutions lead to effects that  are potentially detectable near the horizon of astrophysical black holes [20–22], or in quantum black holes’ formation  [23, 24], and may provide hints for the quantum path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  One of the major challenges in general relativity is finding physically relevant solutions to Einstein’s field equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  On the other hand, deriving new solutions from other previously known ones is a widespread technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' This approach  is precisely what the so-called gravitational decoupling (GD) method intends to achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' It has recently commanded  the community’s attention due to its simplicity and effectiveness [25–27] in generating new, exact analytical solutions  by considering additional sources to the stress-energy tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The recent description of anisotropic stellar distribu-  tions [28, 29], whose predictions might be tested in astrophysical observations [30–33], as well as the hairy black hole  solutions by gravitational decoupling, are particularly interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The latter describes a black hole with hair sourced  by generic fields, possibly of quantum nature, surrounding the vacuum Schwarzschild solution [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Exciting results  have been found during investigation of this solution [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  From the quantum side, one of the key features of quantum gravity phenomenology is the generalized uncertainty  principle (GUP), which modifies the Heisenberg uncertainty principle accordingly  ∆x∆p ≳ ℏ  �  1 + ϵ(∆p)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (1)  This expression of the GUP, which stems from different approaches to quantum gravity [37–46], characterizes a  minimum scale length ∆x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' This feature emerges quite naturally in the horizon quantum mechanics formalism (HQM)  [16, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In addition to the GUP, HQM also provides an estimation of the probability of quantum black hole formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  In a scenario of extra-dimensional spacetimes, the HQM gave an explanation for the null results of quantum black  hole formation in current colliders [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Could it also tell us something about a mechanism for decreasing the  fundamental scale to something near the scale of current colliders?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Our aim is to investigate the quantitative and  qualitative effects of black hole hair, regarding the probability of black hole formation and the GUP by applying the  horizon quantum mechanics formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  This paper is organized as follows: Section II is dedicated to reviewing the gravitational decoupling procedure, the  metric for GD hairy black holes, and an approximation for the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In Section III, we apply the horizon  ∗Electronic address: rogerio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='cavalcanti@unesp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='br  †Electronic address: julio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='hoff@unesp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='br  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='00319v1  [gr-qc]  1 Jan 2023  2  quantum mechanics formalism to the hairy black hole solution of the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' We compare the probability of  quantum black hole formation and the GUPs of hairy black holes for a range of hair parameters, unveiling the effects  of the hair fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Finally, Section IV is dedicated to conclusions and discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  HAIRY BLACK HOLES AND HORIZON RADIUS  Starting from Einstein’s field equations  Gµν = 8π ˇTµν,  (2)  where Gµν = Rµν − 1  2Rgµν denotes the Einstein tensor, the gravitational decoupling (GD) [25] method takes the  energy–momentum tensor decomposed as  ˇTµν = Tµν + Θµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (3)  Here, Tµν is the source of a known solution to general relativity, while Θµν introduces a new field or extension of the  gravitational sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' From ∇µ Gµν = 0, we also have ∇µ ˇT µν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The effective density and the tangential and radial  pressures can be determined by examining the field equations  ˇρ = ρ + Θ 0  0 ,  (4a)  ˇpt = p − Θ 2  2 ,  (4b)  ˇpr = p − Θ 1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (4c)  The idea is to deform a known solution to split the field equations in a sector containing the known solution with  source Tµν and a decoupled one governing the deformation, encompassing Θµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In fact,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' assuming a known spherically  symmetric metric,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  ds2 = −eκ(r)dt2 + eζ(r)dr2 + r2dΩ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (5)  and deforming κ(r) and ζ(r) as  κ(r) �→ κ(r) + αf2(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (6a)  e−ζ(r) �→ e−ζ(r) + αf1(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (6b)  the resulting decoupled field equations read  8π Θ 0  0  = α  �f1  r2 + f ′  1  r  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (7a)  8π Θ 1  1 − α e−ζ f ′  2  r  = α f1  � 1  r2 + κ′(r) + αf ′  2(r)  r  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (7b)  8πΘ 2  2 −αf1Z1(r) =αf ′  1  4  �  κ′(r) + αf ′  2(r)+ 2  r  �  +αZ2(r),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (7c)  where [25]  Z1(r) = α2f ′  2 (r)2 + 2 α  �  f ′  2 (r) κ′ (r) + f ′  2 (r)  r  + f ′′  2 (r)  �  + κ′ (r)2 + 2 κ′ (r)  r  + 2 κ′′ (r) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (8a)  Z2(r) = αe−ζ  �  2f ′′  2 + f 2′  2 + 2f ′  2  r  + 2κ′f ′  2 − ζ′f ′  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (8b)  The above equations state that if the deformation parameter α goes to zero, then Θµν must go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' It is worth  mentioning that for extended geometric deformation, that is, for f2 ̸= 0, the sources are not individually conserved  in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' However, as discussed in [26], in this case, the decoupling of the field equations without an exchange of  energy is allowed in two scenarios: (a) when Tµν is a barotropic fluid whose equation of state is T00 = T11 or (b) for  vacuum regions of the first system Tµν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' When minimal geometric deformation is applied, on the other hand, the  sources are shown to be individually conserved [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Assuming the Schwarzschild solution to be the known one and requiring a well-defined horizon structure [27], from  grr = − 1  gtt follows  �  1 − 2M  r  � �  eαf2(r) − 1  �  = αf1(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (9)  3  Therefore, one is able to write  ds2 = −  �  1 − 2M  r  �  eαf2(r)dt2+  �  1 − 2M  r  �−1  e−α f2(r)dr2 + r2 dΩ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (10)  Further, assuming strong energy conditions,  ˇρ + ˇpr + 2 ˇpt ≥ 0,  (11a)  ˇρ + ˇpr ≥ 0,  (11b)  ˇρ + ˇpt ≥ 0,  (11c)  and managing the field equations, a new hairy black hole solution was found [27]  ds2 = −f(r)dt2 +  1  f(r)dr2 + r2dΩ2,  (12)  where  f(r) = 1 − 2GM + αℓ  r  + αe−  r  GM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (13)  The dimensionless parameter 0 ≤ α ≤ 1 tracks the deformation of the Schwarzschild black hole, e is the Euler constant,  and ℓ is the direct effect of the nonvanishing additional font Θµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Notice that by taking α = 0, the Schwarzschild  solution is restored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Further, the ℓ parameter is limited to 2GM/e2 ≤ ℓ ≤ 1 due to the assumption of a strong energy  condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In extreme cases, ℓ = 2GM/e2 and  fe(r) = 1 − 2GM  r  + α  �  e−  r  GM − 2GM  e2 r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (14)  The hairy black hole has a single horizon, located at r = rH, such that  �  1 + αe− rH  GM  �  rH = 2GM + αℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (15)  Such an equation has no analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Nevertheless, a very accurate analytical approximation is found by Taylor  expanding it around the Schwarzschild horizon radius rS = 2GM,  rH  GM ≈ 4  �  αℓe2/GM − 3 α + e2�  αℓe2/GM − 4 α + 2 e2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (16)  Figure 1 shows a comparison between the exact and approximated horizon radii for different values of the hairy  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In the following section, we are going to use Equation (16) for the analytical expression of the hairy black  hole’s horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='0  ℓ  GM  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='8  rH  GM  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='00  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='20  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='40  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='60  α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='80  α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='0  Exact  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 1: The radius of the hairy black hole horizon rH as a function of ℓ for different values of the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The colored  dashed lines represent the approximated radius, and the gray lines are the exact ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' It shows how the hairy horizon deviates  from the Schwarzschild horizon for an increasing α and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The ranges for α and ℓ were fixed due to the assumption of a strong  energy condition [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  4  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  THE HORIZON QUANTUM MECHANICS FORMALISM  Horizon quantum mechanics (also known as horizon wave function formalism) is an effective approach capable of  providing the signatures of black hole physics to the Planck scale [48–51] (see [47] for a comprehensive review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The  main idea is to extend quantum mechanics and gravity further than the current experimental limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In such an  approach, we face the conceptual challenge of consistently describing classical and quantum mechanical objects, such  as horizons and particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' This is achieved by assigning wave functions to the quantum black hole horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' This  association allows the use of quantum mechanical machinery to distinguish between particles and quantum black  holes and to estimate the GUPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Nevertheless, first, we must choose a model describing the particle wave function to  derive the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Due to the previous results’ simplicity and efficiency, we shall use the Gaussian model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  From classical general relativity, we know that the horizons of black holes are described by trapping surfaces, whose  locations are determined by  gij∇ir∇jr = 0 ,  (17)  where ∇ir is orthogonal to the surfaces of the constant area A = 4πr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' A trapping surface then exists if there are  values of r and t such that the gravitational radius RH satisfies  RH(r, t) ≥ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (18)  Considering a spinless point-particle of mass m, an uncertainty in the spatial particle localization of the same order  of the Compton scale λm ≃ ℏ/m = lp mp/m follows from the uncertainty principle, where lp and mp are the Planck  length and mass, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Arguing that quantum mechanics gives a more precise description of physics, RH makes  sense only if it is larger than the Compton wavelength associated with the same mass, namely RH ≳ λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Thus, for  the Schwarzschild radius RS = 2Gm = 2 lp  mp m,  lp m/mp ≳ lp mp/m  =⇒  m ≳ mp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (19)  This suggests that the Planck mass is the minimum mass such that the Schwarzchild radius can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  From quantum mechanics, the spectral decomposition of a spherically symmetric matter distribution is given by  the expression  |ψS⟩ =  �  E  C(E) |ψE⟩ ,  (20)  with the usual eigenfunction equation  ˆH |ψE⟩ = E |ψE⟩ ,  (21)  regardless of the specific form of the actual Hamiltonian operator ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Using the energy spectrum and inverting the  expression of the Schwarzschild radius, we have  E = mp  rH  2lp  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (22)  Putting it back into the wave function, one can define the (unnormalized) horizon wave function as  ψH(rH) = C  �  mp  rH  2lp  �  (23)  whose normalization is fixed, as usual, by the inner product  ⟨ψH | φH⟩ = 4π  � ∞  0  ψ∗  H(rH)φH(rH)r2  HdrH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (24)  However, the classical radius RH is thus replaced by the expected value of the operator ˆRH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' From the uncertainty  of the expectation value, it follows that the radius will necessarily be “fuzzy”, similar to the position of the source  itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The next aspect one has to approach to establish a criterion for deciding if a mass distribution does or does  not form a black hole is if it lies inside its horizon of radius r = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' From quantum mechanics, one finds that it is  given by the product  P<(r < rH) = PS(r < rH)PH(rH),  (25)  5  where the first term,  PS(r < rH) = 4π  � rH  0  |ψS(r)|2r2dr,  (26)  is the probability that the particle resides inside the sphere of radius r = rH, while the second term,  PH(rH) = 4πr2  H|ψH(rH)|2  (27)  is the probability density that the value of the gravitational radius is rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Finally, the probability that the particle  described by the wave function ψS is a BH will be given by the integral of (25) over all possible values of the horizon  radius rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Namely,  PBH =  � ∞  0  P<(r < rH)drH,  (28)  which is one of the main outcomes of the formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Gaussian Sources  The previous construction can be made explicit by applying the Gaussian model for the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' To implement  this idea, let us recall that spectral decomposition is also assumed to be valid for momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Therefore, from (20),  ⟨p |ψS⟩ = C(p) ≡ ψH(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The Gaussian wave function for ψS scales as r2 in the position space and leads to a Gaussian  wave function in the momentum space, scaling as p2, naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Finally, since the dispersion relation relates p2 with  energy, we are able to have ⟨p |ψS⟩ = ψH(rH) via (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Hence, starting with a Gaussian wave function, we can describe  a spherically symmetric massive particle at rest, such as  ψS(r) =  e− r2  2 l2  (l √π)3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (29)  The corresponding function in momentum space is thus given by  ˜ψS(p) = 4π  � ∞  0  sin(rp)  √  8π3rp  e− r2  2 l2  (l √π)3/2 r2dr  =  e−  p2  2 ∆2  (∆ √π)3/2 ,  (30)  where ∆ = mp lp/l is the spread of the wave packet in momentum space, whose width l the Compton length of the  particle should diminish,  l ≥ λm ∼ mp lp  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (31)  In addition to the straightforward handling of a Gaussian wave packet, it is also relevant to recall that the Gaussian  wave function leads to a minimal uncertainty for the expected values computed with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Had we used another wave  function, it would certainly imply a worsening uncertainty, eventually leading to unnecessary extra difficulties relating  to the HQM and GUP (see next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Back to our problem, assuming the relativistic mass-shell relation in flat  space [48]  p2 = E2 − m2 ,  (32)  the energy E of the particle is expressed in terms of the related horizon radius rH = RH(E), following from Equation  (16),  E = αmpℓe2 +  �  α − e2�  mprH  2 (2 α − e2)lp  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (33)  6  Thus, from Equations (30) and (33), one finds the the horizon wave function of the hairy black hole  ψH(rH) = NHΘ(rH − RH) e(C2r2  H+C1rH+C0),  where  C0 = − α2l2m2  pℓ2e4  8 (2 α − e2)2l2p  ,  C1 = −  �  α − e2�  αl2m2  pℓe2  4 (2 α − e2)2l2p  ,  C2 = −  �  α − e2�2l2m2  p  8 (2 α − e2)2l2p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (34)  The Heaviside step function Θ appears above due to the imposition E ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The normalisation factor NH is fixed  according to  N −2  H  = 4π  � ∞  0  |ψH(rH)|2 r2  H drH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  The normalized horizon wave function is thus given as follows  ψH(rH) = −  2 C  3  2  2 e  A(rH )  2  √π  �  4 C1C2eA(RH) −  �  2  √  2C2Γ  � 3  2 , −A(RH)  �  +  √  2πC2  1  �  erf  � √  2(2 C2RH+C1)  2 √−C2  �  − 1  ��√−C2  ,  (35)  A(x) = 4 C2  2x2 + 4 C1C2x + C2  1  2 C2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Here, Γ(s, x) denotes the upper incomplete Euler–Gamma function and erf(x) the error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The expression above  has two classes of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Two of these, α and ℓ, are related to the hairy black hole, and two are non-fixed a priori:  the particle mass m, encoded in RH, and the Gaussian width l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The resulting probability PBH = PBH(l, m, ℓ, α) will  also depend on the same parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  According to the previous discussion, before finding the probability distribution, we have first to find the probability  that the particle resides inside a sphere with the radius r = rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' From Equations (26) and (29), one obtains  PS(r < rH) = 4π  � rH  0  |ψS(r)|2r2dr =  2  √π γ  �3  2, r2  H  l2  �  ,  with γ(s, x) = Γ(s) − Γ(s, x), the lower incomplete Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Equations (27) and (35) yield PH(rH), as  depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  0  1  2  3  4  5  mprH  lpm  0  1  PH(rH)  l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='50  l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='0  l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='5  l = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 2: The probability density for the value of the gravitational radius is rH for α = ℓ/(GM) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='5 and different values of  the Gaussian width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Combining the previous results, one finds that the probability density for the particle resides within its own  gravitational radius  P<(r < rH) = 8√πγ  �3  2, r2  H  l2  �  r2  H|ψH(rH)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  7  The probability of the particle described by the Gaussian to be a black hole is finally given by  PBH(l, m, ℓ, α) = 8√π  � ∞  RH  γ  �3  2, r2  H  l2  �  r2  H|ψH(rH)|2,  (36)  which has to be calculated numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Assuming the Gaussian width has the same order as the particle Compton  length, we could set l ∼ m−1 on Equation (36) and find the probability depending on either l or m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' On the other  hand, by departing again from Equation (31), we may set values for m in terms of the Planck mass and find the  probability in this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Applying l ∼ m−1 yields  PBH(l, ℓ, α) = 8√π  � ∞  RH  γ  �3  2, r2  H  l2  �  r2  H|ψH(rH)|2,  (37)  or  PBH(m, ℓ, α) = 8√π  � ∞  RH  γ  �3  2, r2  Hm2  �  r2  H|ψH(rH)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (38)  The resulting probabilities are shown in Figure 3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Figure 4 displays the probability for m given as a fraction of  the Planck mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  0  1  2  3  4  5  l  lp  0  1  PBH  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='00  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='30  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='60  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='90  1  2  m  mp  0  1  PBH  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='00  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='30  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='60  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='90  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 3: The probability of a ”particle” being a black hole depending on the Gaussian width or mass, assuming l ∼ m−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  1  2  3  4  5  l  lp  0  1  PBH  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='00  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='30  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='60  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='90  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 4: The probability of a ”particle” being a black hole depending on the Gaussian width and mass m given as a fraction of  the Planck mass, with m = mp (solid), m = 3mp/4 (dashed), and m = mp/2 (dotted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  HQM and GUP  Since the horizon quantum mechanics formalism applies the standard wave function description for particles, a  natural question is whether it affects the Heisenberg uncertainty principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' As mentioned, it produces a GUP similar  to that produced by Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In quantum mechanics, the uncertainty principle may be derived by calculating  the uncertainty associated with the wave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Here, we start from the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' From the Gaussian wave  function (29), the particle size uncertainty is given by  ∆r2  0 = ⟨r2⟩ − ⟨r⟩2  = 4π  � ∞  0  |ψS(r)|2r4dr −  �  4π  � ∞  0  |ψS(r)|2r3dr  �2  = 3π − 8  2π  l2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (39)  One might find the uncertainty of the horizon radius in an analogous way,1  ∆r2  H = ⟨r2  H⟩ − ⟨rH⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  (40)  The total radial uncertainty can now be taken as a linear combination of the quantities calculated above, ∆r =  ∆r0 + ϵ∆rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' For the uncertainty in momentum, we have  ∆p2 = ⟨p2⟩ − ⟨p⟩2 = 3π − 8  2π  m2  pl2  p  l2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Note that the momentum uncertainty and the width l are related such that ∆p ∼ 1/l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Using this fact in ∆r =  ∆r0 + ϵ∆rH, one is able to find  ∆r  lp  = 3π − 8  2π  mp  ∆p + ϵ∆H  �∆p  mp  �  ,  (41)  which is similar to the GUP discussed previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The function ∆H also depends on the wave function and hairy black  hole parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Figure 5 shows the behavior of the GUP as a function of the momentum uncertainty, taking ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  There, we can see a minimum ∆r placed around the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' From the GUP expression, it is straightforward  to see that a larger ϵ means significant correction to the quantum mechanics’ uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The hairy parameters,  however, have a small qualitative effect on fixing the minimum scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' As shown in Figure 5, their effects become  prominent for a large ∆p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  1  2  ∆p  mp  1  2  ∆r  lp  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='00  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='30  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='60  ℓmp  lpm = α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='90  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 5: GUP profile emerged from the horizon wave function formalism for ϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' The dotted line represents the particle size  uncertainty ∆r0, the dashed line represents the uncertainty of the horizon radius ∆rH, and the solid lines describe the GUP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  1 The analytical expression of ∆r2  H is huge and little enlightening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  9  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  DISCUSSION  A few years ago, effective theories suggested lowering the scale of quantum black hole formation to TeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Thus, in  principle, it became experimentally accessible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' In spite of no quantum black holes being detected, solid theoretical  results point out that such objects should exist in nature [7, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' They could give us valuable hints about quantum  gravity features [7, 13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' One of this paper’s motivating questions was whether a generic black hole hair could  significantly change the scale of quantum black hole formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' However, regarding the analysis carried out here, the  hairy black holes look qualitatively similar to the Schwarzschild one, with a probability PBH of a similar shape and a  related GUP, leading to the existence of a minimum length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Nevertheless, one of the main results of the present  paper is that the existence of hair increases the probability PBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' This is indeed a point to be stressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Its explanation  rests upon the fact that the hairy black hole radius is slightly larger than the one for Schwarzschild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' This implies  that, although the scale of quantum black hole formation is still beyond the current experimental scale, additional  fields may lower such scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Those results might impact future colliders’ estimations of quantum black holes coming  from alternative theories of gravity and potentially stimulate investigations of specific models of quantum hairy black  holes [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Acknowledgements  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' thanks Unesp—AGRUP for the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' thanks CNPq (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 303561/2018-1) for  the financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [1] Hawking, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=' Cambridge Monographs on Mathematical Physics,  Cambridge University Press: Cambridge, UK, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='  The  Mathematical  Theory  of  Black  Holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Fundam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1007/978-94-009-6469-3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=', Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Black Hole Physics: Basic Concepts and New Developments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Kluwer Academic Publishers:  Dordrecht, The Netherlands, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1007/978-94-011-5139-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [4] Abbott, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Observation of Gravitational Waves from a Binary Black Hole Merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='061102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [5] Cardoso, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Pani, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Testing the nature of dark compact objects: A status report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Living Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 2019, 22, 4,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1007/s41114-019-0020-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [6] Barack, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Black holes, gravitational waves and fundamental physics: A roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1088/1361-6382/ab0587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [7] Calmet,  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=',  Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Quantum  Aspects  of  Black  Holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Springer:  Berlin/Heidelberg,  Germany,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1007/978-3-319-10852-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [8] Wald,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  General  Relativity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Chicago  Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' :  Chicago,  IL,  USA,  1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='7208/chicago/9780226870373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [9] Faraoni, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Cosmological and Black Hole Apparent Horizons;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Springer: Berlin/Heidelberg, Germany, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Volume 907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='  Krishnan,  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Dynamical  horizons  and  their  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=' Jaramillo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='  New Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='  What is the final state of a black hole merger?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='  Quantum  Harmonic  Black  Holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  JHEP  2013,  08,  025,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=' Orlandi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Black holes as self-sustained quantum states, and Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='084040.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=' Hsu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Sotomayor, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Hairy black holes by gravitational decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 2019,  51, 103, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1007/s10714-019-2587-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [50] Casadio,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Micu,  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Horizon Quantum Mechanics of collapsing shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' C 2018,  78, 852,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1140/ epjc/s10052-018-6326-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='  [51] Casadio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Giugno, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Giusti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Global and Local Horizon Quantum Mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content=' 2017, 49, 32,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
+page_content='1007/s10714-017-2198-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdAyT4oBgHgl3EQfePjS/content/2301.00319v1.pdf'}
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+arXiv:2301.03111v1  [math.PR]  8 Jan 2023
+Stochastic Reservoir Calculations
+Steven Finch
+January 8, 2023
+Abstract.
+Prabhu (1958) obtained the stationary distribution of storage
+level Zt in a reservoir of finite volume v, given an inflow Xt and an outflow
+Yt.
+Time t is assumed to be discrete, Xt ∼ Gamma(p, µ) are independent
+and p is a positive integer.
+The mean inflow is p/µ; the target outflow is
+m (constant).
+We attempt to clarify intricate details, often omitted in the
+literature, by working through several examples.
+Of special interest are the
+probabilities of depletion (Zt = 0) and spillage (Zt = v).
+For prescribed
+{v, p, µ}, what value of m minimizes both of these?
+Let v > 0, p be a positive integer, µ > 0 and m > 0. At each time t = 1, 2, 3, . . .,
+a reservoir of volume v absorbs an inflow Xt ∼ Gamma(p, µ) and simultaneously
+releases an outflow 0 ≤ Yt ≤ m, depending on availablity. More precisely,
+Yt = min {Xt + Zt, m}
+where 0 ≤ Zt ≤ v is the storage level.
+Independence across time is assumed.
+Our
+interest is in the probability density function of Zt in the limit as t → ∞. We need
+not explicitly refer to Yt again, as Zt+1 can be defined recursively without it:
+Zt+1 = max {0, min {Xt + Zt − m, v}} ,
+Z1 = v/2.
+Let n = ⌊v/m⌋ and δ = v − m n.
+In words, δ is 0 if and only if v is an integer
+multiple of m, and δ is otherwise > 0. Let
+λ = (−1)p−1µp exp(−µ m).
+The graph of the PDF for Zt is piecewise smooth and contains at most n + 1 arcs, as
+well as point masses at z = 0 and z = v. The arcs are identified by j = 0, 1, 2, . . ., n
+from left to right, and correspond to open subintervals
+max {(j − 1)m + δ, 0} < z < min {j m + δ, v}
+of 0 < z < v. Prabhu [1] impressively obtained the cumulative distribution function
+F(z) = 1 − exp [µ(v − z)]
+p−1
+�
+r=0
+αr
+n−j
+�
+q=0
+(−λ)q (v − q m − z)q p+r
+(q p + r)!
+0Copyright © 2023 by Steven R. Finch. All rights reserved.
+1
+
+Stochastic Reservoir Calculations
+2
+that shall occupy us for the remainder of this paper.
+The αr coefficients are found
+by solving a system of p linear equations with coefficients drs for r, s = 0, 1, . . . , p−1.
+These will be defined shortly.
+Special considerations apply to endpoints.
+Let κ = n − 1 if δ = 0 and κ = n
+otherwise. The depletion probability, i.e., odds for the reservoir to be dry, is
+F(0) = 1 − exp(µ v)
+p−1
+�
+r=0
+αr
+κ
+�
+q=0
+(−λ)q (v − q m)q p+r
+(q p + r)!
+.
+In contrast, the spillage probability, i.e., odds for the reservoir to be full, is just
+1 − F(v) = α0.
+Minimizing the chance of both zero supply (harmful) and oversupply (wasteful) is
+clearly important. Other quantities of interest include the total deficit, i.e., unsatis-
+fied demand, over a specified time duration; and total surplus, i.e., unwanted supply
+(because v < ∞) that necessarily leaks into the environment.
+For r = 0, 1, . . . , p − 1, the linear system
+αr − λ
+p−1
+�
+s=0
+drs αs = (−µ)r exp [−µ(v + m)]
+p−r−1
+�
+s=0
+[µ(v + m)]s
+s!
+requires solution, where
+drs = (−1)p+r−1
+n
+�
+q=0
+(−λ)q
+v
+�
+q m
+(t − q m)q p+s(t + m)p−r−1
+(q p + s)!(p − r − 1)!
+dt.
+The integral can be easily expressed in closed-form.
+Prabhu’s CDF formula, given gamma-distributed inflow, extends a PDF formula
+discovered earlier by Moran [2], given exponentially distributed inflow (p = 1).
+We
+have not studied [2] in depth.
+More discussion of [1] appears in [3, 4, 5, 6].
+The
+treatment in [7, 8] is, however, most pragmatic and useful for our purposes.
+Henceforth we fix v = 1 and explore results for selected {p, µ, m}. It is surprising,
+more than fifty years after the publication of Prabhu’s work, that greater attention
+has not been paid to this research [7].
+We can only imagine that intricate details,
+often lost in theoretical summaries, have conspired to prevent greater understanding
+and widespread recognition.
+Our hope is that working through a few examples will
+help to improve matters.
+
+Stochastic Reservoir Calculations
+3
+1.
+{p, µ, m} =
+�
+1, 2, 1
+2
+�
+The mean inflow is p/µ = 1/2 and the target outflow is m = 1/2.
+Clearly n =
+⌊1/m⌋ = 2 and δ = 1 − m n = 0, i.e., there is no offset.
+The arcs j = 0, 1, 2
+correspond to intervals
+0 < z < 0,
+0 < z < 1/2,
+1/2 < z < 1
+and thus j = 0 can be ignored (being empty). Prabhu’s formula gives F(z) as
+1 − 1
+2 exp [µ(1 − z)] [2 − (1 − 2z) λ] α0
+for j = 1 and
+1 − exp [µ(1 − z)] α0
+for j = 2. The linear equation
+(1 − λ d00)α0 = exp
+�
+−3
+2µ
+�
+coupled with
+d00 = 1 − 1
+8λ
+and λ = 2e−1 give
+α0 =
+8
+8 − 8λ + λ2 exp
+�
+−3
+2µ
+�
+= 0.15000227...
+as the spillage probability. Because κ = n − 1 = 1,
+F(0) = 1 − 1
+2 exp(µ) (2 − λ) α0 = 0.29937324...
+is the depletion probability.
+One may have expected these two probabilities to be
+almost equal (since 1/µ = 1/2 = m and by a certain symmetry), but this is not true.
+The derivative f(z) of F(z) is plotted in Figure 1.
+2.
+{p, µ, m} =
+�
+1, 2, 1
+3
+�
+The mean inflow is p/µ = 1/2 and the target outflow is m = 1/3.
+Clearly n =
+⌊1/m⌋ = 3 and δ = 1 − m n = 0, i.e., there is no offset.
+The arcs j = 0, 1, 2, 3
+correspond to intervals
+0 < z < 0,
+0 < z < 1/3,
+1/3 < z < 2/3,
+2/3 < z < 1
+
+Stochastic Reservoir Calculations
+4
+and thus j = 0 can be ignored (being empty). Prabhu’s formula gives F(z) as
+1 − 1
+18 exp [µ(1 − z)]
+�
+18 − 6 (2 − 3z) λ + (1 − 3z)2λ2�
+α0
+for j = 1,
+1 − 1
+3 exp [µ(1 − z)] [3 − (2 − 3z) λ] α0
+for j = 2 and
+1 − exp [µ(1 − z)] α0
+for j = 3. The linear equation
+(1 − λ d00)α0 = exp
+�
+−4
+3µ
+�
+coupled with
+d00 = 1 − 2
+9λ +
+1
+162λ2
+and λ = 2e−2/3 give
+α0 =
+162
+162 − 162λ + 36λ2 − λ3 exp
+�
+−4
+3µ
+�
+= 0.34604845...
+as the spillage probability. Because κ = n − 1 = 1,
+F(0) = 1 − 1
+18 exp(µ)
+�
+18 − 12λ + λ2�
+α0 = 0.04363903...
+is the depletion probability. While α0 < F(0) in Section 1, we have α0 > F(0) here.
+This outcome suggests examining a value of m between 1/3 and 1/2. The derivative
+f(z) of F(z) is plotted in Figure 2.
+3.
+{p, µ, m} =
+�
+1, 2, 2
+5
+�
+The mean inflow is p/µ = 1/2 and the target outflow is m = 2/5.
+Clearly n =
+⌊1/m⌋ = 2 and δ = 1 − m n = 1/5, i.e., the offset is nonzero.
+The arcs j = 0, 1, 2
+correspond to intervals
+0 < z < 1/5,
+1/5 < z < 3/5,
+3/5 < z < 1;
+note that j = 0 has length only 1/5. Prabhu’s formula gives F(z) as
+1 − 1
+50 exp [µ(1 − z)]
+�
+50 − 10 (3 − 5z) λ + (1 − 5z)2λ2�
+α0
+
+Stochastic Reservoir Calculations
+5
+for j = 0,
+1 − 1
+5 exp [µ(1 − z)] [5 − (3 − 5z) λ] α0
+for j = 1 and
+1 − exp [µ(1 − z)] α0
+for j = 2. The linear equation
+(1 − λ d00)α0 = exp
+�
+−7
+5µ
+�
+coupled with
+d00 = 1 − 9
+50λ +
+1
+750λ2
+and λ = 2e−4/5 give
+α0 =
+750
+750 − 750λ + 135λ2 − λ3 exp
+�
+−7
+5µ
+�
+= 0.24745701...
+as the spillage probability. Because κ = n = 2,
+F(0) = 1 − 1
+50 exp(µ)
+�
+50 − 30λ + λ2�
+α0 = 0.12789671...
+is the depletion probability. The values α0 and F(0) are closer than in the previous
+two sections; a choice of m that is intermediate to 2/5 and 1/2 should make these
+coincident.
+We estimate that m = 0.44276 meets this objective (with 0.199 as the
+common probability). On the other hand, if our goal is to minimize the unweighted
+combination α0 + F(0), then m = 0.38 achieves the goal (with sum 0.372).
+The
+derivative f(z) of F(z) is plotted in Figure 3.
+4.
+{p, µ, m} =
+�
+2, 4, 1
+2
+�
+The mean inflow is p/µ = 1/2 and the target outflow is m = 1/2.
+Clearly n =
+⌊1/m⌋ = 2 and δ = 1 − m n = 0, i.e., there is no offset.
+The arcs j = 0, 1, 2
+correspond to intervals
+0 < z < 0,
+0 < z < 1/2,
+1/2 < z < 1
+and thus j = 0 can be ignored (being empty). Prabhu’s formula gives F(z) as
+1 − 1
+48 exp [µ(1 − z)]
+��
+48 − 6 (1 − 2z)2 λ
+�
+α0 +
+�
+48 − 48z − (1 − 2z)3 λ
+�
+α1
+�
+for j = 1 and
+1 − exp [µ(1 − z)] {α0 + (1 − z)α1}
+
+Stochastic Reservoir Calculations
+6
+for j = 2. The linear equations
+(1 − λ d00)α0 − λ d01α1 = exp
+�
+−3
+2µ
+� �
+1 + 3
+2µ
+�
+,
+λ d10α0 − (1 − λ d11)α1 = exp
+�
+−3
+2µ
+�
+µ
+coupled with
+d00 = −1 + 11
+384λ,
+d01 = − 7
+12 +
+7
+1920λ,
+d10 = 1 − 1
+48λ,
+d11 = 1
+2 −
+1
+384λ
+and λ = −16e−2 give
+α0 = 1
+2
+2 + 3µ − 2λ µ d01 − λ (2 + 3µ) d11
+1 − λ (d00 + d11) + λ2 (d00d11 − d01d10) exp
+�
+−3
+2µ
+�
+,
+α1 = 1
+2
+−2µ + 2λ µ d00 + λ (2 + 3µ) d10
+1 − λ (d00 + d11) + λ2 (d00d11 − d01d10) exp
+�
+−3
+2µ
+�
+;
+the spillage probability is hence α0 = 0.13554701.... Because κ = n − 1 = 1,
+F(0) = 1 − 1
+48 exp(µ) [(48 − 6λ) α0 + (48 − λ)α1] = 0.22163253...
+is the depletion probability. The mode of Gamma(2, µ) is 1/µ > 0 whereas the mode
+of Gamma(1, µ) is 0; a small inflow is less likely for p = 2 than for p = 1, thus F(0)
+is noticeably smaller than in Section 1.
+The tail of Gamma(2, µ) is fatter than the
+tail of Gamma(1, µ); a large inflow is more likely for p = 2 than for p = 1, however
+α0 is paradoxically smaller than in Section 1 (but only slightly). The derivative f(z)
+of F(z) is plotted in Figure 4.
+5.
+Invariance
+One verification of Prabhu’s formula is based on simulation (easily programmed, since
+the recurrence for Zt is straightforward).
+Another verification is more esoteric: to
+confirm that the formula is invariant under the transformation
+�
+v, p
+µ, m
+�
+�−→
+�
+˜v, p
+˜µ, ˜m
+�
+=
+� v
+m,
+p
+m µ, 1
+�
+in the sense that spillage & depletion probabilities should remain constant and storage
+level CDF arguments should simply scale by m. First,
+˜n =
+� ˜v
+˜m
+�
+=
+� v
+m
+�
+= n,
+
+Stochastic Reservoir Calculations
+7
+˜λ = (−1)p−1˜µp exp[−˜µ ˜m] = (−1)p−1(m µ)p exp[−m µ · 1] = mpλ
+and
+˜drs = (−1)p+r−1
+n
+�
+q=0
+(−˜λ)q
+˜v
+�
+q ˜m
+(t − q ˜m)q p+s(t + ˜m)p−r−1
+(q p + s)!(p − r − 1)!
+dt
+= (−1)p+r−1
+n
+�
+q=0
+mp q(−λ)q
+v/m
+�
+q
+(t − q)q p+s(t + 1)p−r−1
+(q p + s)!(p − r − 1)! dt
+= (−1)p+r−1
+n
+�
+q=0
+mp q(−λ)q
+v
+�
+q m
+( u
+m − q)q p+s( u
+m + 1)p−r−1
+(q p + s)!(p − r − 1)!
+du
+m
+upon setting u = m t, du = m dt; thus
+˜drs = (−1)p+r−1
+n
+�
+q=0
+mp q(−λ)q
+mp q+s+p−r−1+1
+v
+�
+q m
+(u − q m)q p+s(u + m)p−r−1
+(q p + s)!(p − r − 1)!
+du
+= m−(p−r+s)drs.
+Second,
+˜αr − ˜λ
+p−1
+�
+s=0
+˜drs ˜αs = (−˜µ)r exp [−˜µ(˜v + ˜m)]
+p−r−1
+�
+s=0
+[˜µ(˜v + ˜m)]s
+s!
+implies
+˜αr − mpλ
+p−1
+�
+s=0
+m−(p−r+s)drs ˜αs = mr(−µ)r exp [−µ(v + m)]
+p−r−1
+�
+s=0
+[µ(v + m)]s
+s!
+because ˜µ(˜v + ˜m) = m µ
+� v
+m + 1
+�
+= µ(v + m); therefore
+m−r ˜αr − λ
+p−1
+�
+s=0
+m−sdrs ˜αs = (−µ)r exp [−µ(v + m)]
+p−r−1
+�
+s=0
+[µ(v + m)]s
+s!
+which is immediately satisfied by ˜αr = mrαr. In particular, ˜α0 = α0. Third,
+˜δ = ˜v − ˜m n = v − m n
+m
+= δ
+m.
+
+Stochastic Reservoir Calculations
+8
+Figure 1: Plot of storage level density w = f(z) for {p, µ, m} =
+�
+1, 2, 1
+2
+�
+.
+Finally, given j,
+˜F(z) = 1 − exp [˜µ(˜v − z)]
+p−1
+�
+r=0
+˜αr
+n−j
+�
+q=0
+(−˜λ)q (˜v − q ˜m − z)q p+r
+(q p + r)!
+= 1 − exp
+�
+(m µ)
+� v
+m − z
+�� p−1
+�
+r=0
+mrαr
+n−j
+�
+q=0
+mp q(−λ)q ( v
+m − q − z)q p+r
+(q p + r)!
+= 1 − exp [µ(v − m z)]
+p−1
+�
+r=0
+mrαr
+n−j
+�
+q=0
+mp q(−λ)q
+mp q+r
+(v − q m − m z)q p+r
+(q p + r)!
+= F(m z)
+for (j − 1) ˜m + ˜δ < z < j ˜m + ˜δ, i.e., (j − 1)m + δ < m z < j m + δ. In the same way,
+˜F(0) = F(0), with the upper summation limit n − j replaced by ˜κ = κ.
+6.
+Inquiry
+Moran [9, 10] introduced a different model – in continuous time – for an infinite
+volume reservoir. Let X(t) ∼ Gamma(t, 1/ρ) denote the total inflow over the interval
+(0, t], assumed to be a nonnegative stochastic process with stationary independent
+increments, where 0 < ρ < 1 is constant.
+In particular, E(X(T)) = ρ t.
+Let the
+
+M
+8'0
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+8:0
+1.0Stochastic Reservoir Calculations
+9
+Figure 2: Plot of storage level density w = f(z) for {p, µ, m} =
+�
+1, 2, 1
+3
+�
+.
+Figure 3: Plot of storage level density w = f(z) for {p, µ, m} =
+�
+1, 2, 2
+5
+�
+.
+
+w
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0M
+1.2
+1.0
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0Stochastic Reservoir Calculations
+10
+Figure 4: Plot of storage level density w = f(z) for {p, µ, m} =
+�
+2, 4, 1
+2
+�
+.
+outflow be continuous and at unit rate except when the reservoir is empty. We have
+Z(t) = Z(0) + X(t) − t +
+t
+�
+0
+1{Z(τ)=0} dτ
+where 1Ω is the indicator function of Ω ⊆ R. By a limiting argument (from discrete
+to continuous), the PDF of Z(t) as t → ∞ has Laplace transform [11]
+(1 − ρ)θ
+θ − ln (1 + ρ θ),
+Re(θ) > 0
+which Daniels [12] inverted to yield
+f(z) = −(1 − ρ)
+∞
+�
+0
+d
+dz
+(z + w)w−1 exp [−(z + w)/ρ]
+ρwΓ(w)
+dw,
+z > 0
+with a point mass 1 − ρ at z = 0.
+We seek an experimental approach to verify
+this PDF.
+How might one efficiently simulate Z(t) for suitably large t?
+Offers of
+assistance would be most appreciated.
+We wonder too if Prabhu’s formula could
+possibly be reconfigured to play a role in this inquiry.
+The fact that v < ∞ earlier
+but v = ∞ here is an issue; the fact that Xt was the precise inflow at time t whereas
+X(t) is an accumulated inflow over (0, t] is another issue.
+
+0.8
+0.6
+0.4
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0Stochastic Reservoir Calculations
+11
+7.
+Acknowledgements
+Khaled Hamed was so kind to answer several questions of mine; this paper would not
+have been possible without his very helpful articles [7, 8]. In particular, he appears
+to be the first author to specify the role of the offset δ when v is not an integer
+multiple of m.
+I am grateful to innumerable software developers.
+The symbolic
+manipulations described here are tailor-made for Mathematica, and the simulations
+employed here to check predictions are ideal for R.
+References
+[1] N. U. Prabhu, On the integral equation for the finite dam, Quart. J. Math. 9
+(1958) 183–188; MR0099726.
+[2] P. A. P. Moran, A probability theory of dams and storage systems: modifications
+of the release rules, Austral. J. Appl. Sci. 6 (1955) 117–130; MR0077807.
+[3] P. A. P. Moran, The Theory of Storage, Wiley, 1959, pp. 39–51; MR0114254.
+[4] N. U. Prabhu, Queues and Inventories, Wiley, 1965, pp. 209–213; MR0211494.
+[5] P. Lochert and R. M. Phatarfod, On the problem of discretization in dam theory,
+Water Resources Research 15 (1979) 1593-1597.
+[6] E. H. Lloyd, The stochastic reservoir: exact and approximate evaluations of
+storage distribution, J. Hydrology 151 (1993) 65–107.
+[7] K. H. Hamed, On the implementation of Prabhu’s exact solution of the stochastic
+reservoir equation, Adv. in Water Resources 32 (2009) 594–606.
+[8] K. H. Hamed, Stochastic reservoir analysis, from Handbook of Engineering Hy-
+drology, ed. S. Eslamian, CRC Press, 2014, pp. 531–548.
+[9] P. A. P. Moran, A probability theory of a dam with a continuous release, Quart.
+J. Math. 7 (1956) 130–137; MR0101573.
+[10] J. Gani, Problems in the probability theory of storage systems, J. Royal Statist.
+Soc. Ser. B 19 (1957) 181–206; MR0092289.
+[11] D. G. Kendall, Some problems in the theory of dams, J. Royal Statist. Soc. Ser.
+B 19 (1957) 207–212.
+[12] H. E. Daniels, Discussion on the papers by Dr. Gani and Mr. Kendall, J. Royal
+Statist. Soc. Ser. B 19 (1957) 224–225.
+
+Stochastic Reservoir Calculations
+12
+Steven Finch
+MIT Sloan School of Management
+Cambridge, MA, USA
+steven finch@harvard.edu
+
diff --git a/DNE1T4oBgHgl3EQfWQQt/content/tmp_files/load_file.txt b/DNE1T4oBgHgl3EQfWQQt/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,319 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf,len=318
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='03111v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='PR]  8 Jan 2023  Stochastic Reservoir Calculations  Steven Finch  January 8, 2023  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Prabhu (1958) obtained the stationary distribution of storage  level Zt in a reservoir of finite volume v, given an inflow Xt and an outflow  Yt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Time t is assumed to be discrete, Xt ∼ Gamma(p, µ) are independent  and p is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The mean inflow is p/µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' the target outflow is  m (constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  We attempt to clarify intricate details, often omitted in the  literature, by working through several examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Of special interest are the  probabilities of depletion (Zt = 0) and spillage (Zt = v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  For prescribed  {v, p, µ}, what value of m minimizes both of these?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Let v > 0, p be a positive integer, µ > 0 and m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' At each time t = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=',  a reservoir of volume v absorbs an inflow Xt ∼ Gamma(p, µ) and simultaneously  releases an outflow 0 ≤ Yt ≤ m, depending on availablity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' More precisely,  Yt = min {Xt + Zt, m}  where 0 ≤ Zt ≤ v is the storage level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Independence across time is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Our  interest is in the probability density function of Zt in the limit as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' We need  not explicitly refer to Yt again, as Zt+1 can be defined recursively without it:  Zt+1 = max {0, min {Xt + Zt − m, v}} ,  Z1 = v/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Let n = ⌊v/m⌋ and δ = v − m n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  In words, δ is 0 if and only if v is an integer  multiple of m, and δ is otherwise > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Let  λ = (−1)p−1µp exp(−µ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The graph of the PDF for Zt is piecewise smooth and contains at most n + 1 arcs, as  well as point masses at z = 0 and z = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The arcs are identified by j = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', n  from left to right, and correspond to open subintervals  max {(j − 1)m + δ, 0} < z < min {j m + δ, v}  of 0 < z < v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Prabhu [1] impressively obtained the cumulative distribution function  F(z) = 1 − exp [µ(v − z)]  p−1  �  r=0  αr  n−j  �  q=0  (−λ)q (v − q m − z)q p+r  (q p + r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  0Copyright © 2023 by Steven R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Finch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  1  Stochastic Reservoir Calculations  2  that shall occupy us for the remainder of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The αr coefficients are found  by solving a system of p linear equations with coefficients drs for r, s = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' , p−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  These will be defined shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Special considerations apply to endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Let κ = n − 1 if δ = 0 and κ = n  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The depletion probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', odds for the reservoir to be dry, is  F(0) = 1 − exp(µ v)  p−1  �  r=0  αr  κ  �  q=0  (−λ)q (v − q m)q p+r  (q p + r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  In contrast, the spillage probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', odds for the reservoir to be full, is just  1 − F(v) = α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Minimizing the chance of both zero supply (harmful) and oversupply (wasteful) is  clearly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Other quantities of interest include the total deficit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', unsatis-  fied demand, over a specified time duration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' and total surplus, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', unwanted supply  (because v < ∞) that necessarily leaks into the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  For r = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' , p − 1, the linear system  αr − λ  p−1  �  s=0  drs αs = (−µ)r exp [−µ(v + m)]  p−r−1  �  s=0  [µ(v + m)]s  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  requires solution, where  drs = (−1)p+r−1  n  �  q=0  (−λ)q  v  �  q m  (t − q m)q p+s(t + m)p−r−1  (q p + s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' (p − r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The integral can be easily expressed in closed-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Prabhu’s CDF formula, given gamma-distributed inflow, extends a PDF formula  discovered earlier by Moran [2], given exponentially distributed inflow (p = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  We  have not studied [2] in depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  More discussion of [1] appears in [3, 4, 5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The  treatment in [7, 8] is, however, most pragmatic and useful for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Henceforth we fix v = 1 and explore results for selected {p, µ, m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' It is surprising,  more than fifty years after the publication of Prabhu’s work, that greater attention  has not been paid to this research [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  We can only imagine that intricate details,  often lost in theoretical summaries, have conspired to prevent greater understanding  and widespread recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Our hope is that working through a few examples will  help to improve matters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Stochastic Reservoir Calculations  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  {p, µ, m} =  �  1, 2, 1  2  �  The mean inflow is p/µ = 1/2 and the target outflow is m = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Clearly n =  ⌊1/m⌋ = 2 and δ = 1 − m n = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', there is no offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The arcs j = 0, 1, 2  correspond to intervals  0 < z < 0,  0 < z < 1/2,  1/2 < z < 1  and thus j = 0 can be ignored (being empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Prabhu’s formula gives F(z) as  1 − 1  2 exp [µ(1 − z)] [2 − (1 − 2z) λ] α0  for j = 1 and  1 − exp [µ(1 − z)] α0  for j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The linear equation  (1 − λ d00)α0 = exp  �  −3  2µ  �  coupled with  d00 = 1 − 1  8λ  and λ = 2e−1 give  α0 =  8  8 − 8λ + λ2 exp  �  −3  2µ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='15000227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  as the spillage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Because κ = n − 1 = 1,  F(0) = 1 − 1  2 exp(µ) (2 − λ) α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='29937324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  is the depletion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  One may have expected these two probabilities to be  almost equal (since 1/µ = 1/2 = m and by a certain symmetry), but this is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The derivative f(z) of F(z) is plotted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  {p, µ, m} =  �  1, 2, 1  3  �  The mean inflow is p/µ = 1/2 and the target outflow is m = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Clearly n =  ⌊1/m⌋ = 3 and δ = 1 − m n = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', there is no offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The arcs j = 0, 1, 2, 3  correspond to intervals  0 < z < 0,  0 < z < 1/3,  1/3 < z < 2/3,  2/3 < z < 1  Stochastic Reservoir Calculations  4  and thus j = 0 can be ignored (being empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Prabhu’s formula gives F(z) as  1 − 1  18 exp [µ(1 − z)]  �  18 − 6 (2 − 3z) λ + (1 − 3z)2λ2�  α0  for j = 1,  1 − 1  3 exp [µ(1 − z)] [3 − (2 − 3z) λ] α0  for j = 2 and  1 − exp [µ(1 − z)] α0  for j = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The linear equation  (1 − λ d00)α0 = exp  �  −4  3µ  �  coupled with  d00 = 1 − 2  9λ +  1  162λ2  and λ = 2e−2/3 give  α0 =  162  162 − 162λ + 36λ2 − λ3 exp  �  −4  3µ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='34604845.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  as the spillage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Because κ = n − 1 = 1,  F(0) = 1 − 1  18 exp(µ)  �  18 − 12λ + λ2�  α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='04363903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  is the depletion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' While α0 < F(0) in Section 1, we have α0 > F(0) here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  This outcome suggests examining a value of m between 1/3 and 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The derivative  f(z) of F(z) is plotted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  {p, µ, m} =  �  1, 2, 2  5  �  The mean inflow is p/µ = 1/2 and the target outflow is m = 2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Clearly n =  ⌊1/m⌋ = 2 and δ = 1 − m n = 1/5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', the offset is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The arcs j = 0, 1, 2  correspond to intervals  0 < z < 1/5,  1/5 < z < 3/5,  3/5 < z < 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  note that j = 0 has length only 1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Prabhu’s formula gives F(z) as  1 − 1  50 exp [µ(1 − z)]  �  50 − 10 (3 − 5z) λ + (1 − 5z)2λ2�  α0  Stochastic Reservoir Calculations  5  for j = 0,  1 − 1  5 exp [µ(1 − z)] [5 − (3 − 5z) λ] α0  for j = 1 and  1 − exp [µ(1 − z)] α0  for j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The linear equation  (1 − λ d00)α0 = exp  �  −7  5µ  �  coupled with  d00 = 1 − 9  50λ +  1  750λ2  and λ = 2e−4/5 give  α0 =  750  750 − 750λ + 135λ2 − λ3 exp  �  −7  5µ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='24745701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  as the spillage probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Because κ = n = 2,  F(0) = 1 − 1  50 exp(µ)  �  50 − 30λ + λ2�  α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='12789671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  is the depletion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The values α0 and F(0) are closer than in the previous  two sections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' a choice of m that is intermediate to 2/5 and 1/2 should make these  coincident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  We estimate that m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='44276 meets this objective (with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='199 as the  common probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' On the other hand, if our goal is to minimize the unweighted  combination α0 + F(0), then m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='38 achieves the goal (with sum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='372).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The  derivative f(z) of F(z) is plotted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  {p, µ, m} =  �  2, 4, 1  2  �  The mean inflow is p/µ = 1/2 and the target outflow is m = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Clearly n =  ⌊1/m⌋ = 2 and δ = 1 − m n = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', there is no offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The arcs j = 0, 1, 2  correspond to intervals  0 < z < 0,  0 < z < 1/2,  1/2 < z < 1  and thus j = 0 can be ignored (being empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Prabhu’s formula gives F(z) as  1 − 1  48 exp [µ(1 − z)]  ��  48 − 6 (1 − 2z)2 λ  �  α0 +  �  48 − 48z − (1 − 2z)3 λ  �  α1  �  for j = 1 and  1 − exp [µ(1 − z)] {α0 + (1 − z)α1}  Stochastic Reservoir Calculations  6  for j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The linear equations  (1 − λ d00)α0 − λ d01α1 = exp  �  −3  2µ  � �  1 + 3  2µ  �  ,  λ d10α0 − (1 − λ d11)α1 = exp  �  −3  2µ  �  µ  coupled with  d00 = −1 + 11  384λ,  d01 = − 7  12 +  7  1920λ,  d10 = 1 − 1  48λ,  d11 = 1  2 −  1  384λ  and λ = −16e−2 give  α0 = 1  2  2 + 3µ − 2λ µ d01 − λ (2 + 3µ) d11  1 − λ (d00 + d11) + λ2 (d00d11 − d01d10) exp  �  −3  2µ  �  ,  α1 = 1  2  −2µ + 2λ µ d00 + λ (2 + 3µ) d10  1 − λ (d00 + d11) + λ2 (d00d11 − d01d10) exp  �  −3  2µ  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  the spillage probability is hence α0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='13554701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='. Because κ = n − 1 = 1,  F(0) = 1 − 1  48 exp(µ) [(48 − 6λ) α0 + (48 − λ)α1] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='22163253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  is the depletion probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The mode of Gamma(2, µ) is 1/µ > 0 whereas the mode  of Gamma(1, µ) is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' a small inflow is less likely for p = 2 than for p = 1, thus F(0)  is noticeably smaller than in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The tail of Gamma(2, µ) is fatter than the  tail of Gamma(1, µ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' a large inflow is more likely for p = 2 than for p = 1, however  α0 is paradoxically smaller than in Section 1 (but only slightly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' The derivative f(z)  of F(z) is plotted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Invariance  One verification of Prabhu’s formula is based on simulation (easily programmed, since  the recurrence for Zt is straightforward).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Another verification is more esoteric: to  confirm that the formula is invariant under the transformation  �  v, p  µ, m  �  �−→  �  ˜v, p  ˜µ, ˜m  �  =  � v  m,  p  m µ, 1  �  in the sense that spillage & depletion probabilities should remain constant and storage  level CDF arguments should simply scale by m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' First,  ˜n =  � ˜v  ˜m  �  =  � v  m  �  = n,  Stochastic Reservoir Calculations  7  ˜λ = (−1)p−1˜µp exp[−˜µ ˜m] = (−1)p−1(m µ)p exp[−m µ · 1] = mpλ  and  ˜drs = (−1)p+r−1  n  �  q=0  (−˜λ)q  ˜v  �  q ˜m  (t − q ˜m)q p+s(t + ˜m)p−r−1  (q p + s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' (p − r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  dt  = (−1)p+r−1  n  �  q=0  mp q(−λ)q  v/m  �  q  (t − q)q p+s(t + 1)p−r−1  (q p + s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' (p − r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' dt  = (−1)p+r−1  n  �  q=0  mp q(−λ)q  v  �  q m  ( u  m − q)q p+s( u  m + 1)p−r−1  (q p + s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' (p − r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  du  m  upon setting u = m t, du = m dt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' thus  ˜drs = (−1)p+r−1  n  �  q=0  mp q(−λ)q  mp q+s+p−r−1+1  v  �  q m  (u − q m)q p+s(u + m)p−r−1  (q p + s)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' (p − r − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  du  = m−(p−r+s)drs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Second,  ˜αr − ˜λ  p−1  �  s=0  ˜drs ˜αs = (−˜µ)r exp [−˜µ(˜v + ˜m)]  p−r−1  �  s=0  [˜µ(˜v + ˜m)]s  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  implies  ˜αr − mpλ  p−1  �  s=0  m−(p−r+s)drs ˜αs = mr(−µ)r exp [−µ(v + m)]  p−r−1  �  s=0  [µ(v + m)]s  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  because ˜µ(˜v + ˜m) = m µ  � v  m + 1  �  = µ(v + m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' therefore  m−r ˜αr − λ  p−1  �  s=0  m−sdrs ˜αs = (−µ)r exp [−µ(v + m)]  p−r−1  �  s=0  [µ(v + m)]s  s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  which is immediately satisfied by ˜αr = mrαr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' In particular, ˜α0 = α0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Third,  ˜δ = ˜v − ˜m n = v − m n  m  = δ  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Stochastic Reservoir Calculations  8  Figure 1: Plot of storage level density w = f(z) for {p, µ, m} =  �  1, 2, 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Finally, given j,  ˜F(z) = 1 − exp [˜µ(˜v − z)]  p−1  �  r=0  ˜αr  n−j  �  q=0  (−˜λ)q (˜v − q ˜m − z)q p+r  (q p + r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  = 1 − exp  �  (m µ)  � v  m − z  �� p−1  �  r=0  mrαr  n−j  �  q=0  mp q(−λ)q ( v  m − q − z)q p+r  (q p + r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  = 1 − exp [µ(v − m z)]  p−1  �  r=0  mrαr  n−j  �  q=0  mp q(−λ)q  mp q+r  (v − q m − m z)q p+r  (q p + r)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  = F(m z)  for (j − 1) ˜m + ˜δ < z < j ˜m + ˜δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=', (j − 1)m + δ < m z < j m + δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' In the same way,  ˜F(0) = F(0), with the upper summation limit n − j replaced by ˜κ = κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Inquiry  Moran [9, 10] introduced a different model – in continuous time – for an infinite  volume reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Let X(t) ∼ Gamma(t, 1/ρ) denote the total inflow over the interval  (0, t], assumed to be a nonnegative stochastic process with stationary independent  increments, where 0 < ρ < 1 is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  In particular, E(X(T)) = ρ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content="  Let the  M  8'0  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content='6  8:0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='0Stochastic Reservoir Calculations  9  Figure 2: Plot of storage level density w = f(z) for {p, µ, m} =  �  1, 2, 1  3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Figure 3: Plot of storage level density w = f(z) for {p, µ, m} =  �  1, 2, 2  5  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  w  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='0Stochastic Reservoir Calculations  10  Figure 4: Plot of storage level density w = f(z) for {p, µ, m} =  �  2, 4, 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  outflow be continuous and at unit rate except when the reservoir is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' We have  Z(t) = Z(0) + X(t) − t +  t  �  0  1{Z(τ)=0} dτ  where 1Ω is the indicator function of Ω ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' By a limiting argument (from discrete  to continuous), the PDF of Z(t) as t → ∞ has Laplace transform [11]  (1 − ρ)θ  θ − ln (1 + ρ θ),  Re(θ) > 0  which Daniels [12] inverted to yield  f(z) = −(1 − ρ)  ∞  �  0  d  dz  (z + w)w−1 exp [−(z + w)/ρ]  ρwΓ(w)  dw,  z > 0  with a point mass 1 − ρ at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  We seek an experimental approach to verify  this PDF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  How might one efficiently simulate Z(t) for suitably large t?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Offers of  assistance would be most appreciated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  We wonder too if Prabhu’s formula could  possibly be reconfigured to play a role in this inquiry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The fact that v < ∞ earlier  but v = ∞ here is an issue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' the fact that Xt was the precise inflow at time t whereas  X(t) is an accumulated inflow over (0, t] is another issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content='0Stochastic Reservoir Calculations  11  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  Acknowledgements  Khaled Hamed was so kind to answer several questions of mine;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' this paper would not  have been possible without his very helpful articles [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' In particular, he appears  to be the first author to specify the role of the offset δ when v is not an integer  multiple of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  I am grateful to innumerable software developers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  The symbolic  manipulations described here are tailor-made for Mathematica, and the simulations  employed here to check predictions are ideal for R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  References  [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content=' Prabhu, On the integral equation for the finite dam, Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content=' MR0211494.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content=' Lochert and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Lloyd, The stochastic reservoir: exact and approximate evaluations of  storage distribution, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' Hydrology 151 (1993) 65–107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content='  [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNE1T4oBgHgl3EQfWQQt/content/2301.03111v1.pdf'}
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+Geometric theory of topological defects: methodological
+developments and new trends
+Sébastien Fumeron‡, Bertrand Berche‡, and Fernando Moraes†
+‡Laboratoire de Physique et Chimie Théoriques,
+UMR Université de Lorraine - CNRS 7019, 54000 Nancy, France and
+†Departamento de Física, Universidade Federal Rural
+de Pernambuco, 52171-900, Recife, PE, Brazil
+Abstract
+Liquid crystals generally support orientational singularities of the director field known as topo-
+logical defects. These latter modifiy transport properties in their vicinity as if the geometry was
+non-Euclidean. We present a state of the art of the differential geometry of nematic liquid crystals,
+with a special emphasis on linear defects. We then discuss unexpected but deep connections with
+cosmology and high-energy-physics, and conclude with a review on defect engineering for transport
+phenomena.
+1
+arXiv:2301.01050v1  [cond-mat.soft]  3 Jan 2023
+
+I.
+INTRODUCTION
+One of Pierre-Gilles de Gennes’s greatest breakthrough was to realize that methods and
+concepts borrowed from superconductivity also apply to describe smectic-A phases [1]. His
+work is a striking example of cross-fertilization between different areas of physics and it
+highlights how progress arises at the crossroads of various scientific fields. In an article
+that has not been translated in English [2], he took the example of line singularities as a
+common denominator between liquid crystals, quark physics and superconductors [3]. The
+same observation was also made by William Brinkman and Patricia Cladis [4], and most
+notably by the two-Nobel-Prize winner John Bardeen in a review written as a plea for
+interdisciplinarity: “Line defects in three-dimensional systems, quantized vortex lines or flux
+lines, and dislocations account for similarities of behavior in superconductors, liquid crystals,
+and, it is hoped, color confinement of quarks“[5].
+In this spirit, the objective of this paper is to perform an in-depth survey of geometrical
+methods useful for investigating topological defects and to describe some of its modern
+applications, either as a playground to test fundamental ideas in high-energy physics or
+gravitational physics, or as high-performance tools to taylor transport phenomena from soft
+matter devices. We will be mainly concerned with nematic liquid crystals and topological
+line defects.
+In section II, we provide a self-contained introduction to phase transitions, geometry,
+topology and optics in such systems. We show how a metric description of defect lines in
+terms of Riemann manifolds naturally arises in nematics, before addressing the question of
+analogue gravity which may be less familiar to the liquid crystals community.
+Section III is an introduction to some of the ideas borrowed to cosmology which can
+be dealt with liquid crystals, such as the Kibble mechanism, which rules the formation of
+defects in the early universe but also in nematics. We review several outstanding problems
+involving line singularities, such as cosmic strings, wormholes and bouncing cosmologies,
+and discuss their connections with disclinations in liquid crystals.
+Section IV eventually discusses applications of the geometric formalism previously in-
+troduced for the description of acoustics, optics and heat transfer in the presence of such
+defects. The main idea is to show how the curvature carried by the topological defects
+can be used to design specific propagation patterns, a possible step forward towards the
+2
+
+defect-engineering of transport phenomena.
+II.
+TOPOLOGICAL DEFECTS IN NEMATIC LIQUID CRYSTALS
+A.
+Basics of the isotropic-nematic phase transition
+a.
+Historical milestones
+The story of liquid crystal has met with a shaky start. The
+first observations reported of what is today understood as a liquid crystal belong to the
+realm of biology. Georges-Louis Buffon (1707-1788) and later Rudolf Virchow (1854) and
+Carl Mettenheimer (1855) reported about the strange behavior of lecithins, a family of
+phospholipid substances contained in plants (wheat, rye...) and in animals (yolks, myelin
+- the insulating coating of nerve fibres - ...) [6]. When suspended in water, lecithins form
+birefringent tubular structures like a Iceland spar but that writhed like eels. Julius Planer
+in 1861 [7] and most importantly Friedrich Reinitzer in 1888 discovered similar optical
+behaviors with cholesterol compounds. Reinitzer extracted cholesteryl esters from carrots
+and made an unanticipated observation: contrary to what was known in crystallography,
+cholesteryl benzoate displays two melting points [8]. The lower one occurs at about 145.5◦C
+and correspond to the melting of the solid phase into a turbid fluid. The higher melting
+point corresponds to the clarification of the milky liquid beyond 178.5◦C.
+Such behavior left Reinitzer skeptical: had he discovered a genuinely new behavior of
+matter or was this simply the result of impure and incompletely melted crystals? Reinitzer
+wrote to Otto Lehmann, a leading physicist known for designing the first “crystallization
+microscope”: this latter consists in a microscope equipped with crossed polarizers and a
+thermal deck (a small Bunsen burner and two cooling blasts) to observe how crystals behave
+when the temperature varies. Lehmann reproduced and improved Reinitzer’s observations,
+and promoted these substances as new forms of matter, half-between liquids and crystals. In
+1889, he coined the term “liquid crystals” to account for his discovery (somehow pulling the
+sheet back towards him as he even claimed priority over Reinitzer [9]). Soon afterwards, he
+kept changing its name (including “flowing crystals”, “crystalline liquids”...), which reveals
+his difficulties to grasp the real nature of what he found.
+The years that followed Lehmann’s breakthrough have been critical. On one hand, the
+subject became more and more discussed in the scientific community, even drawing the
+3
+
+attention of future Nobel Prize laureates such as Max Born (in 1916, he conceived the
+first molecular theory of liquid crystals but predicted a generic ferroelectric behavior that
+turned out to be incorrect [10]), Jacobus Henricus van’t Hoff and Walther Nernst (Lehmann
+himself was an unlucky nominee from 1913 to 1922). On the other hand, the subject became
+highly controversial: partly because Nernst and most especially Gustav Tammann led a
+vivid opposition against liquid crystals (suspecting them of being nothing more than poorly
+prepared colloidal mixtures), partly because of Lehmann’s personality (a sparkling mix of
+pretentiousness and mysticism [9]).
+Liquid crystals have stayed a controversial subject, since the decisive contributions of
+Rudolf Schenk (1905), Daniel Vorländer (1907) and George Friedel (1922) to get a clear view
+on this subject. Schenk led a thorough study of the clearing point and unambiguously showed
+the observed properties (density, viscosity) could not be related to mixtures.
+Vorländer
+elucidated the mysterious anisotropic behavior exibited by the fluid: from a microscopic
+standpoint, liquid crystals consist in self-organized assemblies of rod-like molecules.
+As
+such, they exibit both the birefringence property expected from anisotropic uniaxial media
+and the ability to flow. Finally, Friedel realized that such substances should properly be
+understood as new full-blown phases of matter, that he named mesophases. Initially divided
+into three broad families (nematic, cholesteric and smectic), liquid crystals have now been
+enriched by many new mesophases, including columnar phases, cubatic phases, blue phases
+I, II and III...
+b.
+Mesogenic behavior
+The recipe for a molecule to be nematogenic (i.e. to have the
+ability to organize into a nematic phase) is rather simple: take a rod-like molecule and deck
+it with 1) a flexible outer part (an aliphatic chain), 2) a rigid core (phenyl groups most
+generally) and 3) a chemical group bearing a permanent dipole (for instance, a carbonitrile
+group as in 5CB). The resulting substance is a thermotropic nematic, a sensitive compromise
+between the attractive Van der Waals interactions that align rigid cores on average along the
+same direction (anisotropy) and the thermal agitation of the aliphatic chains increasing the
+mean steric hindrance (fluidity). Nematics can also be lyotropic: the nematogens display
+an amphiphilic structure (they have both hydrophilic and hydrophobic parts), the control
+parameter being the concentration of molecules in a solvent such as water. The frontier
+between thermotropic and lyotropic liquid crystalline behavior being not strict, nematogens
+can also behave as amphotropic media.
+4
+
+In the perspective of section III, let us focus on thermotropic nematics. In that case, there
+are different models accounting adequately for the phase transition. The Lebwohl-Lasher
+model [11] is the paradigmatic model in this context, in a sense the liquid crystal analogue
+of the Ising model [12]: the nematogen molecules are represented only by their direction and
+they occupy fixed positions on the sites of a cubic lattice. The different sites of the lattice
+interact only between nearest neighbors through a potential that favors configurations where
+the neighboring molecules point in the same direction. On the contrary, the Maier-Saupe
+approach is a mean-field theory: interactions between particles are replaced by an effective
+field experienced by all particles at the same time. This model considers only London forces
+between instantaneous molecular dipoles and ignores repulsive interactions [13]. This also
+favors alignment of the molecules in the same direction. We will briefly mention that for
+lyotropic nematics, Lars Onsager described the phase transition of an assembly of rod-like
+sticks as an entropic process driven [14]: due to purely steric effects, orientational entropy
+loss is more than offset by positional entropy gain which triggers the transition.
+Depending on the temperature range, three (or more) phases can be observed. At low
+temperatures, the steric hindrance of aliphatic chains is minimal and the nematogens get
+close enough for attractive forces to drive the system. As the dipole-dipole interactions
+prevail over thermal agitation, the assembly of rod-like molecules organize into a molecular
+crystal. This latter displays both a positional and rotational order for each molecule. On
+the contrary, at high temperatures, Van der Waals interactions are dominated by thermal
+effects and the nematogens form an isotropic fluid phase: both positional and orientational
+order are lost. Within the intermediate range of temperature, the two effects are of the same
+order and different kinds of mesophases may appear. In the nematic mesophase, only the
+orientational order is preserved: locally, the nematogens tend to align on average along a
+common direction, which defines the director field n. In usual nematics, the orientational
+order is preserved at long distance, as the correlation length is typically about a few µm,
+compared to the nematogen length around a few nanometers.
+As the phase transition
+involves nucleation, these domains wherein nematogens share a common orientation form
+submicronic bubbles (or spherulites). Then they grow in size and eventually they meet and
+mingle, sometimes leaving relics in the form of long threads.
+c.
+Order parameter
+Within a nematic, a particular molecule does generally not point
+exactly in the direction n and the degree of orientational ordering of the mesophase can thus
+5
+
+be assessed by looking how well the nematogens are aligned along the director field. The
+quadrupolar scalar order parameter S defined by Tsvetkov (1942) provides a quantitative
+criterion to characterize the nematic order: it is normalized (S = 0 in the isotropic fluid phase
+and S = 1 for perfectly aligned rods), and ranges from 0.3 < S < 0.8 in usual nematics (in
+practice, 0.3 < S < 0.4 for thermotropic liquid crystals, whereas 0.6 < S < 0.8 for lyotropic
+ones). The order parameter can be refined to include informations on the local orientation
+of n (Landau – de Gennes tensorial order parameter) or to encompass phase involving more
+complex-shaped mesogens (higher-order mutipole-multipole correlation functions).
+For thermotropic nematics, S can be taken as a function depending only on the temper-
+ature. The behavior of S at the transition can essentially discriminate between two main
+families of phase transitions (in the sense defined by Lev Landau in 1937 [15]): first-order
+phase transitions, for which the order parameter displays a jump at the transition control
+parameter (this class also involves latent heats and nucleation processes), and second-order
+phase transitions, for which the order parameter varies continuously at the transition (this
+class involves pretransitional effects and scaling behaviors). Experimentally, for most com-
+pounds (5CB, MBBA, 5CN...), the isotropic-nematic phase transition is identified as weakly
+first-order phase transition in three dimensions. It combines small discontinuities of S [16],
+nucleation [17] and low latent heats [18], but pretransitional effects of the dielectric proper-
+ties [18].
+B.
+From symmetry to topology
+a.
+Homotopy theory
+The existence of orientational and/or positional orders impart
+each phase about the transition with a specific set of symmetries. As a rule, the higher-
+temperature phase is generally the less-ordered one and its symmetry group is larger. In
+the isotropic-nematic case, the isotropic fluid phase and the nematic liquid crystal are both
+statistically invariant under any translation in space. But for the orientational part, the
+two phases do not share the same symmetry group. Indeed, the isotropic fluid phase is
+statistically invariant under any rotation in three dimensions, that is, under the elements of
+the group SO(3), while in the mesophase the director field plays the role of a symmetry axis,
+restricting the symmetry to statistical invariance under the elements of the group SO(2).
+But for energetic reasons, the dipoles borne by the nematogens tend to align anticollinear,
+6
+
+such that the assembly of rods is unchanged when inverting heads and tails (this dimeric
+structure was confirmed early by X-ray diffraction experiments in 5CB and 7CB [19]). Hence,
+the full symmetry group of the nematic phase is O(2).
+Many important features of a phase transition with a spontaneous symmetry-breaking
+are encompassed within the topology of an abstract object, called the order parameter space
+M. For a phase transition with a symmetry-breaking pattern G → H, the order parameter
+space is a manifold defined from the coset M = G/H. The toolbox of algebraic topology
+(Poincaré’s former analysis situs) can then be used to seek the algebraic invariants (numbers,
+groups, rings. . . ) of M and to classify this space into equivalence classes.
+Among the many entry points, homotopy theory is of particular interest to determine the
+presence of singularities. Two topological spaces are homotopic if they can be mapped into
+each other by a continuous deformation where bijectivity is not necessarily preserved (i.e.
+gluing, shrinking or fattening the space is allowed). Homotopy groups, denoted generically
+as πk(M), have been extensively studied in condensed matter physics, mainly in the pio-
+neering works of Kleman, Lavrentovich, Michel, Toulouse and Volovik [20–26] and they are
+associated to different kinds of topological properties for M. For instance, π1(M) tests the
+simple-connectedness of the order parameter space. Indeed, consider first M = R × R. It
+is simply-connected as all closed loops (dimension 1) are homotopic to a point (dimension
+0): therefore π1(M) = I and the order parameter space is simply connected. Conversely,
+for M = R∗ × R∗, there are two equivalence classes of closed loops: those not encircling
+the origin, which are homotopic to a point, and those encircling the origin which cannot be
+shrunk into a point. Therefore π1(M) ̸= I and the order parameter space is not simply con-
+nected. The 0D-hole has thus changed the homotopy content of π1(M). Interestingly, the
+dimensionality of the manifold is crucial here: a loop trying to lasso a 0D-hole can succeed in
+2D but will always fail in 3D. In its most general form, the fundamental result of homotopy
+analysis states that in dimension n, if the k th homotopy group πk(M) ̸= I, then holes of
+dimension n − 1 − k appear: such singularities are called topological defects. Strictly speak-
+ing, a defect is topological when the singular configuration of the order parameter cannot
+be transformed continuously into a uniform configuration. This process depends not only
+on the order parameter configuration but also on the dimensionality of the order parameter
+space (due to the possibility to “escape in the third dimension”). Therefore, there is also a
+more flexible use for the terminology “topological defect”, referring to the singularity asso-
+7
+
+ciated to any non-trivial homotopy content of the order parameter manifold, whatever its
+topological stability is. In the remainder of this article, we will stick to that latter meaning.
+b.
+Zoology of topological defects in nematics
+For the isotropic-nematic phase, the order
+parameter space is given by M = SO(3)/O(2) ≡ S2/Z2: the resulting space, called the real
+projective plane RP 2, can be pictured as a 2-sphere having its antipodal points identified.
+The manifold corresponding to an immersion of the real projective plane in 3D space is
+called a Boy surface and its topology is encompassed into its first four homotopy groups: for
+uniaxial nematics in 3D, these are π0(RP 2) = I (no domain wall), π1(RP 2) = Z2 (existence
+of linear defects), π2(RP 2) = Z (existence of point defects) and π3(RP 2) = Z (existence of
+textures).
+FIG. 1. Left: Class N = 0 of closed loops homotopic to a point in the order parameter space.
+Right: Class N = 1 of closed loops consisting of paths connecting two antipodal points, which are
+not homotopic to a point.
+Linear defects (or “disclinations” in Frank’s terminology) come from a breaking of the ro-
+tational symmetry group and they are probably the most widespread singularities observed
+in nematics. In optical microscopy, they appear as thread-like structures used by Friedel to
+coin the term nematic (from the greek νηµα="thread"). In polarizing microscopy, disclina-
+tions give rise to the beautiful Schlieren patterns, where dark brushes connect at singular
+8
+
+A'=A
+O
+T,(RP2)=Z2=[0,1)
+unstable
+stablepoints corresponding to the line defects viewed end on. The content of the first homotopy
+group (or Poincaré group) is Z2 = 0, 1, which means that there are two equivalence classes
+for closed loops. The trivial class N = 0 corresponds to defects that are not topologically
+stable (they can relax into a uniform configuration), whereas the second non-trivial class
+N = 1 corresponds to defects that cannot be removed (see Fig. 1). For reasons we will
+clarify later, we retain the terminology of wedge disclinations to the trivial class and the
+terminology of Mœbius disclinations to the non-trivial class. A disclination can stay almost
+straight or form loops. It is generally associated to other disclinations within dipoles (edge
+dislocations), amorphous networks (blue phases), etc. In that case, they have the possibility
+to interconnect [27] and they combine according to the algebra of Z2, that is 0 + 0 = 0,
+0 + 1 = 1 and 1 + 1 = 0. An extensive review on linear defects in the general context of ill-
+ordered condensed matter can be found in [28]. Since our main concern here is disclinations
+we refer the reader interested in point defects and textures (including the exotic skyrmions
+and hopfions) to the the very complete reviews [17] and [29], respectively.
+C.
+Optics in the presence of linear defects
+a.
+Director field of axial disclinations
+A region characterized by a given director field
+can undergo orientational distorsions as a result of external constraints. As n is a unit
+(headless) vector, the distorsions always occur in a plane orthogonal to the director field,
+i.e. δn.n = 0. From a Taylor expansion, one can rewrite the deformed state as the sum of
+three main elastic modes: a splay term in ∇.n, a twist term in n.(∇ × n) and a bend term
+in |n × (∇ × n)|. The Frank-Oseen free energy density is the elastic cost of orientational
+oscillations around n:
+fV = 1
+2K1(∇.n)2 + 1
+2K2(n.(∇ × n))2 + 1
+2K3(n × (∇ × n))2.
+(1)
+Equilibrium state corresponds to configurations such that fV is extremal. Elastic constants
+are of order E0/L, with the interaction energy about E0 ≈ 0.1 eV and L ≈ 1 nm, it is
+customary to perform the one-constant approximation for which K1 ≈ K2 ≈ K3 = K =
+10−11 N. This assumption is fair for most ordinary nematics: for instance, in the case of
+5CB at 298 K, one measures K1 = 6.2 pN, K2 = 6 pN and K3 = 8.2 pN [30, 31].
+The simplest class of linear defects consists in axial disclinations and they were firsly
+9
+
+considered by Oseen [32] and Frank [33] (for the class of perpendicular disclinations, proposed
+by de Gennes, see for instance [34]). Orientation of the director field is ill-defined along a
+line (say the z−axis) and n lies in a plane orthogonal to the defect axis (in our example,
+the x − y plane). In cylindral coordinates, let ψ(r, θ) be the angle between the director field
+and the radial unit vector. Then the Euler-Lagrange equations corresponding to a minimum
+of fV simply writes as ∆ψ = 0. The solutions representing disclination lines are given by
+ψ(θ) = mθ + ψ0, where m is the defect strength or topological charge (a priori in R) and ψ0
+a constant phase term. Around a closed loop, the total change in ψ is thus 2mπ. For the
+director field to be well-valued, this variation is tied by the hodograph rule coming from the
+Z2 symmetry of the nematic phase:
+�
+θ=2π
+dψ = 2mπ = kπ
+(2)
+where k ∈ Z. Hence, the director field writes as
+n =
+�
+�
+�
+�
+�
+cos(mθ + ψ0)
+sin(mθ + ψ0)
+0
+�
+�
+�
+�
+� ,
+(3)
+with the topological charge constrained to be integer and half-integer, i.e. m = ±1/2, ±1,
+±3/2,. . .
+Disclinations with integer strengths are topologically removable and belong to the N = 0
+homotopy class: defects m = +1 and m = −1 are topologically equivalent and can be trans-
+formed into one another by continuous deformations. They appear in optical microscopy
+as thick lines and their core is not singular (possibility to escape into the third dimension).
+Disclinations with half-integer strengths are not topologically removable and belong to the
+N = 1 homotopy class. In this latter case, fibring over a circle about the defect line by
+a line segment containing the director which is met at that point, gives a Mœbius ribbon
+which twists along the loop an odd number of times [26] (on the contrary, for the N = 0
+disclination, one gets an ordinary ribbon with two sides). They appear in optical microscopy
+as thin lines and they display a singular core structure. As the free energy density varies in
+m2 and therefore, it is energetically more favorable for a wedge disclination to decay into two
+Mœbius disclinations, as prescribed by the combination 0 = 1+1. It must be remarked that
+besides |m|, other topological invariants (such as the self-linking number, Poincaré-Hopf’s
+10
+
+index...) are needed to characterize the topology of a linear defect, as a disclination can
+globally self-connect, entangle with itself...
+b.
+The secrets of Fermat-Grandjean principle
+In the geometrical optics limit, light
+propagates along paths that can be traveled within the least time. In the case of isotropic
+media, this variational formulation takes the form of the well-known Fermat’s principle,
+established by Pierre de Fermat in 1662. In anisotropic uniaxial media, the constitutive re-
+lations involve a dielectric tensor that displays two different principal permittivities, namely
+ε⊥ and ε∥ (in a nematic, ε∥ corresponds to the permittivity in the direction of the director
+field, whereas ε⊥ is the permittivity orthogonally to it). Fresnel’s equation then provides
+two modes inside such material: the ordinary mode, behaving similarly as in an isotropic
+medium with refractive index n2 = ε⊥, and the extraordinary mode which experiences a
+direction-dependent refractive ray index given by [35]:
+Ne(r) =
+�
+ε⊥ cos2 β(r) + ε∥ sin2 β(r)
+(4)
+where β is the angle between n and the local tangent vector T. In 1919, Grandjean extended
+Fermat’s principle to uniaxial media and he showed that the energy carried by extraordinary
+light rays propagates along paths obeying [8]
+δ
+��
+Ne(r)dℓ
+�
+= 0
+(5)
+where ℓ is the curvilinear abscissa that parameterizes a ray.
+Because the director field changes from point to point, a nematic generally displays a
+varying refractive index and hence, extraordinary light beams propagate into the medium
+along curves (see Fig. 2). In the case of planar axial disclinations, the direction of n and
+consequently β is known at each point. In that case, it can be shown that the integrand in
+Fermat-Grandjean’s principle can be generally rewritten as [36, 37]
+N 2
+e (r)dℓ2 =
+�
+ε⊥ cos2 [(m − 1)θ + ψ0] + ε∥ sin2 [(m − 1)θ + ψ0]
+�
+dr2
++
+�
+ε⊥ sin2 [(m − 1)θ + ψ0] + ε∥ cos2 [(m − 1)θ + ψ0]
+�
+r2dθ2
+−
+�
+ε∥ − ε⊥
+�
+sin2 [2(m − 1)θ + 2ψ0] rdrdθ + dz2
+(6)
+In a seminal work [38], Walter Gordon pointed out the formal analogy between light
+propagation inside a moving dielectric and light propagation inside a non-Euclidean geome-
+try. This idea was developed by many authors eversince [39–45], as it elegantly replaces the
+11
+
+FIG. 2. Light paths and director fields in the presence of a planar disclination (Up left: m =
+1, ψ0 = π/2.
+Down left: m = −1, ψ0 = π/2.
+Up right: m = 1/2, ψ0 = π/4.
+Down right:
+m = −1/2, ψ0 = 0). Taken from [36].
+resolution of Fermat’s principle in a material medium by the search for the minimum-length
+lines (or geodesics) of an empty curved space. The main asset of that point of view is that
+one can use the toolbox of differential geometry to understand how the defect modifies trans-
+port phenomena in its vicinity. To illustrate how this works, let us consider the example of
+a (m = 1, ψ0 = π/2)-disclination (see Fig. 2). For this defect, Eq. (6) leads to the following
+line element:
+ds2
+3d = N 2
+e (r)dℓ2 = dr2 + α2r2dθ2 + dz2,
+(7)
+where α2 = ε∥/ε⊥. The line element is a fundamental quantity in differential geometry
+and it simply consists in a generalization of Pythagoras’ theorem for computing distances in
+arbitrary geometries. Here, instead of the familiar Euclidean line element ds2
+3d = dr2+r2dθ2+
+dz2, the term in α2 means that the circumference of a closed unit circle about the defect is no
+longer 2π but 2πα instead: in other words, there is a mismatch angle (called Frank angle)
+of value 2π(1 − α) compared to flat geometries. It is customary in differential geometry
+12
+
+to rewrite the line element as ds2
+3d = gijdxidxj, where Einstein’s summation convention on
+repeated indices is used. g is called the metric tensor and it corresponds to a positive definite
+quadratic form. The curvature scalar [46] as computed from the metric is:
+R = 2π(1 − α)
+αr
+δ2 (r)
+(8)
+whereas the torsion tensor is identically zero.
+An alternate approach to describe the influence of defects in optics, also from differential
+geometry, consists in using the formalism introduced by Paul Finsler in his 1918 thesis,
+for which there is no quadratic constraint on the geometry as in the Riemannian case [47].
+As a matter of fact, the arc length is given by a Finsler function F such that ds3d =
+F (x, y, z, dx, dy, dz) instead of ds3d =
+�
+gijdxidxj. In the case of anisotropic media, F(r) =
+Ne(r)dℓ and the metric corresponds to the Hessian of the ray index [48]. Ought to the
+particularly simple dependency of the ray index with respect to coordinates, this formalism
+turns out to be fully equivalent to Riemann’s approach (see discussion at the end of [36]).
+A line element of exactly the same form as (7) appears in the geometric theory of defects
+in elastic media [49], related to the strain field associated to wedge disclinations as will be
+described in Section II D. Such line defects can be formed by either inserting or removing
+a wedge of material of angle 2π(1 − α) with subsequent identification of the edges. In the
+case of a removal wedge disclination of axis z (α < 1), the geometry surrounding the defect
+is conical and can be easily pictured from the Volterra cut-and-weld process of Fig. 3. In
+other words, a disclination can be pictured as a Riemann manifold, for which curvature is
+only located on the disclination axis and vanishes everywhere else.
+c.
+Discussion
+From the elasticity point of view (as opposed to optics) the description
+of an axial wedge disclination by the geometry (7), or more generally by (6), calls for
+several remarks (it is important to stress here that ε∥ and ε⊥ now are related to elastic, not
+optical, anisotropy). First, a liquid crystal consists in an assembly of rod-like molecules and
+modeling it as a continuous medium is not self-explanatory. Rigorously, the continuum limit
+for nematoelasticity should come as a coarse grained approximation of molecular dynamics
+and it should fail at the atomic scale. n(r) is defined statistically, as the average common
+direction of the nematogens at each “point” in space. The “point” actually refers to a small
+volume of space that includes enough molecules for the averaging process to be physically
+significant. Hence, in practice, it means that the “point-volume” has to be large enough
+13
+
+FIG. 3. Volterra cut-and-weld process for a (wedge) disclination along z.
+compared to the molecular scale a (typically a ≈ 20 Å) and that the variations of the
+director field must occur at much larger scales than a.
+Only then, the distorted liquid
+crystal can be described as a continuous medium, as discussed by Oseen [32], Zöcher [50]
+and Frank [33].
+A second caveat is related to the status of (7), which obviously possesses non-vanishing
+curvature as in three-dimensional gravity.
+Yet, the nematic actually lives in a three-
+dimensional Euclidean space, which means that the background geometry is flat. How to
+reconcile these two standpoints? Following the analysis from De Wit [51], the state described
+by (7) does exist in the flat space, but only in an imaginary space where the medium is
+relaxed: gij comes from the projection of this imaginary space onto the physical flat space,
+in a similar way as a stereographic map projection transfers the geometric properties on
+a 2-sphere (the Earth, with its meridians and parallels) onto a flat plane while deforming
+them (Wulff net). It turns out that the geometric description of defects thus requires two
+metrics: 1) The physical flat metric, δij, will be used to perform operations on tensors such
+as raising/lowering indices... 2) The effective metric gij, which contains the elastic informa-
+tion, will be used to determine the kinematics of low energy perturbations (geodesics, first
+14
+
+disclination axis
+Frank angle
+2=2元(1-α）
+(α<1: removal)integrals...).
+Third, one may naturally wonder what really happens on the defect axis and how to
+refine our zero-width model.
+In soft matter and more especially nematics, defect cores
+are very narrow as well but they still belong to the realm of continuum mechanics. As
+discussed in [8], a disclination line can accurately be described by a “two-phase model”: the
+core consists in a tubular region, filled with the nematogens in isotropic phase (vanishing
+order-parameter), and surrounded by the nematic phase (non-zero order parameter). This
+approach is consistent with exact solutions obtained from the minimization of the Landau
+– Ginzburg – de Gennes free energy. Yet, the last word has probably not been said about
+disclination cores: observations made on lyotropic chromonic liquid crystals revealed that
+the core region has several unexpected features (asymmetric non-circular interfaces between
+the nematic and the isotropic phases, azimuthal and radial dependencies for the phase and
+amplitude of the order parameter...) compared to classic two-phase models [52].
+D.
+Analogue gravity: lessons and pitfalls
+a.
+Physics as geometry
+The geometric description of transport near linear defects does
+not restrict to optics near axial disclinations. Since the pioneering works by Bilby [53] and
+Kröner [54] in the 1950s, this approach has been extended to elasticity theory as well [55, 56].
+In the noteworthy set of works [49, 57, 58], Katanaev proposed a general framework based
+on Riemann-Cartan manifolds for dislocations and disclinations in elastic media but only
+considered the strain tensor field as the relevant degree of freedom. An expression of the
+effective metric gij in the medium rest frame can be obtained in the case of linear elasticity
+as
+gij = δij + 2εij
+(9)
+where εij denotes the strain tensor. Compared to ordinary elasticity theory (OET), the
+geometric theory of defects is in principle more accurate (ordinary elasticity only reproduces
+the first-order approximation of the geometric theory of defects [49]) and it is more versatile
+(changing the kind of defect only requires changing the metric, instead of a complicated set
+of boundary conditions in ordinary elasticity theory). Moreover, the geometric approach
+is also likely to encompass many other kinds of linear defects of interest in liquid crystals
+physics, such as screw dislocations in smectic A and C [59, 60] (in that case, the defect
+15
+
+must be described in terms of a Riemann-Cartan manifold, for which torsion is only located
+on the dislocation axis and vanishes everywhere else [61]), dispirations in antiferroelectric
+SmCA and the dimeric SmC2 [61, 62], edge dislocations in smectics [63, 64] (which are merely
+disclination dipoles)...
+The preceding examples testify that in many condensed matter systems, the effective
+degrees of freedom are represented by specific field excitations that propagate over effective
+Riemann-Cartan manifolds. Geometrization of physics is not a new idea. In Plato’s Timaeus,
+an attempt was made to describe the world in terms of only five regular polyhedra and ever
+since, geometrization of physics has been a dream pursued by many figures in science,
+including René Descartes, Bernhard Riemann, William K. Clifford [65] (for an updated
+account, see [66])... The most successful step forward in merging geometry and physics was
+made in the twentieth century by Albert Einstein with the theory of general relativity: the
+gravitational interaction turns out to be nothing more than a manifestation of the spacetime
+curvature. The possible implications of that theory did not escape the attention of influencial
+physicists such as Hermann Weyl [67], Arthur Stanley Eddington [68] and more especially
+John A. Wheeler. In the seminal paper Classical physics as geometry [69], Wheeler and
+Charles W. Misner borrow tools from cohomology, differential geometry, exterior algebra and
+topology to fully merge gravitation, electrodynamics and geometry. Provided spacetime is
+multiply-connected, Misner and Wheeler showed that similarly to mass, classical charge can
+also be seen as a byproduct of the spacetime geometry. A particularly meaningful example
+is the low-dimensional gravitational model proposed by Gerard ’t Hooft in the context of
+quantum gravity [70] (see below III C): in 2+1 dimensions, ’t Hooft showed that gravitating
+point particles can elegantly be described as conical point-like singularities of space-time,
+each deficit angle being related to the particle’s total energy. Today, this kind of ideas has
+spread out to the point where it has become an area of research on its own: analogue gravity
+(for an extensive review, see [71] and more recently [72]).
+b.
+Pitfalls
+Despite appealing to classical fields, analogue gravity is tricky and must be
+handled with great care. Indeed, it relies simultaneously on two different manifolds: 1) the
+background gravitational metric – which is experienced by all fields (generally Minskowski’s)
+– is the outcome of Einstein-Cartan equations. It is a tensor, used for instance to raise and
+lower indices of tensors, and as such, it is a covariant quantity, and 2) the effective metric –
+which is experienced only by the fields coupled to matter – does not obey Einstein-Cartan
+16
+
+gravitational equations.
+Its purpose is limited (for instance, to determine the geodesics
+followed by the coupled-fields excitations, as it is not a covariant quantity.
+Indeed, the
+effective metric is derived from physical quantities which are defined in a privileged frame,
+the medium rest-frame, and that are not invariant under Lorentz transformations.
+As can be seen from (9), the effective metric superimposes the background Minkowski
+metric and a correction taking into account the couplings between field and matter. In the
+original experiment led by Hippolyte Fizeau, the changes in the velocity field of water (and
+hence of the Gordon metric itself) were obviously ruled by the Navier-Stokes equations (for
+the velocity field) instead of Einstein-Cartan equations. In other words, effective spacetimes
+are generally stationary. Ref [73] pointed out that textures in nematic liquid crystals can
+indeed be described by the space sector of an Einstein-like equation, with the elastic-stress
+tensor replacing the energy-momentum tensor. The relevance of the effective metric is there-
+fore restricted to calculations of properties related to the kinematic properties of the fields
+coupled to matter. This encompasses as we said the geodesics of low-energy excitations
+but also the less obvious cases of Unruh effect or Hawking radiation which are purely kine-
+matic phenomena [74]. Therefore, the analogy between gravitation and condensed matter
+is strictly kinematic but not dynamical. To rephrase Wheeler, analog spacetime tells matter
+how to move... but matter does not tell analog spacetime how to curve.
+What is the purpose of analogue gravity? In cosmology, putting a theory into test is
+always a thorny challenge. In 1992, the great epistemologist Karl Popper already pointed
+out that the “major theoretical problem in cosmology is how the theory of gravitation may be
+further tested and how unified field theories may be further investigated” [75]. If the plentiful
+harvest of low-energy observations (baryonic oscillation spectroscopy, gravitational wave in-
+terferometry, mapping of the cosmic background ...) answered many questions, theoretical
+models involving (trans)planckian scales bloomed even faster, for which experimental con-
+firmations seem almost impossible – even in principle – to reach. A possible way out of this
+conundrum is to take advantage of the richness and flexibility of condensed matter. Within
+certain limits, analogues of gravity can be used to simulate different types of cosmological
+objects (signature transitions events, cosmic strings...) and to investigate the transport of
+bosonic and fermionic quasiparticles in nontrivial spacetimes. The next section reviews a
+series of works dealing with non-standard cosmological models that can be investigated from
+their liquid crystal counterparts.
+17
+
+III.
+UNRAVELING THE UNIVERSE WITH LIQUID CRYSTALS: COSMOLOGY
+IN THE LABORATORY
+A.
+Phase transitions in cosmology
+a.
+Thermal history of the universe
+In many senses, cosmology consists in thermody-
+namics applied to the largest expanding closed system: our universe. Our current under-
+standing of cosmic history is indeed based on the Standard Hot Big Bang Model and it
+originates in the pioneering works of three founding fathers: Albert Einstein, Alexander
+Friedman and George Lemaître. In essence, this model states that about 13.8 billion-years
+ago, the Universe was in an extremely hot dense state, consisting in a quark-gluon plasma,
+and that it has expanded ever since. In the framework of grand unified theory (GUT), the
+four fundamental interactions (the gravitational interaction, the electromagnetic interaction
+and two lesser known forces, the weak nuclear interaction – responsible for radioactive β de-
+cays – and the strong nuclear interaction – which ensures the cohesion of the atomic nuclei)
+were then assumed to be unified at energy scales estimated at about 1016 GeV.
+Each interaction is associated with internal (or gauge) symmetries: for instance, at today’s
+energy scales, the electromagnetic force displays gauge invariance under the elements of U(1),
+the unitary group of dimension 1 (for an accessible review on gauge theories see for instance
+[76]). Above 1016 GeV, the group G containing the internal symmetries of grand unified
+superforce is not known for sure and many candidates with exotic names are considered,
+such SU(5), SU(6), SU(7), SU(8), SU(9), SO(10), SO(14), E6... [77] The universe expansion
+played the role of a gigantic Joule-Thomson expansion, which caused a large temperature
+drop driving cosmological phase transitions. For example, the last of these transitions is
+the electroweak phase transition, occurring at energy scales about 102 GeV. It marks the
+splitting of the electroweak force into an electromagnetic part, described by Maxwell’s theory
+(1865), and the weak nuclear part, the first theory of which being Fermi’s theory (1933).
+This transition involves a spontaneous gauge symmetry breaking: the high temperature
+gauge symmetry group SU(3)c × SU(2)L × U(1)Y broke into SU(3)c × U(1)em [77].
+Let us now examine the topology of vacuum manifold (that is the set of field configurations
+minimizing the free energy modulo gauge transformations), which is the equivalent of the
+order parameter space M in condensed matter physics. In [78], Jeannerot et al determined
+18
+
+the homotopy content corresponding to all eligible groups G likely to decay below 1016GeV
+into SU(3)c × SU(2)L × U(1)Y . Their conclusion leaves no doubt concerning the formation
+of cosmic strings:
+. ..among the SSB schemes which are compatible with high energy physics
+and cosmology, we did not find any without strings after inflation.
+(if one assumes that the universe is topologically multi-connected, cosmic strings and
+monopoles may appear – not single but pairwise –, whereas two-dimensional defects –
+domain walls – cannot form at all [79]).
+b.
+Kibble-Zurek mechanism
+Cosmic inflation is a period of extremely fast expansion
+of the Universe scale factor (typically a factor 1026 within 10−32 seconds) that presumably
+happened at the very beginning of the universe [80]. From the point of view of statistical
+physics, inflation is nothing more than a quench and as such, it is likely to favor the formation
+of topological defects. In 1976 [81], Tom Kibble introduced a three-step mechanism (later
+refined by Zurek [82] who included the sensitivity to the quench speed) to describe the details
+of this quench.
+Basically, the Kibble-Zurek mechanism (KZM) consists in a nucleation
+process very similar to what happens at the isotropic-nematic phase transition, but instead
+of having an order locally described by the director field n, it is here described by the phase
+of a complex scalar field generically called a Higgs field – or an inflaton, because it needs
+not be the Higgs field responsible for the later breaking of electroweak symmetry. First,
+ordered protodomains (analog to the nematic spherulites) with no correlation between each
+other are formed and at the scale of a whole protodomain, the fast temperature drop due
+to inflation causes the Higgs field to locally take a non-vanishing vacuum expectation value
+and hence to make a phase choice. Then the protodomains grow in size until they coalesce.
+But as they were not correlated, the choices for the Higgs phase (technically, its vacuum
+expectation value) do not match in general, and line singularities of the Higgs appear when
+the boundaries of protodomains finally meet. These linear singularities are called cosmic
+strings.
+Besides this qualitative predictions, the KZM also makes quantitative predictions such
+as the scaling dynamics of the cosmic string network, the average density of defects, cor-
+relations between defects and antidefects... In the 1990s, several works [27, 83–87] showed
+that the KZM, originally developped for cosmology, was also perfectly describing line defects
+19
+
+in nematics with the very same scaling coefficients. For instance, this model predicts that
+in 2D the density of strings scales as ρ ∼ (t/τq)α with a critical exponent αth = 0.5, and
+measurements done by [27] with 5CB indeed gave αth = 0.51 ± 0.04. To sum up: defects
+consist in regions that cannot relax into the new vacuum or equivalently that are unable to
+make the transition into the new ordered phase, and they occur during phase transitions
+in cosmology and in liquid crystal physics that seem to belong to the same universality
+class. But the family resemblance goes further. Networks of cosmic strings and networks of
+disclinations also share similar intersection processes: 1) when two line defects intertwine,
+they may reconnect the other way as they cross (intercommutation) [27, 88] and 2) when
+one line defect self-intersects, it creates a loop [89, 90].
+c.
+An almost perfect analogy?
+Last but not least of these common points: the geometry.
+Nambu-Goto strings, which are the simplest cosmic defects one may expect in cosmology,
+consist in linear concentrations of energy and as such, they are considered as infinitely
+thin objects (as the thickness of a cosmic string is estimated at 10−28 cm, this is a fair
+approximation[91]). As required by thermal field theory and general relativity, the geometry
+around a Nambu-Goto string is described by the Vilenkin’s line element [88]:
+ds2 = −dt2 + dr2 + (1 − 4Gµ)2 r2dθ2 + dz2
+(10)
+where µ is the string energy density estimated at about 10 million billion tons per meter
+(we adopt hereafter the customary unit system of cosmology where c = 1). The space part
+of this element is identical to (7): it is a conical geometry corresponding to a removed Frank
+angle [92] (typically, for a GUT scale string, this angle is a few seconds of arc). The reader
+interested in the classical gauge theory of string interactions in curved spacetimes can refer
+to Ref.[93]. From the standpoint of the soft-matter physicist, Nambu-Goto strings can be
+understood as the cosmic counterparts of wedge disclinations. How to make sense of such
+incredible similarity? For the most part, this question is still open, but a noticeable attempt
+to address it was done in [73]: in essence, the reason is that equations of nematoelasticity
+have the form as the spatial sector of Einstein’s field equations, with the elastic-stress tensor
+playing the role of the energy momentum tensor.
+As the analogy between gravity and nematoelasticity does not concern time components,
+one expects that the dynamics of a cosmic defect cannot be directly mapped with those
+of a disclination. There are other discrepancies between cosmic and elastic defects that
+20
+
+one must bear in mind to avoid fallacies. Obviously, the motion of disclinations is classical
+(typically a few µm per second) whereas cosmic strings are ultra-relativistic. Dissipation
+mechanisms for cosmic strings are due to radiation of gravitational waves, while those in
+liquid crystals are friction-dominated. What is the outcome on the dynamics of the defects?
+In cosmology, monopoles annihilate in pairs (Langacker-Pi mechanism), but they do not
+annihilate fast and early enough to avoid that the Universe becomes monopole dominated
+(which is why inflation is necessary, as it drives monopoles very far away from each other).
+On the contrary, elastic hedgehogs in a nematic annihilate rapidly according to a scaling
+law. At a more fundamental level, this is linked to the fact that in high energy physics,
+broken symmetries are gauged (or internal) whereas in liquid crystals, broken symmetries
+are geometrical: in the first case, one is dealing with “gauged defects” and in the second
+case, one is dealing with “global defects”.
+B.
+Beyond cosmic wedge disclinations
+a.
+The way out of an observational dead-end
+Cosmic wedge disclinations exist either as
+stable infinite straight lines (their equation of state simply equates the string energy density
+to its tension µ = T) or as closed loops that radiate away gravitational waves until they
+vanish. When moving, strings happen to distort spacetime such that at all scales, matter
+accretes along its wake into sheet-like structures.
+They may account for the formation
+of large-scale structures in our universe (including the Great Wall) and they have several
+expected observable signatures such as the Kaiser-Stebbins effect [94, 95] (an asymmetric
+Doppler shift giving rise to anisotropies of the cosmic microwave background), gravitational
+lensing [96] (not in the form of an Einstein ring, but as a double image instead), geometric
+phase (Aharonov-Bohm effect but with a cosmic string replacing the flux tube [97])... Up to
+now, data collected by the PLANCK mission (2014) only settle upper bounds on the string
+parameter µ [98] and in 2020, observations of the stochastic gravitational wave background
+(NANOGrav experiment) may have provided with first evidences for cosmic strings [99–101].
+The non-conclusive observations of Nambu-Goto strings call for the search of refined mod-
+els for linear defects. In fact, the zero-width approximation and the straightness of cosmic
+strings are probably too coarse to account for realistic defects. Hiscock [102] and indepen-
+dently Gott [103] suggest to smoothen this singularity by introducing two string models with
+21
+
+a core structure of constant curvature: the flower-pot model (with zero curvature) and the
+ballpoint-pen model (with non-vanishing curvature). In the Gott-Hiscock thick cosmic string
+spacetime, the metric tensor is piecewise-defined and it must obey matchings conditions at
+the core radius [104]: the extrinsic curvature of the boundary should be the same whether
+measured in the interior or exterior region (O’Brien-Synge-Lichnerowicz jump condition).
+In contrast, thanks to experiments [52, 105] and molecular simulations [106, 107], much is
+known about the NLC disclination core. In particular, there is strong evidence for biaxiality
+and that strength +1 disclinations are in fact bound pairs of strength +1/2 ones, which
+may be manipulated by electrical fields [108]. We note that this rich structure may serve
+as an inspiration for novel cosmic string core models. In the same line, instead of being
+perfectly straight, linear defects can present cusps, kinks and wiggles: the averaged effect
+of these perturbations increases the linear mass density µ and decreases the string tension
+T, as prescribed by the equation of state µ T = µ2
+0 [109–111]. Compared to straight string,
+the geometry remains conical but the deficit angle is larger than in the straight string case,
+which increases polarization anisotropies of the cosmic background radiation [112].
+There are many other ways to dress a Nambu-Goto string such that it may account
+for observational results [113]. From an extension of Volterra process to 3+1 dimensions,
+Puntingam and Soleng showed that there was only 10 ways to modify a Minkowski spacetime
+into different pseudo-Riemann–Cartan geometries with respect to the Poincaré group. For
+example, a cosmic linear defect can display chirality [114–119]: in that case, the defect
+carries torsion along its axis and one gets the cosmic counterpart of a screw dislocation in
+a smectic liquid crystal. Twisted Nambu-Goto strings (or cosmic dispirations), consisting
+in spacetimes with delta function-valued curvature and torsion distributions have also been
+considered, as they combine both rotational and translational anholonomy [120–123]: as
+mentioned earlier, their effects on light could be tested from experiments done with elastic
+dispirations SmCA and SmC2.
+b.
+Going further
+Rotating disclinations are not likely to be stable but it is worth
+mentioning here that their cosmic counterparts have been long predicted in the literature
+[124]. A metamaterial analogue of the rotating cosmic string spacetime has been proposed
+[125], as well as a superfluid vortex analogy [126]. One of the most interesting properties of
+the spinning string is its association to closed timelike curves which may find applications
+in time travel [127]. Incidentally, parallel cosmic strings moving in opposite directions have
+22
+
+been suggested [128] as a prototype time machine. Related to, but not really a model for
+spinning strings, is the case of a hyperbolic nematic-based metamaterial with a disclination
+which was addressed in [129]. Along the same line, in [130] it was proposed a disclination
+model for the compactified Milne model of a cyclic universe. More details on this model in
+Section III C.
+Among the gauge strings, a very interesting possibility is the semilocal string [131]. Like
+Dirac’s string of magnetic dipoles, semilocal strings end on gauge monopoles. They are
+analogous to real disclinations in liquid crystals which, due to the finite size of a liquid
+crystalline sample, must end somewhere (hedgehog, 2D disclination on the liquid surface or
+on receptacle wall) or else, form a loop. Apropos, disclination loops in active nematics have
+very complex dynamics (including chaos) and may present recombination episodes [90].
+Defects in liquid crystals have inspired many other proposals in cosmology. GUT allows
+for discrete gauge symmetry groups, the standard Z2 parity and one Z3 parity, which are the
+only anomaly free groups that remain unbroken at low energy [96, 132]. The corresponding
+cosmic strings are generically called Zn-cosmic strings.
+Based on the known physics of
+Moebius disclinations, which are commonly observed in nematics, Satiro and Moraes have
+investigated some cosmological outcomes of Z2 cosmic strings [133]: in particular, they
+showed that Z2-cosmic strings display both positive and negative mass density regions.
+Bearing in mind the back and forth interplay between cosmology and soft matter, one
+cannot avoid to mention the latest works of Maurice Kleman, who imported homotopy
+theory from condensed matter to astrophysics and cosmology [134–136]. In particular, he
+conjectured and classified new families of cosmic defects (such as r-cosmic forms) allowed in
+a four-dimensional maximally symmetric spacetime [136].
+C.
+Black holes and Early universe
+a.
+Black holes, white holes, wormholes
+This is sometimes referred to as the “cosmology
+in the laboratory” game plan and it covers topics such as classical black holes [137]-[138],
+Hawking radiation [139]-[140], wormholes [141]-[142]... For instance, in Haller’s approxima-
+tion, the hydrodynamics of a nematic liquid crystal radially flowing down a drainhole is
+23
+
+experienced by light beams as the equatorial section of the Schwarzschild’s metric
+ds2 = −
+�
+1 − 2M
+r
+�
+dt2 +
+dr2
+�
+1 − 2M
+r
+� + r2(dθ2 + sin2 θdφ2)
+(11)
+for a specific velocity profile. The ordinary and extraordinary indexes of the NLC depend on
+the scalar order parameter of the liquid crystal. So, it was possible to taylor those indexes
+to get the proper optical metric. In order to achieve this, the Beris-Edwards hydrodynamic
+theory wass used to connect the order parameter with the velocity of the liquid crystal flow
+at each point. This was done in Ref. [143].
+More recently, an optical analogue of a wormhole threaded by a cosmic string was de-
+scribed in [142].
+Wormholes are solutions of Einstein’s equations that connect different
+regions of the spacetime. For instance, a spherically symmetric wormhole can be obtained
+by joining two Schwartzschild black hole spacetimes by a spherical hole carved around each
+singularity.
+Wormholes are usually represented by “embedding diagrams”, which are 2D
+slices of the 4D structure immersed in Euclidean 3D space. The embedding diagram of the
+notorious Morris-Thorne [144] wormhole is obtained by taking a t = const., θ = π/2 section
+of the spherically symmetric spacetime described by the metric
+ds2 = −c2dt2 +
+dr2
+1 − b2
+0/r2 + r2(dθ2 + sin2 θdφ2).
+(12)
+The restricted metric, ds2 =
+dr2
+1−b2
+0/r2 +r2dφ2, can be embedded in a 3D Euclidean space with
+metric ds2 = dz2 + dr2 + r2dφ2 such that z = z(r) is the equation of the embedded surface
+of revolution. For metric (12) the result is the catenoid.
+A thin nematic film on a catenoid with director field aligned either circularly or radially
+(see Fig. 4) has an optical metric given by [142]
+ds2 = dτ 2 + α2(τ 2 + b2
+0)dφ2,
+(13)
+where α = no/ne for the circular case, and α = ne/no for the radial one. The coordinate τ is
+the arc length of the catenary that under rotation gives rise to the surface. The parameter
+b0 is the radius of the wormhole “throat”. For α = 1, Eq. (13) reduces to the catenoid
+metric. It is clear from Eq. (13) that, asymptotically (τ >> b0), the optical metric of the
+disclination is recovered. This is also evident from the top view of the catenoids of Fig. 4.
+This optical model simulates the conical spacetime of a Morris-Thorne wormhole threaded
+by a cosmic string. The geodesics as obtained in [142] are represented in Fig. 5.
+24
+
+FIG. 4. Director field for circular and radial +1 disclinations on the catenoid, respectively. Taken
+from [142].
+(a)
+(b)
+(c)
+FIG. 5.
+Assorted geodesics for the circularly decorated catenoid.
+The blue lines represent the
+isotropic case α = 1. The red and black lines represent, respectively, circular (deficit angle) and
+radial (surplus angle) disclinations with α = 0.85 for (a) and (b), and with α = 0.98 for (c). Taken
+from [142].
+From Fig. 5 it is clear that the two parts of the wormhole joined by its throat act as a
+black hole/white hole pair.
+b.
+Road to quantum gravity
+A major contemporary challenge in physics is to find an
+extension of General Relativity able to describe gravity at all energy scales, in particular at
+the very beginning of the universe. This is the mission devoted to quantum gravity theories,
+which have the daunting task of reconciling Einstein’s general relativity and quantum field
+theory. Despite promising attempts including superstring theories, M-theory or quantum
+loop gravity, no proposal is entirely satisfactory up to now, and even so, the energy scales
+required to test these theories are far beyond our current scientific capabilities. A way out
+of this gridlock is to rely on simpler models that capture the essential features of quantum
+gravity but remain connected to low-energy-physics systems, i.e. analogue gravity. The
+rare pearl was first introduced in a seminal paper by Deser, Jackiw and ’t Hooft [145]: 2+1
+25
+
+gravity with point-particle sources.
+The main point is that there is no gravitational degrees of freedom in three dimensions
+[146], which drastically simplifies general relativity (now an exactly solvable model [147]).
+Within this framework, the geometry surrounding a point-particle is a conical singularity,
+the mismatch angle being proportional to the particle’s mass. In other words, conical de-
+fects represent point particles coupled to gravity in 2+1 spacetimes. After Katanaev [57]
+first pointed out that the theory of linear disclinations was isomorphic to the 2+1-gravity,
+Kholodenko [148] used the apparatus of quadratic differentials to establish the connection
+between Deser, Jackiw and ’t Hooft model and defects in liquid crystals. In essence, the exis-
+tence of massive particles considered as field singularities is directly related to the topology of
+the underlying manifold (the Euler characteristic) and to the emergence of the induced sad-
+dles: this means that 2+1 Einstein’s equations are strictly equivalent to the Poincaré-Hopf
+theorem (see section 5.2 in [148]), the Hopf quantization rule making the direct connection
+between particles masses Mi and the defect topological charge m, 4GMi = m [149].
+The 2+1 gravity model can therefore be experimentally investigated from a network of
+parallel disclinations lines in a 3D nematic sample. Geometry of disclinations networks has
+been theoretically investigated in the literature, sometimes allowing for analytical expres-
+sions for the metric tensor [150–152]. Several authors have shown the possibility to design
+arrays of topological linear defects from photopatterning techniques [153–157] and even to
+manipulate them [158, 159]. If this last point opens the possibility to emulate collisions be-
+tween particles in the 2+1 model, it is even more interesting for the extension of the Deser,
+Jackiw and ’t Hooft model to 3+1 dimensions [70]: matter particles are represented by a
+gas of piecewise straight string segments that are likely to collide with a higher frequency.
+The strings display both positive and negative mass densities, i.e. they are associated to
+α < 1 and α > 1 Frank angles, which makes liquid-crystal-based experiments particularly
+promising to investigate such models. This model may also have deep connections with
+Regge calculus in quantum gravity, where the smooth curved spacetime is replaced by a
+piecewise-flat simplicial manifold. This is like the triangulation of a surface in 3D where the
+local curvature is described by the dihedral angle between adjacent triangles (the triangle is
+a 2D simplex). The effect of gluing the edges of the simplexes generate a network of cone-like
+singularities (Regge cones) which are analogs to wedge disclinations [160, 161] (see Fig. 6).
+26
+
+FIG. 6. Triangulation of a sphere at the Itapetinga radiotelescope (Brazil). Covering a curved
+surface by the triangles generates dihedral angles all around the sphere.
+c.
+Non-standard cosmology
+Cosmology at transplanckian scales is a thorny problem,
+both theoretically and of course observationally. For instance has the universe popped up
+from a unique singular event, the Big-Bang? And if so, how not to wonder what could have
+happened before it and how to design experiments to test these theories? Today, many
+high-energy physics theories such as quantum loop gravity and supertring theories entice
+the search for cyclic universe models, that is an endless repetition of big crunches followed
+by big bounces, along the same line of thought as the Stoics’ concept of palingenesia. A
+safe transition has been proposed [162, 163], where the singularity is nothing more than the
+temporary collapse of a fifth dimension, the three space dimensions remain large and time
+keeps flowing smoothly. A toy model for the geometry of this transition is the compactified
+2D Milne universe MC [164, 165]. The Milne universe metric is given by
+ds2 = −dt2 + t2dχ2 + t2 sinh2 χ
+�
+dθ2 + sin2 θdφ2�
+(14)
+and it was proposed by E.A. Milne in 1933. It represents a homogeneous, isotropic and
+expanding model for the universe with a negative curvature. In order to compactify the
+27
+
+Milne universe on hypersurfaces of fixed solid angle, let the variable χ acquire some period
+denoted as 2πκ: here, 0 < κ ∈ R1 is a constant parameter for compactifications. After
+reparametrization, the line element corresponding to the compactified Milne universe finally
+writes as
+ds2 = −dt2 + κ2t2dφ2
+(15)
+where t ∈ R1. As can be seen from the disclination line element (7), the presence of κ2 in
+the above metric indicates a conical singularity of the curvature at the origin (see Fig. 7).
+The passage through the initial singularity has several unusual features [130, 166]: the
+singularity acts as a filter for classical particles and a phase-eraser for quantum ones. The
+timelike geodesics of (15) reveal that depending on their angular momentum J, particles have
+two ways of crossing the singularity: 1) For non-vanishing J, a two-step dynamics consisting
+in an inward stable motion before the singularity, followed by outward stable motion on the
+other side, with a memory loss of the particle kinematical properties (quantum mechanically,
+this effect simply comes from strong oscillations of the phase of the wave function at the
+singularity. 2) For J = 0, a one-step dynamics consisting in a straight line through the
+collapse: yet such trajectories are very unstable since small perturbations in the value of J
+causes large deviations on the trajectories.
+Probing how particles behave in MC can be tested in the laboratory from hyperbolic
+liquid crystal metamaterials (HLCM): this means that the permittivity along the director
+axis ε∥ < 0 and the permittivity perpendicular to the director axis ε⊥ > 0 are of opposite
+sign [167]. Such media can be made from a host nematic liquid crystal that includes an
+admixture of metallic nanorods [168] or coated core-shell nanospheres [169]. To retrieve the
+Kleinian double-cone geometry, the HCLM must be endowed with an hyperbolic disclination:
+the line element writes as ds2 = ε⊥dρ2 − ε∥ρ2dφ2 + ε⊥dz2, which after a rescaling becomes
+ds2 = −γ2r2φ2 + dr2 + dz2. This line element is relevant only by the extraordinary modes
+and for radial injection conditions (planar trajectories z = Cst), the geometry experienced
+by extraordinary rays is perfectly identical to that of the compactified Milne universe.
+A stable configuration for the director field may be obtained from a cylindrical shell
+of HLCM with homeotropic anchoring at the boundaries. In the geometrical optics limit,
+extraordinary light paths turn out to be Poinsot’s spirals as for the compactified Milne
+universe. The practical realization sets limits to the efficiency of such analog device for
+the classical particles. First, the analysis holds only within a limited frequency bandwidth
+28
+
+FIG. 7. Timelike geodesics with the radial time t given in units of t0 and κ = 1/3. The blue and
+green (orange and red) lines are moving away (towards) from (to) the singularity. Furthermore,
+particles following the trajectories in the first (third) and second (fourth) quadrants are spinning
+clockwise (counterclockwise). The blank point at the origin is just to emphasize that the curves do
+not reach the singularity. Taken from [130].
+due to the resonant nature of the used core-shell spheres. Second, as previous phenomena
+concern the extraordinary mode, an efficient optical absorber should include a filter to shut
+off the ordinary wave. Finally, it should be noticed that the present model concerns optics
+inside a bulk hyperbolic material: to design a perfect optical analog, the hyperbolic medium
+must be impedance matched to avoid sizable reflections at the interfaces. The analogy was
+also extended to quantum particles by investigating light in the scalar wave approximation
+in the same device.
+29
+
+t
+3
+f+, J <0
+f+, J > 0.
+f-, J >0.
+J<O
+BigBounce
+2
+f, J <0.
+-1
+-2
+Big Crunch
+3
+X
+-2
+-1
+0
+1
+3
+J>0
+3
+J=0FIG. 8. Geodesics of the channel of defects in the potential landscape. Taken from [129].
+IV.
+TAYLORING TRANSPORT WITH LIQUID CRYSTALS: DEFECT-ENGINEERED
+MATERIALS
+A.
+Acoustics
+a.
+Ballistic guiding
+The geometric description of linear defects revealed wedge discli-
+nations carry curvature along their axis: a positive Frank angle corresponds to a conical
+geometry that focuses incoming light rays, in similar fashion to what happens with a con-
+verging lens. Conversely, a negative Frank angle corresponds to a saddle-like geometry that
+scatters geodesics, in the same fashion as a diverging lens. It is worth noticing that the effect
+of a disclination does not limit to the eikonal approximation, as it also diffracts incoming
+waves: from the present geometric approach, computing the differential scattering cross sec-
+tion of a wedge disclination showed good agreements with theoretical results obtained by
+Grandjean from standard acoustics [170, 171].
+The effect of a single wedge disclination on rays being clarified, one can legitimately won-
+der if a well-suited arrangement of such defects can be used to taylor the propagation of
+sound (we recall that photopatterning techniques now allow for the practical realization of
+almost any kind of arrays). In [151], a channel of disclination dipoles was considered, con-
+30
+
+V(x,y)
+10sisting in two infinite rows made of alternate disclinations separated by distance 2a (a kind
+of von Kármán alley), the distance between the rows being 2b. The positive disclinations
+are at points located at (na, (−1)nb), n ∈ Z, while negative disclinations have coordinates
+(na, (−1)(n+1)b), n ∈ Z. As always done in geometric models, the bulk medium is consid-
+ered in the continuum limit (i.e. limit of a vanishing lattice spacing). The corresponding
+background geometry writes as
+ds2 = −c2dt2 + e−4V (x,y) �
+dx2 + dy2�
++ dz2 = gµνdxµdxν
+(16)
+where c is the local speed of wave packet and V is the acoustic potential given by [151]:
+V (x, y) =
+|F|
+4π ln
+��
+cosh2 � π
+2a(y − b)
+�
+− cos2 � πx
+2a
+�
+cosh2 � π
+2a(y − b)
+�
+− sin2 � πx
+2a
+�
+� �
+cosh2 � π
+2a(y + b)
+�
+− sin2 � πx
+2a
+�
+cosh2 � π
+2a(y + b)
+�
+− cos2 � πx
+2a
+�
+��
+(17)
+where |F| is the absolute value of the Frank angle that characterizes the ±F defects. The
+sound paths are attracted by the positive defects while repelled by the negative ones (see
+Fig. 8 and 9). Adjusting the defect strengths along with parameters governing the geometry
+of a cell (namely a,b) opens the possibility to taylor material properties of the sheets, but
+this will only be achieved numerically considering the complexity of analytical expressions.
+Sound paths are yet very sensitive to the shooting angle. Hence, a thorough optimization of
+the distribution of defects deserves an additional treatment of chaos involving the statistical
+tools of dynamic hamiltonian systems.
+b.
+Acoustic rectification
+Differential geometry models for acoustics originates from
+[172] where vorticity effects in isotropic fluids were investigated. In nematics, for a pla-
+nar horizontal configuration, the acoustic metric experienced by the extraordinary mode is
+formally identical to (6) with the substitutions ε⊥ ↔ ρv2/C33 and ε∥ ↔ ρv2/C11, where ρ
+is the mass density, v is the velocity of sound in the isotropic phase, C33 and C11 are elas-
+tic constants respectively along the director’s direction and in directions orthogonal to it.
+When the nematic medium is confined inside a capillary tube (radius R) with homeotropic
+boundary conditions, the director field tends to be radially oriented everywhere but on the
+central axis where an orientational singularity lies, but to reduce the elastic energy of this
+configuration, the director escapes in the third dimension [63, 173] and the nematic relaxes
+31
+
+FIG. 9. Different geodesics, shot from the origin, in the channel of disclinations geometry. The
+positive disclinations correspond to red contours while negative disclinations correspond to blue
+contours. Depending on the shooting angle, the propagation of phonons may be guided by the
+street of topological defects. Taken from [129].
+into a funnel-shaped configuration, known as the escaped radial disclination (ERD) where
+the delta-distributed Ricci scalar becomes an extended smooth one [174].
+In general, rectification effects come from an asymmetry of the system in the direction
+along which the transport phenomenon occurs. Usually, this is generally achieved by relying
+on gradients of physical properties (e.g. pore density [175], distribution of compositional
+defects [176]...), asymmetric geometries [177, 178]... all examples of situations implying hard
+and non-flexible systems. Liquid crystals naturally provide a stable but flexible configuration
+corresponding to such asymmetry, the ERD. In the spirit of a low-cost soft-matter-based
+solution, satisfying levels of acoustic rectification have been obtained by combining the
+asymmetry of the ERD configuration to that of a container consisting in conical frustum
+of varying radius R(z) [179]. The inner surface prepared to produce the desired anchoring
+angle[180]. The anchoring angle adapted to maintain the ERD configuration. α depends on
+the surface geometry (radius), on the liquid crystal nature (elastic constant K, saddle-play
+32
+
+0.5
+0.5 constant K24) and on the surface treatment. Reference [179] considered acoustic waves in a
+frequency range between 20 Hz and 20 kHz (the average human audible range) propagating
+in 5CB for the ERD configuration. The rectification parameter used to estimate the acoustic
+device’s efficiency is the percent standard deviation of the lowest variation on the acoustic
+intensity
+Acoustic rectification(%) =
+����
+∆Ibt − ∆Itb
+min (∆Ibt, ∆Itb)
+���� × 100
+(18)
+where ∆Ibt = It − Ib is the acoustic intensity variation if the wave comes from the bot-
+tom to the top of the conical frustum and ∆Itb = Ib − It is the analogous variation for the
+counterpropagating case. Numerical simulations show a rectification effect for a longitudinal
+plane wave propagating along the conical frustum axis (see Figure 10). The optimization
+of the device parameters (geometry of the conical frustum, anchoring conditions the rectifi-
+cation effects...) allows to reach rectification levels up to 1300% for a continuous frequency
+bandwidth [181].
+FIG. 10. Left: Conical frustum with an ERD. Middle: Pressure field for the incoming waves from
+bottom to top. Right:Pressure field for the incoming waves from top to bottom. Taken from [181].
+33
+
+mPa
+mPa
+7.47237
+0.58721
+7.47236
+0.5872
+7.47235
+7.47234
+0.58719
+a)
+20
+Jm
+b)
+20
+Ars
+7.47233
+0.58718
+50
+50
+7.47232
+50
+50
+0.58717
+s) <0
+0
+μm
+-50-50
+m
+0
+-
+Aim
+7.47231
+-50-50
+μm
+0.58716
+7.4723
+0.58715B.
+Optics
+a.
+Waveguiding and light concentration
+Manipulation of light has become a major issue
+in a large number of applications ranging from solar energy harvesting to optical sensing.
+The beam steering technique relies on a modulation of refractive indices to guide light
+in a given direction ([182–185]). More recently, the possibility to continuously deflect in
+arbitrary directions and/or to simultaneously focus/defocus an incoming light beam has
+been demonstrated from multiple stacked nematic liquid crystal cells—building blocks [186].
+Another possibility to guide light beams is to use optical waveguides with nematic cores
+in radial escaped disclination configurations: the defocusing due to natural diffraction is
+compensated by the converging lens effect resulting form negative birefringence [187–191].
+Light focusing has long been suspected of suffering severe limitations, the Rayleigh crite-
+rion forbidding beam sizes below half of a wavelength. Recently, the advent of metamaterials
+provides new hopes for overcoming the diffraction limit using superlenses [192]. Another
+promising possibility is to use an hyperbolic liquid crystal metamaterial (HLCM), obtained
+from an admixture of metallic nanoobjects to nematics [193]. As seen in the previous sec-
+tion, the permittivity along the director axis ε∥ and the permittivity perpendicular to the
+director axis ε⊥ have opposite signs. For an orthoradial director field, the effective metric
+writes as gij = diag (1, −α2ρ2, 1) and in the eikonal approximation, the light paths are the
+aforementioned Poinsot spirals: the hyperbolic defect behaves as a sink for light paths [194].
+The smaller the value of α, the stronger is the spiraling behavior, 1/α corresponding to the
+defect vorticity. Concentration of light by an hyperbolic disclination extends beyond the
+geometrical optics limit. In the scalar wave approximation, the complex amplitude Φ of the
+wave is governed by the generalized form of the d’Alembert equation, involving the Laplace-
+Beltrami operator instead of the ordinary Laplace operator. The wave equation writes as
+a modified Bessel differential equation of imaginary order iℓ/α and the solutions are linear
+combinations of the modified Bessel functions of first and second kind, respectively. The
+intensity distribution for the propagating fields shows 1) that the electromagnetic field con-
+centrates along the axis of the device, and 2) that the bigger the value of the frequency, the
+smaller the light rings are (see Fig. 11).
+34
+
+FIG. 11. Examples of intensity profiles, representing the transverse field distributions concentrated
+in the vicinity of the hyperbolic defect. Taken from [194].
+b.
+Optical vorticity
+Active beam shaping has also emerged as a major trend in modern
+optics. The idea is to taylor the amplitude, the phase or the polarization of an optical
+wavefront from real-time driven systems that ideally must be compact enough, highly-flexible
+but yet have low manufacturing costs. Ought to their high response functions, liquid crystal-
+based devices naturally fulfill all these requirements and have emerged as promising low-
+cost and easy-to-manufacture alternatives to MEMS, moving opto-mechanical devices and
+photonic crystals [195, 196].
+In nematics, the orbital angular momentum carried by light beams can be tuned from
+q-plates. A q-plate consists in an inhomogeneous liquid crystal cell endowed with a discli-
+nation of topological charge q (for details regarding the dielectric tensor, see [197]). When
+an electromagnetic wave propagates inside the medium, its components acquire designed
+phase shifts (Pancharatnam-Berry phase) that trigger spin-to-angular momentum conver-
+sions. This results in light beams displaying optical vorticity, i.e. the wavefronts are helical
+and the intensities profiles distribute in doghnut-like shapes [198–203]. [204] demonstrated
+from FDTD simulations that disclination lines can transform the state of polarization of
+beams propagating along their axis. Umbilics ending disclination lines are also identified as
+35
+
+structures generating optical vortex arrays at predetermined wavelength [205].
+Nematics are not the only contenders in the contest for optical vorticity. In Ref. [206]
+the Raman-Nath diffraction was used to generate optical vortices from edge dislocations in
+the stripe pattern of a cholesteric liquid crystal (cholesterics are chiral nematics for which
+the director whirls around a well-defined direction). Cholesterics were also considered in
+[207], where it has been theoretically and experimentally demonstrated that optical vortices
+were generated from a Bragg-reflection-based device, therefore likely to operate at multiple
+wavelengths. Recently, charged particles crossing a cholesteric plate were reported to radiate
+purely twisted photons [208]. Chiral nematics are also likely to generate defect textures of
+a more complex kind than that optical dislocations [209]. Beside cholesterics, smectics also
+showed their potentialities for optical vorticity. [210] produced an optical vortex from focal
+conic domains, whereas in [166], screw dislocations in smectics were shown to imprint their
+torsion onto wavefronts (see also [211, 212] for a solid-state-oriented context).
+C.
+Heat transfer
+a.
+Principles of thermal design
+Manipulation of heat flux raises intensive research ef-
+forts because of the abundant wealth of potential applications including thermal shielding or
+stealth of objects, concentrated photovoltaics or thermal information processing (heat-flux
+modulators, thermal diodes, thermal transistors and thermal memories). These prospects
+come from the possibility of designing energy paths in a fashion similar to that of light in
+transformation optics. To do so, the first step is to understand the main peculiarities of
+heat transfer in the presence of a non-Euclidean geometry. Generally speaking, diffusion of
+a passive scalar (for instance the temperature field) can be seen as a collection of Markov
+processes obeying the stochastic Fokker-Planck equation. In the case of Brownian motion,
+the Fokker-Planck equation reduces to the well-known parabolic heat equation [213]. When
+considering diffusion processes in the presence of a non-Euclidean space, the problem is ad-
+dressed, as already discussed, by replacing the Laplace operator with the Laplace-Betrami
+operator ∆LB [214]:
+∂T
+∂t = D∆LBT.
+(19)
+Here, D is the diffusivity and its value depends on the material properties. Ought to the
+form of the metric of a wedge disclination, heat conduction locally occurs as in a monoclinic-
+36
+
+like crystal with no internal source [215]: the heat flux vectors are no longer perpendicular
+to the isothermal surfaces, which are bent depending on the value of the Frank angle. In
+other words, disclinations in nematics generate thermal lensing effects.
+FIG. 12. Top: Temperature field (a) and heat flux field (b) for a radial director field (homeotropic
+anchoring) and α =
+�
+C33/C11 = 0.5 Bottom: Temperature field (a) and heat flux field (b) for an
+orthoradial director field (parallel anchoring) such that α =
+�
+C11/C33 = 2. Taken from [216]
+Once the basic effects of single line defects are understood, the next step is to taylor them
+to guide heat. To do so, let us consider a hollow cylinder, inside which there is the core region
+where one aims at controlling the conductive heat flux. The cylinder is inserted inside a
+conducting solid sandwiched between two heated vertical plates. The host material consists
+37
+
+(a)
+(b)
+(a)
+(b)of a homogenous isotropic medium, whereas the intermediate thick cylinder consists of a
+nematic liquid crystal in a disclination-like configuration (no disclination core). For thermal
+management, mesophases with low melting and high clearing temperatures are required: a
+range of about 100 K can be reached by using eutectic liquid crystal mixtures (or “guest-
+host systems”). Numerical simulations [216] confirm the possibility of a strong heat guiding
+phenomenon: depending on the value of elastic constants C33 (along the director) and C11
+(along any direction perpendicular to the director), the device can either cloak the core region
+from the heat flux or concentrate heat there (see Fig. 12). Switching from the concentrator
+to the cloaking device is achieved by an electric-field-driven bistable anchoring with dye-
+doped mematics (sufficiently high values of the electrical potential difference between the
+two sides of the hollow cylinder were indeed shown to induce stable anchoring transitions
+between homeotropic and parallel states [217]). To avoid thermoconvective instabilities in
+the annulus domain, the device must be thin enough and the heat flux and temperature levels
+must be moderate. For instance, using 5CB and MBBA, if the temperature is about a few
+tens of degrees (thermoelectric applications) and the external radius of a few centimeters,
+the device handles heat flux that typically varies from 5 W/m2 (repeller) to 103 W/m2
+(concentrator).
+b.
+Thermal diodes
+The previous study is now refined in order to investigate thermal
+rectification. Thermal rectification is a very active subject in nanoscience and solid-state
+physics, as testified by the abundant litterature dealing with this subject (extensive reviews
+treating thermal rectification can be found in [218, 219]. More seldomly is heat conduction
+rectification considered from soft-matter-based devices. In liquid crystals, the macroscopic
+thermal properties of the nematic phase depend on temperature according to Haller’s ap-
+proximation [220]:
+λ∥(T) = λ0 + λ1 × (T − TNI) + λ1∥ × (T − TNI)α∥
+(20)
+λ⊥(T) = λ0 + λ1 × (T − TNI) + λ1⊥ × (T − TNI)α⊥
+(21)
+where λ0, λ1, λ1∥, λ1⊥, α∥ and α⊥ are material-depending constants, whereas TNI and TC
+are, respectively, the nematic-to-isotropic temperature and the clearing-point temperature
+of the liquid crystal. As discussed before, liquid crystals naturally provide a stable and
+flexible configuration corresponding to such asymmetry, the ERD, which already turns out
+to provide high levels of acoustic rectification. To achieve high rectification levels, the same
+38
+
+conical frustum of varying radius R(z) with anchoring conditions can be used. In analogy
+with the acoustic case discussed earlier, the rectification parameter used to estimate the
+thermal diode efficiency can be defined as
+Thermal rectification(%) =
+����
+∆Ti − ∆Td
+∆Td
+���� × 100
+(22)
+where ∆Td = Td,h − T0 is the difference between Td,h, the high temperature on one base
+produced by the heat pumped in the cylinder when working in the direct setup (i.e. when
+the heat is flowing from the narrow region to the wider one, i.e. the −z direction), and
+T0, the temperature at the other base, which is also the initial temperature.
+Similarly,
+∆Ti = Ti,h − T0 when working in the inverse setup.
+Numerical simulations show thermal rectification rates around 1266% [221]. On the shape
+parameters, alterations on the ratio Rr ∈ [0, 28; 0, 75] produced a percentage variation on
+the thermal rectification around 1273%, while modifications of the height h ∈ [50; 75] µm
+and on the larger radius Rl ∈ [50; 70] µm produced percentage changes lower than 5%.
+This indicates that the anisotropy of the conical frustum tube has a strong influence on the
+rectification. Other non-geometrical parameters such as the anchoring angle (in the range
+[0; 90◦]) and the inward pumped heat flux (in the range [5; 10] kW/m2) give percentage
+variations on the rectification around, respectively, 3, 8 and 1, 7%.
+Such characteristics
+enable this improved thermal diode to be miniaturized, applied on well-determined areas,
+while robust against variations of the inward pumped heat flux.
+The identical forms of the geometry experienced by light and by sound strongly suggests
+that devices using liquid crystals may be used to manipulate simultaneously optical and
+thermodynamical transport. Indeed, Ref. [222] reports the control of both electromagnetic
+propagation and heat flow by a liquid crystal device similar to the one depicted in Fig. 12,
+while Ref. [223] uses an escaped disclination configuration to rectify at the same time both
+heat and light, thus a thermo-optical diode.
+V.
+CONCLUSION AND PERSPECTIVES
+Since their early discovery in the XIXth century, liquid crystals have been the magic bullet
+in physics and engineering. Being in-between anisotropic solids and isotropic fluids, they
+extended our conception of condensed matter to an area where geometry and topology can
+39
+
+FIG. 13. Left: Isothermal surfaces of a liquid crystalline thermal diode in (up) inverse thermal
+setup and (bottom) direct thermal setup. The frustum diode has larger radius Rl = 70 µm, ratio
+between the radii is Rr = Rsm/Rl = 0, 28, the height is h = 50 µm, anchoring is 60◦ and the
+inward heat flux is Q = 5 kW/m2 on the base with the higher temperature and T0=296 K. Right:
+Rectification rate versus temperature T0 of the base for different larger radii Rl. Taken from [179].
+be almost as useful as in General Relativity. An orientationally ordered fluid, as the nematic
+liquid crystal, is a vivid representation of a Riemannian manifold where the director field
+can be associated to a local vector basis (triad or dreibein). The relative rotation of the
+director/triad associated to neighboring points indicates the presence of curvature. Bound-
+ary conditions like vessel shape, immersed objects, anchoring angle, etc., impose restrictions
+to the effective geometry whose eventual incompatibility with the nematic order (ground
+state or zero curvature everywhere) leads to the appearance of topological defects which
+accommodate the incompatibilities.
+This geometric view of the elastic distortions in the NLC is complemented by the optical
+effective geometry that appears naturally by comparing Fermat’s law of least time to the
+geodesic variational principle. Similar effective geometries can be obtained for acoustics
+and heat transport as well. In all these cases the defects, besides being the consequence of
+40
+
+μm
+ 309
+0
+308.58
+306.77
+307.67
+40
+305.87
+304.97
+304.06
+303.16
+20
+1.270 
+301.35
+0
+7 300
+1.260
+-50
+0
+50
+Rectification [%]
+1.250
+R, = 50 μm
+R; = 57 jm
+R; = 63 jm
+1.240 -
+R=70m
+μm
+ 301
+1.230
+300.64
+300.57
+1.220 
+40
+300.5
+300.44
+300.37
+1.210 
+300.3
+295
+296
+297
+298
+299
+300
+TOE
+20
+300.23
+300.17
+Te [K]
+300.1
+0
+300.03
+V 300
+-50
+0
+50the topology (boundary conditions), are the source of the geometry. One might then say
+that, as soon as there is real or effective curved geometry to describe a physical system with
+orientational order, one can expect defects. And, if there is curved geometry, one can relate
+NLC to gravitation and cosmology. In this article we reviewed, not only this relationship,
+but also the physical applications obtained with the help of the geometric tools.
+Many
+open problems both in gravitation and cosmology and in NLC certainly may benefit from
+the analogies derived by the (partially) common geometry. For instance, the experimental
+knowledge about the inner structure of disclinations may be an inspiration for cosmic string
+core models. Active matter, being a dynamic medium, may be described by time-depending
+metrics. Furthermore, being a dissipative medium, active matter (or its effective geometry)
+might obey a geometric flow like Ricci’s [224] in its way to the equilibrium.
+ACKNOWLEDGMENTS
+For the purpose of Open Access, a CC-BY public copyright licence,
+, has been
+applied by the authors to the present document and will be applied to all subsequent versions
+up to the Author Accepted Manuscript arising from this submission.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf,len=1446
+page_content='Geometric theory of topological defects: methodological  developments and new trends  Sébastien Fumeron‡, Bertrand Berche‡, and Fernando Moraes†  ‡Laboratoire de Physique et Chimie Théoriques,  UMR Université de Lorraine - CNRS 7019, 54000 Nancy, France and  †Departamento de Física, Universidade Federal Rural  de Pernambuco, 52171-900, Recife, PE, Brazil  Abstract  Liquid crystals generally support orientational singularities of the director field known as topo-  logical defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' These latter modifiy transport properties in their vicinity as if the geometry was  non-Euclidean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' We present a state of the art of the differential geometry of nematic liquid crystals,  with a special emphasis on linear defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' We then discuss unexpected but deep connections with  cosmology and high-energy-physics, and conclude with a review on defect engineering for transport  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='01050v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='soft]  3 Jan 2023  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  INTRODUCTION  One of Pierre-Gilles de Gennes’s greatest breakthrough was to realize that methods and  concepts borrowed from superconductivity also apply to describe smectic-A phases [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' His  work is a striking example of cross-fertilization between different areas of physics and it  highlights how progress arises at the crossroads of various scientific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In an article  that has not been translated in English [2], he took the example of line singularities as a  common denominator between liquid crystals, quark physics and superconductors [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  same observation was also made by William Brinkman and Patricia Cladis [4], and most  notably by the two-Nobel-Prize winner John Bardeen in a review written as a plea for  interdisciplinarity: “Line defects in three-dimensional systems, quantized vortex lines or flux  lines, and dislocations account for similarities of behavior in superconductors, liquid crystals,  and, it is hoped, color confinement of quarks“[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In this spirit, the objective of this paper is to perform an in-depth survey of geometrical  methods useful for investigating topological defects and to describe some of its modern  applications, either as a playground to test fundamental ideas in high-energy physics or  gravitational physics, or as high-performance tools to taylor transport phenomena from soft  matter devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' We will be mainly concerned with nematic liquid crystals and topological  line defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In section II, we provide a self-contained introduction to phase transitions, geometry,  topology and optics in such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' We show how a metric description of defect lines in  terms of Riemann manifolds naturally arises in nematics, before addressing the question of  analogue gravity which may be less familiar to the liquid crystals community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Section III is an introduction to some of the ideas borrowed to cosmology which can  be dealt with liquid crystals, such as the Kibble mechanism, which rules the formation of  defects in the early universe but also in nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' We review several outstanding problems  involving line singularities, such as cosmic strings, wormholes and bouncing cosmologies,  and discuss their connections with disclinations in liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Section IV eventually discusses applications of the geometric formalism previously in-  troduced for the description of acoustics, optics and heat transfer in the presence of such  defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The main idea is to show how the curvature carried by the topological defects  can be used to design specific propagation patterns, a possible step forward towards the  2  defect-engineering of transport phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  TOPOLOGICAL DEFECTS IN NEMATIC LIQUID CRYSTALS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Basics of the isotropic-nematic phase transition  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Historical milestones  The story of liquid crystal has met with a shaky start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  first observations reported of what is today understood as a liquid crystal belong to the  realm of biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Georges-Louis Buffon (1707-1788) and later Rudolf Virchow (1854) and  Carl Mettenheimer (1855) reported about the strange behavior of lecithins, a family of  phospholipid substances contained in plants (wheat, rye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=') and in animals (yolks, myelin  the insulating coating of nerve fibres - .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=') [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' When suspended in water, lecithins form  birefringent tubular structures like a Iceland spar but that writhed like eels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Julius Planer  in 1861 [7] and most importantly Friedrich Reinitzer in 1888 discovered similar optical  behaviors with cholesterol compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Reinitzer extracted cholesteryl esters from carrots  and made an unanticipated observation: contrary to what was known in crystallography,  cholesteryl benzoate displays two melting points [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The lower one occurs at about 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5◦C  and correspond to the melting of the solid phase into a turbid fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The higher melting  point corresponds to the clarification of the milky liquid beyond 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5◦C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Such behavior left Reinitzer skeptical: had he discovered a genuinely new behavior of  matter or was this simply the result of impure and incompletely melted crystals?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Reinitzer  wrote to Otto Lehmann, a leading physicist known for designing the first “crystallization  microscope”: this latter consists in a microscope equipped with crossed polarizers and a  thermal deck (a small Bunsen burner and two cooling blasts) to observe how crystals behave  when the temperature varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Lehmann reproduced and improved Reinitzer’s observations,  and promoted these substances as new forms of matter, half-between liquids and crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In  1889, he coined the term “liquid crystals” to account for his discovery (somehow pulling the  sheet back towards him as he even claimed priority over Reinitzer [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Soon afterwards, he  kept changing its name (including “flowing crystals”, “crystalline liquids”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='), which reveals  his difficulties to grasp the real nature of what he found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The years that followed Lehmann’s breakthrough have been critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' On one hand, the  subject became more and more discussed in the scientific community, even drawing the  3  attention of future Nobel Prize laureates such as Max Born (in 1916, he conceived the  first molecular theory of liquid crystals but predicted a generic ferroelectric behavior that  turned out to be incorrect [10]), Jacobus Henricus van’t Hoff and Walther Nernst (Lehmann  himself was an unlucky nominee from 1913 to 1922).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' On the other hand, the subject became  highly controversial: partly because Nernst and most especially Gustav Tammann led a  vivid opposition against liquid crystals (suspecting them of being nothing more than poorly  prepared colloidal mixtures), partly because of Lehmann’s personality (a sparkling mix of  pretentiousness and mysticism [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Liquid crystals have stayed a controversial subject, since the decisive contributions of  Rudolf Schenk (1905), Daniel Vorländer (1907) and George Friedel (1922) to get a clear view  on this subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Schenk led a thorough study of the clearing point and unambiguously showed  the observed properties (density, viscosity) could not be related to mixtures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Vorländer  elucidated the mysterious anisotropic behavior exibited by the fluid: from a microscopic  standpoint, liquid crystals consist in self-organized assemblies of rod-like molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  As  such, they exibit both the birefringence property expected from anisotropic uniaxial media  and the ability to flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Finally, Friedel realized that such substances should properly be  understood as new full-blown phases of matter, that he named mesophases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Initially divided  into three broad families (nematic, cholesteric and smectic), liquid crystals have now been  enriched by many new mesophases, including columnar phases, cubatic phases, blue phases  I, II and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Mesogenic behavior  The recipe for a molecule to be nematogenic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' to have the  ability to organize into a nematic phase) is rather simple: take a rod-like molecule and deck  it with 1) a flexible outer part (an aliphatic chain), 2) a rigid core (phenyl groups most  generally) and 3) a chemical group bearing a permanent dipole (for instance, a carbonitrile  group as in 5CB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The resulting substance is a thermotropic nematic, a sensitive compromise  between the attractive Van der Waals interactions that align rigid cores on average along the  same direction (anisotropy) and the thermal agitation of the aliphatic chains increasing the  mean steric hindrance (fluidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Nematics can also be lyotropic: the nematogens display  an amphiphilic structure (they have both hydrophilic and hydrophobic parts), the control  parameter being the concentration of molecules in a solvent such as water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The frontier  between thermotropic and lyotropic liquid crystalline behavior being not strict, nematogens  can also behave as amphotropic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  4  In the perspective of section III, let us focus on thermotropic nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In that case, there  are different models accounting adequately for the phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The Lebwohl-Lasher  model [11] is the paradigmatic model in this context, in a sense the liquid crystal analogue  of the Ising model [12]: the nematogen molecules are represented only by their direction and  they occupy fixed positions on the sites of a cubic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The different sites of the lattice  interact only between nearest neighbors through a potential that favors configurations where  the neighboring molecules point in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' On the contrary, the Maier-Saupe  approach is a mean-field theory: interactions between particles are replaced by an effective  field experienced by all particles at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This model considers only London forces  between instantaneous molecular dipoles and ignores repulsive interactions [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This also  favors alignment of the molecules in the same direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' We will briefly mention that for  lyotropic nematics, Lars Onsager described the phase transition of an assembly of rod-like  sticks as an entropic process driven [14]: due to purely steric effects, orientational entropy  loss is more than offset by positional entropy gain which triggers the transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Depending on the temperature range, three (or more) phases can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' At low  temperatures, the steric hindrance of aliphatic chains is minimal and the nematogens get  close enough for attractive forces to drive the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As the dipole-dipole interactions  prevail over thermal agitation, the assembly of rod-like molecules organize into a molecular  crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This latter displays both a positional and rotational order for each molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' On  the contrary, at high temperatures, Van der Waals interactions are dominated by thermal  effects and the nematogens form an isotropic fluid phase: both positional and orientational  order are lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Within the intermediate range of temperature, the two effects are of the same  order and different kinds of mesophases may appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the nematic mesophase, only the  orientational order is preserved: locally, the nematogens tend to align on average along a  common direction, which defines the director field n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In usual nematics, the orientational  order is preserved at long distance, as the correlation length is typically about a few µm,  compared to the nematogen length around a few nanometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  As the phase transition  involves nucleation, these domains wherein nematogens share a common orientation form  submicronic bubbles (or spherulites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Then they grow in size and eventually they meet and  mingle, sometimes leaving relics in the form of long threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Order parameter  Within a nematic, a particular molecule does generally not point  exactly in the direction n and the degree of orientational ordering of the mesophase can thus  5  be assessed by looking how well the nematogens are aligned along the director field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  quadrupolar scalar order parameter S defined by Tsvetkov (1942) provides a quantitative  criterion to characterize the nematic order: it is normalized (S = 0 in the isotropic fluid phase  and S = 1 for perfectly aligned rods), and ranges from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='3 < S < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='8 in usual nematics (in  practice, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='3 < S < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='4 for thermotropic liquid crystals, whereas 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='6 < S < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='8 for lyotropic  ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The order parameter can be refined to include informations on the local orientation  of n (Landau – de Gennes tensorial order parameter) or to encompass phase involving more  complex-shaped mesogens (higher-order mutipole-multipole correlation functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  For thermotropic nematics, S can be taken as a function depending only on the temper-  ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The behavior of S at the transition can essentially discriminate between two main  families of phase transitions (in the sense defined by Lev Landau in 1937 [15]): first-order  phase transitions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' for which the order parameter displays a jump at the transition control  parameter (this class also involves latent heats and nucleation processes),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' and second-order  phase transitions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' for which the order parameter varies continuously at the transition (this  class involves pretransitional effects and scaling behaviors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Experimentally, for most com-  pounds (5CB, MBBA, 5CN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='), the isotropic-nematic phase transition is identified as weakly  first-order phase transition in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It combines small discontinuities of S [16],  nucleation [17] and low latent heats [18], but pretransitional effects of the dielectric proper-  ties [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  From symmetry to topology  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Homotopy theory  The existence of orientational and/or positional orders impart  each phase about the transition with a specific set of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As a rule, the higher-  temperature phase is generally the less-ordered one and its symmetry group is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In  the isotropic-nematic case, the isotropic fluid phase and the nematic liquid crystal are both  statistically invariant under any translation in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' But for the orientational part, the  two phases do not share the same symmetry group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Indeed, the isotropic fluid phase is  statistically invariant under any rotation in three dimensions, that is, under the elements of  the group SO(3), while in the mesophase the director field plays the role of a symmetry axis,  restricting the symmetry to statistical invariance under the elements of the group SO(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  But for energetic reasons, the dipoles borne by the nematogens tend to align anticollinear,  6  such that the assembly of rods is unchanged when inverting heads and tails (this dimeric  structure was confirmed early by X-ray diffraction experiments in 5CB and 7CB [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Hence,  the full symmetry group of the nematic phase is O(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Many important features of a phase transition with a spontaneous symmetry-breaking  are encompassed within the topology of an abstract object, called the order parameter space  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For a phase transition with a symmetry-breaking pattern G → H, the order parameter  space is a manifold defined from the coset M = G/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The toolbox of algebraic topology  (Poincaré’s former analysis situs) can then be used to seek the algebraic invariants (numbers,  groups, rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' ) of M and to classify this space into equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Among the many entry points, homotopy theory is of particular interest to determine the  presence of singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Two topological spaces are homotopic if they can be mapped into  each other by a continuous deformation where bijectivity is not necessarily preserved (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  gluing, shrinking or fattening the space is allowed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Homotopy groups, denoted generically  as πk(M), have been extensively studied in condensed matter physics, mainly in the pio-  neering works of Kleman, Lavrentovich, Michel, Toulouse and Volovik [20–26] and they are  associated to different kinds of topological properties for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For instance, π1(M) tests the  simple-connectedness of the order parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Indeed, consider first M = R × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It  is simply-connected as all closed loops (dimension 1) are homotopic to a point (dimension  0): therefore π1(M) = I and the order parameter space is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Conversely,  for M = R∗ × R∗, there are two equivalence classes of closed loops: those not encircling  the origin, which are homotopic to a point, and those encircling the origin which cannot be  shrunk into a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Therefore π1(M) ̸= I and the order parameter space is not simply con-  nected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The 0D-hole has thus changed the homotopy content of π1(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Interestingly, the  dimensionality of the manifold is crucial here: a loop trying to lasso a 0D-hole can succeed in  2D but will always fail in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In its most general form, the fundamental result of homotopy  analysis states that in dimension n, if the k th homotopy group πk(M) ̸= I, then holes of  dimension n − 1 − k appear: such singularities are called topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Strictly speak-  ing, a defect is topological when the singular configuration of the order parameter cannot  be transformed continuously into a uniform configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This process depends not only  on the order parameter configuration but also on the dimensionality of the order parameter  space (due to the possibility to “escape in the third dimension”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Therefore, there is also a  more flexible use for the terminology “topological defect”, referring to the singularity asso-  7  ciated to any non-trivial homotopy content of the order parameter manifold, whatever its  topological stability is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the remainder of this article, we will stick to that latter meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Zoology of topological defects in nematics  For the isotropic-nematic phase, the order  parameter space is given by M = SO(3)/O(2) ≡ S2/Z2: the resulting space, called the real  projective plane RP 2, can be pictured as a 2-sphere having its antipodal points identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The manifold corresponding to an immersion of the real projective plane in 3D space is  called a Boy surface and its topology is encompassed into its first four homotopy groups: for  uniaxial nematics in 3D, these are π0(RP 2) = I (no domain wall), π1(RP 2) = Z2 (existence  of linear defects), π2(RP 2) = Z (existence of point defects) and π3(RP 2) = Z (existence of  textures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Left: Class N = 0 of closed loops homotopic to a point in the order parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Right: Class N = 1 of closed loops consisting of paths connecting two antipodal points, which are  not homotopic to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Linear defects (or “disclinations” in Frank’s terminology) come from a breaking of the ro-  tational symmetry group and they are probably the most widespread singularities observed  in nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In optical microscopy, they appear as thread-like structures used by Friedel to  coin the term nematic (from the greek νηµα="thread").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=" In polarizing microscopy, disclina-  tions give rise to the beautiful Schlieren patterns, where dark brushes connect at singular  8  A'=A  O  T,(RP2)=Z2=[0,1)  unstable  stablepoints corresponding to the line defects viewed end on." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The content of the first homotopy  group (or Poincaré group) is Z2 = 0, 1, which means that there are two equivalence classes  for closed loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The trivial class N = 0 corresponds to defects that are not topologically  stable (they can relax into a uniform configuration), whereas the second non-trivial class  N = 1 corresponds to defects that cannot be removed (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For reasons we will  clarify later, we retain the terminology of wedge disclinations to the trivial class and the  terminology of Mœbius disclinations to the non-trivial class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A disclination can stay almost  straight or form loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It is generally associated to other disclinations within dipoles (edge  dislocations), amorphous networks (blue phases), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In that case, they have the possibility  to interconnect [27] and they combine according to the algebra of Z2, that is 0 + 0 = 0,  0 + 1 = 1 and 1 + 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' An extensive review on linear defects in the general context of ill-  ordered condensed matter can be found in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Since our main concern here is disclinations  we refer the reader interested in point defects and textures (including the exotic skyrmions  and hopfions) to the the very complete reviews [17] and [29], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Optics in the presence of linear defects  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Director field of axial disclinations  A region characterized by a given director field  can undergo orientational distorsions as a result of external constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As n is a unit  (headless) vector, the distorsions always occur in a plane orthogonal to the director field,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' From a Taylor expansion, one can rewrite the deformed state as the sum of  three main elastic modes: a splay term in ∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='n, a twist term in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='(∇ × n) and a bend term  in |n × (∇ × n)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The Frank-Oseen free energy density is the elastic cost of orientational  oscillations around n:  fV = 1  2K1(∇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='n)2 + 1  2K2(n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='(∇ × n))2 + 1  2K3(n × (∇ × n))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  (1)  Equilibrium state corresponds to configurations such that fV is extremal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Elastic constants  are of order E0/L, with the interaction energy about E0 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='1 eV and L ≈ 1 nm, it is  customary to perform the one-constant approximation for which K1 ≈ K2 ≈ K3 = K =  10−11 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This assumption is fair for most ordinary nematics: for instance, in the case of  5CB at 298 K, one measures K1 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='2 pN, K2 = 6 pN and K3 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='2 pN [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The simplest class of linear defects consists in axial disclinations and they were firsly  9  considered by Oseen [32] and Frank [33] (for the class of perpendicular disclinations, proposed  by de Gennes, see for instance [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Orientation of the director field is ill-defined along a  line (say the z−axis) and n lies in a plane orthogonal to the defect axis (in our example,  the x − y plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In cylindral coordinates, let ψ(r, θ) be the angle between the director field  and the radial unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Then the Euler-Lagrange equations corresponding to a minimum  of fV simply writes as ∆ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The solutions representing disclination lines are given by  ψ(θ) = mθ + ψ0, where m is the defect strength or topological charge (a priori in R) and ψ0  a constant phase term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Around a closed loop, the total change in ψ is thus 2mπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For the  director field to be well-valued, this variation is tied by the hodograph rule coming from the  Z2 symmetry of the nematic phase:  �  θ=2π  dψ = 2mπ = kπ  (2)  where k ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Hence, the director field writes as  n =  �  �  �  �  �  cos(mθ + ψ0)  sin(mθ + ψ0)  0  �  �  �  �  � ,  (3)  with the topological charge constrained to be integer and half-integer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' m = ±1/2, ±1,  ±3/2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Disclinations with integer strengths are topologically removable and belong to the N = 0  homotopy class: defects m = +1 and m = −1 are topologically equivalent and can be trans-  formed into one another by continuous deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' They appear in optical microscopy  as thick lines and their core is not singular (possibility to escape into the third dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Disclinations with half-integer strengths are not topologically removable and belong to the  N = 1 homotopy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In this latter case, fibring over a circle about the defect line by  a line segment containing the director which is met at that point, gives a Mœbius ribbon  which twists along the loop an odd number of times [26] (on the contrary, for the N = 0  disclination, one gets an ordinary ribbon with two sides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' They appear in optical microscopy  as thin lines and they display a singular core structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As the free energy density varies in  m2 and therefore, it is energetically more favorable for a wedge disclination to decay into two  Mœbius disclinations, as prescribed by the combination 0 = 1+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It must be remarked that  besides |m|, other topological invariants (such as the self-linking number, Poincaré-Hopf’s  10  index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=') are needed to characterize the topology of a linear defect, as a disclination can  globally self-connect, entangle with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The secrets of Fermat-Grandjean principle  In the geometrical optics limit, light  propagates along paths that can be traveled within the least time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the case of isotropic  media, this variational formulation takes the form of the well-known Fermat’s principle,  established by Pierre de Fermat in 1662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In anisotropic uniaxial media, the constitutive re-  lations involve a dielectric tensor that displays two different principal permittivities, namely  ε⊥ and ε∥ (in a nematic, ε∥ corresponds to the permittivity in the direction of the director  field, whereas ε⊥ is the permittivity orthogonally to it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Fresnel’s equation then provides  two modes inside such material: the ordinary mode, behaving similarly as in an isotropic  medium with refractive index n2 = ε⊥, and the extraordinary mode which experiences a  direction-dependent refractive ray index given by [35]:  Ne(r) =  �  ε⊥ cos2 β(r) + ε∥ sin2 β(r)  (4)  where β is the angle between n and the local tangent vector T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In 1919, Grandjean extended  Fermat’s principle to uniaxial media and he showed that the energy carried by extraordinary  light rays propagates along paths obeying [8]  δ  ��  Ne(r)dℓ  �  = 0  (5)  where ℓ is the curvilinear abscissa that parameterizes a ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Because the director field changes from point to point, a nematic generally displays a  varying refractive index and hence, extraordinary light beams propagate into the medium  along curves (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the case of planar axial disclinations, the direction of n and  consequently β is known at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In that case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' it can be shown that the integrand in  Fermat-Grandjean’s principle can be generally rewritten as [36,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 37]  N 2  e (r)dℓ2 =  �  ε⊥ cos2 [(m − 1)θ + ψ0] + ε∥ sin2 [(m − 1)θ + ψ0]  �  dr2  +  �  ε⊥ sin2 [(m − 1)θ + ψ0] + ε∥ cos2 [(m − 1)θ + ψ0]  �  r2dθ2  −  �  ε∥ − ε⊥  �  sin2 [2(m − 1)θ + 2ψ0] rdrdθ + dz2  (6)  In a seminal work [38],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Walter Gordon pointed out the formal analogy between light  propagation inside a moving dielectric and light propagation inside a non-Euclidean geome-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This idea was developed by many authors eversince [39–45], as it elegantly replaces the  11  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Light paths and director fields in the presence of a planar disclination (Up left: m =  1, ψ0 = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Down left: m = −1, ψ0 = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Up right: m = 1/2, ψ0 = π/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Down right:  m = −1/2, ψ0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  resolution of Fermat’s principle in a material medium by the search for the minimum-length  lines (or geodesics) of an empty curved space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The main asset of that point of view is that  one can use the toolbox of differential geometry to understand how the defect modifies trans-  port phenomena in its vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To illustrate how this works, let us consider the example of  a (m = 1, ψ0 = π/2)-disclination (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For this defect, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' (6) leads to the following  line element:  ds2  3d = N 2  e (r)dℓ2 = dr2 + α2r2dθ2 + dz2,  (7)  where α2 = ε∥/ε⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The line element is a fundamental quantity in differential geometry  and it simply consists in a generalization of Pythagoras’ theorem for computing distances in  arbitrary geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Here, instead of the familiar Euclidean line element ds2  3d = dr2+r2dθ2+  dz2, the term in α2 means that the circumference of a closed unit circle about the defect is no  longer 2π but 2πα instead: in other words, there is a mismatch angle (called Frank angle)  of value 2π(1 − α) compared to flat geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It is customary in differential geometry  12  to rewrite the line element as ds2  3d = gijdxidxj, where Einstein’s summation convention on  repeated indices is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' g is called the metric tensor and it corresponds to a positive definite  quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The curvature scalar [46] as computed from the metric is:  R = 2π(1 − α)  αr  δ2 (r)  (8)  whereas the torsion tensor is identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  An alternate approach to describe the influence of defects in optics, also from differential  geometry, consists in using the formalism introduced by Paul Finsler in his 1918 thesis,  for which there is no quadratic constraint on the geometry as in the Riemannian case [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  As a matter of fact, the arc length is given by a Finsler function F such that ds3d =  F (x, y, z, dx, dy, dz) instead of ds3d =  �  gijdxidxj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the case of anisotropic media, F(r) =  Ne(r)dℓ and the metric corresponds to the Hessian of the ray index [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Ought to the  particularly simple dependency of the ray index with respect to coordinates, this formalism  turns out to be fully equivalent to Riemann’s approach (see discussion at the end of [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  A line element of exactly the same form as (7) appears in the geometric theory of defects  in elastic media [49], related to the strain field associated to wedge disclinations as will be  described in Section II D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Such line defects can be formed by either inserting or removing  a wedge of material of angle 2π(1 − α) with subsequent identification of the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the  case of a removal wedge disclination of axis z (α < 1), the geometry surrounding the defect  is conical and can be easily pictured from the Volterra cut-and-weld process of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In  other words, a disclination can be pictured as a Riemann manifold, for which curvature is  only located on the disclination axis and vanishes everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Discussion  From the elasticity point of view (as opposed to optics) the description  of an axial wedge disclination by the geometry (7), or more generally by (6), calls for  several remarks (it is important to stress here that ε∥ and ε⊥ now are related to elastic, not  optical, anisotropy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' First, a liquid crystal consists in an assembly of rod-like molecules and  modeling it as a continuous medium is not self-explanatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Rigorously, the continuum limit  for nematoelasticity should come as a coarse grained approximation of molecular dynamics  and it should fail at the atomic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' n(r) is defined statistically, as the average common  direction of the nematogens at each “point” in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The “point” actually refers to a small  volume of space that includes enough molecules for the averaging process to be physically  significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Hence, in practice, it means that the “point-volume” has to be large enough  13  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Volterra cut-and-weld process for a (wedge) disclination along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  compared to the molecular scale a (typically a ≈ 20 Å) and that the variations of the  director field must occur at much larger scales than a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Only then, the distorted liquid  crystal can be described as a continuous medium, as discussed by Oseen [32], Zöcher [50]  and Frank [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  A second caveat is related to the status of (7), which obviously possesses non-vanishing  curvature as in three-dimensional gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Yet, the nematic actually lives in a three-  dimensional Euclidean space, which means that the background geometry is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' How to  reconcile these two standpoints?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Following the analysis from De Wit [51], the state described  by (7) does exist in the flat space, but only in an imaginary space where the medium is  relaxed: gij comes from the projection of this imaginary space onto the physical flat space,  in a similar way as a stereographic map projection transfers the geometric properties on  a 2-sphere (the Earth, with its meridians and parallels) onto a flat plane while deforming  them (Wulff net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It turns out that the geometric description of defects thus requires two  metrics: 1) The physical flat metric, δij, will be used to perform operations on tensors such  as raising/lowering indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 2) The effective metric gij, which contains the elastic informa-  tion, will be used to determine the kinematics of low energy perturbations (geodesics, first  14  disclination axis  Frank angle  2=2元(1-α）  (α<1: removal)integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Third, one may naturally wonder what really happens on the defect axis and how to  refine our zero-width model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In soft matter and more especially nematics, defect cores  are very narrow as well but they still belong to the realm of continuum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As  discussed in [8], a disclination line can accurately be described by a “two-phase model”: the  core consists in a tubular region, filled with the nematogens in isotropic phase (vanishing  order-parameter), and surrounded by the nematic phase (non-zero order parameter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This  approach is consistent with exact solutions obtained from the minimization of the Landau  – Ginzburg – de Gennes free energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Yet, the last word has probably not been said about  disclination cores: observations made on lyotropic chromonic liquid crystals revealed that  the core region has several unexpected features (asymmetric non-circular interfaces between  the nematic and the isotropic phases, azimuthal and radial dependencies for the phase and  amplitude of the order parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=') compared to classic two-phase models [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Analogue gravity: lessons and pitfalls  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Physics as geometry  The geometric description of transport near linear defects does  not restrict to optics near axial disclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Since the pioneering works by Bilby [53] and  Kröner [54] in the 1950s, this approach has been extended to elasticity theory as well [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In the noteworthy set of works [49, 57, 58], Katanaev proposed a general framework based  on Riemann-Cartan manifolds for dislocations and disclinations in elastic media but only  considered the strain tensor field as the relevant degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' An expression of the  effective metric gij in the medium rest frame can be obtained in the case of linear elasticity  as  gij = δij + 2εij  (9)  where εij denotes the strain tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Compared to ordinary elasticity theory (OET), the  geometric theory of defects is in principle more accurate (ordinary elasticity only reproduces  the first-order approximation of the geometric theory of defects [49]) and it is more versatile  (changing the kind of defect only requires changing the metric, instead of a complicated set  of boundary conditions in ordinary elasticity theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' the geometric approach  is also likely to encompass many other kinds of linear defects of interest in liquid crystals  physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' such as screw dislocations in smectic A and C [59,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 60] (in that case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' the defect  15  must be described in terms of a Riemann-Cartan manifold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' for which torsion is only located  on the dislocation axis and vanishes everywhere else [61]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' dispirations in antiferroelectric  SmCA and the dimeric SmC2 [61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 62],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' edge dislocations in smectics [63,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 64] (which are merely  disclination dipoles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The preceding examples testify that in many condensed matter systems, the effective  degrees of freedom are represented by specific field excitations that propagate over effective  Riemann-Cartan manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Geometrization of physics is not a new idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In Plato’s Timaeus,  an attempt was made to describe the world in terms of only five regular polyhedra and ever  since, geometrization of physics has been a dream pursued by many figures in science,  including René Descartes, Bernhard Riemann, William K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Clifford [65] (for an updated  account, see [66]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The most successful step forward in merging geometry and physics was  made in the twentieth century by Albert Einstein with the theory of general relativity: the  gravitational interaction turns out to be nothing more than a manifestation of the spacetime  curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The possible implications of that theory did not escape the attention of influencial  physicists such as Hermann Weyl [67], Arthur Stanley Eddington [68] and more especially  John A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Wheeler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the seminal paper Classical physics as geometry [69], Wheeler and  Charles W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Misner borrow tools from cohomology, differential geometry, exterior algebra and  topology to fully merge gravitation, electrodynamics and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Provided spacetime is  multiply-connected, Misner and Wheeler showed that similarly to mass, classical charge can  also be seen as a byproduct of the spacetime geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A particularly meaningful example  is the low-dimensional gravitational model proposed by Gerard ’t Hooft in the context of  quantum gravity [70] (see below III C): in 2+1 dimensions, ’t Hooft showed that gravitating  point particles can elegantly be described as conical point-like singularities of space-time,  each deficit angle being related to the particle’s total energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Today, this kind of ideas has  spread out to the point where it has become an area of research on its own: analogue gravity  (for an extensive review, see [71] and more recently [72]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Pitfalls  Despite appealing to classical fields, analogue gravity is tricky and must be  handled with great care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Indeed, it relies simultaneously on two different manifolds: 1) the  background gravitational metric – which is experienced by all fields (generally Minskowski’s)  – is the outcome of Einstein-Cartan equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It is a tensor, used for instance to raise and  lower indices of tensors, and as such, it is a covariant quantity, and 2) the effective metric –  which is experienced only by the fields coupled to matter – does not obey Einstein-Cartan  16  gravitational equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Its purpose is limited (for instance, to determine the geodesics  followed by the coupled-fields excitations, as it is not a covariant quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Indeed, the  effective metric is derived from physical quantities which are defined in a privileged frame,  the medium rest-frame, and that are not invariant under Lorentz transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  As can be seen from (9), the effective metric superimposes the background Minkowski  metric and a correction taking into account the couplings between field and matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the  original experiment led by Hippolyte Fizeau, the changes in the velocity field of water (and  hence of the Gordon metric itself) were obviously ruled by the Navier-Stokes equations (for  the velocity field) instead of Einstein-Cartan equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In other words, effective spacetimes  are generally stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Ref [73] pointed out that textures in nematic liquid crystals can  indeed be described by the space sector of an Einstein-like equation, with the elastic-stress  tensor replacing the energy-momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The relevance of the effective metric is there-  fore restricted to calculations of properties related to the kinematic properties of the fields  coupled to matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This encompasses as we said the geodesics of low-energy excitations  but also the less obvious cases of Unruh effect or Hawking radiation which are purely kine-  matic phenomena [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Therefore, the analogy between gravitation and condensed matter  is strictly kinematic but not dynamical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To rephrase Wheeler, analog spacetime tells matter  how to move.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' but matter does not tell analog spacetime how to curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  What is the purpose of analogue gravity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In cosmology, putting a theory into test is  always a thorny challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In 1992, the great epistemologist Karl Popper already pointed  out that the “major theoretical problem in cosmology is how the theory of gravitation may be  further tested and how unified field theories may be further investigated” [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' If the plentiful  harvest of low-energy observations (baryonic oscillation spectroscopy, gravitational wave in-  terferometry, mapping of the cosmic background .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=') answered many questions, theoretical  models involving (trans)planckian scales bloomed even faster, for which experimental con-  firmations seem almost impossible – even in principle – to reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A possible way out of this  conundrum is to take advantage of the richness and flexibility of condensed matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Within  certain limits, analogues of gravity can be used to simulate different types of cosmological  objects (signature transitions events, cosmic strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=') and to investigate the transport of  bosonic and fermionic quasiparticles in nontrivial spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The next section reviews a  series of works dealing with non-standard cosmological models that can be investigated from  their liquid crystal counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  17  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  UNRAVELING THE UNIVERSE WITH LIQUID CRYSTALS: COSMOLOGY  IN THE LABORATORY  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Phase transitions in cosmology  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Thermal history of the universe  In many senses, cosmology consists in thermody-  namics applied to the largest expanding closed system: our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Our current under-  standing of cosmic history is indeed based on the Standard Hot Big Bang Model and it  originates in the pioneering works of three founding fathers: Albert Einstein, Alexander  Friedman and George Lemaître.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In essence, this model states that about 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='8 billion-years  ago, the Universe was in an extremely hot dense state, consisting in a quark-gluon plasma,  and that it has expanded ever since.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the framework of grand unified theory (GUT), the  four fundamental interactions (the gravitational interaction, the electromagnetic interaction  and two lesser known forces, the weak nuclear interaction – responsible for radioactive β de-  cays – and the strong nuclear interaction – which ensures the cohesion of the atomic nuclei)  were then assumed to be unified at energy scales estimated at about 1016 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Each interaction is associated with internal (or gauge) symmetries: for instance, at today’s  energy scales, the electromagnetic force displays gauge invariance under the elements of U(1),  the unitary group of dimension 1 (for an accessible review on gauge theories see for instance  [76]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Above 1016 GeV, the group G containing the internal symmetries of grand unified  superforce is not known for sure and many candidates with exotic names are considered,  such SU(5), SU(6), SU(7), SU(8), SU(9), SO(10), SO(14), E6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' [77] The universe expansion  played the role of a gigantic Joule-Thomson expansion, which caused a large temperature  drop driving cosmological phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For example, the last of these transitions is  the electroweak phase transition, occurring at energy scales about 102 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It marks the  splitting of the electroweak force into an electromagnetic part, described by Maxwell’s theory  (1865), and the weak nuclear part, the first theory of which being Fermi’s theory (1933).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  This transition involves a spontaneous gauge symmetry breaking: the high temperature  gauge symmetry group SU(3)c × SU(2)L × U(1)Y broke into SU(3)c × U(1)em [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Let us now examine the topology of vacuum manifold (that is the set of field configurations  minimizing the free energy modulo gauge transformations), which is the equivalent of the  order parameter space M in condensed matter physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In [78], Jeannerot et al determined  18  the homotopy content corresponding to all eligible groups G likely to decay below 1016GeV  into SU(3)c × SU(2)L × U(1)Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Their conclusion leaves no doubt concerning the formation  of cosmic strings:  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='.among the SSB schemes which are compatible with high energy physics  and cosmology, we did not find any without strings after inflation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  (if one assumes that the universe is topologically multi-connected, cosmic strings and  monopoles may appear – not single but pairwise –, whereas two-dimensional defects –  domain walls – cannot form at all [79]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Kibble-Zurek mechanism  Cosmic inflation is a period of extremely fast expansion  of the Universe scale factor (typically a factor 1026 within 10−32 seconds) that presumably  happened at the very beginning of the universe [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' From the point of view of statistical  physics, inflation is nothing more than a quench and as such, it is likely to favor the formation  of topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In 1976 [81], Tom Kibble introduced a three-step mechanism (later  refined by Zurek [82] who included the sensitivity to the quench speed) to describe the details  of this quench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Basically, the Kibble-Zurek mechanism (KZM) consists in a nucleation  process very similar to what happens at the isotropic-nematic phase transition, but instead  of having an order locally described by the director field n, it is here described by the phase  of a complex scalar field generically called a Higgs field – or an inflaton, because it needs  not be the Higgs field responsible for the later breaking of electroweak symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' First,  ordered protodomains (analog to the nematic spherulites) with no correlation between each  other are formed and at the scale of a whole protodomain, the fast temperature drop due  to inflation causes the Higgs field to locally take a non-vanishing vacuum expectation value  and hence to make a phase choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Then the protodomains grow in size until they coalesce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  But as they were not correlated, the choices for the Higgs phase (technically, its vacuum  expectation value) do not match in general, and line singularities of the Higgs appear when  the boundaries of protodomains finally meet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' These linear singularities are called cosmic  strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Besides this qualitative predictions, the KZM also makes quantitative predictions such  as the scaling dynamics of the cosmic string network, the average density of defects, cor-  relations between defects and antidefects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the 1990s, several works [27, 83–87] showed  that the KZM, originally developped for cosmology, was also perfectly describing line defects  19  in nematics with the very same scaling coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For instance, this model predicts that  in 2D the density of strings scales as ρ ∼ (t/τq)α with a critical exponent αth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5, and  measurements done by [27] with 5CB indeed gave αth = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To sum up: defects  consist in regions that cannot relax into the new vacuum or equivalently that are unable to  make the transition into the new ordered phase, and they occur during phase transitions  in cosmology and in liquid crystal physics that seem to belong to the same universality  class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' But the family resemblance goes further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Networks of cosmic strings and networks of  disclinations also share similar intersection processes: 1) when two line defects intertwine,  they may reconnect the other way as they cross (intercommutation) [27, 88] and 2) when  one line defect self-intersects, it creates a loop [89, 90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  An almost perfect analogy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Last but not least of these common points: the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Nambu-Goto strings, which are the simplest cosmic defects one may expect in cosmology,  consist in linear concentrations of energy and as such, they are considered as infinitely  thin objects (as the thickness of a cosmic string is estimated at 10−28 cm, this is a fair  approximation[91]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As required by thermal field theory and general relativity, the geometry  around a Nambu-Goto string is described by the Vilenkin’s line element [88]:  ds2 = −dt2 + dr2 + (1 − 4Gµ)2 r2dθ2 + dz2  (10)  where µ is the string energy density estimated at about 10 million billion tons per meter  (we adopt hereafter the customary unit system of cosmology where c = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The space part  of this element is identical to (7): it is a conical geometry corresponding to a removed Frank  angle [92] (typically, for a GUT scale string, this angle is a few seconds of arc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The reader  interested in the classical gauge theory of string interactions in curved spacetimes can refer  to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='[93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' From the standpoint of the soft-matter physicist, Nambu-Goto strings can be  understood as the cosmic counterparts of wedge disclinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' How to make sense of such  incredible similarity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For the most part, this question is still open, but a noticeable attempt  to address it was done in [73]: in essence, the reason is that equations of nematoelasticity  have the form as the spatial sector of Einstein’s field equations, with the elastic-stress tensor  playing the role of the energy momentum tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  As the analogy between gravity and nematoelasticity does not concern time components,  one expects that the dynamics of a cosmic defect cannot be directly mapped with those  of a disclination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' There are other discrepancies between cosmic and elastic defects that  20  one must bear in mind to avoid fallacies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Obviously, the motion of disclinations is classical  (typically a few µm per second) whereas cosmic strings are ultra-relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Dissipation  mechanisms for cosmic strings are due to radiation of gravitational waves, while those in  liquid crystals are friction-dominated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' What is the outcome on the dynamics of the defects?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In cosmology, monopoles annihilate in pairs (Langacker-Pi mechanism), but they do not  annihilate fast and early enough to avoid that the Universe becomes monopole dominated  (which is why inflation is necessary, as it drives monopoles very far away from each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  On the contrary, elastic hedgehogs in a nematic annihilate rapidly according to a scaling  law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' At a more fundamental level, this is linked to the fact that in high energy physics,  broken symmetries are gauged (or internal) whereas in liquid crystals, broken symmetries  are geometrical: in the first case, one is dealing with “gauged defects” and in the second  case, one is dealing with “global defects”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Beyond cosmic wedge disclinations  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The way out of an observational dead-end  Cosmic wedge disclinations exist either as  stable infinite straight lines (their equation of state simply equates the string energy density  to its tension µ = T) or as closed loops that radiate away gravitational waves until they  vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' When moving, strings happen to distort spacetime such that at all scales, matter  accretes along its wake into sheet-like structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  They may account for the formation  of large-scale structures in our universe (including the Great Wall) and they have several  expected observable signatures such as the Kaiser-Stebbins effect [94, 95] (an asymmetric  Doppler shift giving rise to anisotropies of the cosmic microwave background), gravitational  lensing [96] (not in the form of an Einstein ring, but as a double image instead), geometric  phase (Aharonov-Bohm effect but with a cosmic string replacing the flux tube [97]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Up to  now, data collected by the PLANCK mission (2014) only settle upper bounds on the string  parameter µ [98] and in 2020, observations of the stochastic gravitational wave background  (NANOGrav experiment) may have provided with first evidences for cosmic strings [99–101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The non-conclusive observations of Nambu-Goto strings call for the search of refined mod-  els for linear defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In fact, the zero-width approximation and the straightness of cosmic  strings are probably too coarse to account for realistic defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Hiscock [102] and indepen-  dently Gott [103] suggest to smoothen this singularity by introducing two string models with  21  a core structure of constant curvature: the flower-pot model (with zero curvature) and the  ballpoint-pen model (with non-vanishing curvature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the Gott-Hiscock thick cosmic string  spacetime, the metric tensor is piecewise-defined and it must obey matchings conditions at  the core radius [104]: the extrinsic curvature of the boundary should be the same whether  measured in the interior or exterior region (O’Brien-Synge-Lichnerowicz jump condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In contrast, thanks to experiments [52, 105] and molecular simulations [106, 107], much is  known about the NLC disclination core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In particular, there is strong evidence for biaxiality  and that strength +1 disclinations are in fact bound pairs of strength +1/2 ones, which  may be manipulated by electrical fields [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' We note that this rich structure may serve  as an inspiration for novel cosmic string core models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the same line, instead of being  perfectly straight, linear defects can present cusps, kinks and wiggles: the averaged effect  of these perturbations increases the linear mass density µ and decreases the string tension  T, as prescribed by the equation of state µ T = µ2  0 [109–111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Compared to straight string,  the geometry remains conical but the deficit angle is larger than in the straight string case,  which increases polarization anisotropies of the cosmic background radiation [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  There are many other ways to dress a Nambu-Goto string such that it may account  for observational results [113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' From an extension of Volterra process to 3+1 dimensions,  Puntingam and Soleng showed that there was only 10 ways to modify a Minkowski spacetime  into different pseudo-Riemann–Cartan geometries with respect to the Poincaré group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For  example, a cosmic linear defect can display chirality [114–119]: in that case, the defect  carries torsion along its axis and one gets the cosmic counterpart of a screw dislocation in  a smectic liquid crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Twisted Nambu-Goto strings (or cosmic dispirations), consisting  in spacetimes with delta function-valued curvature and torsion distributions have also been  considered, as they combine both rotational and translational anholonomy [120–123]: as  mentioned earlier, their effects on light could be tested from experiments done with elastic  dispirations SmCA and SmC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Going further  Rotating disclinations are not likely to be stable but it is worth  mentioning here that their cosmic counterparts have been long predicted in the literature  [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A metamaterial analogue of the rotating cosmic string spacetime has been proposed  [125], as well as a superfluid vortex analogy [126].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' One of the most interesting properties of  the spinning string is its association to closed timelike curves which may find applications  in time travel [127].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Incidentally, parallel cosmic strings moving in opposite directions have  22  been suggested [128] as a prototype time machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Related to, but not really a model for  spinning strings, is the case of a hyperbolic nematic-based metamaterial with a disclination  which was addressed in [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Along the same line, in [130] it was proposed a disclination  model for the compactified Milne model of a cyclic universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' More details on this model in  Section III C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Among the gauge strings, a very interesting possibility is the semilocal string [131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Like  Dirac’s string of magnetic dipoles, semilocal strings end on gauge monopoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' They are  analogous to real disclinations in liquid crystals which, due to the finite size of a liquid  crystalline sample, must end somewhere (hedgehog, 2D disclination on the liquid surface or  on receptacle wall) or else, form a loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Apropos, disclination loops in active nematics have  very complex dynamics (including chaos) and may present recombination episodes [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Defects in liquid crystals have inspired many other proposals in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' GUT allows  for discrete gauge symmetry groups, the standard Z2 parity and one Z3 parity, which are the  only anomaly free groups that remain unbroken at low energy [96, 132].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The corresponding  cosmic strings are generically called Zn-cosmic strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Based on the known physics of  Moebius disclinations, which are commonly observed in nematics, Satiro and Moraes have  investigated some cosmological outcomes of Z2 cosmic strings [133]: in particular, they  showed that Z2-cosmic strings display both positive and negative mass density regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Bearing in mind the back and forth interplay between cosmology and soft matter, one  cannot avoid to mention the latest works of Maurice Kleman, who imported homotopy  theory from condensed matter to astrophysics and cosmology [134–136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In particular, he  conjectured and classified new families of cosmic defects (such as r-cosmic forms) allowed in  a four-dimensional maximally symmetric spacetime [136].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Black holes and Early universe  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Black holes, white holes, wormholes  This is sometimes referred to as the “cosmology  in the laboratory” game plan and it covers topics such as classical black holes [137]-[138],  Hawking radiation [139]-[140], wormholes [141]-[142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For instance, in Haller’s approxima-  tion, the hydrodynamics of a nematic liquid crystal radially flowing down a drainhole is  23  experienced by light beams as the equatorial section of the Schwarzschild’s metric  ds2 = −  �  1 − 2M  r  �  dt2 +  dr2  �  1 − 2M  r  � + r2(dθ2 + sin2 θdφ2)  (11)  for a specific velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The ordinary and extraordinary indexes of the NLC depend on  the scalar order parameter of the liquid crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' So, it was possible to taylor those indexes  to get the proper optical metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In order to achieve this, the Beris-Edwards hydrodynamic  theory wass used to connect the order parameter with the velocity of the liquid crystal flow  at each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This was done in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' [143].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  More recently, an optical analogue of a wormhole threaded by a cosmic string was de-  scribed in [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Wormholes are solutions of Einstein’s equations that connect different  regions of the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For instance, a spherically symmetric wormhole can be obtained  by joining two Schwartzschild black hole spacetimes by a spherical hole carved around each  singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Wormholes are usually represented by “embedding diagrams”, which are 2D  slices of the 4D structure immersed in Euclidean 3D space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The embedding diagram of the  notorious Morris-Thorne [144] wormhole is obtained by taking a t = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=', θ = π/2 section  of the spherically symmetric spacetime described by the metric  ds2 = −c2dt2 +  dr2  1 − b2  0/r2 + r2(dθ2 + sin2 θdφ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  (12)  The restricted metric, ds2 =  dr2  1−b2  0/r2 +r2dφ2, can be embedded in a 3D Euclidean space with  metric ds2 = dz2 + dr2 + r2dφ2 such that z = z(r) is the equation of the embedded surface  of revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For metric (12) the result is the catenoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  A thin nematic film on a catenoid with director field aligned either circularly or radially  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 4) has an optical metric given by [142]  ds2 = dτ 2 + α2(τ 2 + b2  0)dφ2,  (13)  where α = no/ne for the circular case, and α = ne/no for the radial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The coordinate τ is  the arc length of the catenary that under rotation gives rise to the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The parameter  b0 is the radius of the wormhole “throat”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For α = 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' (13) reduces to the catenoid  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It is clear from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' (13) that, asymptotically (τ >> b0), the optical metric of the  disclination is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This is also evident from the top view of the catenoids of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  This optical model simulates the conical spacetime of a Morris-Thorne wormhole threaded  by a cosmic string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The geodesics as obtained in [142] are represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  24  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Director field for circular and radial +1 disclinations on the catenoid, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken  from [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  (a)  (b)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Assorted geodesics for the circularly decorated catenoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The blue lines represent the  isotropic case α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The red and black lines represent, respectively, circular (deficit angle) and  radial (surplus angle) disclinations with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='85 for (a) and (b), and with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='98 for (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken  from [142].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 5 it is clear that the two parts of the wormhole joined by its throat act as a  black hole/white hole pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Road to quantum gravity  A major contemporary challenge in physics is to find an  extension of General Relativity able to describe gravity at all energy scales, in particular at  the very beginning of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This is the mission devoted to quantum gravity theories,  which have the daunting task of reconciling Einstein’s general relativity and quantum field  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Despite promising attempts including superstring theories, M-theory or quantum  loop gravity, no proposal is entirely satisfactory up to now, and even so, the energy scales  required to test these theories are far beyond our current scientific capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A way out  of this gridlock is to rely on simpler models that capture the essential features of quantum  gravity but remain connected to low-energy-physics systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' analogue gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  rare pearl was first introduced in a seminal paper by Deser, Jackiw and ’t Hooft [145]: 2+1  25  gravity with point-particle sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The main point is that there is no gravitational degrees of freedom in three dimensions  [146], which drastically simplifies general relativity (now an exactly solvable model [147]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Within this framework, the geometry surrounding a point-particle is a conical singularity,  the mismatch angle being proportional to the particle’s mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In other words, conical de-  fects represent point particles coupled to gravity in 2+1 spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' After Katanaev [57]  first pointed out that the theory of linear disclinations was isomorphic to the 2+1-gravity,  Kholodenko [148] used the apparatus of quadratic differentials to establish the connection  between Deser, Jackiw and ’t Hooft model and defects in liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In essence, the exis-  tence of massive particles considered as field singularities is directly related to the topology of  the underlying manifold (the Euler characteristic) and to the emergence of the induced sad-  dles: this means that 2+1 Einstein’s equations are strictly equivalent to the Poincaré-Hopf  theorem (see section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='2 in [148]), the Hopf quantization rule making the direct connection  between particles masses Mi and the defect topological charge m, 4GMi = m [149].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The 2+1 gravity model can therefore be experimentally investigated from a network of  parallel disclinations lines in a 3D nematic sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Geometry of disclinations networks has  been theoretically investigated in the literature, sometimes allowing for analytical expres-  sions for the metric tensor [150–152].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Several authors have shown the possibility to design  arrays of topological linear defects from photopatterning techniques [153–157] and even to  manipulate them [158, 159].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' If this last point opens the possibility to emulate collisions be-  tween particles in the 2+1 model, it is even more interesting for the extension of the Deser,  Jackiw and ’t Hooft model to 3+1 dimensions [70]: matter particles are represented by a  gas of piecewise straight string segments that are likely to collide with a higher frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The strings display both positive and negative mass densities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' they are associated to  α < 1 and α > 1 Frank angles, which makes liquid-crystal-based experiments particularly  promising to investigate such models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This model may also have deep connections with  Regge calculus in quantum gravity, where the smooth curved spacetime is replaced by a  piecewise-flat simplicial manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This is like the triangulation of a surface in 3D where the  local curvature is described by the dihedral angle between adjacent triangles (the triangle is  a 2D simplex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The effect of gluing the edges of the simplexes generate a network of cone-like  singularities (Regge cones) which are analogs to wedge disclinations [160, 161] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  26  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Triangulation of a sphere at the Itapetinga radiotelescope (Brazil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Covering a curved  surface by the triangles generates dihedral angles all around the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Non-standard cosmology  Cosmology at transplanckian scales is a thorny problem,  both theoretically and of course observationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For instance has the universe popped up  from a unique singular event, the Big-Bang?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' And if so, how not to wonder what could have  happened before it and how to design experiments to test these theories?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Today, many  high-energy physics theories such as quantum loop gravity and supertring theories entice  the search for cyclic universe models, that is an endless repetition of big crunches followed  by big bounces, along the same line of thought as the Stoics’ concept of palingenesia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A  safe transition has been proposed [162, 163], where the singularity is nothing more than the  temporary collapse of a fifth dimension, the three space dimensions remain large and time  keeps flowing smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A toy model for the geometry of this transition is the compactified  2D Milne universe MC [164, 165].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The Milne universe metric is given by  ds2 = −dt2 + t2dχ2 + t2 sinh2 χ  �  dθ2 + sin2 θdφ2�  (14)  and it was proposed by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Milne in 1933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It represents a homogeneous, isotropic and  expanding model for the universe with a negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In order to compactify the  27  Milne universe on hypersurfaces of fixed solid angle, let the variable χ acquire some period  denoted as 2πκ: here, 0 < κ ∈ R1 is a constant parameter for compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' After  reparametrization, the line element corresponding to the compactified Milne universe finally  writes as  ds2 = −dt2 + κ2t2dφ2  (15)  where t ∈ R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As can be seen from the disclination line element (7), the presence of κ2 in  the above metric indicates a conical singularity of the curvature at the origin (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The passage through the initial singularity has several unusual features [130, 166]: the  singularity acts as a filter for classical particles and a phase-eraser for quantum ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  timelike geodesics of (15) reveal that depending on their angular momentum J, particles have  two ways of crossing the singularity: 1) For non-vanishing J, a two-step dynamics consisting  in an inward stable motion before the singularity, followed by outward stable motion on the  other side, with a memory loss of the particle kinematical properties (quantum mechanically,  this effect simply comes from strong oscillations of the phase of the wave function at the  singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 2) For J = 0, a one-step dynamics consisting in a straight line through the  collapse: yet such trajectories are very unstable since small perturbations in the value of J  causes large deviations on the trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Probing how particles behave in MC can be tested in the laboratory from hyperbolic  liquid crystal metamaterials (HLCM): this means that the permittivity along the director  axis ε∥ < 0 and the permittivity perpendicular to the director axis ε⊥ > 0 are of opposite  sign [167].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Such media can be made from a host nematic liquid crystal that includes an  admixture of metallic nanorods [168] or coated core-shell nanospheres [169].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To retrieve the  Kleinian double-cone geometry, the HCLM must be endowed with an hyperbolic disclination:  the line element writes as ds2 = ε⊥dρ2 − ε∥ρ2dφ2 + ε⊥dz2, which after a rescaling becomes  ds2 = −γ2r2φ2 + dr2 + dz2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This line element is relevant only by the extraordinary modes  and for radial injection conditions (planar trajectories z = Cst), the geometry experienced  by extraordinary rays is perfectly identical to that of the compactified Milne universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  A stable configuration for the director field may be obtained from a cylindrical shell  of HLCM with homeotropic anchoring at the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the geometrical optics limit,  extraordinary light paths turn out to be Poinsot’s spirals as for the compactified Milne  universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The practical realization sets limits to the efficiency of such analog device for  the classical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' First, the analysis holds only within a limited frequency bandwidth  28  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Timelike geodesics with the radial time t given in units of t0 and κ = 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The blue and  green (orange and red) lines are moving away (towards) from (to) the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Furthermore,  particles following the trajectories in the first (third) and second (fourth) quadrants are spinning  clockwise (counterclockwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The blank point at the origin is just to emphasize that the curves do  not reach the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [130].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  due to the resonant nature of the used core-shell spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Second, as previous phenomena  concern the extraordinary mode, an efficient optical absorber should include a filter to shut  off the ordinary wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Finally, it should be noticed that the present model concerns optics  inside a bulk hyperbolic material: to design a perfect optical analog, the hyperbolic medium  must be impedance matched to avoid sizable reflections at the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The analogy was  also extended to quantum particles by investigating light in the scalar wave approximation  in the same device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  29  t  3  f+, J <0  f+, J > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  f-, J >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  J<O  BigBounce  2  f, J <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  1  2  Big Crunch  3  X  2  1  0  1  3  J>0  3  J=0FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Geodesics of the channel of defects in the potential landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  TAYLORING TRANSPORT WITH LIQUID CRYSTALS: DEFECT-ENGINEERED  MATERIALS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Acoustics  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Ballistic guiding  The geometric description of linear defects revealed wedge discli-  nations carry curvature along their axis: a positive Frank angle corresponds to a conical  geometry that focuses incoming light rays, in similar fashion to what happens with a con-  verging lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Conversely, a negative Frank angle corresponds to a saddle-like geometry that  scatters geodesics, in the same fashion as a diverging lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' It is worth noticing that the effect  of a disclination does not limit to the eikonal approximation, as it also diffracts incoming  waves: from the present geometric approach, computing the differential scattering cross sec-  tion of a wedge disclination showed good agreements with theoretical results obtained by  Grandjean from standard acoustics [170, 171].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The effect of a single wedge disclination on rays being clarified, one can legitimately won-  der if a well-suited arrangement of such defects can be used to taylor the propagation of  sound (we recall that photopatterning techniques now allow for the practical realization of  almost any kind of arrays).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In [151], a channel of disclination dipoles was considered, con-  30  V(x,y)  10sisting in two infinite rows made of alternate disclinations separated by distance 2a (a kind  of von Kármán alley), the distance between the rows being 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The positive disclinations  are at points located at (na, (−1)nb), n ∈ Z, while negative disclinations have coordinates  (na, (−1)(n+1)b), n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As always done in geometric models, the bulk medium is consid-  ered in the continuum limit (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' limit of a vanishing lattice spacing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The corresponding  background geometry writes as  ds2 = −c2dt2 + e−4V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='y) �  dx2 + dy2�  + dz2 = gµνdxµdxν  (16)  where c is the local speed of wave packet and V is the acoustic potential given by [151]:  V (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' y) =  |F|  4π ln  ��  cosh2 � π  2a(y − b)  �  − cos2 � πx  2a  �  cosh2 � π  2a(y − b)  �  − sin2 � πx  2a  �  � �  cosh2 � π  2a(y + b)  �  − sin2 � πx  2a  �  cosh2 � π  2a(y + b)  �  − cos2 � πx  2a  �  ��  (17)  where |F| is the absolute value of the Frank angle that characterizes the ±F defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  sound paths are attracted by the positive defects while repelled by the negative ones (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 8 and 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Adjusting the defect strengths along with parameters governing the geometry  of a cell (namely a,b) opens the possibility to taylor material properties of the sheets, but  this will only be achieved numerically considering the complexity of analytical expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Sound paths are yet very sensitive to the shooting angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Hence, a thorough optimization of  the distribution of defects deserves an additional treatment of chaos involving the statistical  tools of dynamic hamiltonian systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Acoustic rectification  Differential geometry models for acoustics originates from  [172] where vorticity effects in isotropic fluids were investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In nematics, for a pla-  nar horizontal configuration, the acoustic metric experienced by the extraordinary mode is  formally identical to (6) with the substitutions ε⊥ ↔ ρv2/C33 and ε∥ ↔ ρv2/C11, where ρ  is the mass density, v is the velocity of sound in the isotropic phase, C33 and C11 are elas-  tic constants respectively along the director’s direction and in directions orthogonal to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  When the nematic medium is confined inside a capillary tube (radius R) with homeotropic  boundary conditions, the director field tends to be radially oriented everywhere but on the  central axis where an orientational singularity lies, but to reduce the elastic energy of this  configuration, the director escapes in the third dimension [63, 173] and the nematic relaxes  31  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Different geodesics, shot from the origin, in the channel of disclinations geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  positive disclinations correspond to red contours while negative disclinations correspond to blue  contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Depending on the shooting angle, the propagation of phonons may be guided by the  street of topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [129].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  into a funnel-shaped configuration, known as the escaped radial disclination (ERD) where  the delta-distributed Ricci scalar becomes an extended smooth one [174].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In general, rectification effects come from an asymmetry of the system in the direction  along which the transport phenomenon occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Usually, this is generally achieved by relying  on gradients of physical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' pore density [175], distribution of compositional  defects [176].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='), asymmetric geometries [177, 178].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' all examples of situations implying hard  and non-flexible systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Liquid crystals naturally provide a stable but flexible configuration  corresponding to such asymmetry, the ERD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the spirit of a low-cost soft-matter-based  solution, satisfying levels of acoustic rectification have been obtained by combining the  asymmetry of the ERD configuration to that of a container consisting in conical frustum  of varying radius R(z) [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The inner surface prepared to produce the desired anchoring  angle[180].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The anchoring angle adapted to maintain the ERD configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' α depends on  the surface geometry (radius), on the liquid crystal nature (elastic constant K, saddle-play  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5 constant K24) and on the surface treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Reference [179] considered acoustic waves in a  frequency range between 20 Hz and 20 kHz (the average human audible range) propagating  in 5CB for the ERD configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The rectification parameter used to estimate the acoustic  device’s efficiency is the percent standard deviation of the lowest variation on the acoustic  intensity  Acoustic rectification(%) =  ����  ∆Ibt − ∆Itb  min (∆Ibt, ∆Itb)  ���� × 100  (18)  where ∆Ibt = It − Ib is the acoustic intensity variation if the wave comes from the bot-  tom to the top of the conical frustum and ∆Itb = Ib − It is the analogous variation for the  counterpropagating case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Numerical simulations show a rectification effect for a longitudinal  plane wave propagating along the conical frustum axis (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The optimization  of the device parameters (geometry of the conical frustum, anchoring conditions the rectifi-  cation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=') allows to reach rectification levels up to 1300% for a continuous frequency  bandwidth [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Left: Conical frustum with an ERD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Middle: Pressure field for the incoming waves from  bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Right:Pressure field for the incoming waves from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [181].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  33  mPa  mPa  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='47237  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='58721  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='47236  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5872  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='47235  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='47234  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='58719  a)  20  Jm  b)  20  Ars  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='47233  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='58718  50  50  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='47232  50  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='58717  s) <0  0  μm  50-50  m  0 Aim  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='47231  50-50  μm  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='58716  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='4723  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='58715B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Optics  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Waveguiding and light concentration  Manipulation of light has become a major issue  in a large number of applications ranging from solar energy harvesting to optical sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The beam steering technique relies on a modulation of refractive indices to guide light  in a given direction ([182–185]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' More recently, the possibility to continuously deflect in  arbitrary directions and/or to simultaneously focus/defocus an incoming light beam has  been demonstrated from multiple stacked nematic liquid crystal cells—building blocks [186].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Another possibility to guide light beams is to use optical waveguides with nematic cores  in radial escaped disclination configurations: the defocusing due to natural diffraction is  compensated by the converging lens effect resulting form negative birefringence [187–191].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Light focusing has long been suspected of suffering severe limitations, the Rayleigh crite-  rion forbidding beam sizes below half of a wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Recently, the advent of metamaterials  provides new hopes for overcoming the diffraction limit using superlenses [192].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Another  promising possibility is to use an hyperbolic liquid crystal metamaterial (HLCM), obtained  from an admixture of metallic nanoobjects to nematics [193].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As seen in the previous sec-  tion, the permittivity along the director axis ε∥ and the permittivity perpendicular to the  director axis ε⊥ have opposite signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For an orthoradial director field, the effective metric  writes as gij = diag (1, −α2ρ2, 1) and in the eikonal approximation, the light paths are the  aforementioned Poinsot spirals: the hyperbolic defect behaves as a sink for light paths [194].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The smaller the value of α, the stronger is the spiraling behavior, 1/α corresponding to the  defect vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Concentration of light by an hyperbolic disclination extends beyond the  geometrical optics limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the scalar wave approximation, the complex amplitude Φ of the  wave is governed by the generalized form of the d’Alembert equation, involving the Laplace-  Beltrami operator instead of the ordinary Laplace operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The wave equation writes as  a modified Bessel differential equation of imaginary order iℓ/α and the solutions are linear  combinations of the modified Bessel functions of first and second kind, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The  intensity distribution for the propagating fields shows 1) that the electromagnetic field con-  centrates along the axis of the device, and 2) that the bigger the value of the frequency, the  smaller the light rings are (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  34  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Examples of intensity profiles, representing the transverse field distributions concentrated  in the vicinity of the hyperbolic defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [194].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Optical vorticity  Active beam shaping has also emerged as a major trend in modern  optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The idea is to taylor the amplitude, the phase or the polarization of an optical  wavefront from real-time driven systems that ideally must be compact enough, highly-flexible  but yet have low manufacturing costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Ought to their high response functions, liquid crystal-  based devices naturally fulfill all these requirements and have emerged as promising low-  cost and easy-to-manufacture alternatives to MEMS, moving opto-mechanical devices and  photonic crystals [195, 196].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  In nematics, the orbital angular momentum carried by light beams can be tuned from  q-plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' A q-plate consists in an inhomogeneous liquid crystal cell endowed with a discli-  nation of topological charge q (for details regarding the dielectric tensor, see [197]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' When  an electromagnetic wave propagates inside the medium, its components acquire designed  phase shifts (Pancharatnam-Berry phase) that trigger spin-to-angular momentum conver-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' This results in light beams displaying optical vorticity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' the wavefronts are helical  and the intensities profiles distribute in doghnut-like shapes [198–203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' [204] demonstrated  from FDTD simulations that disclination lines can transform the state of polarization of  beams propagating along their axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Umbilics ending disclination lines are also identified as  35  structures generating optical vortex arrays at predetermined wavelength [205].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Nematics are not the only contenders in the contest for optical vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' [206]  the Raman-Nath diffraction was used to generate optical vortices from edge dislocations in  the stripe pattern of a cholesteric liquid crystal (cholesterics are chiral nematics for which  the director whirls around a well-defined direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Cholesterics were also considered in  [207], where it has been theoretically and experimentally demonstrated that optical vortices  were generated from a Bragg-reflection-based device, therefore likely to operate at multiple  wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Recently, charged particles crossing a cholesteric plate were reported to radiate  purely twisted photons [208].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Chiral nematics are also likely to generate defect textures of  a more complex kind than that optical dislocations [209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Beside cholesterics, smectics also  showed their potentialities for optical vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' [210] produced an optical vortex from focal  conic domains, whereas in [166], screw dislocations in smectics were shown to imprint their  torsion onto wavefronts (see also [211, 212] for a solid-state-oriented context).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Heat transfer  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Principles of thermal design  Manipulation of heat flux raises intensive research ef-  forts because of the abundant wealth of potential applications including thermal shielding or  stealth of objects, concentrated photovoltaics or thermal information processing (heat-flux  modulators, thermal diodes, thermal transistors and thermal memories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' These prospects  come from the possibility of designing energy paths in a fashion similar to that of light in  transformation optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To do so, the first step is to understand the main peculiarities of  heat transfer in the presence of a non-Euclidean geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Generally speaking, diffusion of  a passive scalar (for instance the temperature field) can be seen as a collection of Markov  processes obeying the stochastic Fokker-Planck equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In the case of Brownian motion,  the Fokker-Planck equation reduces to the well-known parabolic heat equation [213].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' When  considering diffusion processes in the presence of a non-Euclidean space, the problem is ad-  dressed, as already discussed, by replacing the Laplace operator with the Laplace-Betrami  operator ∆LB [214]:  ∂T  ∂t = D∆LBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  (19)  Here, D is the diffusivity and its value depends on the material properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Ought to the  form of the metric of a wedge disclination, heat conduction locally occurs as in a monoclinic-  36  like crystal with no internal source [215]: the heat flux vectors are no longer perpendicular  to the isothermal surfaces, which are bent depending on the value of the Frank angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In  other words, disclinations in nematics generate thermal lensing effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Top: Temperature field (a) and heat flux field (b) for a radial director field (homeotropic  anchoring) and α =  �  C33/C11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5 Bottom: Temperature field (a) and heat flux field (b) for an  orthoradial director field (parallel anchoring) such that α =  �  C11/C33 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [216]  Once the basic effects of single line defects are understood, the next step is to taylor them  to guide heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To do so, let us consider a hollow cylinder, inside which there is the core region  where one aims at controlling the conductive heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The cylinder is inserted inside a  conducting solid sandwiched between two heated vertical plates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The host material consists  37  (a)  (b)  (a)  (b)of a homogenous isotropic medium, whereas the intermediate thick cylinder consists of a  nematic liquid crystal in a disclination-like configuration (no disclination core).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For thermal  management, mesophases with low melting and high clearing temperatures are required: a  range of about 100 K can be reached by using eutectic liquid crystal mixtures (or “guest-  host systems”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Numerical simulations [216] confirm the possibility of a strong heat guiding  phenomenon: depending on the value of elastic constants C33 (along the director) and C11  (along any direction perpendicular to the director), the device can either cloak the core region  from the heat flux or concentrate heat there (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Switching from the concentrator  to the cloaking device is achieved by an electric-field-driven bistable anchoring with dye-  doped mematics (sufficiently high values of the electrical potential difference between the  two sides of the hollow cylinder were indeed shown to induce stable anchoring transitions  between homeotropic and parallel states [217]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To avoid thermoconvective instabilities in  the annulus domain, the device must be thin enough and the heat flux and temperature levels  must be moderate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For instance, using 5CB and MBBA, if the temperature is about a few  tens of degrees (thermoelectric applications) and the external radius of a few centimeters,  the device handles heat flux that typically varies from 5 W/m2 (repeller) to 103 W/m2  (concentrator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Thermal diodes  The previous study is now refined in order to investigate thermal  rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Thermal rectification is a very active subject in nanoscience and solid-state  physics, as testified by the abundant litterature dealing with this subject (extensive reviews  treating thermal rectification can be found in [218, 219].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' More seldomly is heat conduction  rectification considered from soft-matter-based devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In liquid crystals, the macroscopic  thermal properties of the nematic phase depend on temperature according to Haller’s ap-  proximation [220]:  λ∥(T) = λ0 + λ1 × (T − TNI) + λ1∥ × (T − TNI)α∥  (20)  λ⊥(T) = λ0 + λ1 × (T − TNI) + λ1⊥ × (T − TNI)α⊥  (21)  where λ0, λ1, λ1∥, λ1⊥, α∥ and α⊥ are material-depending constants, whereas TNI and TC  are, respectively, the nematic-to-isotropic temperature and the clearing-point temperature  of the liquid crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' As discussed before, liquid crystals naturally provide a stable and  flexible configuration corresponding to such asymmetry, the ERD, which already turns out  to provide high levels of acoustic rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' To achieve high rectification levels, the same  38  conical frustum of varying radius R(z) with anchoring conditions can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In analogy  with the acoustic case discussed earlier, the rectification parameter used to estimate the  thermal diode efficiency can be defined as  Thermal rectification(%) =  ����  ∆Ti − ∆Td  ∆Td  ���� × 100  (22)  where ∆Td = Td,h − T0 is the difference between Td,h, the high temperature on one base  produced by the heat pumped in the cylinder when working in the direct setup (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' when  the heat is flowing from the narrow region to the wider one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' the −z direction), and  T0, the temperature at the other base, which is also the initial temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Similarly,  ∆Ti = Ti,h − T0 when working in the inverse setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Numerical simulations show thermal rectification rates around 1266% [221].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' On the shape  parameters, alterations on the ratio Rr ∈ [0, 28;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 0, 75] produced a percentage variation on  the thermal rectification around 1273%, while modifications of the height h ∈ [50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 75] µm  and on the larger radius Rl ∈ [50;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 70] µm produced percentage changes lower than 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  This indicates that the anisotropy of the conical frustum tube has a strong influence on the  rectification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Other non-geometrical parameters such as the anchoring angle (in the range  [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 90◦]) and the inward pumped heat flux (in the range [5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 10] kW/m2) give percentage  variations on the rectification around, respectively, 3, 8 and 1, 7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Such characteristics  enable this improved thermal diode to be miniaturized, applied on well-determined areas,  while robust against variations of the inward pumped heat flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  The identical forms of the geometry experienced by light and by sound strongly suggests  that devices using liquid crystals may be used to manipulate simultaneously optical and  thermodynamical transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Indeed, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' [222] reports the control of both electromagnetic  propagation and heat flow by a liquid crystal device similar to the one depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 12,  while Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' [223] uses an escaped disclination configuration to rectify at the same time both  heat and light, thus a thermo-optical diode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  CONCLUSION AND PERSPECTIVES  Since their early discovery in the XIXth century, liquid crystals have been the magic bullet  in physics and engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Being in-between anisotropic solids and isotropic fluids, they  extended our conception of condensed matter to an area where geometry and topology can  39  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Left: Isothermal surfaces of a liquid crystalline thermal diode in (up) inverse thermal  setup and (bottom) direct thermal setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The frustum diode has larger radius Rl = 70 µm, ratio  between the radii is Rr = Rsm/Rl = 0, 28, the height is h = 50 µm, anchoring is 60◦ and the  inward heat flux is Q = 5 kW/m2 on the base with the higher temperature and T0=296 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Right:  Rectification rate versus temperature T0 of the base for different larger radii Rl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Taken from [179].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  be almost as useful as in General Relativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' An orientationally ordered fluid, as the nematic  liquid crystal, is a vivid representation of a Riemannian manifold where the director field  can be associated to a local vector basis (triad or dreibein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' The relative rotation of the  director/triad associated to neighboring points indicates the presence of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Bound-  ary conditions like vessel shape, immersed objects, anchoring angle, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=', impose restrictions  to the effective geometry whose eventual incompatibility with the nematic order (ground  state or zero curvature everywhere) leads to the appearance of topological defects which  accommodate the incompatibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  This geometric view of the elastic distortions in the NLC is complemented by the optical  effective geometry that appears naturally by comparing Fermat’s law of least time to the  geodesic variational principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Similar effective geometries can be obtained for acoustics  and heat transport as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In all these cases the defects, besides being the consequence of  40  μm  309  0  308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='58  306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='77  307.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='67  40  305.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='87  304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='97  304.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='06  303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='16  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='270  301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='35  0  7 300  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='260  50  0  50  Rectification [%]  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='250  R, = 50 μm  R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' = 57 jm  R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' = 63 jm  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='240 -  R=70m  μm  301  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='230  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='64  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='220  40  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='5  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='44  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='37  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='210  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='3  295  296  297  298  299  300  TOE  20  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='23  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='17  Te [K]  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='1  0  300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='03  V 300  50  0  50the topology (boundary conditions), are the source of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' One might then say  that, as soon as there is real or effective curved geometry to describe a physical system with  orientational order, one can expect defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' And, if there is curved geometry, one can relate  NLC to gravitation and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' In this article we reviewed, not only this relationship,  but also the physical applications obtained with the help of the geometric tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  Many  open problems both in gravitation and cosmology and in NLC certainly may benefit from  the analogies derived by the (partially) common geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' For instance, the experimental  knowledge about the inner structure of disclinations may be an inspiration for cosmic string  core models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Active matter, being a dynamic medium, may be described by time-depending  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content=' Furthermore, being a dissipative medium, active matter (or its effective geometry)  might obey a geometric flow like Ricci’s [224] in its way to the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  For the purpose of Open Access, a CC-BY public copyright licence,  , has been  applied by the authors to the present document and will be applied to all subsequent versions  up to the Author Accepted Manuscript arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
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+page_content=' The diverse world of liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
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+page_content=' Cosmology in the laboratory:  Defect dynamics in liquid crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9AzT4oBgHgl3EQfHvt9/content/2301.01050v1.pdf'}
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+A quantum interferometer for quartets in superconducting three-terminal Josephson junctions
+R´egis M´elin1, 2 and Denis Feinberg1, 2
+1Universit´e Grenoble-Alpes, Institut N´eel, BP 166, F-38042 Grenoble Cedex 9, France
+2CNRS, Institut N´eel, BP 166, F-38042 Grenoble Cedex 9, France
+(Dated: January 30, 2023)
+An interferometric device is proposed in order to analyze the quartet mode in biased three-terminal Josephson
+junctions (TTJs), and to provide experimental evidence for emergence of a single stationary phase, the so-called
+quartet phase. In such a quartet-Superconducting Quantum Interference Device (quartet-SQUID), the flux sen-
+sitivity exhibits period hc/4e, which is the fingerprint of a transient intermediate state involving two entangled
+Cooper pairs. The quartet-SQUID provides two informations: an amplitude that measures a total “quartet crit-
+ical current”, and a phase lapse coming from the superposition of the following two current components: the
+quartet supercurrent that is odd in the quartet phase, and the phase-sensitive multiple Andreev reflection (phase-
+MAR) quasiparticle current, which occurs in transparent enough TTJs and is even in the quartet phase. Evidence
+for the latter plays against conservative scenarii involving synchronization of AC Josephson currents, based on
+“adiabatic” phase dynamics and RSJ-like models.
+Introduction:
+Multiterminal Josephson junctions (MTJs)
+[1–4] appear as a very fertile evolution in the field of super-
+conductivity. While unbiased MTJs offer prospects as plat-
+forms for controllable topological properties [5–23], biased
+MTJs reveal new channels for both superconducting phase-
+sensitive and quantum mechanical DC currents, as predicted
+by theory [22, 24–41] and confirmed in experiments [42–
+54]. A paradigm of multiterminal Josephson junction [26]
+involves three superconductors biased at the opposite volt-
+ages 0,V,−V, this making the junction host Cooper quartets
+[26]. Those transient quartets are made of entangled pairs
+of Cooper pairs and flowing from the unbiased terminal to-
+wards the two others simultaneously. This voltage configu-
+ration ensures energy conservation, a necessary condition for
+having DC Josephson currents. The quartet mechanism goes
+together with emergence of a stationary phase combination
+for the three terminal phases, the so-called quartet phase ϕQ.
+At the microscopic level, the minimal process appearing in
+perturbation theory in the tunnel amplitudes consists of four
+Andreev reflections. Quartets (as well as higher-order multi-
+pairs such as sextets, octets, ....) therefore constitute a genuine
+quantum mechanical mesoscopic phenomenon, not occurring
+in simple classical Josephson arrays but instead in truly mul-
+titerminal junctions.
+Besides this quartet supercurrent, another current compo-
+nent happens to depend on the quartet phase. It is related to
+multiple Andreev reflections (MAR), which promote quasi-
+particles across the superconducting gap 2∆ with the help of
+Cooper pair transfers, each one gaining energy 2eV [55]. New
+channels open in a three-terminal Josephson junction (TTJ)
+[34], where all pairs of terminals are simultaneously involved.
+Among those, specific processes involve emission of quartets
+at zero energy but with phase ϕQ: the energy cost for promot-
+ing a quasiparticle between two terminals, say S1, S0 (with
+V1 −V0 = V), instead of transferring a pair between termi-
+nals S1, S0, can be provided by transferring a pair between
+terminals S0, S2 (with V0 −V2 = V) plus absorbing simulta-
+neously a quartet from (S1, S2) to S0 (Figure 1). This quartet
+carries a phase ϕQ and these MAR processes become phase-
+dependent subgap quasiparticle currents [27]. Detailed calcu-
+lations about the phase and voltage sensitivities of both quar-
+tet and phase-MAR currents can be found in Refs. 31 and
+56. While quartet supercurrents are truly nondissipative, the
+phase-MAR currents are dissipative. Both of them depend
+on the control variables (ϕQ,V) but with different symmetries
+[27, 31]. Owing to time inversion symmetry, the quartet and
+phase-MAR currents have to be antisymmetric with respect to
+inverting both variables ϕQ and V. The quartet current is anti-
+symmetric in phase and symmetric in voltage, but the phase-
+MAR current is instead symmetric in phase and antisymmetric
+in voltage. This duality is reminiscent of the tunnel junction
+treated by Josephson [57] in his seminal work, concerning the
+DC current and the phase-sensitive quasiparticle current. The
+latter is AC in a two-terminal junction, but can become DC in
+a multiterminal one.
+Regarding experiments, an important question is about the
+interpretation of transport anomalies observed when a TTJ is
+biased at the voltages 0,V,−V [42, 44, 46, 52–54]. A conser-
+vative explanation involves the synchronization of AC Joseph-
+son currents flowing across each of the junctions polarized at
+V and −V respectively [58]. This mechanism is electromag-
+netic in nature and it involves the impedance (or the photon
+modes) of the whole circuit including the junction.
+Minimal models involve an adiabatic dependence of the
+currents with time-dependent phases, in a way similar to the
+standard treatment of Shapiro resonances [59]. This can be
+done in the presence of an external environment described
+by a circuit impedance, that includes the resistive part of the
+junction itself, within RSJ-related models [58]. This qualita-
+tively accounts for the DC-current features observed in TTJs
+[45, 47–49, 52], but is not a proof of the physical relevance of
+such a description. For instance, the zero-frequency current-
+current cross-correlations [44] can hardly receive interpreta-
+tion in terms of the RSJ model, and quite specific frequency
+dependence of the device external circuit impedance should
+be advocated to interpret a recent four-terminal experiment
+as originating from a RSJ model [46]. Still, complementary
+experiments would be important to ascertain the mesoscopic
+nature of multipair processes.
+A first requisite is the control over the quartet phase, which
+can be used to prove coherence of the multipair supercur-
+rent. Such a phase coherence might be present in an extrin-
+arXiv:2301.11633v1  [cond-mat.supr-con]  27 Jan 2023
+
+2
+(a)
+(b)
+(c)
+CP
+CP
+AR
+AR
+AR
+CAR
+CAR
+AR
+CAR
+S1
+S0
+S2
+S1
+S0
+S2
+S1
+S0
+S2
+Q
++
+=
+V
+-V
+AR
+FIG. 1. Diagram (a) pictures a quasiparticle promoted
+through the gap and a Cooper pair (CP) transferred
+from S1 to S0, via two Andreev reflections (AR). Di-
+agram (b) pictures a quartet (Q) formed of two entan-
+gled Cooper pairs, transferred from S0 to S1 and S2 si-
+multaneously, with two AR and two crossed Andreev
+reflections (CAR). Diagram (c) pictures a quasiparticle
+promoted through the gap, transferred from S1 to S0,
+while a Cooper pair is instead transferred from S0 to
+S2. Diagram (c) can be formally obtained by superim-
+posing the lines of diagrams (a) and (b), showing that
+phase-dependent MARs involve quartets.
+sic synchronization scenario, although hampered by decoher-
+ence mechanisms due to the environment itself. On the con-
+trary, phase coherence of the quartets is expected to be much
+more robust. To go further in the discrimination between ex-
+trinsic and intrinsic mechanisms, one must take into account
+the high transparency of the junctions, necessary to produce a
+mesoscopic multipair transport. The consequence is the exis-
+tence of MAR processes, which in the standard two-terminal
+case have no explanation but with the help of subgap An-
+dreev reflections, and thus go well beyond a phenomenolog-
+ical RSJ modeling. Specifically, in MTJs, the observation of
+the phase-sensitive MARs can be taken as evidence for truly
+mesoscopic processes involving quartets, thus disproving any
+classical synchronization scenario.
+In this work, we propose an interferometric scheme able
+to control the quartet phase and, at the same time, reveal the
+phase-MAR component, thus proving both the phase coher-
+ence of multipair processes and their truly subgap mesoscopic
+nature.
+Following Josephson’s discovery that a current must flow
+in an unbiased junction and depends on the phase difference
+between the contacts [57], Superconducting Quantum Inter-
+ference Devices (SQUIDs) were invented in order to control
+and analyze this phase sensitivity [59]. The flux dependence
+exhibits period hc/2e that directly proves supercurrents car-
+ried by Cooper pairs with charge 2e. Similarly, one expects
+that interferometry also helps elucidating the mechanism of
+quartets in TTJs, in particular proving that they carry a charge
+4e. Yet, this simple expectation meets a difficulty : a TTJ
+involves three terminals, two of them being biased. This pre-
+vents from building a trivial generalization of the original two-
+terminal SQUID which is fully equipotential. Such a device
+must necessarily be different from those already proposed for
+multijunctions at equilibrium [28, 43].
+In this work, we describe a four-terminal scheme building a
+true quartet-SQUID. The clue is to connect two TTJs in paral-
+lel by their unbiased as well as their biased terminals, in order
+to close them in a double-TTJ loop. Cooper pairs injected in
+the quartet-SQUID at voltage V = 0 can cross either TTJ as
+quartets, picking up the quartet phase of each TTJ, and recom-
+bine in the common outputs at voltages V and −V. The design
+encloses two loops instead of one. Generalizing the standard
+SQUID argument in the presence of magnetic flux shows that
+this imposes a difference between the quartet phases of the
+two TTJs, thus achieving a perfect parallel with an ordinary
+SQUID.
+This scheme allows analyzing the sensitivity of the quartet
+mode on voltage, as a new control parameter for a DC super-
+current. Microscopic models show that it is not monotonous,
+owing to nonadiabatic transitions between Andreev levels.
+Moreover it can switch from a generic π-junction behavior
+(perturbative and low voltage case) to a 0-junction one. Such
+an evidence goes beyond classical synchronization scenarii
+unless assuming ad hoc an unlikely voltage (i.e. AC Joseph-
+son frequency) dependence of the circuit impedance.
+The proposed quartet-SQUID also allows exploiting the in-
+terplay between quartets and phase-sensitive MARs. Separa-
+tion between those two distinct processes could in principle
+be achieved in ideally symmetrical TJJs. More generally, the
+different phase symmetry of quartet and phase-sensitive MAR
+currents results in a phase lapse in the periodic flux response
+of the quartet-SQUID. Measuring this phase lapse quantifies
+the presence of phase-MARs in transparent enough junctions.
+Phase-MARs are mesoscopic and they involve quartet excita-
+tion amplitudes, therefore they bring the necessary proof of a
+truly new physics being involved in TTJs.
+Three-terminal junction quartet-SQUID: The principle of
+the quartet-SQUID is to make two TTJs interfere with each
+other by joining their biased arms in a secondary circuit, as
+pictured in Figure 2. The two TTJs thus enclose a secondary
+loop with two branches respectively at the voltages V (here-
+after denoted as “L-branch”), −V (hereafter denoted as “R-
+branch”), threaded by a flux φ. The main loop is threaded by
+a flux Φ. Both loops are separated by the L branch (see Fig-
+ure 2). The total current is injected as Itot = I1M + I2M where
+I1M and I2M are the currents entering each of the TTJs from the
+
+3
+Ι1M
+Ι1R
+Ι2M
+Ι1L
+Ι2R
+Ι2L
+ΙL
+ΙR
+Ιtot
+V
+-V
+Φ
+Φ*	+	φ*/2
+ϕ1L
+φ
+ϕ1R
+0
+ϕ2L
+ϕ2R
+FIG. 2. Scheme of a quartet-SQUID based on two TTJs,
+with a main loop threaded by a flux Φ and a secondary
+loop threaded by a flux φ. The current is injected into
+the large loop such as to make the two TTJs interfere
+with each other. The current exits through the biased
+leads at voltages V,−V of the branches L,R of the sec-
+ondary loop. The phases are mentioned in red within a
+simple gauge convention, see text. Here ϕ1M = 0 and
+ϕ2M = Φ∗ +φ∗/2.
+unbiased branch, and eventually exiting in the biased branches
+(second circuit) as IL and IR. Current conservation reads:
+Itot = I1M +I2M = IL +IR.
+(1)
+Let us define the phases at the unbiased branch of TTJ1 and
+TTJ2 as ϕ1M,ϕ2M respectively, and the phases at the biased
+branches of the TTJs as (ϕ1L,ϕ1R) and (ϕ2L,ϕ2R) respectively.
+From previous works [26] one knows that the stationary quar-
+tet phase components are
+ϕQi = ϕiL +ϕiR −2ϕiM,
+(2)
+while the oscillating phase components (at frequency 4eV/¯h)
+are ϕiL − ϕiR (i = 1,2). Let us define the normalized fluxes
+between 0 and 2π as Φ∗ = (2π/φ0)Φ and φ ∗ = (2π/φ0)φ,
+with φ0 = hc/2e. The fluxoid argument is applied to the main
+loop containing the L branch, then to the main plus secondary
+loop containing the R branch. This is perfectly allowed, in
+spite of the main loop and the biased branches not being at the
+same potential. In fact, the fluxoid argument takes care of the
+phase variation inside each superconductor, whatever its po-
+tential. The supercurrent circulation in the bulk of each super-
+conductor is assumed to be zero as for a thick superconductor,
+see Ref. 59. The presence of voltage biases between the dif-
+ferent superconductors only enters in the phase difference at
+the junctions, that can depend on time in the present scheme,
+with frequency 2eV/¯h. The fluxoid argument [59] amounts
+to equating on both paths the sum of the phase differences at
+the junctions to the normalized flux in the loop (modulo 2π),
+which yields:
+Φ∗ = ϕ1L −ϕ1M +ϕ2M −ϕ2L
+(3)
+Φ∗ +φ ∗ = ϕ1R −ϕ1M +ϕ2M −ϕ2R.
+(4)
+Taking the difference between these two equations, one ob-
+tains a relation between the oscillating phases components at
+the two TTJs:
+(ϕ1R −ϕ1L)−(ϕ2R −ϕ2L) = φ ∗,
+(5)
+expressing that these time-dependent components are per-
+fectly synchronized. On the other hand, taking the sum of
+Eqs. (3)-(4) yields a relation between the quartet phases of the
+two TTJs [see Eq. (2)]:
+ϕ1Q −ϕ2Q = 2(Φ∗ +φ ∗/2)
+(6)
+This central result shows that, like an ordinary SQUID, the
+interferometer imposes a phase difference between the sta-
+tionary quartet phases at the two TTJs. Because of the (L,
+R) symmetry of the quartet current, the corresponding flux is
+the arithmetic mean of the fluxes delimited by the L (i.e. Φ∗)
+and the R (i.e. Φ∗ +φ ∗) branches.
+Interestingly, if the TTJs are symmetric by exchanging their
+contacts to branches L,R, the currents I1,2M entering the TTJs
+are pure quartet currents i.e. I1M = I1Q(ϕ1Q),I2M = I2Q(ϕ2Q).
+In turn, a pure MAR current IL −IR flows between branches L
+and R thus in the secondary circuit, and one can write:
+IL = 1
+2
+�
+I1Q(ϕ1Q)+I2Q(ϕ2Q)
+�
+(7)
++ I1MAR(ϕ1Q)+I2MAR(ϕ2Q)
+IR = 1
+2
+�
+I1Q(ϕ1Q)+I2Q(ϕ2Q)
+�
+(8)
+− I1MAR(ϕ1Q)−I2MAR(ϕ2Q).
+In this case, a Cooper pair current circulates in the main loop,
+while the secondary one contains a superposition of a quartet
+current – flowing as parallel Cooper pair currents in the L and
+R branches, thus insensitive to the flux φ ∗ – and a circulating
+MAR current – sensitive to φ ∗ via its phase-MAR component.
+In the general case of asymmetric TTJs, all currents
+I1,2M,IL,R contain components of both quartet and MAR ori-
+gins. One can carry the analysis further in the simplifying
+case of weak transparencies.
+Noting that the quartet and
+MAR components are respectively odd and even in the quartet
+phases, and one can write:
+I1M = IQc1(V)sinϕ1Q + ¯IMAR1(V)+IMARc1(V)cosϕ1Q (9)
+I2M = IQc2(V)sinϕ2Q + ¯IMAR2(V)+IMARc2(V)cosϕ2Q,(10)
+
+4
+where the first terms in Eqs. (9)-(10) are the quartet currents,
+with “critical currents” IQci. The second terms are the phase-
+averaged MAR components, including the phase-independent
+two-terminal MAR processes, and the last terms contain the
+phase-sensitive MAR components, with “critical currents”
+IMARci. The critical current values defining the amplitude of
+the phase oscillations of the quartet and MAR currents are
+voltage-sensitive, and have in general nonmonotonous vari-
+ations with V [31, 37, 38, 56]. The sine and cosine depen-
+dences of the respective quartet and MAR currents stem from
+their symmetry in phase. Such expressions can be checked by
+microscopic calculations in the low transparency case [31].
+From Eqs. (1), (6), (9) and (10), the total current injected
+in this quartet-SQUID can be written as (omitting the voltage
+sensitivities):
+Itot = ¯IMAR1 + ¯IMAR2 +Ic1 sin(ϕ1Q +α1)
+(11)
++ Ic2 sin
+�
+ϕ1Q −2(Φ∗ +φ ∗/2)+α2
+�
+,
+(12)
+with (i = 1,2):
+Ici =
+�
+I2
+Qci +I2
+MARci
+tan(αi) = IMARci/IQci.
+(13)
+The total current appears as the sum of (i) a phase-
+independent MAR current and (ii) a typical SQUID current,
+which depends on the quartet phase ϕ1Q, and on the effective
+flux Φ∗ +φ ∗/2, with phase lapses α1,2 that measure the ratio
+of phase-sensitive MAR currents to quartet currents. As in a
+usual SQUID, maximizing the total current with respect to the
+(quartet) phase yields the following expression for the critical
+current:
+Itot = ¯IMAR1 + ¯IMAR2
+(14)
++
+�
+I2
+c1 +I2
+c2 +2Ic1Ic2 cos
+�
+2(Φ∗ +φ ∗/2)+α1 −α2
+��
+.
+This relation achieves the goal of building a quartet-
+SQUID. As a first result, the factor 2 in the flux sensitivity, that
+results in a hc/4e periodicity, manifests the fact that quartets
+are made of two entangled Cooper pairs and carry charge 4e.
+Second, the phase lapses α1,2 directly contain the information
+about the presence or not of phase-MARs. These phase lapses
+disappear in the case of TTJs with symmetric branches V,−V
+(α1,2 = 0) or in the unlikely case of identical TTJs (α1 = α2).
+In experiments performed at low voltage and in incoher-
+ent diffusive regimes, the MAR currents are negligible, and
+the quartet-SQUID gives direct access to the pure quartet cur-
+rents.
+The above discussion is not restricted to harmonic depen-
+dences of the quartet and MAR current with phase. In reso-
+nant dot models, nonharmonic behavior is easily obtained and
+the quartet current can be quite large, actually comparable to
+the ordinary Josephson current of a two-terminal junction in
+the same conditions [27, 31].
+Exploring the voltage dependence, from “0−” to “π−”
+junction: Having a quartet-SQUID in hands allows a thorough
+study of the dependence of the quartet (and phase-MAR) cur-
+rents with voltage, as a new control parameter for DC Joseph-
+son currents. Focusing on the quartet current, different mod-
+els, suited to different types of junctions (single or many level
+quantum dot, or diffusive metallic) lead to the same conclu-
+sions: the quartet current-phase characteristics changes sign
+several times with voltage, owing to nonadiabatic transitions
+between Andreev levels, triggered by the voltage via the run-
+ning phase (ϕL −ϕR)(t) at frequency 4eV/¯h [31, 37, 38, 56].
+This means that, in terms of the quartet current component, a
+TTJ can be either a “0−” or a “π−” junction, with respect to
+the quartet phase ϕQ. The same occurs with the phase-MAR
+current component that also changes sign but at different volt-
+ages. Superposition of quartet and phase-MAR components
+actually makes a generic TTJ a “θ-junction”[56].
+More generally, the characteristics of a TTJ (transparency,
+asymmetries between the three contacts, degree of decoher-
+ence) all conspire to shift or even suppress the sign changes.
+For instance, if the couplings to the biased terminals are much
+smaller than the one to the unbiased terminal, and the quar-
+tet TTJ current keeps a “π−” junction character at low and
+intermediate voltages [56]. Focusing on quartets only, in the
+case where backgates allow to separately control the trans-
+parency of the different contacts, one can reach a situation
+where, for a given voltage, the pair of TTJs of the SQUID
+can be both “0−” (or both “π−”) junctions, or one being
+a “0−” and the other a “π−” junction.
+This strongly re-
+minds the experiments performed with carbon nanotubes [60]
+(nanoSQUID) where the mechanism for “0−” to “π−” tran-
+sition is instead the Coulomb interaction and the gate control
+of the nanotube junctions. In addition, “0−” to “π−” transi-
+tions have also been observed in superconductor-ferromagnet-
+superconductor Josephson junctions [61].
+To illustrate the possibilities of such a quartet-SQUID, let
+us assume that TTJ1 is fully symmetric and resonant, with
+high quartet critical currents and several sign changes as V is
+increased from 0 to 2∆. On the contrary, TTJ2 couples weakly
+but equally to the L, R terminals. This suppresses the MAR
+component in the SQUID current, and leaves us with a very
+asymmetric quartet-SQUID, with (neglecting the anharmonic-
+ity in this example):
+Itot(V) = Ic1(V)sin(ϕ1Q)+Ic2(V)sin
+�
+ϕ1Q −2(Φ∗ +φ ∗/2)
+�
+(15)
+and |Ic1(V)| ≫ |Ic2(V)|. As said above, TTJ2 remains a “π−”
+junction so that Ic2 < 0, while the sign of Ic1 depends on V.
+Following the classical argument of an asymmetric SQUID,
+one first maximizes Itot ∼ Ic1 sin(ϕ1Q) with respect to ϕ1Q,
+which yields ϕ1Q ∼ π/2 if Ic1 > 0 and ϕ1Q ∼ 3π/2 if Ic1 < 0.
+Inserting this value into the – small – second term of Eq. (15)
+yields
+Itot ∼ |Ic1|±Ic2 sin
+�π
+2 −2(Φ∗ +φ ∗/2)
+�
+,
+(16)
+with ± sign depending on the “0−” or “π−” character of
+TTJ1.
+First, this reconstructs the current-phase relation of
+TTJ2, including the sign of Ic2. Second, as V is swept up-
+wards from 0, the sign changes of TTJ1 reflect themselves in
+π− shifts in the flux dependence of Itot.
+As another example, Eq. (14) shows that phase-MARs can
+be investigated in TTJ1 only, provided TTJ2 is symmetric in
+
+5
+(L, R) thus α2 = 0. The relative amplitude of phase-MARs
+and quartets in TTJ1 reflects directly in the shift α1 of the
+total current vs flux dependence.
+Conclusions: In conclusion, we have proposed a quartet-
+SQUID generalizing the standard SQUID geometry to make
+quartet and phase-sensitive MAR current interfere under con-
+trol of a magnetic flux. The periodicity in the flux depen-
+dence of the total critical current through the SQUID reflects
+the quartet charge 4e. In addition, the distinguishing phase
+symmetries of both current components imply a phase lapse
+in the flux sensitivity of the critical current of the interferome-
+ter, which allows to quantify the phase-MARs with respect to
+the quartet current. Finally, phase-MARs are a consequence
+of both quartet emission and coherent subgap transport. Thus,
+they provide evidence against scenarii based on extrinsic syn-
+chronization via the outer circuit of junctions described by an
+adiabatic current-phase relation. The principle of the present
+quartet-SQUID can obviously be generalized to higher order
+multipair transport in MTJs with four or more terminals.
+R.M. acknowledges support from the French National Re-
+search Agency (ANR) in the framework of the Graphmon
+project (ANR-19-CE47-0007).
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new file mode 100644
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+page_content='A quantum interferometer for quartets in superconducting three-terminal Josephson junctions  R´egis M´elin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' 2 and Denis Feinberg1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' 2  1Universit´e Grenoble-Alpes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Institut N´eel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' BP 166,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' F-38042 Grenoble Cedex 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' France  2CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Institut N´eel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' BP 166,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' F-38042 Grenoble Cedex 9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' France  (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' 2023)  An interferometric device is proposed in order to analyze the quartet mode in biased three-terminal Josephson  junctions (TTJs),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' and to provide experimental evidence for emergence of a single stationary phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' the so-called  quartet phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' In such a quartet-Superconducting Quantum Interference Device (quartet-SQUID), the flux sen-  sitivity exhibits period hc/4e, which is the fingerprint of a transient intermediate state involving two entangled  Cooper pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The quartet-SQUID provides two informations: an amplitude that measures a total “quartet crit-  ical current”, and a phase lapse coming from the superposition of the following two current components: the  quartet supercurrent that is odd in the quartet phase, and the phase-sensitive multiple Andreev reflection (phase-  MAR) quasiparticle current, which occurs in transparent enough TTJs and is even in the quartet phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Evidence  for the latter plays against conservative scenarii involving synchronization of AC Josephson currents, based on  “adiabatic” phase dynamics and RSJ-like models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Introduction:  Multiterminal Josephson junctions (MTJs)  [1–4] appear as a very fertile evolution in the field of super-  conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' While unbiased MTJs offer prospects as plat-  forms for controllable topological properties [5–23], biased  MTJs reveal new channels for both superconducting phase-  sensitive and quantum mechanical DC currents, as predicted  by theory [22, 24–41] and confirmed in experiments [42–  54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' A paradigm of multiterminal Josephson junction [26]  involves three superconductors biased at the opposite volt-  ages 0,V,−V, this making the junction host Cooper quartets  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Those transient quartets are made of entangled pairs  of Cooper pairs and flowing from the unbiased terminal to-  wards the two others simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This voltage configu-  ration ensures energy conservation, a necessary condition for  having DC Josephson currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The quartet mechanism goes  together with emergence of a stationary phase combination  for the three terminal phases, the so-called quartet phase ϕQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  At the microscopic level, the minimal process appearing in  perturbation theory in the tunnel amplitudes consists of four  Andreev reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Quartets (as well as higher-order multi-  pairs such as sextets, octets, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='.) therefore constitute a genuine  quantum mechanical mesoscopic phenomenon, not occurring  in simple classical Josephson arrays but instead in truly mul-  titerminal junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Besides this quartet supercurrent, another current compo-  nent happens to depend on the quartet phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' It is related to  multiple Andreev reflections (MAR), which promote quasi-  particles across the superconducting gap 2∆ with the help of  Cooper pair transfers, each one gaining energy 2eV [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' New  channels open in a three-terminal Josephson junction (TTJ)  [34], where all pairs of terminals are simultaneously involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Among those, specific processes involve emission of quartets  at zero energy but with phase ϕQ: the energy cost for promot-  ing a quasiparticle between two terminals, say S1, S0 (with  V1 −V0 = V), instead of transferring a pair between termi-  nals S1, S0, can be provided by transferring a pair between  terminals S0, S2 (with V0 −V2 = V) plus absorbing simulta-  neously a quartet from (S1, S2) to S0 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This quartet  carries a phase ϕQ and these MAR processes become phase-  dependent subgap quasiparticle currents [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Detailed calcu-  lations about the phase and voltage sensitivities of both quar-  tet and phase-MAR currents can be found in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' 31 and  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' While quartet supercurrents are truly nondissipative, the  phase-MAR currents are dissipative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Both of them depend  on the control variables (ϕQ,V) but with different symmetries  [27, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Owing to time inversion symmetry, the quartet and  phase-MAR currents have to be antisymmetric with respect to  inverting both variables ϕQ and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The quartet current is anti-  symmetric in phase and symmetric in voltage, but the phase-  MAR current is instead symmetric in phase and antisymmetric  in voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This duality is reminiscent of the tunnel junction  treated by Josephson [57] in his seminal work, concerning the  DC current and the phase-sensitive quasiparticle current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The  latter is AC in a two-terminal junction, but can become DC in  a multiterminal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Regarding experiments, an important question is about the  interpretation of transport anomalies observed when a TTJ is  biased at the voltages 0,V,−V [42, 44, 46, 52–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' A conser-  vative explanation involves the synchronization of AC Joseph-  son currents flowing across each of the junctions polarized at  V and −V respectively [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This mechanism is electromag-  netic in nature and it involves the impedance (or the photon  modes) of the whole circuit including the junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Minimal models involve an adiabatic dependence of the  currents with time-dependent phases, in a way similar to the  standard treatment of Shapiro resonances [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This can be  done in the presence of an external environment described  by a circuit impedance, that includes the resistive part of the  junction itself, within RSJ-related models [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This qualita-  tively accounts for the DC-current features observed in TTJs  [45, 47–49, 52], but is not a proof of the physical relevance of  such a description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' For instance, the zero-frequency current-  current cross-correlations [44] can hardly receive interpreta-  tion in terms of the RSJ model, and quite specific frequency  dependence of the device external circuit impedance should  be advocated to interpret a recent four-terminal experiment  as originating from a RSJ model [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Still, complementary  experiments would be important to ascertain the mesoscopic  nature of multipair processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  A first requisite is the control over the quartet phase, which  can be used to prove coherence of the multipair supercur-  rent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Such a phase coherence might be present in an extrin-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='11633v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='supr-con]  27 Jan 2023  2  (a)  (b)  (c)  CP  CP  AR  AR  AR  CAR  CAR  AR  CAR  S1  S0  S2  S1  S0  S2  S1  S0  S2  Q  +  =  V  V  AR  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Diagram (a) pictures a quasiparticle promoted  through the gap and a Cooper pair (CP) transferred  from S1 to S0, via two Andreev reflections (AR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Di-  agram (b) pictures a quartet (Q) formed of two entan-  gled Cooper pairs, transferred from S0 to S1 and S2 si-  multaneously, with two AR and two crossed Andreev  reflections (CAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Diagram (c) pictures a quasiparticle  promoted through the gap, transferred from S1 to S0,  while a Cooper pair is instead transferred from S0 to  S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Diagram (c) can be formally obtained by superim-  posing the lines of diagrams (a) and (b), showing that  phase-dependent MARs involve quartets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  sic synchronization scenario, although hampered by decoher-  ence mechanisms due to the environment itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' On the con-  trary, phase coherence of the quartets is expected to be much  more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' To go further in the discrimination between ex-  trinsic and intrinsic mechanisms, one must take into account  the high transparency of the junctions, necessary to produce a  mesoscopic multipair transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The consequence is the exis-  tence of MAR processes, which in the standard two-terminal  case have no explanation but with the help of subgap An-  dreev reflections, and thus go well beyond a phenomenolog-  ical RSJ modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Specifically, in MTJs, the observation of  the phase-sensitive MARs can be taken as evidence for truly  mesoscopic processes involving quartets, thus disproving any  classical synchronization scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  In this work, we propose an interferometric scheme able  to control the quartet phase and, at the same time, reveal the  phase-MAR component, thus proving both the phase coher-  ence of multipair processes and their truly subgap mesoscopic  nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Following Josephson’s discovery that a current must flow  in an unbiased junction and depends on the phase difference  between the contacts [57], Superconducting Quantum Inter-  ference Devices (SQUIDs) were invented in order to control  and analyze this phase sensitivity [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The flux dependence  exhibits period hc/2e that directly proves supercurrents car-  ried by Cooper pairs with charge 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Similarly, one expects  that interferometry also helps elucidating the mechanism of  quartets in TTJs, in particular proving that they carry a charge  4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Yet, this simple expectation meets a difficulty : a TTJ  involves three terminals, two of them being biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This pre-  vents from building a trivial generalization of the original two-  terminal SQUID which is fully equipotential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Such a device  must necessarily be different from those already proposed for  multijunctions at equilibrium [28, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  In this work, we describe a four-terminal scheme building a  true quartet-SQUID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The clue is to connect two TTJs in paral-  lel by their unbiased as well as their biased terminals, in order  to close them in a double-TTJ loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Cooper pairs injected in  the quartet-SQUID at voltage V = 0 can cross either TTJ as  quartets, picking up the quartet phase of each TTJ, and recom-  bine in the common outputs at voltages V and −V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The design  encloses two loops instead of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Generalizing the standard  SQUID argument in the presence of magnetic flux shows that  this imposes a difference between the quartet phases of the  two TTJs, thus achieving a perfect parallel with an ordinary  SQUID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  This scheme allows analyzing the sensitivity of the quartet  mode on voltage, as a new control parameter for a DC super-  current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Microscopic models show that it is not monotonous,  owing to nonadiabatic transitions between Andreev levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Moreover it can switch from a generic π-junction behavior  (perturbative and low voltage case) to a 0-junction one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Such  an evidence goes beyond classical synchronization scenarii  unless assuming ad hoc an unlikely voltage (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' AC Joseph-  son frequency) dependence of the circuit impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  The proposed quartet-SQUID also allows exploiting the in-  terplay between quartets and phase-sensitive MARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Separa-  tion between those two distinct processes could in principle  be achieved in ideally symmetrical TJJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' More generally, the  different phase symmetry of quartet and phase-sensitive MAR  currents results in a phase lapse in the periodic flux response  of the quartet-SQUID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Measuring this phase lapse quantifies  the presence of phase-MARs in transparent enough junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Phase-MARs are mesoscopic and they involve quartet excita-  tion amplitudes, therefore they bring the necessary proof of a  truly new physics being involved in TTJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Three-terminal junction quartet-SQUID: The principle of  the quartet-SQUID is to make two TTJs interfere with each  other by joining their biased arms in a secondary circuit, as  pictured in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The two TTJs thus enclose a secondary  loop with two branches respectively at the voltages V (here-  after denoted as “L-branch”), −V (hereafter denoted as “R-  branch”), threaded by a flux φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The main loop is threaded by  a flux Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Both loops are separated by the L branch (see Fig-  ure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The total current is injected as Itot = I1M + I2M where  I1M and I2M are the currents entering each of the TTJs from the  3  Ι1M  Ι1R  Ι2M  Ι1L  Ι2R  Ι2L  ΙL  ΙR  Ιtot  V  V  Φ  Φ* + φ*/2  ϕ1L  φ  ϕ1R  0  ϕ2L  ϕ2R  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Scheme of a quartet-SQUID based on two TTJs,  with a main loop threaded by a flux Φ and a secondary  loop threaded by a flux φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The current is injected into  the large loop such as to make the two TTJs interfere  with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The current exits through the biased  leads at voltages V,−V of the branches L,R of the sec-  ondary loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The phases are mentioned in red within a  simple gauge convention, see text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Here ϕ1M = 0 and  ϕ2M = Φ∗ +φ∗/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  unbiased branch, and eventually exiting in the biased branches  (second circuit) as IL and IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Current conservation reads:  Itot = I1M +I2M = IL +IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  (1)  Let us define the phases at the unbiased branch of TTJ1 and  TTJ2 as ϕ1M,ϕ2M respectively, and the phases at the biased  branches of the TTJs as (ϕ1L,ϕ1R) and (ϕ2L,ϕ2R) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  From previous works [26] one knows that the stationary quar-  tet phase components are  ϕQi = ϕiL +ϕiR −2ϕiM,  (2)  while the oscillating phase components (at frequency 4eV/¯h)  are ϕiL − ϕiR (i = 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Let us define the normalized fluxes  between 0 and 2π as Φ∗ = (2π/φ0)Φ and φ ∗ = (2π/φ0)φ,  with φ0 = hc/2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The fluxoid argument is applied to the main  loop containing the L branch, then to the main plus secondary  loop containing the R branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This is perfectly allowed, in  spite of the main loop and the biased branches not being at the  same potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' In fact, the fluxoid argument takes care of the  phase variation inside each superconductor, whatever its po-  tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The supercurrent circulation in the bulk of each super-  conductor is assumed to be zero as for a thick superconductor,  see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The presence of voltage biases between the dif-  ferent superconductors only enters in the phase difference at  the junctions, that can depend on time in the present scheme,  with frequency 2eV/¯h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The fluxoid argument [59] amounts  to equating on both paths the sum of the phase differences at  the junctions to the normalized flux in the loop (modulo 2π),  which yields:  Φ∗ = ϕ1L −ϕ1M +ϕ2M −ϕ2L  (3)  Φ∗ +φ ∗ = ϕ1R −ϕ1M +ϕ2M −ϕ2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  (4)  Taking the difference between these two equations, one ob-  tains a relation between the oscillating phases components at  the two TTJs:  (ϕ1R −ϕ1L)−(ϕ2R −ϕ2L) = φ ∗,  (5)  expressing that these time-dependent components are per-  fectly synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' On the other hand, taking the sum of  Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' (3)-(4) yields a relation between the quartet phases of the  two TTJs [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' (2)]:  ϕ1Q −ϕ2Q = 2(Φ∗ +φ ∗/2)  (6)  This central result shows that, like an ordinary SQUID, the  interferometer imposes a phase difference between the sta-  tionary quartet phases at the two TTJs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Because of the (L,  R) symmetry of the quartet current, the corresponding flux is  the arithmetic mean of the fluxes delimited by the L (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Φ∗)  and the R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Φ∗ +φ ∗) branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Interestingly, if the TTJs are symmetric by exchanging their  contacts to branches L,R, the currents I1,2M entering the TTJs  are pure quartet currents i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' I1M = I1Q(ϕ1Q),I2M = I2Q(ϕ2Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  In turn, a pure MAR current IL −IR flows between branches L  and R thus in the secondary circuit, and one can write:  IL = 1  2  �  I1Q(ϕ1Q)+I2Q(ϕ2Q)  �  (7)  + I1MAR(ϕ1Q)+I2MAR(ϕ2Q)  IR = 1  2  �  I1Q(ϕ1Q)+I2Q(ϕ2Q)  �  (8)  − I1MAR(ϕ1Q)−I2MAR(ϕ2Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  In this case, a Cooper pair current circulates in the main loop,  while the secondary one contains a superposition of a quartet  current – flowing as parallel Cooper pair currents in the L and  R branches, thus insensitive to the flux φ ∗ – and a circulating  MAR current – sensitive to φ ∗ via its phase-MAR component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  In the general case of asymmetric TTJs, all currents  I1,2M,IL,R contain components of both quartet and MAR ori-  gins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' One can carry the analysis further in the simplifying  case of weak transparencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Noting that the quartet and  MAR components are respectively odd and even in the quartet  phases, and one can write:  I1M = IQc1(V)sinϕ1Q + ¯IMAR1(V)+IMARc1(V)cosϕ1Q (9)  I2M = IQc2(V)sinϕ2Q + ¯IMAR2(V)+IMARc2(V)cosϕ2Q,(10)  4  where the first terms in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' (9)-(10) are the quartet currents,  with “critical currents” IQci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The second terms are the phase-  averaged MAR components, including the phase-independent  two-terminal MAR processes, and the last terms contain the  phase-sensitive MAR components, with “critical currents”  IMARci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The critical current values defining the amplitude of  the phase oscillations of the quartet and MAR currents are  voltage-sensitive, and have in general nonmonotonous vari-  ations with V [31, 37, 38, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The sine and cosine depen-  dences of the respective quartet and MAR currents stem from  their symmetry in phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Such expressions can be checked by  microscopic calculations in the low transparency case [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' (1), (6), (9) and (10), the total current injected  in this quartet-SQUID can be written as (omitting the voltage  sensitivities):  Itot = ¯IMAR1 + ¯IMAR2 +Ic1 sin(ϕ1Q +α1)  (11)  + Ic2 sin  �  ϕ1Q −2(Φ∗ +φ ∗/2)+α2  �  ,  (12)  with (i = 1,2):  Ici =  �  I2  Qci +I2  MARci  tan(αi) = IMARci/IQci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  (13)  The total current appears as the sum of (i) a phase-  independent MAR current and (ii) a typical SQUID current,  which depends on the quartet phase ϕ1Q, and on the effective  flux Φ∗ +φ ∗/2, with phase lapses α1,2 that measure the ratio  of phase-sensitive MAR currents to quartet currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' As in a  usual SQUID, maximizing the total current with respect to the  (quartet) phase yields the following expression for the critical  current:  Itot = ¯IMAR1 + ¯IMAR2  (14)  +  �  I2  c1 +I2  c2 +2Ic1Ic2 cos  �  2(Φ∗ +φ ∗/2)+α1 −α2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  This relation achieves the goal of building a quartet-  SQUID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' As a first result, the factor 2 in the flux sensitivity, that  results in a hc/4e periodicity, manifests the fact that quartets  are made of two entangled Cooper pairs and carry charge 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Second, the phase lapses α1,2 directly contain the information  about the presence or not of phase-MARs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' These phase lapses  disappear in the case of TTJs with symmetric branches V,−V  (α1,2 = 0) or in the unlikely case of identical TTJs (α1 = α2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  In experiments performed at low voltage and in incoher-  ent diffusive regimes, the MAR currents are negligible, and  the quartet-SQUID gives direct access to the pure quartet cur-  rents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  The above discussion is not restricted to harmonic depen-  dences of the quartet and MAR current with phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' In reso-  nant dot models, nonharmonic behavior is easily obtained and  the quartet current can be quite large, actually comparable to  the ordinary Josephson current of a two-terminal junction in  the same conditions [27, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Exploring the voltage dependence, from “0−” to “π−”  junction: Having a quartet-SQUID in hands allows a thorough  study of the dependence of the quartet (and phase-MAR) cur-  rents with voltage, as a new control parameter for DC Joseph-  son currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Focusing on the quartet current, different mod-  els, suited to different types of junctions (single or many level  quantum dot, or diffusive metallic) lead to the same conclu-  sions: the quartet current-phase characteristics changes sign  several times with voltage, owing to nonadiabatic transitions  between Andreev levels, triggered by the voltage via the run-  ning phase (ϕL −ϕR)(t) at frequency 4eV/¯h [31, 37, 38, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  This means that, in terms of the quartet current component, a  TTJ can be either a “0−” or a “π−” junction, with respect to  the quartet phase ϕQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The same occurs with the phase-MAR  current component that also changes sign but at different volt-  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Superposition of quartet and phase-MAR components  actually makes a generic TTJ a “θ-junction”[56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  More generally, the characteristics of a TTJ (transparency,  asymmetries between the three contacts, degree of decoher-  ence) all conspire to shift or even suppress the sign changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  For instance, if the couplings to the biased terminals are much  smaller than the one to the unbiased terminal, and the quar-  tet TTJ current keeps a “π−” junction character at low and  intermediate voltages [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Focusing on quartets only, in the  case where backgates allow to separately control the trans-  parency of the different contacts, one can reach a situation  where, for a given voltage, the pair of TTJs of the SQUID  can be both “0−” (or both “π−”) junctions, or one being  a “0−” and the other a “π−” junction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  This strongly re-  minds the experiments performed with carbon nanotubes [60]  (nanoSQUID) where the mechanism for “0−” to “π−” tran-  sition is instead the Coulomb interaction and the gate control  of the nanotube junctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' In addition, “0−” to “π−” transi-  tions have also been observed in superconductor-ferromagnet-  superconductor Josephson junctions [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  To illustrate the possibilities of such a quartet-SQUID, let  us assume that TTJ1 is fully symmetric and resonant, with  high quartet critical currents and several sign changes as V is  increased from 0 to 2∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' On the contrary, TTJ2 couples weakly  but equally to the L, R terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' This suppresses the MAR  component in the SQUID current, and leaves us with a very  asymmetric quartet-SQUID, with (neglecting the anharmonic-  ity in this example):  Itot(V) = Ic1(V)sin(ϕ1Q)+Ic2(V)sin  �  ϕ1Q −2(Φ∗ +φ ∗/2)  �  (15)  and |Ic1(V)| ≫ |Ic2(V)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' As said above, TTJ2 remains a “π−”  junction so that Ic2 < 0, while the sign of Ic1 depends on V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Following the classical argument of an asymmetric SQUID,  one first maximizes Itot ∼ Ic1 sin(ϕ1Q) with respect to ϕ1Q,  which yields ϕ1Q ∼ π/2 if Ic1 > 0 and ϕ1Q ∼ 3π/2 if Ic1 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Inserting this value into the – small – second term of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' (15)  yields  Itot ∼ |Ic1|±Ic2 sin  �π  2 −2(Φ∗ +φ ∗/2)  �  ,  (16)  with ± sign depending on the “0−” or “π−” character of  TTJ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  First, this reconstructs the current-phase relation of  TTJ2, including the sign of Ic2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Second, as V is swept up-  wards from 0, the sign changes of TTJ1 reflect themselves in  π− shifts in the flux dependence of Itot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  As another example, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' (14) shows that phase-MARs can  be investigated in TTJ1 only, provided TTJ2 is symmetric in  5  (L, R) thus α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The relative amplitude of phase-MARs  and quartets in TTJ1 reflects directly in the shift α1 of the  total current vs flux dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content='  Conclusions: In conclusion, we have proposed a quartet-  SQUID generalizing the standard SQUID geometry to make  quartet and phase-sensitive MAR current interfere under con-  trol of a magnetic flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The periodicity in the flux depen-  dence of the total critical current through the SQUID reflects  the quartet charge 4e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' In addition, the distinguishing phase  symmetries of both current components imply a phase lapse  in the flux sensitivity of the critical current of the interferome-  ter, which allows to quantify the phase-MARs with respect to  the quartet current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Finally, phase-MARs are a consequence  of both quartet emission and coherent subgap transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' Thus,  they provide evidence against scenarii based on extrinsic syn-  chronization via the outer circuit of junctions described by an  adiabatic current-phase relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' The principle of the present  quartet-SQUID can obviously be generalized to higher order  multipair transport in MTJs with four or more terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
+page_content=' acknowledges support from the French National Re-  search Agency (ANR) in the framework of the Graphmon  project (ANR-19-CE47-0007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdFJT4oBgHgl3EQfxC3t/content/2301.11633v1.pdf'}
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+Sparse neural networks with
+skip-connections for nonlinear system
+identification ⋆
+Erlend Torje Berg Lundby ∗,∗∗∗ Haakon Robinson ∗,∗∗∗
+Adil Rasheed ∗ Ivar Johan Halvorsen ∗∗
+Jan Tommy Gravdahl ∗
+∗ Department of Engineering Cybernetics, Norwegian University of
+Science and Technology, O. S. Bragstads plass 2, Trondheim, NO-7034,
+Norway (e-mail: erlend.t.b.lundby@ntnu.no haakon.robinson@ntnu.no,
+adil.rasheed@ntnu.no, jan.tommy.gravdahl@ntnu.no).
+∗∗ SINTEF Digital, Trondheim, No-7465, Norway
+∗∗∗ Equal contribution
+Abstract: Data-driven models such as neural networks are being applied more and more to
+safety-critical applications, such as the modeling and control of cyber-physical systems. Despite
+the flexibility of the approach, there are still concerns about the safety of these models in this
+context, as well as the need for large amounts of potentially expensive data. In particular,
+when long-term predictions are needed or frequent measurements are not available, the open-
+loop stability of the model becomes important. However, it is difficult to make such guarantees
+for complex black-box models such as neural networks, and prior work has shown that model
+stability is indeed an issue. In this work, we consider an aluminum extraction process where
+measurements of the internal state of the reactor are time-consuming and expensive. We model
+the process using neural networks and investigate the role of including skip connections in the
+network architecture as well as using ℓ1 regularization to induce sparse connection weights. We
+demonstrate that these measures can greatly improve both the accuracy and the stability of the
+models for datasets of varying sizes.
+Keywords: Deep Neural Network, Machine Learning, Timeseries modeling, Modeling and
+Simulation
+1. INTRODUCTION
+There is increasing interest in using machine learning-
+based methods to develop predictive models directly from
+data. The advantage of this compared to standard sys-
+tem identification methods is that it doesn’t require any
+assumptions about the system, but all phenomena that
+are well represented by the data can often be accurately
+captured. One example of such a method that has seen
+widespread popularity in recent years is the neural network
+(NN), which is known to be a universal function approx-
+imator. These are often used in Reinforcement Learning
+(RL) to represent a value function or a model for some
+dynamical system. However, this approach requires a lot
+of data to train effective models, which can be expensive
+to obtain in many domains. One hypothesis for this is that
+NNs are typically overparameterized, and therefore require
+many steps to adjust all of the parameters. However, over-
+training on the same limited dataset will cause the model
+to overfit to the training data, and perform poorly on
+unseen data. While overparameterization has been found
+to aid convergence during training (Allen-Zhu et al., 2019),
+it also introduces redundant information into the weights.
+⋆ This work was supported by the project TAPI: Towards Autonomy
+in Process Industries (grant no. 294544), and EXAIGON: Explain-
+able AI systems for gradual industry adoption (grant no. 304843)
+Recent research has found that using sparser networks
+may be the key to training models that can generalize
+across many situations. In particular, Frankle and Carbin
+(2019) showed empirically that for any dense architecture,
+there is a very high probability that there is a sparse
+subnetwork which will train faster and generalize better
+than the full model. This is known as the Lottery Ticket
+Hypothesis, and many methods of sparsification can be
+seen as attempts to somehow extract such a “winning
+lottery ticket” from an initially dense network. There have
+been numerous advances in this field, and we refer to
+Hoefler et al. (2021) for a recent and comprehensive review.
+In this work we use the well-known ℓ1 regularization for
+sparsification of the model.
+Another challenge related to the use of NNs is the choice
+of architecture and hyperparameters. Typical networks
+have multiple layers which are densely connected, although
+this can vary between domains. Choosing an appropriate
+architecture is an art, generally involving trial and error to
+improve performance and avoid overfitting. It is commonly
+understood that the early layers of a neural network have
+a significant impact on the overall performance of the
+network, but deep networks often suffer from the vanishing
+or exploding gradient problem which prevents effective
+training of these early parameters (Goodfellow et al.,
+arXiv:2301.00582v1  [eess.SY]  2 Jan 2023
+
+2016). Skip-connections were originally proposed by He
+et al. (2016) as a way to circumvent this, by introducing a
+shorter path between the early layers and the output. They
+were not only found to enable the training of significantly
+deeper networks, but Li et al. (2017) also demonstrated
+that they may help improve training convergence.
+In the field of dynamical systems and control, we often
+design a model with a purpose in mind, such as the design
+of a control system or state observer. Crucially, we are in-
+terested in the behavior and performance of the controlled
+system in terms of objectives such as energy efficiency or
+yield. This implies that the model does not need to be
+perfectly accurate for the entire state space, so long as
+the resulting closed-loop performance is sufficient (known
+as identification for control (I4C)). If high-frequency mea-
+surements from the system are available, only the short-
+term behavior of the model is important, since any drift
+out of the operational space is quickly corrected. However,
+if measurements are rarely available, such as in the alu-
+minum electrolysis process that we consider, the long-term
+model behavior and open-loop stability become much more
+important. Stable long-term predictions can be important
+for decision-making, meaning that a model with good long-
+term stability and accuracy is inherently important.
+In this work, we investigate the effects of adding skip
+connections and ℓ1 regularization on the accuracy and
+stability of these models for short, medium, and long
+horizons. We address the following questions:
+• How do skip connections affect the stability and
+generalization error of neural networks trained on
+high-dimensional nonlinear dynamical systems?
+• How does sparsity affect stability and generalization
+error for neural networks with skip connections that
+model nonlinear dynamics?
+• How does the amount of training data affect neural
+networks with skip connections compared to neural
+networks without skip connections?
+We make the following contributions:
+• We perform a black box system identification of an
+aluminum electrolysis cell using different NN archi-
+tectures.
+• We demonstrate that the accuracy and open-loop
+stability of the resulting models is greatly improved
+by using ℓ1 weight regularization and incorporating
+skip connections into the architecture.
+• This advantage is consistent across datasets of vary-
+ing sizes.
+2. THEORY
+2.1 Physics-based model for aluminum extraction
+We evaluate NNs for nonlinear system identification by
+first training them on synthetic data generated from a
+known physics-based models (PBM). The model used in
+this work describes the internal dynamics of an aluminum
+electrolysis cell based on the Hall-H´eroult process. Figure 1
+shows a diagram of the electrolysis cell. Traditional PBMs
+of such systems are generally constructed by studying the
+mass/energy balance of the chemical reactions. Lundby
+Insulation
+Temperature of
+side ledge
+Alumina feed
+Carbon
+anode
+Carbon
+anode
+Mass 
+of  
+Alumina + Aluminium Fluoride + Molten Cryolite  
+Molten Aluminum
+Carbon lining
+Bath
+temperature 
+Side wall temperature
+Alumina
+supply
+Current bar collector
+Current bar collector
+Produced aluminum mass
+Mass of side ledge 
+Line current
+Anode-cathode distance
+Tapped metal  
+flow rate
+Fig. 1. Schematic of the setup
+et al. (2022) presents a more detailed exposition of the
+model that we use in this work. The system is described
+by a set of ordinary differential equations (ODE):
+˙x = f(x, u),
+(1)
+where x ∈ R8 and u ∈ R5 represent the time-varying
+states and inputs of the system respectively. The full set
+of equations are:
+˙x1 = k1(g1 − x7)
+x1k0
+− k2(x6 − g1)
+(2a)
+˙x2 = u1 − k3u2
+(2b)
+˙x3 = u3 − k4u1
+(2c)
+˙x4 = −k1(g1 − x7)
+x1k0
++ k2(x6 − g1) + k5u1
+(2d)
+˙x5 = k6u2 − u4
+(2e)
+˙x6 =
+α
+x2 + x3 + x4
+�
+u2g5 + u2
+2u5
+2620g2
+− k7(x6 − g1)2
+(2f)
++ k8
+(x6 − g1)(g1 − x7)
+k0x1
+− k9
+x6 − x7
+k10 + k11k0x1
+�
+˙x7 = β
+x1
+�k9(g1 − x7)
+k15k0x1
+− k12(x6 − g1)(g1 − x7)
+(2g)
++ k13(g1 − x7)2
+k0x1
+−
+x7 − x8
+k14 + k15k0x1
+�
+˙x8 = k17k9
+�
+x7 − x8
+k14 + k15k0 · x1
+− x8 − k16
+k14 + k18
+�
+,
+(2h)
+where the intrinsic properties gi of the bath mixture are
+given as:
+
+g1 = 991.2 + 112cx3 + 61c1.5
+x3 − 3265.5c2.2
+x3
+(3a)
+−
+793cx2
+−23cx2cx3 − 17c2x3 + 9.36cx3 + 1
+g2 = exp
+�
+2.496 −
+2068.4
+273 + x6
+− 2.07cx2
+�
+(3b)
+g3 = 0.531 + 3.06 · 10−18u3
+1 − 2.51 · 10−12u2
+1
+(3c)
++ 6.96 · 10−7u1 − 14.37(cx2 − cx2,crit) − 0.431
+735.3(cx2 − cx2,crit) + 1
+g4 = 0.5517 + 3.8168 · 10−6u2
+1 + 8.271 · 10−6u2
+(3d)
+g5 = 3.8168 · 10−6g3g4u2
+g2(1 − g3)
+.
+(3e)
+See Table 1 for a description of these quantities.
+The values of these constants can be found in Lundby
+et al. (2022). The dynamics of the system are relatively
+slow. The control inputs u1,
+u3 and u4 are therefore
+well modeled as impulses that represent discrete events
+involving the addition or removal of substances. This
+results in step changes in the linear states x2, x3, x5, which
+act as accumulator states for the mass of the corresponding
+substance (see Table 1). The control inputs u2 and u5 are
+piecewise constant, and always nonzero. The inputs u are
+determined by a simple proportional controller π(x). The
+simulation model is derived in Lundby et al. (2022), and
+we refer to that article for further details.
+2.2 Deep neural network with skip connections
+A NN with L layers can be compactly written as an
+alternating composition of affine transformations Wz + b
+and nonlinear activation functions σ : Rn �→ Rn:
+ˆf(z) = ˆfL ◦ · · · ◦ ˆf2 ◦ ˆf1
+ˆfi(z) = σi(Wiz + bi),
+(4)
+where the activation function σi, weight matrix Wi, and
+bias vector bi correspond to the ith layer of the network.
+The universal approximation property of NNs makes them
+very attractive as a flexible model class when a lot of
+data is available. The representation capacity is generally
+understood to increase with both the depth and the width
+(the number of neurons in each layer), although early at-
+tempts to train very deep networks found them challenging
+to optimize using backpropagation due to the vanishing
+gradients problem. One of the major developments that
+enabled researchers to train deep NNs with many layers
+is the skip connection. A skip connection is simply an
+additional inter-layer connection that bypasses some of the
+layers of the network. This provides alternate pathways
+through which the loss can be backpropagated to the early
+layers of the NN, which helps mitigate the issues of vanish-
+ing and exploding gradients, which were major hurdles to
+training deeper models. In this work, we utilize a modified
+DenseNet architecture as proposed by Huang et al. (2017),
+where the outputs of earlier layers are concatenated to all
+the consecutive layers. We simplify the structure such that
+the model only contains skip connections from the input
+layer to all consecutive layers. We call this architecture
+InputSkip, which has reduced complexity compared to
+DenseNet. This design is motivated by the fact that the
+output of each layer (including the final output) becomes
+a sum of both a linear and a nonlinear transformation of
+Table 1.
+Ta-
+ble
+of
+states,
+in-
+puts,
+and
+other
+quan-
+ti-
+ties
+used
+to
+model
+the
+elec-
+trol-
+y-
+sis
+cell
+Variable
+Physical meaning
+Units
+x1
+Mass side ledge
+kg
+x2
+Mass Al2O3
+kg
+x3
+Mass AlF3
+kg
+x4
+Mass Na3 AlF6
+kg
+x5
+Mass metal
+kg
+x6
+Temperature bath
+°C
+x7
+Temperature side ledge
+°C
+x8
+Temperature wall
+°C
+u1
+Al2O3 feed
+kg/s
+u2
+Line current
+kA
+u3
+AlF3 feed
+kg/s
+u4
+Aluminum tapping
+kg/s
+u5
+Anode-cathode distance
+cm
+cx2
+Al2O3 mass ratio x2/(x2 + x3 + x4)
+-
+cx3
+AlF3
+mass ratio x3/(x2 + x3 + x4)
+-
+g1
+Liquidus temperature
+°C
+g2
+Electrical conductivity
+S m
+g3
+Bubble coverage
+-
+g4
+Bubble thickness
+cm
+g5
+Bubble voltage
+V
+the initial input x. Hence, the skip connections from the
+input layer to consecutive layers facilitate the reuse of the
+input features for modeling different linear and nonlinear
+relationships more independently of each other.
+3. METHOD AND SETUP
+In this section, we present all the details of data generation
+and its preprocessing, and the methods that are required to
+reproduce the work. The steps can be briefly summarized
+as follows:
+• Use Equation (2) with random initial conditions to
+generate 140 trajectories with 5000 timesteps each.
+Set aside 40 for training and 100 for testing. Con-
+struct 3 datasets by selecting 10,20 and 40 trajectories
+respectively.
+• For each model class and dataset, train 10 instances
+on the training data.
+• Repeat all experiments with ℓ1 regularization, see loss
+function in Equation (5).
+
+• Use trained models to generate predicted trajectories
+along the test set and compare them to the 100 test
+trajectories.
+3.1 Data generation
+Equation (2) was discretized using the RK4 scheme with
+a fixed timestep h = 10 s and numerically integrated
+on the interval [0, 5000h]. We used uniformly randomly
+sampled initial conditions from the intervals shown in
+Table 2 to generate 140 unique trajectories. We set aside
+40 trajectories for training and 100 of the trajectories
+as a test set. The 40 training trajectories were used
+to create 3 datasets of varying sizes (small, medium,
+large), namely 10, 20, and 40 trajectories. In total, the
+datasets contained 50000, 100000, and 200000 individual
+data points respectively.
+Equation (2) also depends on the input signal u. In
+practice, this is given by a deterministic control policy
+u = π(x) that stabilizes the system and keeps the state
+x within some region of the state space that is suitable
+for safe operation. We found that this was insufficient
+to successfully train our models, because the controlled
+trajectories showed very little variation after some time,
+despite having different initial conditions. This lack of
+diversity in the dataset resulted in models that could not
+generalize to unseen states, a situation that frequently
+arose during evaluation. To inject more variety into the
+data and sample states x outside of the standard opera-
+tional area, we used a stochastic controller
+πs(x) = π(x) + r(t)
+that introduced random perturbations r(t) to the input.
+These perturbations were sampled using the Amplitude-
+modulated Pseudo-Random Binary Signal (APRBS) method
+proposed by Winter and Breitsamter (2018) for nonlinear
+system identification.
+In system identification it is typical to optimize the model
+to estimate the function ˙x = f(x, u). However, this is not
+feasible for Equation (2) because the inputs u are not
+differentiable. Instead, we discretize the trajectories using
+the forward Euler difference and use this as the regression
+variable:
+yk = xk+1 − xk
+h
+The datasets are then constructed as sets of the pairs
+([xk, uk], yk).
+3.2 Training setup
+We optimize the models by minimizing the following loss
+function using stochastic gradient descent:
+Jθ = 1
+|B|
+�
+i∈B
+(yi − ˆf(xi, ui))2 + λ
+L
+�
+j=1
+|Wj|
+(5)
+where B is a batch of randomly sampled subset of indices
+from the dataset, L is the number of layers of the NN, and
+λ is the regularization parameter. This loss function is the
+sum of the mean squared error (MSE) of the model ˆf with
+respect to the regression variables y, and the ℓ1 norm of
+the connection weight matrices Wi in all layers. We used
+a batch size of |B| = 128. We used the popular ADAM
+solver proposed by Kingma and Ba (2014) with default
+parameters to minimize Equation (5).
+Table 2.
+Ini-
+tial
+con-
+di-
+tions
+in-
+ter-
+vals
+for
+x
+Variable
+Initial condition interval
+x1
+[2060, 4460]
+cx2
+[0.02, 0.05]
+cx3
+[0.09, 0.13]
+x4
+[11500, 16000]
+x5
+[9550, 10600]
+x6
+[940, 990]
+x7
+[790, 850]
+x8
+[555, 610]
+3.3 Evaluation of model accuracy
+As previously mentioned, we are interested in evaluating
+the long-term predictive accuracy of the models. Starting
+from a given initial condition x(t0), the model ˆf(x, u)
+is used to generate an estimated trajectory using the
+recurrence:
+ˆxk+1 = ˆxk + hˆf(ˆxk, uk)
+(6)
+where ˆx0 = x0. Note that the input signal uk is replayed
+directly from the test trajectory. Borrowing a term from
+the field of time-series analysis, we refer to this as a rolling
+forecast. To evaluate the accuracy of a model over multiple
+trajectories, we define the Average Normalized Rolling
+Forecast Mean Squared Error (AN-RFMSE):
+AN-RFMSE = 1
+p
+p
+�
+i=1
+1
+n
+n
+�
+j=1
+� ˆxi(tj) − xi(tj)
+std(xi)
+�2
+,
+(7)
+where ˆxi(tj) is the model estimate of the simulated state
+variable xi at time step tj, std(xi) is the standard deviation
+of variable xi in the training set Strain, p = 8 is the number
+of state variables and n is the number of time steps being
+averaged over.
+3.4 Evaluation of model stability
+A symptom of model instability is that its predictions
+can blow-up, which is characterized by a rapid (often
+exponential) increase in prediction error. More precisely,
+we say that a blow-up occurs when the normalized mean
+absolute error for all system states exceeds three (this
+corresponds to standard deviations). We detect this as
+follows:
+max
+j<n
+�
+1
+p
+p
+�
+i=1
+�|ˆxi(tj) − xi(tj)|
+std(xi)
+��
+> 3
+(8)
+where p = 8 is again the number of state variables and
+n is the number of time steps to consider. This is a
+conservative estimate. However, this does not lead to any
+significant underestimation of the number of blow-ups.
+This is because once a model starts to drift rapidly, it
+very quickly exceeds the normal error of three standard
+deviations.
+
+4. RESULTS AND DISCUSSIONS
+We characterize the different model classes (PlainDense,
+PlainSparse, InputSkipDense, InputSkipSparse) by esti-
+mating their blow-up frequencies and their rolling forecast
+mean squared error (RFMSE) on the validation data. The
+blow-up frequency is an interesting measure since it can
+indicate how stable the model is in practice.
+We perform a Monte Carlo analysis by training 10 in-
+stances of each model class and evaluating these on 100
+trajectories randomly generated using the true model,
+yielding 1000 data points for each model class. We repeat
+the experiments for 3 different dataset sizes to study the
+data efficiency of the models.
+Figure 2 presents the total number of blow-ups recorded
+within each model class after 100h, 2000h, and 5000h
+(short, medium, and long term respectively). For simplic-
+ity, blow-ups were detected by thresholding the computed
+variance of a predicted trajectory and manually inspected.
+It is clear that for short time horizons all the models
+exhibit robust behavior independently of the size of the
+training datasets. However, for medium and long time
+horizons, PlainDense, PlainSparse, and InputSkipDense
+architectures exhibit a significant number of blow-ups and
+therefore instability. Figure 2a - 2c show that PlainDense
+is generally the most unstable, with up to 67% of all tra-
+jectories resulting in a blow-up. For the smallest amount of
+training data (Figure 2a) PlainSparse and InputSkipDense
+have similar blow-up frequencies. For larger datasets, the
+PlainSparse architecture shows significantly better stabil-
+ity than both PlainDense and InputSkipDense. InputSkip-
+Dense and PlainDense both show better stability with
+increasing amounts of training data in terms of fewer blow-
+ups. However, both these dense models still suffer from
+significant amounts of blow-ups.
+In comparison, almost no blow-ups are recorded when
+using the InputSkipSparse architecture, even for the small
+training dataset. In Figure 2, the orange bars correspond-
+ing to the blow-up frequency of InputSkipSparse models
+are not visible for any of the training sets due to the sig-
+nificantly lower number of blow-ups. For InputSkipSparse
+models trained on the smallest dataset, only 3 out of 1000
+possible blow-ups were reported for the longest horizon.
+Apart from that, no blow-ups were reported for the Input-
+SkipSparse models.
+Only a few blow-ups were recorded after 5000h in the
+medium term.
+Figure 3 presents a violin plot of the accuracy of each
+model class, expressed in terms of RFMSE over different
+time horizons. Only the plot for the smallest dataset
+(50000 points) is shown, due to the results being very
+similar. A larger width of the violin indicates a higher
+density of that given RFMSE value, while the error bars
+show the minimum and maximum recorded RFMSE val-
+ues. The model estimates that blew up (see Figure 2) are
+not included. In this way, we estimate the generalization
+performance of the models only within their regions of
+stability. Note that the violin plots for model classes with
+many blow-ups are made using fewer samples, and can be
+seen as slightly “cherry-picked”. Nonetheless, the Input-
+SkipSparse architecture consistently yields more accurate
+100 T
+2500 T
+5000 T
+0
+100
+200
+300
+400
+500
+600
+700
+(a) Trained on smallest dataset with 50000 datapoints
+100 T
+2500 T
+5000 T
+0
+50
+100
+150
+200
+250
+(b) Trained on medium sized dataset with 100000 datapoints
+100 T
+2500 T
+5000 T
+0
+25
+50
+75
+100
+125
+150
+175
+200
+(c) Trained on largest dataset with 200000 datapoints
+PlainDense
+PlainSparse
+InputSkipDense
+InputSkipSparse
+Fig. 2. Divergence plot: Number of trajectories that blow-
+up over different time horizons. The total number of
+trajectories is 1000, so the values can be read as a
+permille.
+results, up to an order of magnitude better than the others
+in the long term.
+5. CONCLUSION AND FUTURE WORK
+In this work, we compared the performance of two different
+model structures trained both with and without sparsity
+promoting ℓ1 regularization. The two model types are
+standard Multi-Layer Perceptrons (MLP), and a more
+specialized architecture that includes skip connections
+from the input layer to all consecutive layers. This yields
+four different model structures, which we call PlainDense,
+PlainSparse, InputSkipDense, and InputSkipSparse. The
+main conclusions of the article are as follows:
+• NNs with skip connections are more stable for predic-
+tions over long time horizons compared to standard
+MLPs. Furthermore, the accuracy of NNs with skip
+
+100 T
+2500 T
+5000 T
+10
+2
+10
+1
+100
+PlainDense
+PlainSparse
+InputSkipDense
+InputSkipSparse
+Fig. 3. Model accuracy expressed in terms of RFMSE over
+different horizons. Ten models of each of the model
+types (PlainDense, PlainSparse, InputSkipDense, In-
+putSkipSparse) are trained on the smallest dataset of
+50000 data points. The model estimates that blow up
+(see Figure 2) are excluded. The error bars for each
+model type The plot shows that sparse models with
+skip connections (InputSkipSparse) are consistently
+more accurate than both sparse and dense models
+without skip connections.
+connections is consistently higher for all forecasting
+horizons.
+• The application of sparsity-promoting ℓ1 regulariza-
+tion significantly improves the stability of both the
+standard MLP and InputSkip architectures. This im-
+provement was more apparent for models with the
+InputSkip architecture.
+• The InputSkipSparse showed satisfactory stability
+characteristics even when the amount of training data
+was restricted. This suggests that this architecture is
+more suitable for system identification tasks than the
+standard MLP structure.
+The case study shows that both sparsity-promoting regu-
+larization and skip connections can result in more stable
+NN models for system identification tasks while requiring
+less data, as well as improving their multi-step generaliza-
+tion for both short, medium, and long prediction horizons.
+Despite the encouraging performance of the sparse-skip
+networks, we can not guarantee similar performance for
+noisy data, as we have only investigated the use of syn-
+thetic data devoid of any noise. However, such a study will
+be an interesting line of future work. This case study also
+has relevance beyond the current setup. In more realistic
+situations, we often have a partial understanding of the
+system we wish to model (see Equation (2)), and only
+wish to use data-driven methods to correct a PBM when
+it disagrees with the observations (e.g. due to a faulty as-
+sumption). As shown in Robinson et al. (2022), combining
+PBMs and data-driven methods in this way also has the
+potential to inject instability into the system. Finding new
+ways to improve or guarantee out-of-sample behavior for
+data-driven methods is therefore of paramount importance
+to improve the safety of such systems.
+ACKNOWLEDGEMENTS
+This work was supported by the industry partners Bor-
+regaard, Elkem, Hydro, Yara and the Research Council of
+Norway through the projects TAPI: Towards Autonomy in
+Process Industries (grant no. 294544) and EXAIGON: Ex-
+plainable AI systems for gradual industry adoption (grant
+no. 304843)
+REFERENCES
+Allen-Zhu, Z., Li, Y., and Song, Z. (2019). A convergence
+theory for deep learning via over-parameterization.
+In
+K.
+Chaudhuri
+and
+R.
+Salakhutdinov
+(eds.),
+Proceedings
+of
+the
+36th
+International
+Conference
+on Machine Learning, volume 97 of Proceedings of
+Machine Learning Research, 242–252. PMLR.
+URL
+https://proceedings.mlr.press/v97/allen-zhu19a.html.
+Frankle, J. and Carbin, M. (2019). The lottery ticket hy-
+pothesis: Finding sparse, trainable neural networks. In
+International Conference on Learning Representations.
+Goodfellow, I., Bengio, Y., and Courville, A. (2016). Deep
+learning. MIT press.
+He, K., Zhang, X., Ren, S., and Sun, J. (2016).
+Deep
+residual learning for image recognition. In Proceedings
+of the IEEE conference on computer vision and pattern
+recognition, 770–778.
+Hoefler, T., Alistarh, D., Ben-Nun, T., Dryden, N., and
+Peste, A. (2021).
+Sparsity in deep learning: Pruning
+and growth for efficient inference and training in neural
+networks. J. Mach. Learn. Res., 22(241), 1–124.
+Huang, G., Liu, Z., Van Der Maaten, L., and Wein-
+berger, K.Q. (2017). Densely connected convolutional
+networks. In 2017 IEEE Conference on Computer Vi-
+sion and Pattern Recognition (CVPR), 2261–2269. doi:
+10.1109/CVPR.2017.243.
+Kingma,
+D.P.
+and
+Ba,
+J.
+(2014).
+Adam:
+A
+method for stochastic optimization.
+arXiv preprint
+arXiv:1412.6980.
+Li, H., Xu, Z., Taylor, G., Studer, C., and Goldstein, T.
+(2017). Visualizing the loss landscape of neural nets.
+URL https://arxiv.org/abs/1712.09913.
+Lundby, E.T.B., Rasheed, A., Halvorsen, I.J., and Grav-
+dahl, J.T. (2022).
+Sparse deep neural networks for
+modeling aluminum electrolysis dynamics. arXiv. doi:
+doi.org/10.48550/arXiv.2209.05832.
+Robinson, H., Lundby, E., Rasheed, A., and Gravdahl,
+J.T. (2022). A novel corrective-source term approach
+to modeling unknown physics in aluminum extraction
+process. arXiv. doi:10.48550/ARXIV.2209.10861. URL
+https://arxiv.org/abs/2209.10861.
+Winter, M. and Breitsamter, C. (2018). Nonlinear identifi-
+cation via connected neural networks for unsteady aero-
+dynamic analysis.
+Aerospace Science and Technology,
+77, 802–818.
+
+0
+2
+4
+6
+8
+10
+Time (hours)
+2000
+4000
+6000
+Mass (kg)
+(a) Side ledge mass x1
+0
+2
+4
+6
+8
+10
+Time (hours)
+1000
+0
+1000
+2000
+Mass (kg)
+(b) Alumina mass x2
+0
+2
+4
+6
+8
+10
+Time (hours)
+1600
+1800
+2000
+2200
+Mass (kg)
+(c) Aluminum fluoride x3
+0
+2
+4
+6
+8
+10
+Time (hours)
+10000
+12000
+14000
+16000
+Mass (kg)
+(d) Molten cryolite x4
+0
+2
+4
+6
+8
+10
+Time (hours)
+9000
+9500
+10000
+10500
+Mass (kg)
+(e) Produced aluminum x5
+0
+2
+4
+6
+8
+10
+Time (hours)
+940
+960
+980
+1000
+Temp ( C)
+(f) Bath temperature x6
+0
+2
+4
+6
+8
+10
+Time (hours)
+650
+700
+750
+800
+850
+Temp ( C)
+(g) Side ledge temperature x7
+0
+2
+4
+6
+8
+10
+Time (hours)
+400
+500
+600
+700
+Temp ( C)
+(h) Side wall temperature x8
+Truth
+InputSkipSparse
+PlainSparse
+99.7% conf. PlainSparse
+99.7% conf. InputSkipSparse
+Fig. 4. Rolling forecast of a representative test trajectory
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf,len=573
+page_content='Sparse neural networks with  skip-connections for nonlinear system  identification ⋆  Erlend Torje Berg Lundby ∗,∗∗∗ Haakon Robinson ∗,∗∗∗  Adil Rasheed ∗ Ivar Johan Halvorsen ∗∗  Jan Tommy Gravdahl ∗  ∗ Department of Engineering Cybernetics, Norwegian University of  Science and Technology, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Bragstads plass 2, Trondheim, NO-7034,  Norway (e-mail: erlend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='lundby@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='no haakon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='robinson@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='no,  adil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='rasheed@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='no, jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='tommy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='gravdahl@ntnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='no).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  ∗∗ SINTEF Digital, Trondheim, No-7465, Norway  ∗∗∗ Equal contribution  Abstract: Data-driven models such as neural networks are being applied more and more to  safety-critical applications, such as the modeling and control of cyber-physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Despite  the flexibility of the approach, there are still concerns about the safety of these models in this  context, as well as the need for large amounts of potentially expensive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In particular,  when long-term predictions are needed or frequent measurements are not available, the open-  loop stability of the model becomes important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, it is difficult to make such guarantees  for complex black-box models such as neural networks, and prior work has shown that model  stability is indeed an issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In this work, we consider an aluminum extraction process where  measurements of the internal state of the reactor are time-consuming and expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We model  the process using neural networks and investigate the role of including skip connections in the  network architecture as well as using ℓ1 regularization to induce sparse connection weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We  demonstrate that these measures can greatly improve both the accuracy and the stability of the  models for datasets of varying sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Keywords: Deep Neural Network, Machine Learning, Timeseries modeling, Modeling and  Simulation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' INTRODUCTION  There is increasing interest in using machine learning-  based methods to develop predictive models directly from  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The advantage of this compared to standard sys-  tem identification methods is that it doesn’t require any  assumptions about the system, but all phenomena that  are well represented by the data can often be accurately  captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' One example of such a method that has seen  widespread popularity in recent years is the neural network  (NN), which is known to be a universal function approx-  imator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' These are often used in Reinforcement Learning  (RL) to represent a value function or a model for some  dynamical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, this approach requires a lot  of data to train effective models, which can be expensive  to obtain in many domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' One hypothesis for this is that  NNs are typically overparameterized, and therefore require  many steps to adjust all of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, over-  training on the same limited dataset will cause the model  to overfit to the training data, and perform poorly on  unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' While overparameterization has been found  to aid convergence during training (Allen-Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=', 2019),  it also introduces redundant information into the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  ⋆ This work was supported by the project TAPI: Towards Autonomy  in Process Industries (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 294544), and EXAIGON: Explain-  able AI systems for gradual industry adoption (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 304843)  Recent research has found that using sparser networks  may be the key to training models that can generalize  across many situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In particular, Frankle and Carbin  (2019) showed empirically that for any dense architecture,  there is a very high probability that there is a sparse  subnetwork which will train faster and generalize better  than the full model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This is known as the Lottery Ticket  Hypothesis, and many methods of sparsification can be  seen as attempts to somehow extract such a “winning  lottery ticket” from an initially dense network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' There have  been numerous advances in this field, and we refer to  Hoefler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2021) for a recent and comprehensive review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  In this work we use the well-known ℓ1 regularization for  sparsification of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Another challenge related to the use of NNs is the choice  of architecture and hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Typical networks  have multiple layers which are densely connected, although  this can vary between domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Choosing an appropriate  architecture is an art, generally involving trial and error to  improve performance and avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' It is commonly  understood that the early layers of a neural network have  a significant impact on the overall performance of the  network, but deep networks often suffer from the vanishing  or exploding gradient problem which prevents effective  training of these early parameters (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=',  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='00582v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='SY]  2 Jan 2023  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Skip-connections were originally proposed by He  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2016) as a way to circumvent this, by introducing a  shorter path between the early layers and the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' They  were not only found to enable the training of significantly  deeper networks, but Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2017) also demonstrated  that they may help improve training convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  In the field of dynamical systems and control, we often  design a model with a purpose in mind, such as the design  of a control system or state observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Crucially, we are in-  terested in the behavior and performance of the controlled  system in terms of objectives such as energy efficiency or  yield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This implies that the model does not need to be  perfectly accurate for the entire state space, so long as  the resulting closed-loop performance is sufficient (known  as identification for control (I4C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' If high-frequency mea-  surements from the system are available, only the short-  term behavior of the model is important, since any drift  out of the operational space is quickly corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However,  if measurements are rarely available, such as in the alu-  minum electrolysis process that we consider, the long-term  model behavior and open-loop stability become much more  important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Stable long-term predictions can be important  for decision-making, meaning that a model with good long-  term stability and accuracy is inherently important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  In this work, we investigate the effects of adding skip  connections and ℓ1 regularization on the accuracy and  stability of these models for short, medium, and long  horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We address the following questions:  How do skip connections affect the stability and  generalization error of neural networks trained on  high-dimensional nonlinear dynamical systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  How does sparsity affect stability and generalization  error for neural networks with skip connections that  model nonlinear dynamics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  How does the amount of training data affect neural  networks with skip connections compared to neural  networks without skip connections?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  We make the following contributions:  We perform a black box system identification of an  aluminum electrolysis cell using different NN archi-  tectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  We demonstrate that the accuracy and open-loop  stability of the resulting models is greatly improved  by using ℓ1 weight regularization and incorporating  skip connections into the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  This advantage is consistent across datasets of vary-  ing sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' THEORY  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='1 Physics-based model for aluminum extraction  We evaluate NNs for nonlinear system identification by  first training them on synthetic data generated from a  known physics-based models (PBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The model used in  this work describes the internal dynamics of an aluminum  electrolysis cell based on the Hall-H´eroult process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Figure 1  shows a diagram of the electrolysis cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Traditional PBMs  of such systems are generally constructed by studying the  mass/energy balance of the chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Lundby  Insulation  Temperature of  side ledge  Alumina feed  Carbon  anode  Carbon  anode  Mass  of  Alumina + Aluminium Fluoride + Molten Cryolite  Molten Aluminum  Carbon lining  Bath  temperature  Side wall temperature  Alumina  supply  Current bar collector  Current bar collector  Produced aluminum mass  Mass of side ledge  Line current  Anode-cathode distance  Tapped metal  flow rate  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Schematic of the setup  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2022) presents a more detailed exposition of the  model that we use in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The system is described  by a set of ordinary differential equations (ODE):  ˙x = f(x, u),  (1)  where x ∈ R8 and u ∈ R5 represent the time-varying  states and inputs of the system respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The full set  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='of equations are:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x1 = k1(g1 − x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x1k0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='− k2(x6 − g1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(2a)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x2 = u1 − k3u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(2b)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x3 = u3 − k4u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(2c)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x4 = −k1(g1 − x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x1k0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='+ k2(x6 − g1) + k5u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(2d)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x5 = k6u2 − u4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(2e)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x6 =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='α  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x2 + x3 + x4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='u2g5 + u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='2u5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='2620g2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='− k7(x6 − g1)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(2f)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='+ k8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(x6 − g1)(g1 − x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='k0x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='− k9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x6 − x7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='k10 + k11k0x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x7 = β  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='�k9(g1 − x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='k15k0x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='− k12(x6 − g1)(g1 − x7)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(2g)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='+ k13(g1 − x7)2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='k0x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x7 − x8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='k14 + k15k0x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='˙x8 = k17k9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x7 − x8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='k14 + k15k0 · x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='− x8 − k16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='k14 + k18  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  (2h)  where the intrinsic properties gi of the bath mixture are  given as:  g1 = 991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='2 + 112cx3 + 61c1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='5  x3 − 3265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='5c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='2  x3  (3a)  −  793cx2  −23cx2cx3 − 17c2x3 + 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='36cx3 + 1  g2 = exp  �  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='496 −  2068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='4  273 + x6  − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='07cx2  �  (3b)  g3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='531 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='06 · 10−18u3  1 − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='51 · 10−12u2  1  (3c)  + 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='96 · 10−7u1 − 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='37(cx2 − cx2,crit) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='431  735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='3(cx2 − cx2,crit) + 1  g4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='5517 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='8168 · 10−6u2  1 + 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='271 · 10−6u2  (3d)  g5 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='8168 · 10−6g3g4u2  g2(1 − g3)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  (3e)  See Table 1 for a description of these quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  The values of these constants can be found in Lundby  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The dynamics of the system are relatively  slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The control inputs u1,  u3 and u4 are therefore  well modeled as impulses that represent discrete events  involving the addition or removal of substances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This  results in step changes in the linear states x2, x3, x5, which  act as accumulator states for the mass of the corresponding  substance (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The control inputs u2 and u5 are  piecewise constant, and always nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The inputs u are  determined by a simple proportional controller π(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The  simulation model is derived in Lundby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2022), and  we refer to that article for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='2 Deep neural network with skip connections  A NN with L layers can be compactly written as an  alternating composition of affine transformations Wz + b  and nonlinear activation functions σ : Rn �→ Rn:  ˆf(z) = ˆfL ◦ · · · ◦ ˆf2 ◦ ˆf1  ˆfi(z) = σi(Wiz + bi),  (4)  where the activation function σi, weight matrix Wi, and  bias vector bi correspond to the ith layer of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  The universal approximation property of NNs makes them  very attractive as a flexible model class when a lot of  data is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The representation capacity is generally  understood to increase with both the depth and the width  (the number of neurons in each layer), although early at-  tempts to train very deep networks found them challenging  to optimize using backpropagation due to the vanishing  gradients problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' One of the major developments that  enabled researchers to train deep NNs with many layers  is the skip connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' A skip connection is simply an  additional inter-layer connection that bypasses some of the  layers of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This provides alternate pathways  through which the loss can be backpropagated to the early  layers of the NN, which helps mitigate the issues of vanish-  ing and exploding gradients, which were major hurdles to  training deeper models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In this work, we utilize a modified  DenseNet architecture as proposed by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2017),  where the outputs of earlier layers are concatenated to all  the consecutive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We simplify the structure such that  the model only contains skip connections from the input  layer to all consecutive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We call this architecture  InputSkip, which has reduced complexity compared to  DenseNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This design is motivated by the fact that the  output of each layer (including the final output) becomes  a sum of both a linear and a nonlinear transformation of  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Ta-  ble  of  states,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  in-  puts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='and  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='other  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='quan-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='ti-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='ties  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='used  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='to  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='elec-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='trol-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='y-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='sis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='cell  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Variable  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Physical meaning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Units  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Mass side ledge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Mass Al2O3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Mass AlF3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Mass Na3 AlF6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Mass metal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Temperature bath  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='°C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Temperature side ledge  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='°C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='x8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Temperature wall  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='°C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='u1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Al2O3 feed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg/s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='u2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Line current  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='u3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='AlF3 feed  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg/s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='u4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Aluminum tapping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='kg/s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='u5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Anode-cathode distance  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='cm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='cx2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Al2O3 mass ratio x2/(x2 + x3 + x4) cx3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='AlF3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='mass ratio x3/(x2 + x3 + x4) g1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Liquidus temperature  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='°C  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='g2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Electrical conductivity  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='S m  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='g3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Bubble coverage g4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Bubble thickness  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='cm  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='g5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Bubble voltage  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='V  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='the initial input x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Hence, the skip connections from the  input layer to consecutive layers facilitate the reuse of the  input features for modeling different linear and nonlinear  relationships more independently of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' METHOD AND SETUP  In this section, we present all the details of data generation  and its preprocessing, and the methods that are required to  reproduce the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The steps can be briefly summarized  as follows:  Use Equation (2) with random initial conditions to  generate 140 trajectories with 5000 timesteps each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Set aside 40 for training and 100 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Con-  struct 3 datasets by selecting 10,20 and 40 trajectories  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  For each model class and dataset, train 10 instances  on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Repeat all experiments with ℓ1 regularization, see loss  function in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Use trained models to generate predicted trajectories  along the test set and compare them to the 100 test  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='1 Data generation  Equation (2) was discretized using the RK4 scheme with  a fixed timestep h = 10 s and numerically integrated  on the interval [0, 5000h].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We used uniformly randomly  sampled initial conditions from the intervals shown in  Table 2 to generate 140 unique trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We set aside  40 trajectories for training and 100 of the trajectories  as a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The 40 training trajectories were used  to create 3 datasets of varying sizes (small, medium,  large), namely 10, 20, and 40 trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In total, the  datasets contained 50000, 100000, and 200000 individual  data points respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Equation (2) also depends on the input signal u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In  practice, this is given by a deterministic control policy  u = π(x) that stabilizes the system and keeps the state  x within some region of the state space that is suitable  for safe operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We found that this was insufficient  to successfully train our models, because the controlled  trajectories showed very little variation after some time,  despite having different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This lack of  diversity in the dataset resulted in models that could not  generalize to unseen states, a situation that frequently  arose during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' To inject more variety into the  data and sample states x outside of the standard opera-  tional area, we used a stochastic controller  πs(x) = π(x) + r(t)  that introduced random perturbations r(t) to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  These perturbations were sampled using the Amplitude-  modulated Pseudo-Random Binary Signal (APRBS) method  proposed by Winter and Breitsamter (2018) for nonlinear  system identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  In system identification it is typical to optimize the model  to estimate the function ˙x = f(x, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, this is not  feasible for Equation (2) because the inputs u are not  differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Instead, we discretize the trajectories using  the forward Euler difference and use this as the regression  variable:  yk = xk+1 − xk  h  The datasets are then constructed as sets of the pairs  ([xk, uk], yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='2 Training setup  We optimize the models by minimizing the following loss  function using stochastic gradient descent:  Jθ = 1  |B|  �  i∈B  (yi − ˆf(xi, ui))2 + λ  L  �  j=1  |Wj|  (5)  where B is a batch of randomly sampled subset of indices  from the dataset, L is the number of layers of the NN, and  λ is the regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This loss function is the  sum of the mean squared error (MSE) of the model ˆf with  respect to the regression variables y, and the ℓ1 norm of  the connection weight matrices Wi in all layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We used  a batch size of |B| = 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We used the popular ADAM  solver proposed by Kingma and Ba (2014) with default  parameters to minimize Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Ini-  tial  con-  di-  tions  in-  ter-  vals  for  x  Variable  Initial condition interval  x1  [2060, 4460]  cx2  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='02, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='05]  cx3  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='09, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='13]  x4  [11500, 16000]  x5  [9550, 10600]  x6  [940, 990]  x7  [790, 850]  x8  [555, 610]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='3 Evaluation of model accuracy  As previously mentioned, we are interested in evaluating  the long-term predictive accuracy of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Starting  from a given initial condition x(t0), the model ˆf(x, u)  is used to generate an estimated trajectory using the  recurrence:  ˆxk+1 = ˆxk + hˆf(ˆxk, uk)  (6)  where ˆx0 = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Note that the input signal uk is replayed  directly from the test trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Borrowing a term from  the field of time-series analysis, we refer to this as a rolling  forecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' To evaluate the accuracy of a model over multiple  trajectories, we define the Average Normalized Rolling  Forecast Mean Squared Error (AN-RFMSE):  AN-RFMSE = 1  p  p  �  i=1  1  n  n  �  j=1  � ˆxi(tj) − xi(tj)  std(xi)  �2  ,  (7)  where ˆxi(tj) is the model estimate of the simulated state  variable xi at time step tj, std(xi) is the standard deviation  of variable xi in the training set Strain, p = 8 is the number  of state variables and n is the number of time steps being  averaged over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='4 Evaluation of model stability  A symptom of model instability is that its predictions  can blow-up, which is characterized by a rapid (often  exponential) increase in prediction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' More precisely,  we say that a blow-up occurs when the normalized mean  absolute error for all system states exceeds three (this  corresponds to standard deviations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We detect this as  follows:  max  j<n  �  1  p  p  �  i=1  �|ˆxi(tj) − xi(tj)|  std(xi)  ��  > 3  (8)  where p = 8 is again the number of state variables and  n is the number of time steps to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This is a  conservative estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, this does not lead to any  significant underestimation of the number of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  This is because once a model starts to drift rapidly, it  very quickly exceeds the normal error of three standard  deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' RESULTS AND DISCUSSIONS  We characterize the different model classes (PlainDense,  PlainSparse, InputSkipDense, InputSkipSparse) by esti-  mating their blow-up frequencies and their rolling forecast  mean squared error (RFMSE) on the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The  blow-up frequency is an interesting measure since it can  indicate how stable the model is in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  We perform a Monte Carlo analysis by training 10 in-  stances of each model class and evaluating these on 100  trajectories randomly generated using the true model,  yielding 1000 data points for each model class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' We repeat  the experiments for 3 different dataset sizes to study the  data efficiency of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Figure 2 presents the total number of blow-ups recorded  within each model class after 100h, 2000h, and 5000h  (short, medium, and long term respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' For simplic-  ity, blow-ups were detected by thresholding the computed  variance of a predicted trajectory and manually inspected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  It is clear that for short time horizons all the models  exhibit robust behavior independently of the size of the  training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, for medium and long time  horizons, PlainDense, PlainSparse, and InputSkipDense  architectures exhibit a significant number of blow-ups and  therefore instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Figure 2a - 2c show that PlainDense  is generally the most unstable, with up to 67% of all tra-  jectories resulting in a blow-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' For the smallest amount of  training data (Figure 2a) PlainSparse and InputSkipDense  have similar blow-up frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' For larger datasets, the  PlainSparse architecture shows significantly better stabil-  ity than both PlainDense and InputSkipDense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' InputSkip-  Dense and PlainDense both show better stability with  increasing amounts of training data in terms of fewer blow-  ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, both these dense models still suffer from  significant amounts of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  In comparison, almost no blow-ups are recorded when  using the InputSkipSparse architecture, even for the small  training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In Figure 2, the orange bars correspond-  ing to the blow-up frequency of InputSkipSparse models  are not visible for any of the training sets due to the sig-  nificantly lower number of blow-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' For InputSkipSparse  models trained on the smallest dataset, only 3 out of 1000  possible blow-ups were reported for the longest horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Apart from that, no blow-ups were reported for the Input-  SkipSparse models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Only a few blow-ups were recorded after 5000h in the  medium term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Figure 3 presents a violin plot of the accuracy of each  model class, expressed in terms of RFMSE over different  time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Only the plot for the smallest dataset  (50000 points) is shown, due to the results being very  similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' A larger width of the violin indicates a higher  density of that given RFMSE value, while the error bars  show the minimum and maximum recorded RFMSE val-  ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The model estimates that blew up (see Figure 2) are  not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In this way, we estimate the generalization  performance of the models only within their regions of  stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Note that the violin plots for model classes with  many blow-ups are made using fewer samples, and can be  seen as slightly “cherry-picked”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Nonetheless, the Input-  SkipSparse architecture consistently yields more accurate  100 T  2500 T  5000 T  0  100  200  300  400  500  600  700  (a) Trained on smallest dataset with 50000 datapoints  100 T  2500 T  5000 T  0  50  100  150  200  250  (b) Trained on medium sized dataset with 100000 datapoints  100 T  2500 T  5000 T  0  25  50  75  100  125  150  175  200  (c) Trained on largest dataset with 200000 datapoints  PlainDense  PlainSparse  InputSkipDense  InputSkipSparse  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Divergence plot: Number of trajectories that blow-  up over different time horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The total number of  trajectories is 1000, so the values can be read as a  permille.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  results, up to an order of magnitude better than the others  in the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' CONCLUSION AND FUTURE WORK  In this work, we compared the performance of two different  model structures trained both with and without sparsity  promoting ℓ1 regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The two model types are  standard Multi-Layer Perceptrons (MLP), and a more  specialized architecture that includes skip connections  from the input layer to all consecutive layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This yields  four different model structures, which we call PlainDense,  PlainSparse, InputSkipDense, and InputSkipSparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The  main conclusions of the article are as follows:  NNs with skip connections are more stable for predic-  tions over long time horizons compared to standard  MLPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Furthermore, the accuracy of NNs with skip  100 T  2500 T  5000 T  10  2  10  1  100  PlainDense  PlainSparse  InputSkipDense  InputSkipSparse  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Model accuracy expressed in terms of RFMSE over  different horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Ten models of each of the model  types (PlainDense, PlainSparse, InputSkipDense, In-  putSkipSparse) are trained on the smallest dataset of  50000 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The model estimates that blow up  (see Figure 2) are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' The error bars for each  model type The plot shows that sparse models with  skip connections (InputSkipSparse) are consistently  more accurate than both sparse and dense models  without skip connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  connections is consistently higher for all forecasting  horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  The application of sparsity-promoting ℓ1 regulariza-  tion significantly improves the stability of both the  standard MLP and InputSkip architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This im-  provement was more apparent for models with the  InputSkip architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  The InputSkipSparse showed satisfactory stability  characteristics even when the amount of training data  was restricted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This suggests that this architecture is  more suitable for system identification tasks than the  standard MLP structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  The case study shows that both sparsity-promoting regu-  larization and skip connections can result in more stable  NN models for system identification tasks while requiring  less data, as well as improving their multi-step generaliza-  tion for both short, medium, and long prediction horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Despite the encouraging performance of the sparse-skip  networks, we can not guarantee similar performance for  noisy data, as we have only investigated the use of syn-  thetic data devoid of any noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' However, such a study will  be an interesting line of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' This case study also  has relevance beyond the current setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' In more realistic  situations, we often have a partial understanding of the  system we wish to model (see Equation (2)), and only  wish to use data-driven methods to correct a PBM when  it disagrees with the observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' due to a faulty as-  sumption).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' As shown in Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2022), combining  PBMs and data-driven methods in this way also has the  potential to inject instability into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Finding new  ways to improve or guarantee out-of-sample behavior for  data-driven methods is therefore of paramount importance  to improve the safety of such systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This work was supported by the industry partners Bor-  regaard, Elkem, Hydro, Yara and the Research Council of  Norway through the projects TAPI: Towards Autonomy in  Process Industries (grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 294544) and EXAIGON: Ex-  plainable AI systems for gradual industry adoption (grant  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 304843)  REFERENCES  Allen-Zhu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='  In  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Chaudhuri  and  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  Salakhutdinov  (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' ),  Proceedings  of  the  36th  International  Conference  on Machine Learning, volume 97 of Proceedings of  Machine Learning Research, 242–252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content=' and Carbin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content=' The lottery ticket hy-  pothesis: Finding sparse, trainable neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='  Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=', Bengio, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=', and Courville, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Deep  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  He, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content=', Van Der Maaten, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=', and Wein-  berger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  and  Ba,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content=', Xu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='(d) Molten cryolite x4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='Time (hours)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='9000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='(e) Produced aluminum x5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='Time (hours)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='940  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='960  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='Temp ( C)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(f) Bath temperature x6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Time (hours)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='650  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='(g) Side ledge temperature x7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='Time (hours)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
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+page_content='Temp ( C)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='(h) Side wall temperature x8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='Truth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='InputSkipSparse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='PlainSparse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='7% conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' PlainSparse  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content='7% conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' InputSkipSparse  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
+page_content=' Rolling forecast of a representative test trajectory' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtAyT4oBgHgl3EQfsvmN/content/2301.00582v1.pdf'}
diff --git a/LtE1T4oBgHgl3EQfGwPf/content/tmp_files/2301.02919v1.pdf.txt b/LtE1T4oBgHgl3EQfGwPf/content/tmp_files/2301.02919v1.pdf.txt
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@@ -0,0 +1,1081 @@
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+Charles	Babbage,	Ada	Lovelace,	and	the	Bernoulli	Numbers	
+Thomas	J.	Misa	
+University	of	Minnesota	
+ABSTRACT: This chapter assembles the pertinent sources to suggest several corrections to an 
+unduly negative scholarly view of Ada Lovelace.  I suggest that the Lovelace-Babbage question is 
+not a zero-sum game, where any credit added to Lovelace somehow detracts from Babbage.  
+Ample evidence indicates Babbage and Lovelace each had important contributions to the 1843 
+Sketch and the accompanying Notes.  Further, prior claims about her lack of mathematical 
+background seem doubtful after consulting Lovelace’s detailed correspondence with two highly 
+accomplished figures in 19th century mathematics, Charles Babbage and Augustus De Morgan.  
+Babbage and Lovelace’s treatment of the Bernoulli numbers in note “G” spotlights the 
+mathematical sophistication their collaboration.  Finally, acknowledging significant definitional 
+problems in calling Lovelace the world’s “first computer programmer,” I conclude that Lovelace 
+created an step-by-step elemental sequence of instructions—that is, an algorithm—for computing 
+the series of Bernoulli numbers that was intended for Babbage’s Analytical Engine. 
+Few figures in the long history of computing generate more passion and sometimes more enmity 
+than Charles Babbage and Ada Lovelace.  History has treated Babbage as a brilliant but 
+temperamental pioneer in a half dozen scientific fields, an “irascible genius” in one biographer’s 
+persisting image.  Histories of mathematics typically praise his efforts to bring the modern 
+notation of continental calculus to Cambridge University where the long shadow of Isaac 
+Newton had reigned for more than a century. Babbage was made a Fellow of the Royal Society 
+in 1816, just two years after leaving Cambridge.  In 1820, he founded the Astronomical Society 
+to standardize observational data and improve positional calculations.  Babbage immersed 
+himself in numerous projects and publications during the next two decades.  While histories of 
+insurance credit a early book on actuarial calculations (1826), histories of science debate his 
+polemical On the Decline of Science (1830), histories of industry spotlight On the Economy of 
+Machinery and Manufactures (1832), and histories of religion and science note his Ninth 
+Bridgewater Treatise (1837).  In the midst of this publishing storm, he served as Lucasian 
+Professor of Mathematics at Cambridge (1828-39), Newton’s old post; inherited a sizable fortune 
+from his father; married, raised a family, and lost his wife; and embarked upon designing two 
+mechanical computing machines, the simple but elegant Difference Engine and the complex and 
+enigmatic Analytical Engine.  During these years he hosted a fashionable salon gathering in 
+Page �  of �
+1
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+London, twice ran for Parliament, traveled widely, and corresponded energetically.  He made 
+some powerful friends and a few powerful enemies. 
+In the early 1840s Babbage collaborated closely with Ada Lovelace, and this essay 
+examines their work as an intellectually intimate “creative couple.”   Especially in the popular 
+1
+mind, Ada Lovelace has recently been on a roll.  She is lionized as the founder of scientific 
+computing and hailed as the world’s first computer programmer.  “Readers will recognize Steve 
+Jobs, Charles Babbage, Bill Gates, and Ada Lovelace as appropriate inclusions in [the young-
+adult book] Computer Technology Innovators,” states a 2013 review in School Library Journal.   
+2
+And in his recent best-selling The Innovators, Walter Isaacson employs Ada Lovelace as 
+bookends: in chapter 1, “Ada, Countess of Lovelace,” it is she (above Babbage) who is an 
+engaging founding figure; and in his concluding chapter, “Ada Forever,” he places the promise 
+of computer innovation in the hands of her “spiritual heirs.”  As chapters in this volume amply 
+attest, Ada’s legacy is wide and deep.  There are no other 19th century women who have a 
+programming language named for them (chapters 3-5) and figure prominently in a contemporary 
+science-fiction literary genre (chapters 8-10) and serve as inspiration for contemporary 
+computing reform (chapters 11-13). 
+Scholarship on these two figures is a something of a puzzlement.  A scientific figure like 
+Babbage should have inspired a full-length biography somewhere along the line, but despite 
+numerous essays and several books, we still lack a complete life-and-times biography.   One 
+3
+recent effort by David Alan Grier to explore such a biography confronted a daunting mass of 
+archival materials, some in private hands and difficult to access.  Babbage’s published papers 
+alone run to eleven volumes.   Several popular treatments of Ada Lovelace have appeared, in 
+4
+ Helena M. Pycior, Nancy G. Slack, and Pnina G. Abir-Am., eds, Creative Couples in the Sciences 
+1
+(New Brunswick, NJ: Rutgers University Press, 1996).  “There isn’t a hint of romance in any of their 
+correspondence with one another,” according to Sydney Padua’s The Thrilling Adventures of Lovelace 
+and Babbage (New York: Pantheon, 2015), quote 38 note 16.  Babbage, a widower, was the age of Ada’s 
+mother.
+ Vicki Reutter, “Computer Technology Innovators,” School Library Journal (Oct. 2013): 65.
+2
+ Anthony Hyman’s Charles Babbage: Pioneer of the Computer (Princeton: Princeton University 
+3
+Press, 1982) treats Babbage’s life in 250 pages.  An early study was Maboth Moseley’s Irascible Genius: 
+A Life of Charles Babbage, Inventor (London: Hutchinson, 1964), which is attacked by Dorothy Stein, in 
+Ada: A Life and a Legacy (Cambridge: MIT Press, 1985), p. x, as “almost perversely inaccurate, distorted, 
+and fabricated.”  A specialized and valuable study is J. M. Dubbey, The Mathematical Work of Charles 
+Babbage (Cambridge: Cambridge University Press, 1978). [See also Christopher Hollings, Ursula 
+Martin, and Adrian Rice’s Ada Lovelace: The Making of a Computer Scientist (Bodleian Libraries, 2018) 
+ David Alan Grier, “The Inconsistent Youth of Charles Babbage,” IEEE Annals of the History of 
+4
+Computing 32 no. 4 (2010): 18-31; Martin Campbell Kelly, ed., The Works of Charles Babbage (London: 
+Pickering / New York: New York University Press, 1989; 11 vols.).
+Page �  of �
+2
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+addition to Isaacson’s,  but the existing scholarly consensus on her is a dour one, often highly 
+5
+critical.  Allan Bromley, whose research in the Babbage materials inspired Doron Swade and led 
+to the Science Museum’s project to reconstruct the Difference Engine No. 2, saw the world from 
+a Babbage-centered perspective.  “The [Lovelace] translation has extensive notes, written under 
+Babbage’s supervision, that give an excellent account of his understanding of the mechanization 
+of computational processes and the mathematical powers of the machine,” he wrote in 1982 
+(emphasis added).  Bromley’s dismissal of Lovelace hardened in subsequent years.   Dorothy 
+6
+Stein, in Ada: A Life and a Legacy (1985), sternly cautioned that Lovelace was “a figure whose 
+achievement turns out not to deserve the recognition accorded it.”   Stein’s and Bromley’s 
+7
+gloomy verdict persists in the scholarly survey Computer: History of the Information Machine 
+(1996), now in its third edition (2014), where the key critical passage on Lovelace remains: “the 
+extent of Lovelace’s intellectual contribution to the Sketch has been much exaggerated . . . . Later 
+scholarship has shown that most of the technical content and all of the programs in the Sketch 
+were Babbage’s work.”   The most strident negative verdict derives from Bruce Collier’s 1970 
+8
+Harvard thesis, recently publicized in the Economist magazine on Ada Lovelace day: 
+Ada was as mad as a hatter, and contributed little more to the “Notes” than trouble . . . .  I will 
+retain an open mind on whether Ada was crazy because of her substance abuse . . . or despite 
+ James Essinger, A Female Genius: How Ada Lovelace Started the Computer Age (London: Gibson 
+5
+Square, 2013), which appeared in the United States as Ada’s Algorithm (Brooklyn: Melville House, 2014).  
+Essinger had earlier written Jacquard’s Web: How a Hand-loom Led to the Birth of the Information 
+Age(Oxford: Oxford University Press, 2004).  Contrast Martin Davis and Virginia Davis, “Mistaken 
+Ancestry: The Jacquard and the Computer,” Textile 3 no. 1 (2005): 76-87.  Earlier positive treatments 
+include Betty Alexandra Toole, ed., Ada, The Enchantress of Numbers: A Selection from the Letters of 
+Lord Byron’s Daughter (Mill Valley CA: Strawberry Press, 1992) and Doris Langley Moore, Ada, 
+Countess of Lovelace: Byron’s Legitimate Daughter (New York: Harper & Row, 1977).
+ Doron D. Swade, “Redeeming Charles Babbage’s Mechanical Computer,” Scientific American 
+6
+(February 1993): 86-91; Allan Bromley, “Charles Babbage’s Analytical Engine, 1838,” Annals of the 
+History of Computing 4 no. 3 (1982): 196-217, quote p. 197; Allan Bromley, “Difference and Analytical 
+Engines,” in William Aspray, ed., Computing before Computers (Ames: Iowa State University Press, 
+1990), 59-98, on p. 89.
+ Dorothy Stein, Ada: A Life and a Legacy (Cambridge: MIT Press, 1985), quote xii.  Walter 
+7
+Isaacson’s The Innovators: How a Group of Hackers, Geniuses, and Geeks Created the Digital Revolution 
+(New York: Simon & Schuster, 2014), on p. 493 note 1, praises Stein’s book as “the most scholarly and 
+balanced.” A later negative view is Jay Belanger and Dorothy Stein, “Shadowy Vision: Spanners in the 
+Mechanization of Mathematics,” Historia Mathematica 32 (2005): 76-93.  Stein strongly criticized 
+Dorothy L. Moore, Ada, Countess of Lovelace: Bryon’s Legitimate Daughter (London: Murray, 1977).  
+ Martin Campbell-Kelly, William Aspray, Nathan Ensmenger, and Jeffrey R. Yost, Computer: A 
+8
+History of the Information Machine (Boulder CO: Westview Press, 2014), quote p. 44.  Note that Bromley 
+qualified his claim: that all but one program (Bernoulli numbers) was Babbage’s.  Compare Campbell 
+Kelly and Aspray’s first edition of Computer (p. 57).
+Page �  of �
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+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+it. I hope nobody feels compelled to write another book on the subject. But, then, I guess 
+someone has to be the most overrated figure in the history of computing.  
+9
+In this essay, I assemble the pertinent sources, including correspondence about the Notes 
+to the Menabrea Sketch (printed in this volume: LINK), and suggest several modest corrections 
+to the unduly negative scholarly view.   First, in contrast to much of the existing literature, the 
+10
+Lovelace-Babbage question is not a zero-sum game, where some portion of credit added to 
+Lovelace somehow detracts from Babbage, or vice versa.  There is ample evidence that Babbage 
+and Lovelace each had important contributions to the Sketch and the Notes, and attention to their 
+intellectual collaboration is revealing.  Second, claims about her lack of mathematical 
+background seem doubtful after consulting Lovelace’s detailed correspondence with Babbage 
+and Augustus De Morgan, two highly accomplished figures in 19th century mathematics.  The 
+treatment of the Bernoulli numbers in note “G” spotlights the intellectually intimate 
+collaboration between Babbage and Lovelace and its mathematical sophistication.  Finally, while 
+there may be significant definitional problems in calling Lovelace the world’s “first computer 
+programmer,” the evidence is reasonably clear that Lovelace created an step-by-step elemental 
+sequence of instructions—that is, an algorithm—for computing the series of Bernoulli numbers 
+that was intended for Babbage’s Analytical Engine.  The underlying mathematics might well 
+have been Babbage’s, for he was a distinguished mathematical and scientific figure.  Lovelace 
+transformed an equation for the Bernoulli numbers into a precise series of elemental additions, 
+multiplications, and substitutions. 
+The algorithm specified a sequence of calculations, requiring a real-life computer capable 
+of running a program with a looping structure and conditional testing.  Contemporary computer 
+experts have noted in the large table (representing the Bernoulli number algorithm) in the 
+Sketch’s Note “G” there was one misplaced minus sign which, when corrected, led to the result 
+that Ada’s algorithm correctly computed the series of Bernoulli numbers.  In “The Babbage 
+Machine and the Origins of Programming,” the authors reproduce Lovelace’s table for the 
+Bernoulli numbers and translate the algorithm into a 65-line FORTRAN program that computes 
+ “Ada Lovelace Day: Right Idea. Wrong Woman?” Economist (24 March 2010) at 
+9
+www.economist.com/node/21005551/print (accessed 2015).  I use “derived from” intentionally, since in 
+the printed copy of Collier’s dissertation I examined (CBI QA75.C634x 1970a) I did not find this 
+quotation; see also robroy.dyndns.info/collier (Jan. 2015).  Doron Swade quotes Collier in The Cogwheel 
+Brain: Charles Babbage and the Quest to Build the First Computer (London: Little, Brown, 2000), p. 
+168.
+ Correspondence in the Toole, Stein, Moore, and Swade volumes as well as the document-centered 
+10
+accounts in Velma R. Huskey and Harry D. Huskey, “Lady Lovelace and Charles Babbage,” Annals of the 
+History of Computing 2 no. 4 (1980): 299-329; and John Fuegi and Jo Francis, “Lovelace & Babbage and 
+the Creation of the 1843 ‘Notes’,” IEEE Annals of the History of Computing 25 no. 4 (2003): 16-26.
+Page �  of �
+4
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+them.  The program has eight “if . . . [then] go-to” statements and a simple structure, straight 
+from Lovelace’s table-algorithm, that builds up algebraic statements one mathematical operation 
+at a time: for example, computing the expression (2n - 1) / (2n + 1) requires 4 program steps.   
+11
+In the original 1843 publication of Note “G,” there are clearly two nested loops, embedded in a 
+larger looping structure (see below).  There is direct documentary evidence that Ada Lovelace 
+created this table (writing it out in pencil).  She and Babbage corresponded intensively in the 
+weeks and days prior to its publication in Taylor’s Scientific Memoirs.  This series, published in 
+London between 1837 and 1852 by Richard Taylor in cooperation with the British Association 
+for the Advancement of Science (BAAS), printed English translations of prominent European 
+scientific papers.  “Here was to be found . . . such European leaders” as Bessel, Bunsen, Gauss, 
+Ohm and many others.   The Analytical Engine was Babbage’s creation while the Sketch and 
+12
+Notes are best understood as the product of an intense intellectual collaboration between 
+Babbage and Lovelace. 
+Babbage and Lovelace
+It is with good reason that computing history scholars have praised the research of Allan G. 
+Bromley (1947-2002).  Bromley, a computer scientist at the University of Sydney, made careful 
+studies of the Babbage letters and notebooks that led to the Science Museum’s reconstruction of 
+the Babbage Difference Engine No. 2.  In 2000 Tim Bergin, editor-in-chief of IEEE Annals of the 
+History of Computing, introduced a special issue of the journal dedicated to Bromley’s 
+scholarship by noting his “fundamental contributions,” former Annals editor-in-chief Michael 
+Williams called his work “groundbreaking,” and none other than computer pioneer Maurice 
+Wilkes stated “it is to Bromley that we owe nearly all our present knowledge of Babbage’s work 
+ Campbell Kelly, Works of Charles Babbage, volume 3: 159 note a.  Garry J. Tee, in reviewing a 
+11
+1979 Russian publication by A. K. Petrenko and O. L. Petrenko, wrote this: “The most advanced 
+illustration given in Lovelace’s paper is an elaborate program for computing the sequence of Bernoulli 
+numbers, which was written by Babbage but corrected by her. The present authors have transcribed that 
+program into FORTRAN, detecting thereby a few misprints but only one significant error, with one 
+variable having the wrong sign. Their transcribed version is easier for a modern reader to understand than 
+the original program of 1843, and it does compute correctly the sequence of Bernoulli numbers.” See 
+<www.ams.org/mathscinet-getitem?mr=83b:01045> (accessed 2015) reviewing Petrenko and Petrenko’s 
+“The Babbage Machine and the Origins of Programming,” [in Russian] Istoriko-matematicheskie 
+issledovaniíà 24 (1979): 340-360, 389.  I examined the FORTRAN program in the Russian original.
+ W. H. Brock and A. J. Meadows, The Lamp of Learning: Taylor & Francis and Two Centuries Of 
+12
+Publishing (London: Taylor & Francis, 1998; 2nd edition), esp. chapter 4 “Taylor and the Commercial 
+Science Journal,” quote p. 105.
+Page �  of �
+5
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+on computing machinery at the detailed mechanical level.”  Wilkes noted Bromley’s determined 
+insistence that “Babbage’s work on the Analytical Engine was completely original.”  
+13
+By contrast, Bromley’s view on Lovelace was strongly critical. “All but one of the 
+programs cited in her notes had been prepared by Babbage from three to seven years earlier. The 
+exception [on Bernoulli numbers] was prepared by Babbage for her, although she did detect a 
+‘bug’ in it,” he wrote. “Not only is there no evidence that Ada Lovelace ever prepared a program 
+for the Analytical Engine but her correspondence with Babbage shows that she did not have the 
+knowledge to do so.”   It is Bromley’s viewpoint, slightly modified, that finds its way into the 
+14
+recent edition of the highly regarded Computer: A History of the Information Machine (2014) 
+with its undue assertion that “all of the programs in the Sketch were Babbage’s work.”  
+Reviewing the evidence about Lovelace’s mathematical knowledge and the writing of the notes 
+to her translation of Menabrea’s sketch might modify these overly negative assertions.  In later 
+correspondence with Wilkes, Bromley did allow, in comparison with fellow Babbage scholar 
+Doron Swade, “I have been known to express my views more intemperately.”  
+15
+One must acknowledge that Ada Lovelace in her energetic, imaginative, self-absorbed, 
+and at times grandiloquent correspondence gives her later-day critics much to aim at.  Her self-
+regarding statements about her own mathematical abilities can be off-putting.  And her pointed 
+remarks sometimes aimed at Charles Babbage might rub the wrong way anyone who thinks 
+distinguished scientists should be treated with dignity and respect.   Science at the time was 
+16
+expanding from its strictly male enclaves at Oxford and Cambridge, with the creation of the 
+BAAS (f. 1831) and other learned societies, but in the new scientific institutions Ada Lovelace 
+and Mary Somerville were treated at best as “second class members.”   One need not make 
+17
+excuses for her imaginative flights of fancy, but it does need to be borne in mind that Lovelace 
+was the daughter of an aristocratic Baron (the poet Lord Byron) and married to a highly ranked 
+Earl.  Her mother had schemes drawing on the family’s network that extended into the royal 
+ Tim Bergin, “About this Issue” (quote p. 2); Michael R. Williams, “Allan Bromley,” (quote p. 3); 
+13
+Maurice V. Wilkes, “Introduction to ‘Babbage’s Analytical Plans 28 and 28a—The Programmer’s 
+Interface’,” (quote p. 4) in IEEE Annals of the History of Computing 22 no. 4 (2000).
+ Allan Bromley, “Difference and Analytical Engines,” in William Aspray, ed., Computing before 
+14
+Computers (Ames: Iowa State University Press, 1990), 59-98, quote p. 89.
+ See “Appreciating Charles Babbage: Emails between Allan Bromley and Maurice Wilkes,” IEEE 
+15
+Annals of the History of Computing 26 no. 4 (2004): 62-70, on 62 (intemperately).
+ In the midst of writing the translation’s notes, when letters and draft manuscripts were passed daily 
+16
+between them, Ada writes Babbage, somewhat curtly: “Now pray attend strictly to my requests; or you 
+will cause me very serious annoyance,” quoted in Huskey and Huskey, “Lady Lovelace and Charles 
+Babbage,” p. 313.  There is a curious mix of defiance and deference in Ada’s correspondence with 
+Babbage, Somerville, De Morgan and other prominent figures.
+ Ruth Watts, Gender, Power and the Unitarians in England, 1760-1860 (New York: Longman, 
+17
+1998), quote p. 153.
+Page �  of �
+6
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+family itself.  Babbage, although he was handsomely wealthy after his banker-father died in 1827 
+and a fortune of £100,000 passed down to him, was all the same a commoner.  Ada sought for 
+years to land Babbage a knighthood.   
+18
+The first line of evidence suggesting an intellectual partnership between Charles Babbage 
+and Ada Lovelace comes from witnesses to their first meeting.  She first met Babbage in 1833, a 
+year after being formally presented to the court, through an introduction by Mary Somerville, the 
+mathematician, scientific popularizer, and notable English translator of Pierre-Simon Laplace’s 
+Mécanique Céleste; the two women kept up a scientific correspondence for many years.   Two 
+19
+years after she met Babbage, Ada married William King, already a Baron, who was within three 
+years created the first Earl of Lovelace, and so it is a convenience to refer to her as Ada Lovelace 
+rather than the more precise but ponderous Augusta Ada King, Right Honourable the Countess of 
+Lovelace. 
+There is eyewitness evidence that, when she saw it, Ada grasped the principles and 
+significance of Babbage’s prototype Difference Engine.  Babbage had started work on it in 1822, 
+and it was in an advanced state of development in 1833; fatefully, the following year Babbage set 
+the Difference Engine aside and focused instead on the conceptually elaborate Analytical Engine, 
+which remained a “brilliant obsession” nearly to the end of his life.   In June 1833 Ada’s mother, 
+20
+Lady Byron, described a visit along with her daughter and a friend to inspect Babbage’s machine 
+in some detail and with great wonder while admitting herself only “faint glimpses of the 
+principles by which it worked.”  The Difference Engine was at the time able to compute 
+polynomial expressions, extract roots to quadratic equations, and count to 10,000.  Ada saw its 
+significance.  “I well remember accompanying her to see Mr. Babbage’s wonderful analytical 
+engine,” wrote Sophia De Morgan, the wife of mathematician Augustus De Morgan and, like 
+Mary Somerville, a long-term correspondent with Ada herself.  “While other visitors gazed at the 
+working of this beautiful instrument with the sort of expression . . . that some savages are said to 
+have shown on first seeing a looking-glass or hearing a gun . . . Miss Byron, young as she was, 
+understood its working and saw the great beauty of the invention.  She had read the Differential 
+Calculus to some extent, and after her marriage she pursued the study and translated a small 
+ Doron Swade, s.v. Babbage, Charles (1791-1871), Oxford Dictionary of National Biography 
+18
+(Oxford University Press, 2004); online edition, May 2009 at www.oxforddnb.com/view/article/962. 
+Babbage died in 1871 with a fortune “under” £40,000.
+ Elizabeth Chambers Patterson, Mary Somerville and the Cultivation of Science, 1815-1840 
+19
+(Boston: Nijhoff, 1983); and Kathryn A. Neeley, Mary Somerville: Science, Illumination, and the Female 
+Mind (Cambridge: Cambridge University Press, 2001).
+ See “Appreciating Charles Babbage: Emails between Allan Bromley and Maurice Wilkes,” IEEE 
+20
+Annals of the History of Computing 26 no. 4 (2004): 62-70, quote p. 63 (obsession).
+Page �  of �
+7
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+work of the Italian mathematician Menabrea, in which the mathematical principles of its 
+construction [were] explained.”  
+21
+Babbage invited Ada with a friend or chaperone to attend his series of “Saturday 
+evenings” where his London home became a fashionable salon, filled with celebrities from the 
+political, cultural, and scientific world.  Charles Dickens and Charles Darwin, among hundreds 
+of others, were happy to mix with the well-cultured crowd.  “One of three qualifications were 
+necessary for those who sought to be invited—intellect, beauty, or rank—without one of these, 
+you might be rich as Croesus—and yet be told, you cannot enter here,” recalled one society 
+figure.  “His calculating machine was an endless subject of monologue.”   Some were deeply 
+22
+impressed by its ability to generate a list of prime numbers.  Since it could solve any second-
+degree polynomial equation, Babbage set it to compute a series of 40 prime numbers by 
+evaluating the expression x2 + x + 41 for the first 40 integers.  A few months after the invitation, 
+she wrote to Mary Somerville asking her to convey to Babbage’s son “how exceedingly obliged I 
+am . . . for his unexpected kindness in sending me the plates & account of the Machine, which is 
+exactly what I was in want of; & is a very great help to me.”   Ada was at the time 19 years old. 
+23
+A significant line of evidence bearing on Bromley’s claim that “she did not have the 
+knowledge” to prepare a program for the Analytical Engine is the substantial depth of Ada’s 
+mathematical studies beginning in the 1830s, which predated her contact with the Menabrea 
+manuscript and the translation project of the early 1840s.  “Ada was much attached to me, and 
+often came to stay with me. It was by my advice that she studied mathematics,” recalled Mary 
+Somerville.  “She always wrote to me for an explanation when she met with any difficulty. 
+Among my papers I lately found many of her notes, asking mathematical questions.”  
+24
+During these years Ada had three tutors in mathematics, in addition to intellectual 
+interchange with Somerville, Babbage, and her scientifically minded husband, who was made a 
+Fellow of the Royal Society in 1841; two of them were distinguished figures.  Her first 
+mathematics tutor was the elderly William Frend, the notable social reformer who had authored 
+ Lady Byron quoted in Moore, Ada, p. 44; Sophia De Morgan, Memoir of Augustus De Morgan 
+21
+(London: Longmans, Green, 1882), quote p. 89 (accompanying her).
+ “John Kenyon and His Friends,” Temple Bar: A London Magazine for Town and Country Readers 
+22
+88 (1890): quote p. 490 (three qualifications); AAL to Mary Somerville 19 March 1834 in Toole, Ada, p. 
+57 (Babbage’s invitation).
+ AAL to Mary Somerville 8 November 1834 quoted Huskey and Huskey, “Lady Lovelace and 
+23
+Charles Babbage,” quote p. 303 (account of the Machine).
+ Martha Somerville, ed., Personal Recollections from early life to old age of Mary Somerville 
+24
+(Boston: Roberts Brothers, 1874), quote p. 154. [See also, two subsequent publications: Hollings et al 
+“The Lovelace-De Morgan Mathematical Correspondence: A Critical Re-Appraisal,” Historia 
+Mathematica 2017 at https://doi.org/10.1016/j.hm. 2017.04.001 and “The early mathematical education 
+of Ada Lovelace” BSHM Bulletin (2017) at https://doi.org/10.1080/17498430.2017.1325297 
+Page �  of �
+8
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+Principles of Algebra (1796) along with many other tracts; among his students had been Ada’s 
+own mother.   It seems likely that Frend introduced Ada to Mary Somerville, connecting her 
+25
+with Babbage.  Ada and Babbage corresponded on mathematical topics following their 1833 
+meeting, again years prior to the translation project.  Perhaps her most important mathematical 
+tutor was her friend Sophia’s husband and William Frend’s son-in-law, Augustus De Morgan.  He 
+gave Ada Lovelace, as Sophia later wrote, “much help in her mathematical studies, which were 
+carried farther than her mother’s had been.”   Even the severely critical study by Dorothy Stein 
+26
+acknowledges the De Morgan-Lovelace letters as “a correspondence course in calculus.”  
+27
+Augustus De Morgan was like Babbage a graduate of Cambridge, a disbeliever in the 
+traditional Church of England, and a distinguished mathematician.  He was named founding 
+professor of mathematics at London University (now University College London), shortly after 
+its founding in 1826, at the age of 22.  It was a secular university, unlike Cambridge and Oxford, 
+and admitted women as regular students.  De Morgan published books on trigonometry, 
+arithmetic, algebra, probability, and logic.  In the early 1840s while exchanging regular letters 
+with De Morgan on the topic, Ada reported that she was “drowning in Calculus.”  During these 
+years De Morgan was working on a book project for the London-based Society for the Diffusion 
+of Useful Knowledge (1826-48), published in 1842 as an 800-page textbook on Differential and 
+Integral Calculus.   Her letters to De Morgan are filled with specific questions about differential 
+28
+calculus, limits, Leibnitz’s notation, three-dimensional geometry, notation of functions, and 
+standards of reasoning and proof.  On 21 November 1841, in the context of her mathematical 
+exercises, she asked him about the “numbers of Bernoulli.”   Ada’s awareness of the Bernoulli 
+29
+numbers thus predated her work on the Menabrea translation and the writing of Note G 
+describing an algorithm for their computation.  Whereas Ada’s first tutor, the elder William 
+ Judith S. Lewis,  “Princess of Parallelograms and Her Daughter: Math and Gender in the 
+25
+Nineteenth Century English Aristocracy,” Women’s Studies International Forum 18 no. 4 (1995): 
+387-394.
+ Sophia De Morgan, Memoir of Augustus De Morgan (London: Longmans, Green, 1882), quote p. 
+26
+89 (much help)
+ Stein, Ada, quote pp. xii (correspondence course)
+27
+ Toole, Ada, quote p. 169 (drowning in Calculus).  See Augustus De Morgan, The Differential and 
+28
+Integral Calculus: Containing Differentiation, Integration, Development, Series, Differential Equations, 
+Differences, Summation, Equations of Differences, Calculus of Variations, Definite Integrals (London: 
+Baldwin & Cradock, 1842).  For further testimony on her mathematical work with De Morgan, see 
+Huskey and Huskey, “Lady Lovelace and Charles Babbage,” quotes p. 309: AAL to her mother “I go on 
+most delightfully with Mr De Morgan.  What can I ever do to repay him?”; AAL to Babbage “I am now 
+studying the Finite Differences . . . And in this I have more particular interest, because I know it bears 
+directly on some of your business”; and AAL to her mother “The Mathematics & Mr. De Morgan going 
+on very well indeed. You would be much pleased to see the heap of papers of my writing.”
+ Toole, Ada, quote p. 173 (numbers of Bernoulli).
+29
+Page �  of �
+9
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+Frend, doubted the existence of negative numbers, De Morgan was a modern mathematician of 
+the first rank.  In his book Trigonometry and Double Algebra (1849) he presented a geometrical 
+interpretation of complex numbers, those with real and imaginary parts.   
+In January 1844 De Morgan wrote a lengthy and detailed confidential letter to Ada’s 
+mother, Lady Byron, making an acute assessment of Ada’s unusual facility with mathematics.  “I 
+never expressed to Lady Lovelace my opinion of her as a student in these matters,” De Morgan 
+began.  “The power of thinking on these matters which Lady L[ovelace] has always shown from 
+the beginning of my correspondence with her, has been something so utterly out of the common 
+way for any beginner, man or woman, that this power must be duly considered by her friends . . . 
+whether they should urge or check her obvious determination . . . to get beyond the present 
+bounds of knowledge.”  De Morgan rated Ada favorably with Maria Agnesi, the Italian author of 
+a pioneering calculus textbook (1748), and far more highly than Mary Somerville.  
+30
+During the same years as his correspondence with Ada, De Morgan also facilitated the 
+mathematical work of George Boole, later author of the landmark Investigation of the Laws of 
+Thought (1854) and today widely hailed as the father of Boolean algebra.  In his Treatise on the 
+Calculus of Finite Differences, Boole suggests a useful historical insight relating the character of 
+mid-nineteenth century mathematics to the capabilities of Babbage’s analytical engine.   
+31
+Originally the Bernoulli numbers were discovered in 1712-13 (by the Swiss mathematician Jacob 
+Bernoulli and by the Japanese mathematician Seki Kōwa), to aid in such calculations as the 
+summation of powers (1n + 2n + 3n + 4n . . .).  With their assistance Bernoulli computed the sum 
+of the first 1,000 integers, each raised to the tenth power, a immense number 
+91,409,924,241,424,243,424,241,924,242,500, in (as he claimed) “less than half of a quarter of 
+an hour.”  
+32
+Subsequent work by the Swiss and Scottish mathematicians Euler and Maclaurin 
+connected the mathematics of integral calculus to the summation of polynomial expressions 
+(which are essentially sums of powers each multiplied by some coefficient).  Transcendental 
+mathematical functions as well as integral calculus could be expressed by polynomial 
+expressions; the Bernoulli numbers appeared as coefficients in some of these polynomial series.  
+ Velma R. Huskey and Harry D. Huskey, “Lady Lovelace and Charles Babbage,” Annals of the 
+30
+History of Computing 2 no. 4 (1980): 299-329, De Morgan quoted p. 326; and Massimo Mazzotti, “Maria 
+Gaetana Agnesi: Mathematics and the Making of the Catholic Enlightenment,” Isis 92 no. 4 (2001): 
+657-683.
+ Originally published in 1860, George Boole’s Treatise on the Calculus of Finite Differences (New 
+31
+York: Dover, 1960) deals with Bernoulli numbers in chapter 6.
+ Jacques [Jacob] Bernoulli, “On the ‘Bernoulli numbers’,” in David Eugene Smith, A Source Book 
+32
+in Mathematics (New York: McGraw-Hill, 1929), 85-90, quote p. 90.  In modern notation, the Bernoulli 
+numbers are defined as the coefficients (Bn) for the series expression for an exponential generating 
+function, as follows (see <mathworld.wolfram.com/BernoulliNumber.html>)
+Page �
+ of �
+10
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+Thus, by summing up the correct polynomial (using repeated multiplications, squaring, cubing, 
+et seq.) Babbage’s analytical engine could calculate transcendental mathematical functions (such 
+as sine and cosine) as well as evaluate integral-calculus expressions, so long as they could be 
+expanded into polynomial series.  The Bernoulli numbers could be used to simplify the notation 
+and computation of certain polynomial series, and thus were a powerful aid to computing.  
+Bernoulli achieved his remarkable computation by transforming the extensive summation of 
+powers (110 + 210 + 310 . . . 100010) into a straightforward seven-term polynomial equation using 
+the Bernoulli numbers (up to B10 in the series).  
+33
+Since Ada’s calculus studies with De Morgan likely drew on the calculus textbook he was 
+writing during these years, it is relevant to review its treatment of the Bernoulli numbers.  De 
+Morgan used the Bernoulli numbers in treating polynomial series expansions for ex, tangent, and 
+cotangent; in the calculus of operations; and in convergent series for definite integrals.   One 
+34
+exercise (page 307 §163) presents a general expression for directly computing a specific 
+Bernoulli number, in terms of its predecessors.   
+�
+ 
+So for n = 7, the expression for Bn+1 or B8 is as follows (where the fractional terms are 
+computations on earlier instances in the series): 
+�
+ 
+So, with Ada Lovelace’s interest in Babbage’s machines, her mathematical studies with 
+Babbage, Somerville, and De Morgan, and her relentless curiosity, she was surprisingly well 
+ Sum of the first 1000 integers raised to the tenth power = 1/11x11 + B1x10 + 5B2x9 + 30B4x7 + 
+33
+42B6x5 + 15B8x3 + B10x, where x = 1000 and Bn are the Bernoulli numbers, viz., B1 = 1/2,  B2 =  1/6, B4 = 
+-1/30, B6 = 1/42, B8 = -1/30, B10 =  5/66.
+ See De Morgan, The Differential and Integral Calculus, pp. 247, 248, 308, 553, 581.
+34
+Page �
+ of �
+11
+20
+
+n+1
+1
+n+1
+10*
+AO*
+*0*
+△*0"
+Ba+1=
+.0*=
++
+2*+1
+2+△
+2*+1
+2
+4
+8
+2*+1For thc value of B, (n=7) we have
+8
+1
+126
+1806
+8400
+16600
+15120
+5040)
+1
+255
+4
+8
+16
+32
+64
+128
+256
+30published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+exposed to the advanced mathematics of the period and had ample background and motivation to 
+delve into the computations that appeared in the Notes to the Sketch.  
+35
+Steps to the Sketch
+Babbage conceived a general computing machine around 1834, setting aside his still-
+uncompleted work on the Difference Engine to take up the challenges of what became known as 
+the Analytical Engine.  Whereas it was necessary to mechanically set up the Difference Engine to 
+do each calculation, such as the prime-number generating polynomial x2 + x + 41, Babbage’s 
+inspiration for the Analytical Engine was a computing machine able to re-configure itself—if not 
+precisely “programmable” in the modern sense of the term.   Babbage’s mature design while 
+36
+remaining a mechanical-age technology had several features in common with modern computers: 
+separating the computation of numbers from their storage (he used the terms ‘mill’ and ‘store’ in 
+an analogy with the industrial factory); adopting a mechanism to do conditional testing on 
+intermediate results and hence permitting the branching of calculations; and using punched cards 
+loosely inspired by the Jacquard loom.  
+37
+Babbage took one of his sets of evolving plans for the Analytical Engine to present at a 
+conference in Turin in 1840.  In the audience was a future prime minister of Italy.  At the time 
+Luigi Menabrea was a professor of mechanics and construction at the university of Turin; he 
+subsequently served as a military engineer, naval minister, and eventually prime minister of Italy 
+ See also Imogen Forbes-Macphail’s chapter in this volume [Ada’s Legacy] exploring the “poetical” 
+35
+nature of mathematics.
+ Babbage spent years working out a notation for expressing how its computations might be 
+36
+expressed.  See Allan G. Bromley, “Charles Babbage’s Analytical Engine, 1838,” IEEE Annals of the 
+History of Computing 20 no. 4 (1998): 29-45; and Allan G.Bromley, “Babbage’s Analytical Engine Plans 
+28 and 28a: The Programmer’s Interface,” IEEE Annals of the History of Computing 22 no. 4 (2000): 
+5-19.
+ The easy one-to-one correspondence between Jacquard looms and Babbage’s cards, posited by 
+37
+such authors as James Essinger (author of popular works on Jacquard and Lovelace), is critically 
+scrutinized by Martin Davis and Virginia Davis, “Mistaken Ancestry: The Jacquard and the Computer,” 
+Textile 3 no. 1 (2005): 76-87.  After years of close study, Allan Bromley felt that the Analytical Engine 
+fell somewhat short of a modern computer, as he wrote (privately) to Wilkes: “Perhaps my 
+disappointment comes from being forced to accept that Babbage did NOT devise a COMPUTER but only 
+a very sophisticated CALCULATOR after the style of the Harvard Mark I or the NCR accounting 
+machines. He did not cross the watershed that marked off the [stored-program computers] EDVAC/
+EDSAC, although many of his technical innovations in implementation were astounding.  It is a little 
+difficult to admit this conclusion after expending so many years studying Babbage’s designs. Perhaps it is 
+why I ‘took a break from Babbage’ in the late 1980s and never really came back.”  See “Appreciating 
+Charles Babbage: Emails between Allan Bromley and Maurice Wilkes,” IEEE Annals of the History of 
+Computing 26 no. 4 (2004): 62-70.
+Page �
+ of �
+12
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+(1867-69).   In October 1842, Menabrea published a short description of Babbage’s Analytical 
+38
+Engine in a Swiss journal (written in French).  By this time, Babbage was well on the way to 
+ruining whatever chance might have remained for support from the British government, 
+especially after a disastrous meeting in November of that year with prime minister Robert Peel.  
+“What shall we do to get rid of Mr. Babbage and his calculating Machine?  Surely if completed it 
+would be worthless as far as science is concerned,” he wrote.  Peel, signaling his displeasure, 
+soon dispatched Babbage’s prime-number-calculating Difference Engine to the King’s College 
+Museum.  
+39
+It was not initially Babbage who encouraged Ada Lovelace to examine the Menabrea 
+manuscript and translate it into English.  Rather the prompting came from the notable scientist 
+and sometime telegraph inventor Charles Wheatstone, who knew both Babbage and Lovelace 
+and recruited contributions for Taylor’s Scientific Memoirs.  Wheatstone had gained fame in 
+1837 with the patenting of a multiple-wire electric telegraph system using the positioning of 
+magnetic needles to encode individual letters, rather than the Morse code system of dots and 
+dashes.  He, too, was intrigued with using electromechanical apparatus for computation.  
+“Yesterday saw Wheatstone’s model for telegraph and his drawings for Multiplication Engine,” 
+wrote Babbage after a visit in 1839.   In 1843, the publication year of the Menabrea translation, 
+40
+Wheatstone improved on and publicized the famous “Wheatstone bridge” used to precisely 
+measure electrical resistances.   
+Over the winter of 1842-43 Lovelace worked on translating the Menabrea manuscript, 
+around 8,000 words, and first showed her results to Babbage in the spring of 1843.  In his 
+memoir, Passages from the Life of a Philosopher, Babbage recalled encouraging Ada to add 
+descriptive notes to her translation.  
+We discussed together the various illustrations that might be introduced: I suggested several, 
+but the selection was entirely her own.  So also was the algebraic working out of the different 
+problems, except, indeed, that relating to the numbers of Bernoulli, which I had offered to do 
+to save Lady Lovelace the trouble.  This she sent back to me for an amendment, having 
+detected a grave mistake which I had made in the process. 
+The notes of the Countess of Lovelace extend to about three times the length of the original 
+memoir.  Their author has entered fully into almost all the very difficult and abstract 
+questions connected with the subject.  
+41
+ See “Luigi Federico Menabrea” at <www-history.mcs.st-andrews.ac.uk/Biographies/
+38
+Menabrea.html>.
+ Fuegi and Francis, “Lovelace & Babbage,” quote p. 16 (Peel on Babbage).
+39
+ Fuegi and Francis, “Lovelace & Babbage,” quote p. 18 (Babbage on Wheatstone); Stein, Ada, 88.
+40
+ Charles Babbage, Passages from the Life of a Philosopher (London: Longman, Green, 1864), 
+41
+quote p. 136.
+Page �
+ of �
+13
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+By this time in his life, while he was certainly writing for posterity and obviously keen on 
+memorializing his computing engines, Babbage had no particular reason to exaggerate Ada’s 
+achievements.  She had died years earlier, at age 36, and left Babbage a modest legacy of £600.  I 
+think we can take him at his word when he admits a “grave mistake” in deriving the Bernoulli 
+numbers (which Bromley labels anachronistically as a “bug”).  This passage, along with the 
+Babbage-Lovelace letters, clearly describes a collaboration where Babbage and Lovelace are 
+working together on the Notes.  It’s also clear from the context that “their author” here refers to 
+Lovelace (rather than Menabrea) and that it is certain praise that she “has entered fully into . . . 
+difficult and abstract questions.”  At the very least, Babbage’s statement challenges the assertion 
+that the notes “give an excellent account of his [that is, Babbage’s alone] understanding of the 
+mechanization of computational processes and the mathematical powers of the machine.” 
+Letters from the exact weeks in the summer of 1843 when Babbage and Lovelace were 
+working on the Notes provide additional detail on their collaboration.  This documentary 
+evidence—from the British Library, the Science Museum, and the Oxford Bodleian Library—has 
+been analyzed by Velma Huskey and Harry Huskey as well as John Fuegi and Jo Francis and 
+published in Annals of the History of Computing.   The picture is clearly of a collaboration 
+42
+where both Babbage and Lovelace are making important contributions.  Lovelace wrote the 
+Notes and most of her letters to Babbage at her Ockham Park estate an hour’s south of London; 
+Babbage received her letters and responded from his London house on Dorset Street; and from 
+time to time they met in person at Lovelace’s London house on aristocratic St. James Square.  
+Again, I emphasize that to point out Lovelace’s knowledge, achievements, and contributions is 
+not to belittle Babbage’s. 
+It is not easy to support the conjecture that the notes are Babbage’s alone or that he 
+directed Lovelace to write them.  Bromley is not the only critic who has aimed to reduce 
+Lovelace to a low-level clerk-assistant to Babbage.  In Ada: A Life and a Legacy, Dorothy Stein, 
+for instance, seized on a misprint that Lovelace, Babbage, and Menabrea before them all failed to 
+spot—supposedly highlighting “the significance of her curiously ignored translation of a 
+printer’s error”—and contrives an argument that Lovelace had only a tenuous grasp of 
+mathematics and a “dubious” understanding of the Analytical Engine’s mechanical and logical 
+operations.  Stein contends, on this line of conjecture, that Lovelace was “completely dependent 
+ Velma R. Huskey and Harry D. Huskey, “Lady Lovelace and Charles Babbage,” Annals of the 
+42
+History of Computing 2 no. 4 (1980): 299-329; and John Fuegi and Jo Francis, “Lovelace & Babbage and 
+the Creation of the 1843 ‘Notes’,” IEEE Annals of the History of Computing 25 no. 4 (2003): 16-26.
+Page �
+ of �
+14
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+on [Babbage] for information and claims about the Analytical Engine.”   Stein’s use of evidence 
+43
+is rather thin and highly selective.  In their analysis, Fuegi and Francis point to contemporaneous 
+letters from Babbage, Lovelace, Wheatstone, the editors of Taylor’s Scientific Memoirs and other 
+scientific colleagues, concluding “all contemporaries of Lovelace and Babbage, having first-hand 
+knowledge of how the ‘Notes’ came into being, acknowledged Lovelace at the time as the 
+primary author.”  
+44
+The evidence from Babbage’s letters points to a collaboration between them, while their 
+informal manner of addressing each other indicates a degree of collegiality.  Ada writes to “My 
+Dear Babbage” while he responds to “My Dear Lady Lovelace.”  On 30 June 1843 he writes: 
+I am delighted with Note D.  It is in your usual clear style and required only one trifling 
+alteration which I will make.  This arises from our not having yet had time to examine the 
+outline of the mechanical part . . . I enclose a copy of the integration.  I am still working at 
+some most entangled notations of Division but see my way through them at the expense of 
+heavy labour . . . . Your latest information was the most agreeable.  
+45
+Let us turn to Note G on the computation of the Bernoulli numbers.  Recall that they were 
+a common topic in 19th century mathematics, and that Lovelace herself had previously inquired 
+about them to De Morgan.  The topic first appears in the correspondence when Ada wrote to 
+Babbage, in a letter dated simply “Monday,” as follows:  
+I am working very hard for you . . . .  I think you will be pleased.  I have made what appears 
+to me some very important extensions & improvements . . . . 
+I want to put something about Bernoulli’s Numbers, in one of my Notes, as an example of 
+how an implicit function may be worked out by the engine, without having been worked our 
+by human head & hands first [as the Difference Engine required].  Give me the necessary 
+data & formulae.  
+46
+Even if Babbage provided Ada with the mathematical expressions for the Bernoulli 
+numbers, and assisted with the derivation of a general formula, the transformation of the general 
+formula into a step-by-step algorithm remains Ada’s achievement, as the letters clearly indicate.  
+The mathematics is somewhat involved, beginning with the basic equation where the Bernoulli 
+ Stein, Ada, quote pp. xi (printer’s error) 90-91 (tenuousness, dubious, completely dependent).  The 
+43
+printer’s error was made in Menabrea’s original publication where the French word cas (as in the case 
+where N goes to infinity in a math expression) was mistakenly printed as cos. (as in the abbreviation for 
+cosine).
+ Fuegi and Francis, p. 26 note 29.
+44
+ Babbage to AAL 30 June 1843, quoted in Huskey and Huskey, pp. 312-313.
+45
+ AAL to Babbage, “Monday,” quoted in Huskey and Huskey, pp. 311.
+46
+Page �
+ of �
+15
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+numbers (note the odd-numbered notation B1, B3, B5) appear as coefficients for the exponential 
+function: 
+�
+ 
+and then involving expansion, division, derivation, rounds of intricate multiplication, and finally 
+writing the equation in general form: 
+�
+ 
+This equation (as note G explains, introducing the odd-numbered notation) “enables us to 
+find . . . any nth Number of Bernoulli B2n-1, in terms of all the preceding ones, if we but know the 
+values of B1, B3 . . . B2n-3.”  If n = 1, then the numerators for each of the higher terms (B3 et seq.) 
+contain a zero (2n-2) and so they drop out, permitting the direct calculation of B1.  Similarly, for 
+n = 2 one substitutes the already computed value for B1 while the higher terms (B5 et seq.) again 
+drop out and permit the direct computation of B3.  With n = 3, the process is repeated to compute 
+B5.  The Notes explicitly make the general argument: “And so on, to any extent.” 
+Ada was determined to get the advanced mathematics correct.  The two often exchanged 
+letters daily, and sometimes even more frequently, as when a servant came into London and then 
+waited for Babbage’s reply.  “I am doggedly attacking . . . all the ways of deducing the Bernoulli 
+Numbers,” she wrote at one moment.  On the mathematics Ada wrote to Babbage (in a letter 
+dated simply Tuesday morning) “the few lines I enclosed you last night about the connexion of 
+(8) [the Bernoulli number equation in general form] with the famous Integral, I by no means 
+intend you to insert, unless you fully approve the doing so.”  
+47
+The table and diagram that contained and expressed the algorithm were of special 
+concern.  Ada wrote Babbage (Saturday 6 o’clock), in connection with the note on the Bernoulli 
+numbers, “Think of my horror then at just discovering that the Table & Diagram, (over which I 
+have been spending infinite patience and pains) are seriously wrong, in one or two points.  I have 
+done them however in a beautiful manner, much improved upon our first edition of a Table and 
+ Toole, Ada, quote 204 (doggedly attacking); AAL to Babbage, “Tuesday morng,” quoted in 
+47
+Huskey and Huskey, pp. 314.
+Page �
+ of �
+16
+20
+
+2n-1
++ B1
+2n
++ B3
+2n-(2n
+21
+2
+2n+
+2.3.4
++B5
+2n(2n-1)
+2.3.4.5.6published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+Diagram.  But unluckily I have made some errors.  I send you this final note [G] excepting the 
+Table & Diagrams.”  As a postscript, Ada adds: “Let me know how you like my finishing up of 
+[G].  Mind you scrutinise all the n’s very carefully.  I mean those of Sheets 4 and 5.”  Then, after 
+a full day’s work on Sunday, “you will admire the Table & Diagram extremely.  They have been 
+made out with extreme care & and all the indices most minutely & scrupulously attended to.  
+Lord L[ovelace] is at this moment inking it all over for me.  I had to do it in pencil.”  
+48
+Ada tackled the question of the looping structure with some care.  In a letter dated simply 
+Tuesday, she writes to Babbage: “I hope you will approve of what I send.  I have taken much 
+pains with it.  I have explained that there would be, in this instance & in many others, a recurring 
+group or cycle of Variable as well as of Operation cards . . . .”  She specifically identifies the 
+main or outer loop as the “repetitions of (13 . . . 23),” and explains “as the variations follow a 
+regular rule, they would be easily provided for.”  
+49
+In his correspondence with Ada, Babbage directly admires her work on the Notes.  His 
+praise of Note D, “in your usual clear style,” was cited earlier.  In response to the Saturday letter 
+quoted in the above paragraph, and clearly written before he received the Sunday letter, Babbage 
+wrote: “I like much the improved form of the Bernoulli Note but can judge of it better when I 
+have the diagram and Notation.  I am very reluctant to return the admirable and philosophic view 
+of the Anal. Engine contained in Note A.  Pray do not alter it and do let me have it returned on 
+Monday.”   (Babbage was assembling the entire set of Notes in preparation for handing them to 
+50
+Charles Wheatstone for publication.) 
+As the final versions were going to the printer, Lovelace and Babbage were still revising 
+the Notes’ treatments of the variable cards and the operation cards, which specified the elemental 
+arithmetical operators (+ - x ÷).  Ada made clear that handling the operations cards was at the 
+center of the Analytical Engine’s capability for looping, even while the mechanical apparatus for 
+effecting conditional tests was at the time not entirely clear.   Note D presents a straightforward 
+51
+computation—similar to one that Menabrea had presented in the original essay, also in tabular 
+form—with six variable cards (for constants), nine working variables (for intermediate results), 
+and two variables for results.  There were 11 sequential steps (no looping structures), and the 
+entirety fits onto the page (figure 1). 
+ Huskey and Huskey, pp. 313-14.  In the earlier [Saturday] letter, Lovelace mistakenly labeled the 
+48
+final note “H” rather than “G,” which was corrected in the later Sunday letter.
+ Huskey and Huskey, pp. 316 (recurring group).
+49
+ Huskey and Huskey, pp. 313 (Babbage on improved form of the Bernoulli note).
+50
+ For a mechanical visualization of the Analytical Engine’s “Logic and Loops,” see Sydney Padua’s 
+51
+The Thrilling Adventures of Lovelace and Babbage, pp. 306-308.
+Page �
+ of �
+17
+20
+
+published in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+Figure	1:	Table-algorithm	derived	from	Menabrea	original	(Note	D)	
+�
+ 
+For Note G on the Bernoulli numbers, the table–algorithm has ten data variables, three 
+working variables, and four result variables.  The computation has just 25 operations, but there 
+are in addition two nested loops: an outer loop consisting of steps 13-23, and two inner loops 
+consisting of steps 13-16 and 17-20.  This form of calculation could not possibly have been 
+completed with Babbage’s Difference Engine, since it entirely lacked the ability to do 
+conditional tests and create looping or branching structures.  Nothing like it appeared in 
+Menabrea’s original.  Lady Lovelace chose well when she identified the Bernoulli numbers as a 
+means to show “how an implicit function may be worked out by the engine.” 
+Page �
+ of �
+18
+20
+
+Variables
+forData
+Working
+Variables
+Variablesfor Results
+Number of Operations
+Nature of Operations
+0V12
+0V13
+0V14
+0V15
+0V16
++
++
++
++
++
++
++
+U
+0
+0
+U
+0
+0
+0
+0
+0
+0
+0
+U
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+U
+0
+0
+U
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+U
+0
+0
+U
+U
+0
+0
+0
+0
+0
+m2
+u
+dn"d'n
+d
+mn'm'n
+1
+2
+...
+uu
+L
+dn
+up
+b
+u,p
+6
+dm"
+7
+m'n
+8
+9
+0
+10
+0
+mn'-m'n
+11
+0
+0
+-dmpublished in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+Figure	2:	Original	table-algorithm	for	Bernoulli	numbers	(Note	G)	
+�
+ 
+Oversize table representing Ada Lovelace’s algorithm for computing Bernoulli numbers.  The 
+nested looping structure—“here follows a repetition of operations thirteen to twenty-three”—is 
+clearly visible at lower left (outer loop steps 13-23 and inner loops steps 13-16 and 17-20).  
+Printed oversize and interleaved in Taylor’s Scientific Memoirs (high-res image here and here). 
+This essay assesses Ada Lovelace’s contribution to computing.  It points out her mathematical 
+studies with Babbage and De Morgan, her translation of Menabrea’s Sketch, and her joint 
+authorship with Babbage of the explanatory Notes.  One element, the table-algorithm for 
+computing the series of Bernoulli numbers, is by available evidence her work (written in pencil 
+and inked in by her husband).  Her correspondence with Babbage evinces a direct collegiality; 
+the two figures were jointly grappling with how to communicate the details of a computing 
+machine that did not physically exist.  For this reason, the claim that Ada Lovelace was the 
+world’s first computer programmer might need to be carefully qualified.  At the least, we can 
+grant her primary authorship of the first algorithm intended for a computing machine.  
+52
+ See ACM’s Collected Algorithms at <netlib.org/toms>.
+52
+Page �
+ of �
+19
+20
+
+Diagram for the computation by the Engine of the Numbers of Bernoulli. See Note G. (page 722 et seq.)
+Working Variables
+Data
+Variable
+Nature of Operation.
+10000
+0000
+10000
+10000
+MO0O
+10000
+30000
+10000
+10000
+000
+F0000
+000
+Indication of
+Tariables
+Variables
+receiving
+change in the
+value on any
+Statement of Results.
+upon.
+results.
+Variable.
+2n
++IV
+2n+1
+21
+.2n-
+21
+2n+
+2n-1
+0
+A.
+2n+1
+02
+一
+10
+B,
+= B, Al
+-B,A1
+21
+5
+2n+1
+2(-2)
++V
+2V
+2n2n-1
+V
+2n
+BV
+4V
+a Variable-card.
+Variablepublished in Hammerman and Russell, eds., Ada’s Legacy (ACM Books 2015) DOI
+It is surprising how widely and warmly she was recognized among early figures in 
+computing.  Alan Turing for instance felt obliged to deal with “Lady Lovelace’s objection” to 
+artificial intelligence since (in Turing’s quotation) she had maintained “The Analytical Engine 
+has no pretensions to originate anything. It can do whatever we know how to order it to 
+perform.”   A pioneering conference volume, Faster Than Thought: A Symposium of Digital 
+53
+Computing Machines (1953) was the source, according to the History of Programming 
+Languages chapter on the Ada computer language, “from which we all learned about Lady 
+Lovelace.”   “Lady Lovelace had undoubtedly a profound understanding of the principles of the 
+54
+machine, and she added greatly to the value of her translation by some comprehensive notes 
+about the machine . . . including what we should now call a programme for computing the 
+Bernoulli numbers by a very sophisticated method,” wrote a computer pioneer in 1953.  The 
+statement accords well with my assessment, after several decades of possibly fanciful writing 
+about Ada Lovelace as well as unjustifiably critical blasts against her.  Perhaps in the coming 
+generation, we can come back to the measured appreciation of her achievements that originated, 
+like the computer age itself, in the early 1950s.  After all, as Ada Lovelace put it, “we may 
+consider the [analytical] engine as the material and mechanical representative of analysis.”  
+55
+ Andrew Hodges, “Turing: A Natural Philosopher,” (1997) at https://www.turing.org.uk/
+53
+publications/philobook.html ; and Darren Abramson, “Turing’s Responses to Two Objections,” Minds and 
+Machines 18, no. 2 (2008): 147-167.
+ Thomas J. Bergin, Jr. and Richard G. Gibson, Jr., eds., History of Programming Languages---II 
+54
+(New York ACM, 1996), “Ada Session,” quote p. 207; B. V. Bowden, ed., Faster Than Thought: A 
+Symposium of Digital Computing Machines (London: Isaac Pitman, 1953).
+ B. V. Bowden, ed., Faster Than Thought: A Symposium of Digital Computing Machines (London: 
+55
+Isaac Pitman, 1953), 18-22, 70-75, 341-408, quote p. 18 (profound understanding); Lovelace, 
+“Translator’s Notes,” quote p. 696 (representative of analysis).  Compare Larry Owens, “Vannevar Bush 
+and the Differential Analyzer: The Text and Context of an Early Computer,” Technology and Culture 27 
+no. 1 (1986): 63-95.
+Page �
+ of �
+20
+20
+
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@@ -0,0 +1,642 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf,len=641
+page_content='published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  Charles Babbage, Ada Lovelace, and the Bernoulli Numbers  Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Misa  University of Minnesota  ABSTRACT: This chapter assembles the pertinent sources to suggest several corrections to an  unduly negative scholarly view of Ada Lovelace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I suggest that the Lovelace-Babbage question is  not a zero-sum game, where any credit added to Lovelace somehow detracts from Babbage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Ample evidence indicates Babbage and Lovelace each had important contributions to the 1843  Sketch and the accompanying Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Further, prior claims about her lack of mathematical  background seem doubtful after consulting Lovelace’s detailed correspondence with two highly  accomplished figures in 19th century mathematics, Charles Babbage and Augustus De Morgan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Babbage and Lovelace’s treatment of the Bernoulli numbers in note “G” spotlights the  mathematical sophistication their collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Finally, acknowledging significant definitional  problems in calling Lovelace the world’s “first computer programmer,” I conclude that Lovelace  created an step-by-step elemental sequence of instructions—that is, an algorithm—for computing  the series of Bernoulli numbers that was intended for Babbage’s Analytical Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Few figures in the long history of computing generate more passion and sometimes more enmity  than Charles Babbage and Ada Lovelace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' History has treated Babbage as a brilliant but  temperamental pioneer in a half dozen scientific fields, an “irascible genius” in one biographer’s  persisting image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Histories of mathematics typically praise his efforts to bring the modern  notation of continental calculus to Cambridge University where the long shadow of Isaac  Newton had reigned for more than a century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage was made a Fellow of the Royal Society  in 1816, just two years after leaving Cambridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In 1820, he founded the Astronomical Society  to standardize observational data and improve positional calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage immersed  himself in numerous projects and publications during the next two decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' While histories of  insurance credit a early book on actuarial calculations (1826), histories of science debate his  polemical On the Decline of Science (1830), histories of industry spotlight On the Economy of  Machinery and Manufactures (1832), and histories of religion and science note his Ninth  Bridgewater Treatise (1837).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In the midst of this publishing storm, he served as Lucasian  Professor of Mathematics at Cambridge (1828-39), Newton’s old post;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' inherited a sizable fortune  from his father;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' married, raised a family, and lost his wife;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and embarked upon designing two  mechanical computing machines, the simple but elegant Difference Engine and the complex and  enigmatic Analytical Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' During these years he hosted a fashionable salon gathering in  Page �  of �  1  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  London, twice ran for Parliament, traveled widely, and corresponded energetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' He made  some powerful friends and a few powerful enemies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  In the early 1840s Babbage collaborated closely with Ada Lovelace, and this essay  examines their work as an intellectually intimate “creative couple.”   Especially in the popular  1  mind, Ada Lovelace has recently been on a roll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' She is lionized as the founder of scientific  computing and hailed as the world’s first computer programmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “Readers will recognize Steve  Jobs, Charles Babbage, Bill Gates, and Ada Lovelace as appropriate inclusions in [the young-  adult book] Computer Technology Innovators,” states a 2013 review in School Library Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  2  And in his recent best-selling The Innovators, Walter Isaacson employs Ada Lovelace as  bookends: in chapter 1, “Ada, Countess of Lovelace,” it is she (above Babbage) who is an  engaging founding figure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and in his concluding chapter, “Ada Forever,” he places the promise  of computer innovation in the hands of her “spiritual heirs.”  As chapters in this volume amply  attest, Ada’s legacy is wide and deep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' There are no other 19th century women who have a  programming language named for them (chapters 3-5) and figure prominently in a contemporary  science-fiction literary genre (chapters 8-10) and serve as inspiration for contemporary  computing reform (chapters 11-13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Scholarship on these two figures is a something of a puzzlement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' A scientific figure like  Babbage should have inspired a full-length biography somewhere along the line, but despite  numerous essays and several books, we still lack a complete life-and-times biography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' One  3  recent effort by David Alan Grier to explore such a biography confronted a daunting mass of  archival materials, some in private hands and difficult to access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage’s published papers  alone run to eleven volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Several popular treatments of Ada Lovelace have appeared, in  4  Helena M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Pycior, Nancy G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Slack, and Pnina G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Abir-Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', eds, Creative Couples in the Sciences  1  (New Brunswick, NJ: Rutgers University Press, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “There isn’t a hint of romance in any of their  correspondence with one another,” according to Sydney Padua’s The Thrilling Adventures of Lovelace  and Babbage (New York: Pantheon, 2015), quote 38 note 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage, a widower, was the age of Ada’s  mother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Vicki Reutter, “Computer Technology Innovators,” School Library Journal (Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2013): 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  2  Anthony Hyman’s Charles Babbage: Pioneer of the Computer (Princeton: Princeton University  3  Press, 1982) treats Babbage’s life in 250 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' An early study was Maboth Moseley’s Irascible Genius:  A Life of Charles Babbage, Inventor (London: Hutchinson, 1964), which is attacked by Dorothy Stein, in  Ada: A Life and a Legacy (Cambridge: MIT Press, 1985), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' x, as “almost perversely inaccurate, distorted,  and fabricated.”  A specialized and valuable study is J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Dubbey, The Mathematical Work of Charles  Babbage (Cambridge: Cambridge University Press, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' [See also Christopher Hollings, Ursula  Martin, and Adrian Rice’s Ada Lovelace: The Making of a Computer Scientist (Bodleian Libraries, 2018)  David Alan Grier, “The Inconsistent Youth of Charles Babbage,” IEEE Annals of the History of  4  Computing 32 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2010): 18-31;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Martin Campbell Kelly, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', The Works of Charles Babbage (London:  Pickering / New York: New York University Press, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 11 vols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  2  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  addition to Isaacson’s,  but the existing scholarly consensus on her is a dour one, often highly  5  critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Allan Bromley, whose research in the Babbage materials inspired Doron Swade and led  to the Science Museum’s project to reconstruct the Difference Engine No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2, saw the world from  a Babbage-centered perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “The [Lovelace] translation has extensive notes, written under  Babbage’s supervision, that give an excellent account of his understanding of the mechanization  of computational processes and the mathematical powers of the machine,” he wrote in 1982  (emphasis added).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Bromley’s dismissal of Lovelace hardened in subsequent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Dorothy  6  Stein, in Ada: A Life and a Legacy (1985), sternly cautioned that Lovelace was “a figure whose  achievement turns out not to deserve the recognition accorded it.”   Stein’s and Bromley’s  7  gloomy verdict persists in the scholarly survey Computer: History of the Information Machine  (1996), now in its third edition (2014), where the key critical passage on Lovelace remains: “the  extent of Lovelace’s intellectual contribution to the Sketch has been much exaggerated .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Later  scholarship has shown that most of the technical content and all of the programs in the Sketch  were Babbage’s work.”   The most strident negative verdict derives from Bruce Collier’s 1970  8  Harvard thesis, recently publicized in the Economist magazine on Ada Lovelace day:  Ada was as mad as a hatter, and contributed little more to the “Notes” than trouble .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I will  retain an open mind on whether Ada was crazy because of her substance abuse .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' or despite  James Essinger, A Female Genius: How Ada Lovelace Started the Computer Age (London: Gibson  5  Square, 2013), which appeared in the United States as Ada’s Algorithm (Brooklyn: Melville House, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Essinger had earlier written Jacquard’s Web: How a Hand-loom Led to the Birth of the Information  Age(Oxford: Oxford University Press, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Contrast Martin Davis and Virginia Davis, “Mistaken  Ancestry: The Jacquard and the Computer,” Textile 3 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 1 (2005): 76-87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Earlier positive treatments  include Betty Alexandra Toole, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada, The Enchantress of Numbers: A Selection from the Letters of  Lord Byron’s Daughter (Mill Valley CA: Strawberry Press, 1992) and Doris Langley Moore, Ada,  Countess of Lovelace: Byron’s Legitimate Daughter (New York: Harper & Row, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Doron D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Swade, “Redeeming Charles Babbage’s Mechanical Computer,” Scientific American  6  (February 1993): 86-91;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Allan Bromley, “Charles Babbage’s Analytical Engine, 1838,” Annals of the  History of Computing 4 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 3 (1982): 196-217, quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 197;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Allan Bromley, “Difference and Analytical  Engines,” in William Aspray, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Computing before Computers (Ames: Iowa State University Press,  1990), 59-98, on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Dorothy Stein, Ada: A Life and a Legacy (Cambridge: MIT Press, 1985), quote xii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Walter  7  Isaacson’s The Innovators: How a Group of Hackers, Geniuses, and Geeks Created the Digital Revolution  (New York: Simon & Schuster, 2014), on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 493 note 1, praises Stein’s book as “the most scholarly and  balanced.” A later negative view is Jay Belanger and Dorothy Stein, “Shadowy Vision: Spanners in the  Mechanization of Mathematics,” Historia Mathematica 32 (2005): 76-93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Stein strongly criticized  Dorothy L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Moore, Ada, Countess of Lovelace: Bryon’s Legitimate Daughter (London: Murray, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Martin Campbell-Kelly, William Aspray, Nathan Ensmenger, and Jeffrey R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Yost, Computer: A  8  History of the Information Machine (Boulder CO: Westview Press, 2014), quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Note that Bromley  qualified his claim: that all but one program (Bernoulli numbers) was Babbage’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Compare Campbell  Kelly and Aspray’s first edition of Computer (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  3  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I hope nobody feels compelled to write another book on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' But, then, I guess  someone has to be the most overrated figure in the history of computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  9  In this essay, I assemble the pertinent sources, including correspondence about the Notes  to the Menabrea Sketch (printed in this volume: LINK), and suggest several modest corrections  to the unduly negative scholarly view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' First, in contrast to much of the existing literature, the  10  Lovelace-Babbage question is not a zero-sum game, where some portion of credit added to  Lovelace somehow detracts from Babbage, or vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' There is ample evidence that Babbage  and Lovelace each had important contributions to the Sketch and the Notes, and attention to their  intellectual collaboration is revealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Second, claims about her lack of mathematical  background seem doubtful after consulting Lovelace’s detailed correspondence with Babbage  and Augustus De Morgan, two highly accomplished figures in 19th century mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The  treatment of the Bernoulli numbers in note “G” spotlights the intellectually intimate  collaboration between Babbage and Lovelace and its mathematical sophistication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Finally, while  there may be significant definitional problems in calling Lovelace the world’s “first computer  programmer,” the evidence is reasonably clear that Lovelace created an step-by-step elemental  sequence of instructions—that is, an algorithm—for computing the series of Bernoulli numbers  that was intended for Babbage’s Analytical Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The underlying mathematics might well  have been Babbage’s, for he was a distinguished mathematical and scientific figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Lovelace  transformed an equation for the Bernoulli numbers into a precise series of elemental additions,  multiplications, and substitutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  The algorithm specified a sequence of calculations, requiring a real-life computer capable  of running a program with a looping structure and conditional testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Contemporary computer  experts have noted in the large table (representing the Bernoulli number algorithm) in the  Sketch’s Note “G” there was one misplaced minus sign which, when corrected, led to the result  that Ada’s algorithm correctly computed the series of Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In “The Babbage  Machine and the Origins of Programming,” the authors reproduce Lovelace’s table for the  Bernoulli numbers and translate the algorithm into a 65-line FORTRAN program that computes  “Ada Lovelace Day: Right Idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Wrong Woman?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Economist (24 March 2010) at  9  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='economist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='com/node/21005551/print (accessed 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I use “derived from” intentionally, since in  the printed copy of Collier’s dissertation I examined (CBI QA75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='C634x 1970a) I did not find this  quotation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' see also robroy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='dyndns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='info/collier (Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Doron Swade quotes Collier in The Cogwheel  Brain: Charles Babbage and the Quest to Build the First Computer (London: Little, Brown, 2000), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Correspondence in the Toole, Stein, Moore, and Swade volumes as well as the document-centered  10  accounts in Velma R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Huskey and Harry D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Huskey, “Lady Lovelace and Charles Babbage,” Annals of the  History of Computing 2 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (1980): 299-329;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and John Fuegi and Jo Francis, “Lovelace & Babbage and  the Creation of the 1843 ‘Notes’,” IEEE Annals of the History of Computing 25 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2003): 16-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  4  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The program has eight “if .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' [then] go-to” statements and a simple structure, straight  from Lovelace’s table-algorithm, that builds up algebraic statements one mathematical operation  at a time: for example, computing the expression (2n - 1) / (2n + 1) requires 4 program steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  11  In the original 1843 publication of Note “G,” there are clearly two nested loops, embedded in a  larger looping structure (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' There is direct documentary evidence that Ada Lovelace  created this table (writing it out in pencil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' She and Babbage corresponded intensively in the  weeks and days prior to its publication in Taylor’s Scientific Memoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' This series, published in  London between 1837 and 1852 by Richard Taylor in cooperation with the British Association  for the Advancement of Science (BAAS), printed English translations of prominent European  scientific papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “Here was to be found .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' such European leaders” as Bessel, Bunsen, Gauss,  Ohm and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The Analytical Engine was Babbage’s creation while the Sketch and  12  Notes are best understood as the product of an intense intellectual collaboration between  Babbage and Lovelace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Babbage and Lovelace  It is with good reason that computing history scholars have praised the research of Allan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Bromley (1947-2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Bromley, a computer scientist at the University of Sydney, made careful  studies of the Babbage letters and notebooks that led to the Science Museum’s reconstruction of  the Babbage Difference Engine No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In 2000 Tim Bergin, editor-in-chief of IEEE Annals of the  History of Computing, introduced a special issue of the journal dedicated to Bromley’s  scholarship by noting his “fundamental contributions,” former Annals editor-in-chief Michael  Williams called his work “groundbreaking,” and none other than computer pioneer Maurice  Wilkes stated “it is to Bromley that we owe nearly all our present knowledge of Babbage’s work  Campbell Kelly, Works of Charles Babbage, volume 3: 159 note a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Garry J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Tee, in reviewing a  11  1979 Russian publication by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Petrenko and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Petrenko, wrote this: “The most advanced  illustration given in Lovelace’s paper is an elaborate program for computing the sequence of Bernoulli  numbers, which was written by Babbage but corrected by her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The present authors have transcribed that  program into FORTRAN, detecting thereby a few misprints but only one significant error, with one  variable having the wrong sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Their transcribed version is easier for a modern reader to understand than  the original program of 1843, and it does compute correctly the sequence of Bernoulli numbers.” See  <www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='ams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='org/mathscinet-getitem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='mr=83b:01045> (accessed 2015) reviewing Petrenko and Petrenko’s  “The Babbage Machine and the Origins of Programming,” [in Russian] Istoriko-matematicheskie  issledovaniíà 24 (1979): 340-360, 389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I examined the FORTRAN program in the Russian original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Brock and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Meadows, The Lamp of Learning: Taylor & Francis and Two Centuries Of  12  Publishing (London: Taylor & Francis, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2nd edition), esp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' chapter 4 “Taylor and the Commercial  Science Journal,” quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  5  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  on computing machinery at the detailed mechanical level.”  Wilkes noted Bromley’s determined  insistence that “Babbage’s work on the Analytical Engine was completely original.”  13  By contrast, Bromley’s view on Lovelace was strongly critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “All but one of the  programs cited in her notes had been prepared by Babbage from three to seven years earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The  exception [on Bernoulli numbers] was prepared by Babbage for her, although she did detect a  ‘bug’ in it,” he wrote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “Not only is there no evidence that Ada Lovelace ever prepared a program  for the Analytical Engine but her correspondence with Babbage shows that she did not have the  knowledge to do so.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='   It is Bromley’s viewpoint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' slightly modified,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' that finds its way into the  14  recent edition of the highly regarded Computer: A History of the Information Machine (2014)  with its undue assertion that “all of the programs in the Sketch were Babbage’s work.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Reviewing the evidence about Lovelace’s mathematical knowledge and the writing of the notes  to her translation of Menabrea’s sketch might modify these overly negative assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In later  correspondence with Wilkes, Bromley did allow, in comparison with fellow Babbage scholar  Doron Swade, “I have been known to express my views more intemperately.”  15  One must acknowledge that Ada Lovelace in her energetic, imaginative, self-absorbed,  and at times grandiloquent correspondence gives her later-day critics much to aim at.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Her self-  regarding statements about her own mathematical abilities can be off-putting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' And her pointed  remarks sometimes aimed at Charles Babbage might rub the wrong way anyone who thinks  distinguished scientists should be treated with dignity and respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Science at the time was  16  expanding from its strictly male enclaves at Oxford and Cambridge, with the creation of the  BAAS (f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 1831) and other learned societies, but in the new scientific institutions Ada Lovelace  and Mary Somerville were treated at best as “second class members.”   One need not make  17  excuses for her imaginative flights of fancy, but it does need to be borne in mind that Lovelace  was the daughter of an aristocratic Baron (the poet Lord Byron) and married to a highly ranked  Earl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Her mother had schemes drawing on the family’s network that extended into the royal  Tim Bergin, “About this Issue” (quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Michael R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Williams, “Allan Bromley,” (quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  13  Maurice V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Wilkes, “Introduction to ‘Babbage’s Analytical Plans 28 and 28a—The Programmer’s  Interface’,” (quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4) in IEEE Annals of the History of Computing 22 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Allan Bromley, “Difference and Analytical Engines,” in William Aspray, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Computing before  14  Computers (Ames: Iowa State University Press, 1990), 59-98, quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  See “Appreciating Charles Babbage: Emails between Allan Bromley and Maurice Wilkes,” IEEE  15  Annals of the History of Computing 26 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2004): 62-70, on 62 (intemperately).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  In the midst of writing the translation’s notes, when letters and draft manuscripts were passed daily  16  between them, Ada writes Babbage, somewhat curtly: “Now pray attend strictly to my requests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' or you  will cause me very serious annoyance,” quoted in Huskey and Huskey, “Lady Lovelace and Charles  Babbage,” p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' There is a curious mix of defiance and deference in Ada’s correspondence with  Babbage, Somerville, De Morgan and other prominent figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Ruth Watts, Gender, Power and the Unitarians in England, 1760-1860 (New York: Longman,  17  1998), quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  6  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  family itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage, although he was handsomely wealthy after his banker-father died in 1827  and a fortune of £100,000 passed down to him, was all the same a commoner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Ada sought for  years to land Babbage a knighthood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  18  The first line of evidence suggesting an intellectual partnership between Charles Babbage  and Ada Lovelace comes from witnesses to their first meeting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' She first met Babbage in 1833, a  year after being formally presented to the court, through an introduction by Mary Somerville, the  mathematician, scientific popularizer, and notable English translator of Pierre-Simon Laplace’s  Mécanique Céleste;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' the two women kept up a scientific correspondence for many years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Two  19  years after she met Babbage, Ada married William King, already a Baron, who was within three  years created the first Earl of Lovelace, and so it is a convenience to refer to her as Ada Lovelace  rather than the more precise but ponderous Augusta Ada King, Right Honourable the Countess of  Lovelace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  There is eyewitness evidence that, when she saw it, Ada grasped the principles and  significance of Babbage’s prototype Difference Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage had started work on it in 1822,  and it was in an advanced state of development in 1833;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' fatefully, the following year Babbage set  the Difference Engine aside and focused instead on the conceptually elaborate Analytical Engine,  which remained a “brilliant obsession” nearly to the end of his life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In June 1833 Ada’s mother,  20  Lady Byron, described a visit along with her daughter and a friend to inspect Babbage’s machine  in some detail and with great wonder while admitting herself only “faint glimpses of the  principles by which it worked.”  The Difference Engine was at the time able to compute  polynomial expressions, extract roots to quadratic equations, and count to 10,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Ada saw its  significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “I well remember accompanying her to see Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage’s wonderful analytical  engine,” wrote Sophia De Morgan, the wife of mathematician Augustus De Morgan and, like  Mary Somerville, a long-term correspondent with Ada herself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “While other visitors gazed at the  working of this beautiful instrument with the sort of expression .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' that some savages are said to  have shown on first seeing a looking-glass or hearing a gun .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Miss Byron, young as she was,  understood its working and saw the great beauty of the invention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' She had read the Differential  Calculus to some extent, and after her marriage she pursued the study and translated a small  Doron Swade, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage, Charles (1791-1871), Oxford Dictionary of National Biography  18  (Oxford University Press, 2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' online edition, May 2009 at www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='oxforddnb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='com/view/article/962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Babbage died in 1871 with a fortune “under” £40,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Elizabeth Chambers Patterson, Mary Somerville and the Cultivation of Science, 1815-1840  19  (Boston: Nijhoff, 1983);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and Kathryn A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Neeley, Mary Somerville: Science, Illumination, and the Female  Mind (Cambridge: Cambridge University Press, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  See “Appreciating Charles Babbage: Emails between Allan Bromley and Maurice Wilkes,” IEEE  20  Annals of the History of Computing 26 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2004): 62-70, quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 63 (obsession).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  7  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  work of the Italian mathematician Menabrea, in which the mathematical principles of its  construction [were] explained.”  21  Babbage invited Ada with a friend or chaperone to attend his series of “Saturday  evenings” where his London home became a fashionable salon, filled with celebrities from the  political, cultural, and scientific world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Charles Dickens and Charles Darwin, among hundreds  of others, were happy to mix with the well-cultured crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “One of three qualifications were  necessary for those who sought to be invited—intellect, beauty, or rank—without one of these,  you might be rich as Croesus—and yet be told, you cannot enter here,” recalled one society  figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “His calculating machine was an endless subject of monologue.”   Some were deeply  22  impressed by its ability to generate a list of prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Since it could solve any second-  degree polynomial equation, Babbage set it to compute a series of 40 prime numbers by  evaluating the expression x2 + x + 41 for the first 40 integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' A few months after the invitation,  she wrote to Mary Somerville asking her to convey to Babbage’s son “how exceedingly obliged I  am .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' for his unexpected kindness in sending me the plates & account of the Machine, which is  exactly what I was in want of;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' & is a very great help to me.”   Ada was at the time 19 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  23  A significant line of evidence bearing on Bromley’s claim that “she did not have the  knowledge” to prepare a program for the Analytical Engine is the substantial depth of Ada’s  mathematical studies beginning in the 1830s, which predated her contact with the Menabrea  manuscript and the translation project of the early 1840s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “Ada was much attached to me, and  often came to stay with me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' It was by my advice that she studied mathematics,” recalled Mary  Somerville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “She always wrote to me for an explanation when she met with any difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Among my papers I lately found many of her notes, asking mathematical questions.”  24  During these years Ada had three tutors in mathematics, in addition to intellectual  interchange with Somerville, Babbage, and her scientifically minded husband, who was made a  Fellow of the Royal Society in 1841;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' two of them were distinguished figures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Her first  mathematics tutor was the elderly William Frend, the notable social reformer who had authored  Lady Byron quoted in Moore, Ada, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 44;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Sophia De Morgan, Memoir of Augustus De Morgan  21  (London: Longmans, Green, 1882), quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 89 (accompanying her).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  “John Kenyon and His Friends,” Temple Bar: A London Magazine for Town and Country Readers  22  88 (1890): quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 490 (three qualifications);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' AAL to Mary Somerville 19 March 1834 in Toole, Ada, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  57 (Babbage’s invitation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  AAL to Mary Somerville 8 November 1834 quoted Huskey and Huskey, “Lady Lovelace and  23  Charles Babbage,” quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 303 (account of the Machine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Martha Somerville, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Personal Recollections from early life to old age of Mary Somerville  24  (Boston: Roberts Brothers, 1874), quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' [See also, two subsequent publications: Hollings et al  “The Lovelace-De Morgan Mathematical Correspondence: A Critical Re-Appraisal,” Historia  Mathematica 2017 at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='001 and “The early mathematical education  of Ada Lovelace” BSHM Bulletin (2017) at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='1080/17498430.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='1325297  Page �  of �  8  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  Principles of Algebra (1796) along with many other tracts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' among his students had been Ada’s  own mother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' It seems likely that Frend introduced Ada to Mary Somerville, connecting her  25  with Babbage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Ada and Babbage corresponded on mathematical topics following their 1833  meeting, again years prior to the translation project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Perhaps her most important mathematical  tutor was her friend Sophia’s husband and William Frend’s son-in-law, Augustus De Morgan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' He  gave Ada Lovelace, as Sophia later wrote, “much help in her mathematical studies, which were  carried farther than her mother’s had been.”   Even the severely critical study by Dorothy Stein  26  acknowledges the De Morgan-Lovelace letters as “a correspondence course in calculus.”  27  Augustus De Morgan was like Babbage a graduate of Cambridge, a disbeliever in the  traditional Church of England, and a distinguished mathematician.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' He was named founding  professor of mathematics at London University (now University College London), shortly after  its founding in 1826, at the age of 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' It was a secular university, unlike Cambridge and Oxford,  and admitted women as regular students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' De Morgan published books on trigonometry,  arithmetic, algebra, probability, and logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In the early 1840s while exchanging regular letters  with De Morgan on the topic, Ada reported that she was “drowning in Calculus.”  During these  years De Morgan was working on a book project for the London-based Society for the Diffusion  of Useful Knowledge (1826-48), published in 1842 as an 800-page textbook on Differential and  Integral Calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Her letters to De Morgan are filled with specific questions about differential  28  calculus, limits, Leibnitz’s notation, three-dimensional geometry, notation of functions, and  standards of reasoning and proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' On 21 November 1841, in the context of her mathematical  exercises, she asked him about the “numbers of Bernoulli.”   Ada’s awareness of the Bernoulli  29  numbers thus predated her work on the Menabrea translation and the writing of Note G  describing an algorithm for their computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Whereas Ada’s first tutor, the elder William  Judith S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Lewis,  “Princess of Parallelograms and Her Daughter: Math and Gender in the  25  Nineteenth Century English Aristocracy,” Women’s Studies International Forum 18 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (1995):  387-394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Sophia De Morgan, Memoir of Augustus De Morgan (London: Longmans, Green, 1882), quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  26  89 (much help)  Stein, Ada, quote pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' xii (correspondence course)  27  Toole, Ada, quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 169 (drowning in Calculus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' See Augustus De Morgan, The Differential and  28  Integral Calculus: Containing Differentiation, Integration, Development, Series, Differential Equations,  Differences, Summation, Equations of Differences, Calculus of Variations, Definite Integrals (London:  Baldwin & Cradock, 1842).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' For further testimony on her mathematical work with De Morgan, see  Huskey and Huskey, “Lady Lovelace and Charles Babbage,” quotes p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 309: AAL to her mother “I go on  most delightfully with Mr De Morgan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' What can I ever do to repay him?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' AAL to Babbage “I am now  studying the Finite Differences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' And in this I have more particular interest, because I know it bears  directly on some of your business”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and AAL to her mother “The Mathematics & Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' De Morgan going  on very well indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' You would be much pleased to see the heap of papers of my writing.”  Toole, Ada, quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 173 (numbers of Bernoulli).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  29  Page �  of �  9  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  Frend, doubted the existence of negative numbers, De Morgan was a modern mathematician of  the first rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In his book Trigonometry and Double Algebra (1849) he presented a geometrical  interpretation of complex numbers, those with real and imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  In January 1844 De Morgan wrote a lengthy and detailed confidential letter to Ada’s  mother, Lady Byron, making an acute assessment of Ada’s unusual facility with mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “I  never expressed to Lady Lovelace my opinion of her as a student in these matters,” De Morgan  began.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “The power of thinking on these matters which Lady L[ovelace] has always shown from  the beginning of my correspondence with her, has been something so utterly out of the common  way for any beginner, man or woman, that this power must be duly considered by her friends .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  whether they should urge or check her obvious determination .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' to get beyond the present  bounds of knowledge.”  De Morgan rated Ada favorably with Maria Agnesi, the Italian author of  a pioneering calculus textbook (1748), and far more highly than Mary Somerville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  30  During the same years as his correspondence with Ada, De Morgan also facilitated the  mathematical work of George Boole, later author of the landmark Investigation of the Laws of  Thought (1854) and today widely hailed as the father of Boolean algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In his Treatise on the  Calculus of Finite Differences, Boole suggests a useful historical insight relating the character of  mid-nineteenth century mathematics to the capabilities of Babbage’s analytical engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  31  Originally the Bernoulli numbers were discovered in 1712-13 (by the Swiss mathematician Jacob  Bernoulli and by the Japanese mathematician Seki Kōwa), to aid in such calculations as the  summation of powers (1n + 2n + 3n + 4n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' With their assistance Bernoulli computed the sum  of the first 1,000 integers, each raised to the tenth power, a immense number  91,409,924,241,424,243,424,241,924,242,500, in (as he claimed) “less than half of a quarter of  an hour.”  32  Subsequent work by the Swiss and Scottish mathematicians Euler and Maclaurin  connected the mathematics of integral calculus to the summation of polynomial expressions  (which are essentially sums of powers each multiplied by some coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Transcendental  mathematical functions as well as integral calculus could be expressed by polynomial  expressions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' the Bernoulli numbers appeared as coefficients in some of these polynomial series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Velma R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Huskey and Harry D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Huskey, “Lady Lovelace and Charles Babbage,” Annals of the  30  History of Computing 2 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (1980): 299-329, De Morgan quoted p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 326;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and Massimo Mazzotti, “Maria  Gaetana Agnesi: Mathematics and the Making of the Catholic Enlightenment,” Isis 92 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2001):  657-683.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Originally published in 1860, George Boole’s Treatise on the Calculus of Finite Differences (New  31  York: Dover, 1960) deals with Bernoulli numbers in chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Jacques [Jacob] Bernoulli, “On the ‘Bernoulli numbers’,” in David Eugene Smith, A Source Book  32  in Mathematics (New York: McGraw-Hill, 1929), 85-90, quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In modern notation, the Bernoulli  numbers are defined as the coefficients (Bn) for the series expression for an exponential generating  function, as follows (see <mathworld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='wolfram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='com/BernoulliNumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='html>)  Page �  of �  10  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  Thus, by summing up the correct polynomial (using repeated multiplications, squaring, cubing,  et seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=') Babbage’s analytical engine could calculate transcendental mathematical functions (such  as sine and cosine) as well as evaluate integral-calculus expressions, so long as they could be  expanded into polynomial series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The Bernoulli numbers could be used to simplify the notation  and computation of certain polynomial series, and thus were a powerful aid to computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Bernoulli achieved his remarkable computation by transforming the extensive summation of  powers (110 + 210 + 310 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 100010) into a straightforward seven-term polynomial equation using  the Bernoulli numbers (up to B10 in the series).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  33  Since Ada’s calculus studies with De Morgan likely drew on the calculus textbook he was  writing during these years, it is relevant to review its treatment of the Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' De  Morgan used the Bernoulli numbers in treating polynomial series expansions for ex, tangent, and  cotangent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' in the calculus of operations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and in convergent series for definite integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' One  34  exercise (page 307 §163) presents a general expression for directly computing a specific  Bernoulli number, in terms of its predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  �  So for n = 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' the expression for Bn+1 or B8 is as follows (where the fractional terms are  computations on earlier instances in the series):  �  So,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' with Ada Lovelace’s interest in Babbage’s machines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' her mathematical studies with  Babbage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Somerville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and De Morgan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and her relentless curiosity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' she was surprisingly well  Sum of the first 1000 integers raised to the tenth power = 1/11x11 + B1x10 + 5B2x9 + 30B4x7 +  33  42B6x5 + 15B8x3 + B10x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' where x = 1000 and Bn are the Bernoulli numbers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', B1 = 1/2,  B2 =  1/6, B4 =  1/30, B6 = 1/42, B8 = -1/30, B10 =  5/66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  See De Morgan, The Differential and Integral Calculus, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 247, 248, 308, 553, 581.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  34  Page �  of �  11  20  n+1  1  n+1  10*  AO*  0*  △*0"  Ba+1=  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='0*=  +  2*+1  2+△  2*+1  2  4  8  2*+1For thc value of B, (n=7) we have  8  1  126  1806  8400  16600  15120  5040)  1  255  4  8  16  32  64  128  256  30published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  exposed to the advanced mathematics of the period and had ample background and motivation to  delve into the computations that appeared in the Notes to the Sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  35  Steps to the Sketch  Babbage conceived a general computing machine around 1834, setting aside his still-  uncompleted work on the Difference Engine to take up the challenges of what became known as  the Analytical Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Whereas it was necessary to mechanically set up the Difference Engine to  do each calculation, such as the prime-number generating polynomial x2 + x + 41, Babbage’s  inspiration for the Analytical Engine was a computing machine able to re-configure itself—if not  precisely “programmable” in the modern sense of the term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage’s mature design while  36  remaining a mechanical-age technology had several features in common with modern computers:  separating the computation of numbers from their storage (he used the terms ‘mill’ and ‘store’ in  an analogy with the industrial factory);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' adopting a mechanism to do conditional testing on  intermediate results and hence permitting the branching of calculations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and using punched cards  loosely inspired by the Jacquard loom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  37  Babbage took one of his sets of evolving plans for the Analytical Engine to present at a  conference in Turin in 1840.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In the audience was a future prime minister of Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' At the time  Luigi Menabrea was a professor of mechanics and construction at the university of Turin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' he  subsequently served as a military engineer, naval minister, and eventually prime minister of Italy  See also Imogen Forbes-Macphail’s chapter in this volume [Ada’s Legacy] exploring the “poetical”  35  nature of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Babbage spent years working out a notation for expressing how its computations might be  36  expressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' See Allan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Bromley, “Charles Babbage’s Analytical Engine, 1838,” IEEE Annals of the  History of Computing 20 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (1998): 29-45;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and Allan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='Bromley, “Babbage’s Analytical Engine Plans  28 and 28a: The Programmer’s Interface,” IEEE Annals of the History of Computing 22 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2000):  5-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  The easy one-to-one correspondence between Jacquard looms and Babbage’s cards, posited by  37  such authors as James Essinger (author of popular works on Jacquard and Lovelace), is critically  scrutinized by Martin Davis and Virginia Davis, “Mistaken Ancestry: The Jacquard and the Computer,”  Textile 3 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 1 (2005): 76-87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' After years of close study, Allan Bromley felt that the Analytical Engine  fell somewhat short of a modern computer, as he wrote (privately) to Wilkes: “Perhaps my  disappointment comes from being forced to accept that Babbage did NOT devise a COMPUTER but only  a very sophisticated CALCULATOR after the style of the Harvard Mark I or the NCR accounting  machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' He did not cross the watershed that marked off the [stored-program computers] EDVAC/  EDSAC, although many of his technical innovations in implementation were astounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' It is a little  difficult to admit this conclusion after expending so many years studying Babbage’s designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Perhaps it is  why I ‘took a break from Babbage’ in the late 1980s and never really came back.”  See “Appreciating  Charles Babbage: Emails between Allan Bromley and Maurice Wilkes,” IEEE Annals of the History of  Computing 26 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2004): 62-70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  12  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  (1867-69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In October 1842, Menabrea published a short description of Babbage’s Analytical  38  Engine in a Swiss journal (written in French).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' By this time, Babbage was well on the way to  ruining whatever chance might have remained for support from the British government,  especially after a disastrous meeting in November of that year with prime minister Robert Peel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  “What shall we do to get rid of Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Babbage and his calculating Machine?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Surely if completed it  would be worthless as far as science is concerned,” he wrote.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Peel, signaling his displeasure,  soon dispatched Babbage’s prime-number-calculating Difference Engine to the King’s College  Museum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  39  It was not initially Babbage who encouraged Ada Lovelace to examine the Menabrea  manuscript and translate it into English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Rather the prompting came from the notable scientist  and sometime telegraph inventor Charles Wheatstone, who knew both Babbage and Lovelace  and recruited contributions for Taylor’s Scientific Memoirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Wheatstone had gained fame in  1837 with the patenting of a multiple-wire electric telegraph system using the positioning of  magnetic needles to encode individual letters, rather than the Morse code system of dots and  dashes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' He, too, was intrigued with using electromechanical apparatus for computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  “Yesterday saw Wheatstone’s model for telegraph and his drawings for Multiplication Engine,”  wrote Babbage after a visit in 1839.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In 1843, the publication year of the Menabrea translation,  40  Wheatstone improved on and publicized the famous “Wheatstone bridge” used to precisely  measure electrical resistances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Over the winter of 1842-43 Lovelace worked on translating the Menabrea manuscript,  around 8,000 words, and first showed her results to Babbage in the spring of 1843.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In his  memoir, Passages from the Life of a Philosopher, Babbage recalled encouraging Ada to add  descriptive notes to her translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  We discussed together the various illustrations that might be introduced: I suggested several,  but the selection was entirely her own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' So also was the algebraic working out of the different  problems, except, indeed, that relating to the numbers of Bernoulli, which I had offered to do  to save Lady Lovelace the trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' This she sent back to me for an amendment, having  detected a grave mistake which I had made in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  The notes of the Countess of Lovelace extend to about three times the length of the original  memoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Their author has entered fully into almost all the very difficult and abstract  questions connected with the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  41  See “Luigi Federico Menabrea” at <www-history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='mcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='st-andrews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='uk/Biographies/  38  Menabrea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='html>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Fuegi and Francis, “Lovelace & Babbage,” quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 16 (Peel on Babbage).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  39  Fuegi and Francis, “Lovelace & Babbage,” quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 18 (Babbage on Wheatstone);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Stein, Ada, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  40  Charles Babbage, Passages from the Life of a Philosopher (London: Longman, Green, 1864),  41  quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  13  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  By this time in his life, while he was certainly writing for posterity and obviously keen on  memorializing his computing engines, Babbage had no particular reason to exaggerate Ada’s  achievements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' She had died years earlier, at age 36, and left Babbage a modest legacy of £600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I  think we can take him at his word when he admits a “grave mistake” in deriving the Bernoulli  numbers (which Bromley labels anachronistically as a “bug”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' This passage, along with the  Babbage-Lovelace letters, clearly describes a collaboration where Babbage and Lovelace are  working together on the Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' It’s also clear from the context that “their author” here refers to  Lovelace (rather than Menabrea) and that it is certain praise that she “has entered fully into .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  difficult and abstract questions.”  At the very least, Babbage’s statement challenges the assertion  that the notes “give an excellent account of his [that is, Babbage’s alone] understanding of the  mechanization of computational processes and the mathematical powers of the machine.”  Letters from the exact weeks in the summer of 1843 when Babbage and Lovelace were  working on the Notes provide additional detail on their collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' This documentary  evidence—from the British Library, the Science Museum, and the Oxford Bodleian Library—has  been analyzed by Velma Huskey and Harry Huskey as well as John Fuegi and Jo Francis and  published in Annals of the History of Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The picture is clearly of a collaboration  42  where both Babbage and Lovelace are making important contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Lovelace wrote the  Notes and most of her letters to Babbage at her Ockham Park estate an hour’s south of London;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Babbage received her letters and responded from his London house on Dorset Street;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and from  time to time they met in person at Lovelace’s London house on aristocratic St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' James Square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Again, I emphasize that to point out Lovelace’s knowledge, achievements, and contributions is  not to belittle Babbage’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  It is not easy to support the conjecture that the notes are Babbage’s alone or that he  directed Lovelace to write them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Bromley is not the only critic who has aimed to reduce  Lovelace to a low-level clerk-assistant to Babbage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In Ada: A Life and a Legacy, Dorothy Stein,  for instance, seized on a misprint that Lovelace, Babbage, and Menabrea before them all failed to  spot—supposedly highlighting “the significance of her curiously ignored translation of a  printer’s error”—and contrives an argument that Lovelace had only a tenuous grasp of  mathematics and a “dubious” understanding of the Analytical Engine’s mechanical and logical  operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Stein contends, on this line of conjecture, that Lovelace was “completely dependent  Velma R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Huskey and Harry D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Huskey, “Lady Lovelace and Charles Babbage,” Annals of the  42  History of Computing 2 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (1980): 299-329;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and John Fuegi and Jo Francis, “Lovelace & Babbage and  the Creation of the 1843 ‘Notes’,” IEEE Annals of the History of Computing 25 no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 4 (2003): 16-26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  14  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  on [Babbage] for information and claims about the Analytical Engine.”   Stein’s use of evidence  43  is rather thin and highly selective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In their analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Fuegi and Francis point to contemporaneous  letters from Babbage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Lovelace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Wheatstone,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' the editors of Taylor’s Scientific Memoirs and other  scientific colleagues,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' concluding “all contemporaries of Lovelace and Babbage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' having first-hand  knowledge of how the ‘Notes’ came into being,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' acknowledged Lovelace at the time as the  primary author.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  44  The evidence from Babbage’s letters points to a collaboration between them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' while their  informal manner of addressing each other indicates a degree of collegiality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Ada writes to “My  Dear Babbage” while he responds to “My Dear Lady Lovelace.”  On 30 June 1843 he writes:  I am delighted with Note D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  It is in your usual clear style and required only one trifling  alteration which I will make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' This arises from our not having yet had time to examine the  outline of the mechanical part .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I enclose a copy of the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I am still working at  some most entangled notations of Division but see my way through them at the expense of  heavy labour .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Your latest information was the most agreeable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  45  Let us turn to Note G on the computation of the Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Recall that they were  a common topic in 19th century mathematics, and that Lovelace herself had previously inquired  about them to De Morgan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The topic first appears in the correspondence when Ada wrote to  Babbage, in a letter dated simply “Monday,” as follows:  I am working very hard for you .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I think you will be pleased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I have made what appears  to me some very important extensions & improvements .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  I want to put something about Bernoulli’s Numbers, in one of my Notes, as an example of  how an implicit function may be worked out by the engine, without having been worked our  by human head & hands first [as the Difference Engine required].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Give me the necessary  data & formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  46  Even if Babbage provided Ada with the mathematical expressions for the Bernoulli  numbers, and assisted with the derivation of a general formula, the transformation of the general  formula into a step-by-step algorithm remains Ada’s achievement, as the letters clearly indicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  The mathematics is somewhat involved, beginning with the basic equation where the Bernoulli  Stein, Ada, quote pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' xi (printer’s error) 90-91 (tenuousness, dubious, completely dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The  43  printer’s error was made in Menabrea’s original publication where the French word cas (as in the case  where N goes to infinity in a math expression) was mistakenly printed as cos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' (as in the abbreviation for  cosine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Fuegi and Francis, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 26 note 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  44  Babbage to AAL 30 June 1843, quoted in Huskey and Huskey, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 312-313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  45  AAL to Babbage, “Monday,” quoted in Huskey and Huskey, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  46  Page �  of �  15  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  numbers (note the odd-numbered notation B1, B3, B5) appear as coefficients for the exponential  function:  �  and then involving expansion, division, derivation, rounds of intricate multiplication, and finally  writing the equation in general form:  �  This equation (as note G explains, introducing the odd-numbered notation) “enables us to  find .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' any nth Number of Bernoulli B2n-1, in terms of all the preceding ones, if we but know the  values of B1, B3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' B2n-3.”  If n = 1, then the numerators for each of the higher terms (B3 et seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=')  contain a zero (2n-2) and so they drop out, permitting the direct calculation of B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Similarly, for  n = 2 one substitutes the already computed value for B1 while the higher terms (B5 et seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=') again  drop out and permit the direct computation of B3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' With n = 3, the process is repeated to compute  B5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The Notes explicitly make the general argument: “And so on, to any extent.”  Ada was determined to get the advanced mathematics correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The two often exchanged  letters daily, and sometimes even more frequently, as when a servant came into London and then  waited for Babbage’s reply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “I am doggedly attacking .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' all the ways of deducing the Bernoulli  Numbers,” she wrote at one moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' On the mathematics Ada wrote to Babbage (in a letter  dated simply Tuesday morning) “the few lines I enclosed you last night about the connexion of  (8) [the Bernoulli number equation in general form] with the famous Integral, I by no means  intend you to insert, unless you fully approve the doing so.”  47  The table and diagram that contained and expressed the algorithm were of special  concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Ada wrote Babbage (Saturday 6 o’clock), in connection with the note on the Bernoulli  numbers, “Think of my horror then at just discovering that the Table & Diagram, (over which I  have been spending infinite patience and pains) are seriously wrong, in one or two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I have  done them however in a beautiful manner, much improved upon our first edition of a Table and  Toole, Ada, quote 204 (doggedly attacking);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' AAL to Babbage, “Tuesday morng,” quoted in  47  Huskey and Huskey, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  16  20  2n-1  + B1  2n  + B3  2n-(2n  21  2  2n+  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='4  +B5  2n(2n-1)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='6published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  Diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' But unluckily I have made some errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I send you this final note [G] excepting the  Table & Diagrams.”  As a postscript, Ada adds: “Let me know how you like my finishing up of  [G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Mind you scrutinise all the n’s very carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I mean those of Sheets 4 and 5.”  Then, after  a full day’s work on Sunday, “you will admire the Table & Diagram extremely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' They have been  made out with extreme care & and all the indices most minutely & scrupulously attended to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Lord L[ovelace] is at this moment inking it all over for me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I had to do it in pencil.”  48  Ada tackled the question of the looping structure with some care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In a letter dated simply  Tuesday, she writes to Babbage: “I hope you will approve of what I send.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I have taken much  pains with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I have explained that there would be, in this instance & in many others, a recurring  group or cycle of Variable as well as of Operation cards .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .”  She specifically identifies the  main or outer loop as the “repetitions of (13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 23),” and explains “as the variations follow a  regular rule, they would be easily provided for.”  49  In his correspondence with Ada, Babbage directly admires her work on the Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' His  praise of Note D, “in your usual clear style,” was cited earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In response to the Saturday letter  quoted in the above paragraph, and clearly written before he received the Sunday letter, Babbage  wrote: “I like much the improved form of the Bernoulli Note but can judge of it better when I  have the diagram and Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' I am very reluctant to return the admirable and philosophic view  of the Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Engine contained in Note A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Pray do not alter it and do let me have it returned on  Monday.”   (Babbage was assembling the entire set of Notes in preparation for handing them to  50  Charles Wheatstone for publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=')  As the final versions were going to the printer, Lovelace and Babbage were still revising  the Notes’ treatments of the variable cards and the operation cards, which specified the elemental  arithmetical operators (+ - x ÷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Ada made clear that handling the operations cards was at the  center of the Analytical Engine’s capability for looping, even while the mechanical apparatus for  effecting conditional tests was at the time not entirely clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Note D presents a straightforward  51  computation—similar to one that Menabrea had presented in the original essay, also in tabular  form—with six variable cards (for constants), nine working variables (for intermediate results),  and two variables for results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' There were 11 sequential steps (no looping structures), and the  entirety fits onto the page (figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Huskey and Huskey, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 313-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' In the earlier [Saturday] letter, Lovelace mistakenly labeled the  48  final note “H” rather than “G,” which was corrected in the later Sunday letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Huskey and Huskey, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 316 (recurring group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  49  Huskey and Huskey, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 313 (Babbage on improved form of the Bernoulli note).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  50  For a mechanical visualization of the Analytical Engine’s “Logic and Loops,” see Sydney Padua’s  51  The Thrilling Adventures of Lovelace and Babbage, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 306-308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  17  20  published in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  Figure 1: Table-algorithm derived from Menabrea original (Note D)  �  For Note G on the Bernoulli numbers, the table–algorithm has ten data variables, three  working variables, and four result variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The computation has just 25 operations, but there  are in addition two nested loops: an outer loop consisting of steps 13-23, and two inner loops  consisting of steps 13-16 and 17-20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' This form of calculation could not possibly have been  completed with Babbage’s Difference Engine, since it entirely lacked the ability to do  conditional tests and create looping or branching structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Nothing like it appeared in  Menabrea’s original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Lady Lovelace chose well when she identified the Bernoulli numbers as a  means to show “how an implicit function may be worked out by the engine.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  18  20  Variables  forData  Working  Variables  Variablesfor Results  Number of Operations  Nature of Operations  0V12  0V13  0V14  0V15  0V16  +  +  +  +  +  +  +  U  0  0  U  0  0  0  0  0  0  0  U  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  U  0  0  U  0  0  0  0  0  0  0  0  0  0  0  0  U  0  0  U  U  0  0  0  0  0  m2  u  dn"d\'n  d  mn\'m\'n  1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  uu  L  dn  up  b  u,p  6  dm"  7  m\'n  8  9  0  10  0  mn\'-m\'n  11  0  0  dmpublished in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  Figure 2: Original table-algorithm for Bernoulli numbers (Note G)  �  Oversize table representing Ada Lovelace’s algorithm for computing Bernoulli numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The  nested looping structure—“here follows a repetition of operations thirteen to twenty-three”—is  clearly visible at lower left (outer loop steps 13-23 and inner loops steps 13-16 and 17-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Printed oversize and interleaved in Taylor’s Scientific Memoirs (high-res image here and here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  This essay assesses Ada Lovelace’s contribution to computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' It points out her mathematical  studies with Babbage and De Morgan, her translation of Menabrea’s Sketch, and her joint  authorship with Babbage of the explanatory Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' One element, the table-algorithm for  computing the series of Bernoulli numbers, is by available evidence her work (written in pencil  and inked in by her husband).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Her correspondence with Babbage evinces a direct collegiality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  the two figures were jointly grappling with how to communicate the details of a computing  machine that did not physically exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' For this reason, the claim that Ada Lovelace was the  world’s first computer programmer might need to be carefully qualified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' At the least, we can  grant her primary authorship of the first algorithm intended for a computing machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  52  See ACM’s Collected Algorithms at <netlib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='org/toms>.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  52  Page �  of �  19  20  Diagram for the computation by the Engine of the Numbers of Bernoulli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' See Note G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' (page 722 et seq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=')  Working Variables  Data  Variable  Nature of Operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  10000  0000  10000  10000  MO0O  10000  30000  10000  10000  000  F0000  000  Indication of  Tariables  Variables  receiving  change in the  value on any  Statement of Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  2n  +IV  2n+1  21  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='2n-  21  2n+  2n-1  0  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  2n+1  02  一  10  B,  = B, Al  B,A1  21  5  2n+1  2(-2)  +V  2V  2n2n-1  V  2n  BV  4V  a Variable-card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Variablepublished in Hammerman and Russell, eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Ada’s Legacy (ACM Books 2015) DOI  It is surprising how widely and warmly she was recognized among early figures in  computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Alan Turing for instance felt obliged to deal with “Lady Lovelace’s objection” to  artificial intelligence since (in Turing’s quotation) she had maintained “The Analytical Engine  has no pretensions to originate anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' It can do whatever we know how to order it to  perform.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='   A pioneering conference volume,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Faster Than Thought: A Symposium of Digital  53  Computing Machines (1953) was the source,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' according to the History of Programming  Languages chapter on the Ada computer language,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' “from which we all learned about Lady  Lovelace.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='   “Lady Lovelace had undoubtedly a profound understanding of the principles of the  54  machine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and she added greatly to the value of her translation by some comprehensive notes  about the machine .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' including what we should now call a programme for computing the  Bernoulli numbers by a very sophisticated method,” wrote a computer pioneer in 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' The  statement accords well with my assessment, after several decades of possibly fanciful writing  about Ada Lovelace as well as unjustifiably critical blasts against her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Perhaps in the coming  generation, we can come back to the measured appreciation of her achievements that originated,  like the computer age itself, in the early 1950s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' After all, as Ada Lovelace put it, “we may  consider the [analytical] engine as the material and mechanical representative of analysis.”  55  Andrew Hodges, “Turing: A Natural Philosopher,” (1997) at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='turing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='uk/  53  publications/philobook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='html ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and Darren Abramson, “Turing’s Responses to Two Objections,” Minds and  Machines 18, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 2 (2008): 147-167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Thomas J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Bergin, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' and Richard G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Gibson, Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', History of Programming Languages---II  54  (New York ACM, 1996), “Ada Session,” quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 207;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Bowden, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Faster Than Thought: A  Symposium of Digital Computing Machines (London: Isaac Pitman, 1953).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Bowden, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=', Faster Than Thought: A Symposium of Digital Computing Machines (London:  55  Isaac Pitman, 1953), 18-22, 70-75, 341-408, quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 18 (profound understanding);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Lovelace,  “Translator’s Notes,” quote p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 696 (representative of analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' Compare Larry Owens, “Vannevar Bush  and the Differential Analyzer: The Text and Context of an Early Computer,” Technology and Culture 27  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content=' 1 (1986): 63-95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
+page_content='  Page �  of �  20  20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/LtE1T4oBgHgl3EQfGwPf/content/2301.02919v1.pdf'}
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+All in Tokens: Unifying Output Space of Visual Tasks via Soft Token
+Jia Ning1,4*, Chen Li2,4*, Zheng Zhang4*, Zigang Geng3,4, Qi Dai4, Kun He1, Han Hu4
+1Huazhong University of Science and Technology
+2Xi’an Jiaotong University
+3University of Science and Technology of China
+4Microsoft Research Asia
+{t-jianing,t-chenli1,zhez,t-ziganggeng,qid,hanhu}@microsoft.com
+brooklet60@hust.edu.cn
+Abstract
+Unlike language tasks, where the output space is usu-
+ally limited to a set of tokens, the output space of visual
+tasks is more complicated, making it difficult to build a uni-
+fied visual model for various visual tasks. In this paper, we
+seek to unify the output space of visual tasks, so that we
+can also build a unified model for visual tasks. To this end,
+we demonstrate a single unified model that simultaneously
+handles two typical visual tasks of instance segmentation
+and depth estimation, which have discrete/fixed-length and
+continuous/varied-length outputs, respectively. We propose
+several new techniques that take into account the particu-
+larity of visual tasks: 1) Soft token. We employ soft token
+to represent the task output. Unlike hard tokens in the com-
+mon VQ-VAE which are assigned one-hot to discrete code-
+books/vocabularies, the soft token is assigned softly to the
+codebook embeddings. Soft token can improve the accu-
+racy of both the next token inference and decoding of the
+task output; 2) Mask augmentation. Many visual tasks have
+corruption, undefined or invalid values in label annotations,
+i.e., occluded area of depth maps. We show that a mask aug-
+mentation technique can greatly benefit these tasks. With
+these new techniques and other designs, we show that the
+proposed general-purpose task-solver can perform both in-
+stance segmentation and depth estimation well. Particu-
+larly, we achieve 0.279 RMSE on the specific task of NYUv2
+depth estimation, setting a new record on this benchmark.
+The general-purpose task-solver, dubbed AiT, is available at
+https://github.com/SwinTransformer/AiT.
+1. Introduction
+A unified model for various tasks across modalities is
+one of the most ambitious goals of artificial intelligence
+(AI). However, this is challenging due to the diversity and
+*Equal Contribution. Jia, Chen, and Zigang are interns at MSRA.
+complexity of real-world tasks.
+Despite the difficulties,
+large-scale language models in the field of natural language
+processing (NLP), e.g., GPT-3 [6], have demonstrated im-
+pressive capabilities as a general-purpose solver for lan-
+guage tasks. Encouraged by the success of NLP, this paper
+attempts to investigate the possibility of universal models
+for various computer vision tasks.
+Most existing related research for universal computer vi-
+sion models focuses on the unification of architectures [19,
+39] and pre-training [2]. A few works aim for develop-
+ing one model for multiple visual tasks, yet they are usu-
+ally applicable to limited tasks:
+[1] handles only tasks
+with language as output; CLIP models [32] and the follow-
+ups [47, 48, 51] tackle only retrieval and image classifica-
+tion tasks; [9] deals with tasks that have describable and
+sequential outputs. In this paper, we aim for a truly general-
+purpose solver for various vision tasks. To this end, we
+first notice that a key obstacle was overlooked and under-
+explored: unlike language tasks whose inputs and outputs
+are represented by language tokens of the same form, the
+output forms for different computer vision tasks are more
+diverse. For example, the output of object detection is a set
+of coordinates and labels; the output of semantic segmenta-
+tion is a discrete label map; the output of depth estimation is
+an image with floating-point values; and the output of flow
+estimation is a vector field.
+In this paper, we address this obstacle by unifying the
+output space of various visual tasks through a general to-
+kenizer to encode the output space into a set of tokens.
+Specifically, the proposed framework consists of three com-
+ponents: tokenizer, detokenizer, and task solver, as illus-
+trated in Figure 1. The tokenizer and detokenizer are in-
+stantiated by VQ-VAE [37], which encodes the task output
+into a set of tokens, which are then reconstructed into the
+original output by the decoder. The task solver for different
+vision tasks is instantiated using an auto-regressive encoder-
+decoder model. The encoder-decoder model takes images
+as inputs and predicts a token sequence in a causal mode,
+arXiv:2301.02229v1  [cs.CV]  5 Jan 2023
+
+which is decoded into the original task-specific output form
+by the VQ-VAE decoder.
+To improve the effectiveness, we propose several new
+techniques that take into account the particularity of visual
+tasks. First, we propose soft token instead of hard ones as
+used in language tasks or by the original visual VQ-VAE
+framework. The soft token is represented by probability
+vectors, with each value in a vector denoting the probabil-
+ity belonging to a codebook. When a soft token is fed to
+the input of the detokenizer or to the input of the next token
+prediction network, its input embedding is computed by the
+weighted average of the codebook embeddings based on the
+probability values. This indicates that the soft token embed-
+ding has spanned an interpolable continuous space, which
+may better represent the visual output, especially when it’s
+continuous. The continuous nature of the soft token also al-
+lows the introduction of an auxiliary loss which learns the
+task output end-to-end, back from the detokenizer output to
+the task-solver input. In experiments, we show that the three
+usages of the soft token can all lead to better results.
+Second, we propose a mask augmentation technique to
+deal with visual tasks which have corrupted, undefined, or
+invalid values in annotations. The depth estimation is a typ-
+ical example of such a task, where the occluded area is not
+defined [36], as shown in Figure 3. The undefined regions
+make it difficult to the training of VQ-VAE tokenizers and
+detokenizers because it does not know what to reconstruct
+in the undefined area. To solve this problem, we randomly
+mask several patches of the input depth map when train-
+ing VQ-VAE. Unlike undefined areas, manually masked
+patches have ground truth annotations, which help train the
+VQ-VAE network to be able to recover the ground truth for
+undefined areas. In experiments, we show this technique
+to notably improve the accuracy of depth estimation. The
+mask augmentation technique is also expected to be use-
+ful for other visual problems with similar undefined or cor-
+rupted annotations, and therefore strengthens the generality
+of our unified task solver.
+Third, we systematically study the impact of architec-
+tures in the VQ-VAE model. We find that a very lightweight
+VQ-VAE with up to 5 convolution layers, 2 residual blocks,
+and 128 codebooks can serve as a very powerful tokenizer.
+The number of parameters and computations can be as small
+as 2M and 0.06G FLOPs. This suggests that the overhead
+of the VQ-VAE model is small, which improves the practi-
+cality of our framework.
+We choose two classical visual tasks with varying out-
+put spaces to validate our approach: depth estimation, and
+instance segmentation, whose output spaces are in formats
+of the floating-point maps and binary masks, and have fixed
+size and variable size, respectively. We achieve competitive
+accuracy on COCO instance segmentation [26], and state-
+of-the-art accuracy for NYUv2 depth estimation [36]. The
+proposed framework and techniques are generic and can be
+applied to other visual tasks, which will be our future work.
+2. Related Works
+Unified Frameworks in Computer Vision
+Encouraged
+by the success of T5 [33]/GPT [6] in NLP, the explo-
+ration of a single unified model for various tasks in com-
+puter vision has emerged. However, most existing works
+[1, 39, 47, 48, 51] are focused on the training algorithm or
+model architectures. This makes these models either only
+available as pre-trained models [39,47,48] or only for VL-
+related tasks [1, 51]. Perceiver-IO [19] presents a frame-
+work that can process different vision tasks, and it adopts
+the learned positional encoding or Fourier feature to unify
+the output of different tasks.
+Very recently, Pix2SeqV2 [9] proposed to unify different
+vision tasks into tokens and the tokens in Pix2SeqV2 need
+to be designed manually for different tasks. For example,
+they use the polygon to represent the instance segmentation.
+The most related works with ours are UViM [21] and
+Unified-IO [28]. They also adopt VQ-GAN/VQ-VAE as a
+general tokenizer/detokenizer and an auto-regressive Trans-
+former encoder/decoder to solve different tasks. However,
+they are direct applications of these techniques without an
+in-depth consideration of the particularity of visual prob-
+lems.
+Our study, while concurrently started, takes more in-
+depth consideration of the particularity of visual problems.
+We propose techniques of soft token and mask augmenta-
+tion, which prove beneficial generally for visual tasks or a
+part of them. We also extensively investigate the architec-
+ture of the VQ-VAE, which shows that this part can be made
+very light-weight, and thus make the framework more prac-
+tical.
+Vector-Quantization
+Discretized token output space is
+widely used in generative models, such as DALL-E [34],
+VQGAN [14], and VQ-Diffusion [16], to represent high-
+dimensional complex data.
+Models like VQ-VAE [37],
+dVAE [3] define a discrete latent space with the encoder-
+decoder architecture and a fixed size of codebook, The in-
+put is mapped to the discrete tokens of the codebook. We
+adopt the VQ-VAE framework to build our token space, and
+our soft token approach that treats the token space as a con-
+tinuous one instead of the original discrete one expands the
+usage of VQ-VAE.
+Monocular Depth Estimation
+Monocular depth estima-
+tion is a fundamental task for 3D perception. Deep learning
+dominates the depth estimation since Eigen et al. [13] intro-
+duces it into the depth task. The follow-up works include
+proposing powerful network [23, 24, 35], designing novel
+
+augmentation [18, 20], making use of the geometric con-
+straints [31,46], exploring pairwise relationship [10,22,52],
+combing with conditional random field [27,43,49].
+Some works [29, 35, 38, 45, 50] combine depth estima-
+tion with other tasks, such as semantic segmentation, and
+edge estimation. However, they design different heads and
+loss functions for different tasks respectively. There are also
+some methods [4, 5, 12, 15, 25] discretizing the continuous
+depth and cast the depth estimation as a per-pixel classi-
+fication task. Our approach represents the depth maps as
+a set of tokens, and unifies it with other visual tasks, i.e.
+instance segmentation, in a unified network structure and
+output space. More importantly, we show that a general
+framework for various visual tasks can achieve state-of-the-
+art accuracy on the NYUv2 depth estimation task.
+Instance Segmentation
+Instance segmentation aims to
+predict the segments of each instance.
+There are many
+works [17,40,44] studying how to represent the masks. For
+example, MaskRCNN [17] used a binary mask, Dense Rep-
+Points [44] adopts a set of deformable points to represent
+the segments, and PolarMask [40] models the segments by
+polygons. While their representation is specific to instance
+segmentation, we model the instance segmentation by a set
+of discrete (soft) tokens, which is more general for visual
+representations.
+3. Framework
+The goal of this work is to unify the output space of vi-
+sual tasks into discrete tokens and to build a single model
+that can handle different tasks simultaneously. In this sec-
+tion, we present the framework to achieve this goal, which
+is shown in Figure. 1. The framework consists of three mod-
+ules, a tokenizer that encodes the task output to the discrete
+tokens, a detokenizer that decodes tokens to the task output,
+and a task-solver that predicts tokens from images. In our
+approach, the encoder and decoder of VQ-VAE are used as
+the tokenizer and detokenizer, and the task-solver is imple-
+mented by an auto-regressive encoder-decoder model. Dur-
+ing training, task annotations are first mapped by the tok-
+enizer as discrete tokens and used as supervision to train the
+task-solver. In inference, the tokens predicted by the task-
+solver are decoded by the detokenizer into task outputs.
+3.1. Tokenizer and Detokenizer
+VQ-VAE is an encoder-decoder model with a set of la-
+tent codes. It was originally proposed to learn discrete rep-
+resentation for natural images. In this work, we use its en-
+coder and decoder as the tokenizer and detokenizer. In train-
+ing, the input image is encoded as a set of contiguous em-
+beddings, and these embeddings are assigned to their near-
+est latent codes. In the decoder, the corresponding codes are
+used as inputs instead of contiguous embeddings and then
+decoded into the image. By minimizing the reconstruction
+loss between the input image and decoded images, the en-
+coder, decoder and latent codes can be trained.
+Since we adopt discrete tokens as targets in the task-
+solver, the accuracy of reconstruction has an upper-bound
+on the performance of the whole framework. In addition,
+both training and inference of task-solver require the tok-
+enizer and detokenizer, the fast inference speed is also de-
+sired. The original network architecture of VQ-VAE is de-
+signed for natural images, which have more complex tex-
+tures and colors than the task output, making it not the opti-
+mal design for us. Therefore, we have exhaustively studied
+the effects of VQ-VAE with different design choices in our
+framework.
+Typical reconstruction losses (e.g. l-1 loss, MSE loss,
+etc.) cannot directly reflect the realistic performance of the
+task, we use standard evaluation metrics for different tasks
+to measure VQ-VAE. As shown in Table 2 and Table 4. We
+found that a very lightweight VQ-VAE can achieve promis-
+ing results in depth estimation and instance segmentation.
+Since the input value ranges for depth estimation and in-
+stance segmentation are different. We train two VQ-VAE
+models for two tasks separately, and they have similar archi-
+tecture. For depth estimation, the encoder consists of 5 con-
+volution layers (kernel size is 3 and stride is 2) and follows
+2 residual blocks. The output feature map has a downsam-
+ple ratio of 32, and the channel dimension is progressively
+increased from 16 to 256. The architecture of the decoder
+is symmetrical to the encoder, only replacing the convolu-
+tion layers with the deconvolution layer. For instance seg-
+mentation, we reduce the convolution and deconvolution of
+the encoder and decoder from 5 layers to 4 layers and keep
+all others the same. Subsequently, the downsample ratio is
+changed to 16.
+In addition, compared to the standard VQ-VAE usually
+adopts a large codebook size (e.g. 8192), our codebook size
+can be reduced to 128. We note that though larger codebook
+size consistently improves VQ-VAE reconstruction ability,
+they show no difference when applied to task-solver (see Ta-
+ble 1 and Table 2). There are two speculations on the effec-
+tiveness of a small codebook: 1) the output space of depth
+and segmentation is simple, without the need for a large
+codebook; 2) the large codebook may increase the learning
+difficulty for task-solver.
+3.2. Task-solver
+The task-solver is an auto-regressive encoder-decoder
+network. The encoder is a Swin Transformer with 6 ad-
+ditional standard transformer blocks, each block consists of
+a self-attention and an FFN. The decoder has 6 blocks, each
+block consists of a self-attention, a cross-attention, and an
+FFN. The architecture we used is similar to [8].
+
+Figure 1. Illustration of our unified framework with two stages. In this framework, various vision task outputs are transferred to discrete
+token space by a VQ-VAE tokenizer. In this way, discrete or continuous visual tasks can be converted into one discrete classified task,
+Given an input image, the encoder is first applied to learn
+a generic representation of all tasks. Then, based on the
+given task token, the decoder is used to predict a token se-
+quence in an auto-regressive manner. For different tasks,
+we customize their sequence formats, as described in the
+following:
+Depth Estimation
+The task token of depth estimation is
+denoted as [DEP]. It has a straightforward format, which
+is a token sequence of length HW
+322 , where H and W are the
+height and width of input images, respectively.
+Instance Segmentation
+The sequence format of instance
+segmentation is more complicated than depth estimation,
+which consists of three parts: bounding box coordinates,
+class of bounding box, and a binary mask. We follow the
+practice of Pix2Seq [8] for representing coordinates and the
+class of a bounding box.
+The coordinates are manually
+quantized into 2000 bins, and different classes are repre-
+sented by different tokens (including a background class,
+e.g. COCO dataset has 81 class tokens in total). For the bi-
+nary mask, we use 4 × 4 tokens to represent one mask. It
+is worth noting that since the computational complexity of
+auto-regressive is proportional to the square of the output
+sequence length, we have to use very few tokens to repre-
+sent the mask. Nevertheless, benefitting from our powerful
+detokenizer, these tokens can be decoded to a 64×64 mask.
+Based on these designs, each instance is represented by a
+total of 21 tokens (i.e. 4 tokens for coordinates, 1 token for
+class, 16 tokens for mask), and we use [INS] as the task
+token of instance segmentation, as shown in Figure 2. We
+append the meaningless zero mask for the noise box and do
+not add loss on those mask tokens during training.
+Figure 2. Illustration of instance segmentation and depth estima-
+tion token format. (a) We organize every object by 21 tokens. For
+positive objects, this format includes 4 box, 1 label and 16 mask
+tokens. While for noise tokens, 4 noise, 1 background and 16 zero
+tokens are employed; (b) For dense tasks like depth estimation, we
+treat every patch a token to form a token map.
+3.3. Soft Token
+In a typical auto-regressive prediction procedure, the to-
+ken with the maximal predicted probability is selected as the
+output, and used its embedding as the input to the decoder
+for the next prediction step. This approach is called hard-
+inference. However, since the tokens learned by VQ-VAE
+are not completely independent of each other, the correla-
+
+967353
+VQVAE
+83187
+Soft Token
+9
+Encoder
+7548318
+4,3, 35,3
+3536754
+Backbone
+Transformer
+Transformer
+Encoder
+Decoder
+Position
+Embeddings
+Soft
+433353
+Token
+83188
+VQVAE
+9
+7548318
+Decoder
+3534154
+Decoder Loss4x Box
+1x Label
+16x Mask
+4x Box
+1x Label
+16x Mask
+4x Noise
+1x BG
+16x Zero
+Token
+Token
+Token
+Token
+Token
+Token
+Token
+Token
+Token
+33
+33
+75B
+8
+6 
+35
+Transformer Decoder
+4 
+333539336424
+3534556
+TransformerDecoderFigure 3. There are some corrupted regions (black regions/pixels)
+in the GT depth map. While we have ignored these regions in
+training VQ-VAE as well, the reconstructed regions are still ab-
+normal, which is reflected in the shadows in reconstruction results.
+This phenomenon can be alleviated by adding masked augmenta-
+tion.
+tion between the tokens may affect the token prediction ac-
+curacy, making the hard inference probably not optimal. To
+leverage the correlation, a soft token technique is presented
+in the inference: instead of directly using the embedding
+of a single token, the soft token is the weighted averaged
+embedding of different tokens by their prediction probabil-
+ity. In addition to being applied in task-solver to predict the
+next token more accurately, the same idea can also be used
+in detokenizor to get better reconstruction results.
+Furthermore, the soft token results in the embedding
+space being spanned to an interpolable continuous space.
+Therefore, we can introduce an auxiliary loss that learns
+the task-specific output targets in an end-to-end manner by
+backing from the detokenzior output to the task-solver in-
+put.
+We examined the soft token in both instance segmen-
+tation and depth estimation. It can consistently improve
+the performance without any additional computational cost,
+which is a free-lunch technique in inference.
+3.4. Mask Augmentation in Depth Estimation
+The ground-truth depth maps in the depth estimation
+dataset often have some corrupted regions that are not anno-
+tated with depth information. In the conventional depth esti-
+mation frameworks, these regions are ignored during train-
+ing. However, the same solution cannot be applied to our
+framework. There are two challenges: First, while we have
+ignored these regions in training VQ-VAE as well, the re-
+constructed regions are still abnormal (see Figure 3) and
+further affect the training of the task-solver and make the
+final result also have many artifacts; Second, a token pre-
+dicted by VQ-VAE corresponds to a 322 patch, which may
+contain both normal and corrupted pixels. Therefore, it is
+hard to deal with this issue by ignoring the tokens.
+As shown in Figure 4, we present to introduce mask
+Figure 4. Illustration of our masked augmentation. In order to
+make the tokenizer have a stronger complement capability, we add
+random mask noise under a certain mask ratio and patch size to the
+original GT depth maps as inputs. But we still use the GT depth
+maps to apply reconstruction loss.
+augmentation in the training of VQ-VAE to alleviate this
+challenge. Specifically, we randomly mask some regions
+in the input depth images and then use their original depth
+information as supervision. In this way, the VQ-VAE can
+complete/recover some corrupted regions with reasonable
+results. Figure 3 shows the visualization. In Table 8, we no-
+tice that applying mask augmentation can improve the per-
+formance of depth estimation.
+4. Experiments
+4.1. Tasks and Datasets
+To examine the generalizability of our framework, we
+choose depth estimation and instance segmentation, which
+are two tasks with very different output spaces.
+Instance Segmentation
+The instance segmentation re-
+quires predicting the location, the class, and the mask of
+each instance. The COCO2017 benchmark is one of the
+most challenging datasets for this task. It consists of 117K
+training images, 5K validation images, and 41K test im-
+ages.
+A total of 80 classes annotation are provided.
+In
+our experiments, we follow the common setting of previ-
+ous works [17, 40, 44] that report the performance on the
+validation set for comparison.
+Depth Estimation
+Depth estimation is a fundamental
+problem in computer vision, which requires estimating the
+depth for each pixel. Unlike segmentation whose output is
+a binary mask, the depth map is a floating point image. In
+this work, we use the NYUv2 Depth dataset, which consists
+of 24K training images and 654 validation images, and the
+RMSE is used as the major metric.
+4.2. Implementation Details
+We train two separate VQ-VAE for depth estimation and
+instance segmentation. The input image size used in depth
+
+VQVAE
+Codebook
+Encoder
+Decoder
+0
+1
+2
+N-1 Nis 4802, with a batch size of 8. The Adam optimizer is used
+with the base learning rate of 1e-3, α1 = 0.9, α2 = 0.999.
+An exponential learning rate schedule is applied with the
+learning rate decay of 0.98 and a total of 20 training epochs.
+For instance segmentation. The input image size is 642 with
+a batch size of 512. The Adam optimizer is used with the
+base learning rate of 3e-4, α1 = 0.9, α2 = 0.999. An ex-
+ponential learning rate schedule is applied with the learning
+rate decay of 0.98 and a total of 20 training epochs. By
+default, the EMA model update technique is used for all
+VQ-VAE models.
+For the task-solver,
+we adopt the auto-regressive
+encoder-decoder architecture, which is similar to Pix2Seq
+[8]. It consists of a backbone, 6 encoder layers, and 6 de-
+coder layers. We use the Swin-B as the backbone, which
+is pre-trained with SimMIM [42].
+Most experiments in
+the ablation study are separately trained in depth estima-
+tion and instance segmentation. In depth estimation, we
+use the AdamW optimizer with a base learning rate of 2e-4,
+α1 = 0.9, α2 = 0.999, a weight decay of 0.05, and a layer
+decay of 0.9. The total training length is 25 epochs and the
+batch size is 24. The step learning rate schedule is used and
+the learning rate dropped to 2e-5 at the 18th epoch. For data
+argumentation, the random cropping of 4802 and horizon-
+tal flip with probability 0.5 are employed. We also append
+random brightness contrast, random gamma, and hue sat-
+uration value. In instance segmentation, the AdamW op-
+timizer with a base learning rate of 1e-4, alpha1 = 0.9,
+alpha2 = 0.999, a weight decay of 0.05, and a layer decay
+of 0.85, linear decay learning rate scheduler is applied, and
+the total training length is 50 epochs with a batch size of 16.
+For data augmentation, large-scale jittering with the range
+of 0.1 to 3.0 and crop size of 6402 are used.
+4.3. Ablation Study
+We ablate the key design choices and techniques in this
+section. By default, we train the model separately for each
+task in the ablation study for better illustration, and Swin-B
+is used as the default backbone.
+Table 1. Ablation study on codebook size of VQ-VAE on depth
+and instance segmentation in reconstruction.
+Width #tokens Depth Instance Seg.
+RMSE Mask mAP
+1.0×
+64
+0.1025
+88.94
+1.0×
+128
+0.0918
+89.34
+1.0×
+256
+0.0902
+90.22
+Architecture of VQ-VAE
+We study how different de-
+signs of VQ-VAE affect performance. We first evaluate the
+reconstruction performance of different codebook sizes. To
+more accurately and intuitively observe the reconstruction
+Table 2. Ablation study on codebook size of VQ-VAE on depth
+and instance segmentation in task-solver.
+Width #tokens Depth
+Instance Seg.
+RMSE Box mAP Mask mAP
+1.0×
+64
+0.3105
+43.5
+33.0
+1.0×
+128
+0.3084
+43.6
+33.2
+1.0×
+256
+0.3098
+43.4
+33.4
+Table 3. Ablation study on the width of VQ-VAE on depth and
+instance segmentation in reconstruction.
+Width #tokens Depth Instance Seg.
+RMSE Mask mAP
+0.5×
+128
+0.1196
+88.97
+1.0×
+128
+0.0918
+89.34
+2.0×
+128
+0.1025
+90.92
+Table 4. Ablation study on the width of VQ-VAE on depth and
+instance segmentation in task-solver.
+Width #tokens Depth
+Instance Seg.
+RMSE Box mAP Mask mAP
+0.5×
+128
+0.3171
+43.4
+33.1
+1.0×
+128
+0.3084
+43.6
+33.2
+2.0×
+128
+0.3151
+43.2
+33.1
+performance, our evaluation is performed on the validation
+set of different tasks and adopts mask mAP and RMSE as
+metrics. The results are shown in Table 1. We find that
+although the large codebook size (e.g. 256) benefits the re-
+construction performance, the small codebook size (e.g. 64)
+can also yield sufficiently good reconstruction performance.
+Further applying to the task-solver, we found different code-
+book size has little effect on final performance (see Table 2).
+Using the same evaluation method, we study the effect
+of the width of VQ-VAE. Table 3 shows the reconstruction
+performance and Table 4 shows the performance of apply-
+ing to task-solver. Similar to the observation on codebook
+size, we find that the network width has little effect on the
+final performance.
+The downsample ratio of VQ-VAE is another key that
+may affect network performance. We vary the downsam-
+ple ratio in [8, 16, 32]. We note that the instance segmen-
+tation cannot support a downsample ratio of 8 because it
+results in too long sequences. Table 5 shows the recon-
+struction performance, as the downsample ratio increases,
+the reconstruction performance gets worse, satisfying the
+intuition. However, we find that better reconstruction per-
+formance does not always lead to better performance when
+applying the VQ-VAE in task-solver. In Table 6, the best
+performance is achieved at downsample ratio of 32. We ex-
+plain this phenomenon is that a smaller downsample ratio
+facilitates reconstruction, but it also increases the length of
+the token sequence, which is detrimental to the task-solver.
+
+Table 5. Ablation study on the downsample ratio of VQ-VAE on
+depth and instance segmentation in reconstruction.
+Downsample Ratio Depth Instance Seg.
+RMSE Mask mAP
+32
+0.0918
+70.91
+16
+0.0696
+89.34
+8
+0.0515
+-
+Table 6. Ablation study on the downsample ratio of VQ-VAE on
+depth and instance segmentation in task-solver.
+Downsample Ratio Depth
+Instance Seg.
+RMSE Box mAP Mask mAP
+32
+0.3084
+42.7
+30.3
+16
+0.3280
+43.6
+33.2
+8
+0.3303
+-
+-
+Table 7. Ablation on the effectiveness of soft token.
+Description
+Depth
+Instance Seg.
+RMSE
+box mAP
+mask mAP
+Baseline (hard-inf)
+0.3163
+43.6
+31.1
+On. task-solver
+0.3108
+43.5
+32.1
+On. detokenizor
+0.3113
+43.6
+32.3
+On. both
+0.3084
+43.6
+33.2
+On. both + aux. loss
+0.3071
+43.3
+34.2
+Soft Token
+To leverage the correlation between tokens,
+we introduce the soft token techniques, which can be used to
+improve the performance in the inference stage for free. We
+examined this technique in Table 7. Compared with the hard
+inference baseline, applying the soft token in task-solver
+and detokenzior alone can bring performance gains, and
+further performance improvement is achieved when used in
+two stages at the same time: the depth performance is im-
+proved by +0.008 RMSE and the instance segmentation is
+improved by +2.1 mAP. On top of it, adding the auxiliary
+loss on the output of detokenizor enlarges the gain to +0.009
+RMSE and +3.1 mAP.
+Table 8. Affects of mask augmentation in training VQ-VAE on
+depth. The patch size of all models is set to 16.
+Mask Ratio VQ-VAE task-solver
+0.0
+0.0831
+0.3138
+0.3
+0.0893
+0.3135
+0.5
+0.0918
+0.3084
+0.7
+0.1196
+0.3194
+Mask Augmentation
+We evaluate the effectiveness of
+mask augmentation in depth estimation. The different mask
+ratios vary from 0.3 to 0.7 are used, and Table 8 shows the
+results. The best performance is achieved by using the mask
+ratio of 0.5, which is +0.005 better than the baseline.
+Table 9.
+Results of monocular depth estimation task on
+NYUv2 [36]. Both AiT and AiT-P are our methods, where AiT
+indicates auto-regressive prediction, and AiT-P indicates parallel
+prediction.
+Method
+RMSE ↓
+δ1 ↑
+δ2 ↑
+δ3 ↑
+REL ↓
+log10 ↓
+DORN [15]
+0.509
+0.828
+0.965
+0.992
+0.115
+0.051
+BTS [23]
+0.392
+0.885
+0.978
+0.995
+0.110
+0.047
+AdaBins [4]
+0.364
+0.903
+0.984
+0.997
+0.103
+0.044
+DPT [35]
+0.357
+0.904
+0.988
+0.998
+0.110
+0.045
+LocalBins [5]
+0.357
+0.907
+0.987
+0.998
+0.099
+0.042
+P3Depth [30]
+0.356
+0.898
+0.981
+0.996
+0.104
+0.043
+BinsFormer [25]
+0.339
+0.921
+0.989
+0.998
+0.096
+0.041
+NeWCRFs [49]
+0.334
+0.922
+0.992
+0.998
+0.095
+0.041
+BinsFormer [25]
+0.330
+0.925
+0.989
+0.997
+0.094
+0.040
+SwinV2-B [41]
+0.303
+0.938
+0.992
+0.998
+0.086
+0.037
+SwinV2-L [41]
+0.287
+0.949
+0.994
+0.999
+0.083
+0.035
+UViM [21]
+0.467
+-
+-
+-
+-
+-
+Unified-IOXL [28]
+0.385
+-
+-
+-
+-
+-
+AiT (SwinV2-B)
+0.307
+0.931
+0.992
+0.998
+0.089
+0.037
+AiT (SwinV2-L)
+0.289
+0.946
+0.992
+0.998
+0.079
+0.034
+AiT-P (SwinV2-B)
+0.302
+0.939
+0.990
+0.998
+0.086
+0.036
+AiT-P (SwinV2-L)
+0.279
+0.953
+0.993
+0.999
+0.076
+0.033
+AiT-P (SwinV2-L)
+w/o soft token
+0.284
+0.951
+0.993
+0.999
+0.078
+0.035
+4.4. Preliminary Study on Parallel Prediction
+In previous sections, we mainly demonstrate the use of
+auto-regressive models to unify various visual tasks. Note
+the bi-directional or even qua-directional natural may en-
+courage parallel decoding approaches. Therefore, we make
+some preliminary studies on the parallel decoder instead of
+the auto-regressive decoder.
+As parallel decoder has been prevalent in object de-
+tection and instance segmentation, i.e. DETR [7] and
+Mask2Former [11], we mainly studied the parallel decoder
+for the depth estimation problem. As shown in Table 9,
+the parallel decoder performs better than the auto-regressive
+counterpart by 0.01 RMSE (0.279 vs. 0.289). This indicates
+a strong potential than the auto-regressive models.
+Our techniques such as soft token also benefit the parallel
+decoder, as shown in Table 9. This implies the generality of
+the proposed techniques.
+4.5. Comparison with Other Unified Frameworks
+and State-of-the-arts in Depth Estimation
+UViM and Unified-IO are the most relevant works to
+ours.
+We compare the performance with these methods
+on the overlap task, i.e. NYUv2 depth estimation. The re-
+sults are shown in Table. 9. Our auto-regressive approach
+achieves 0.289 RMSE, which is 0.178 and 0.096 better than
+UViM and Unified-IO. Moreover, our parallel approach fur-
+ther improves the performance to 0.279 RMSE. This re-
+
+Figure 5. Visualization on instance segmentation task of our method.
+Figure 6. Visualization of depth estimation task. With the well-designed structure and techniques, our model is capable for this task.
+sult surpasses previous state-of-the-arts by +0.008 RMSE.
+While UViM and Unified-IO mainly conceptually propose
+unified frameworks for various visual tasks, we push more
+solid steps through in-depth study of the general visual task-
+solver.
+4.6. One Model for Multiple Tasks
+We train the instance segmentation and depth jointly us-
+ing a shared task-solver. Table 10 shows that the joint train-
+ing with shared model weights has marginal performance
+gradation compared to using separate task solvers.
+5. Conclusion
+In this work, we investigate the unification of output
+spaces for various vision tasks by a set of visual tokens, and
+Table 10. Joint training of depth estimation and instance segmen-
+tation using a single task-solver. The performance of joint training
+is slightly worse compared to using separate task-solvers for each
+task.
+Description.
+Depth
+Instance Seg.
+RMSE Box mAP Mask mAP
+separate training 0.3071
+43.3
+34.2
+joint training
+0.3103
+42.2
+34.1
+further develop a unified auto-regressive encoder-decoder
+model. Two new techniques are proposed which take the
+particularity of visual tasks into account to improve the sys-
+tem: 1) Soft token can leverage the correlation between
+
+RGB Image
+GT Depth Map
+OursTable 11. Ablation study on increasing the number of residual
+blocks on the 16× downsampling setting of depth estimation.
+#Resblock Tokenizor Task-solver
+RMSE
+RMSE
+2
+0.0676
+0.3229
+3
+0.0683
+0.3251
+4
+0.0695
+0.3277
+5
+0.0690
+0.3293
+tokens to improve performance in the inference stage and
+enables end-to-end learning for the final visual targets; 2)
+Mask augmentation is used to alleviate the issue of cor-
+rupted/undefined areas of visual tasks, i.e. depth estima-
+tion. With these two techniques, our general method set a
+new state-of-the-art on the NYUv2 depth dataset, as well as
+competitive accuracy on the COCO 2017 object detection
+and instance segmentation benchmark. We also conducted
+preliminary studies on the parallel decoder which is show
+promising results on the depth estimation problem.
+A. Ablation on the Depths of VQ-VAE
+In Table 3 and Table 4, we have studied how different
+widths of VQ-VAE affect the performance. In Table 11,
+we further examine the effect of depth, i.e. different num-
+ber of residual blocks in VQ-VAE. The result shows that
+the increasing residual blocks lead to worse performance in
+VQ-VAE and task-solver.
+B. VQ-VAE vs. Interpolation Tokenizer
+We use a lightweight VQ-VAE as the tokenizer to reduce
+the resolution and computation for the task-solver. A intu-
+itive baseline to perform down-sampling is the interpolation
+method. For example, in instance segmentation, we can
+use a nearest-neighbor interpolation method to up-sample
+the binary masks. For depth estimation, we use a bilin-
+ear interpolation method and discretize the floating num-
+bers between 0 to 10 by 128 bins. We select the same in-
+put and target resolution as our VQ-VAE tokenizer for a
+fair comparison. As shown in Table 12, the tokenizer using
+nearest-neighbor interpolation is much worse than our VQ-
+VAE method, showing the effectiveness of using VQ-VAE
+to represent the task output space.
+C. Details for Joint Training
+The AdamW optimizer with a base learning rate of 8e-
+4, α1 = 0.9, α2 = 0.999, a weight decay of 0.05, and
+a layer decay of 0.85, linear decay learning rate scheduler
+is applied. We train the model with 300k iterations and
+batch size 64. The decoder loss weights of 5.0 and 1.0 are
+Table 12. Comparison of VQ-VAE and interpolation as the tok-
+enizer.
+Description. Depth
+Instance Seg.
+RMSE Box mAP Mask mAP
+Ours
+0.3084
+43.6
+33.2
+interpolation 0.7544
+43.8
+4.2
+used for instance segmentation and depth estimation respec-
+tively. A depth loss weight of 0.2 is used. We process these
+batch data individually i.e. different tasks do task-specific
+augmentation and generate their discrete training token se-
+quence.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf,len=821
+page_content='All in Tokens: Unifying Output Space of Visual Tasks via Soft Token  Jia Ning1,4*, Chen Li2,4*, Zheng Zhang4*, Zigang Geng3,4, Qi Dai4, Kun He1, Han Hu4  1Huazhong University of Science and Technology  2Xi’an Jiaotong University  3University of Science and Technology of China  4Microsoft Research Asia  {t-jianing,t-chenli1,zhez,t-ziganggeng,qid,hanhu}@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='com  brooklet60@hust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='cn  Abstract  Unlike language tasks, where the output space is usu-  ally limited to a set of tokens, the output space of visual  tasks is more complicated, making it difficult to build a uni-  fied visual model for various visual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In this paper, we  seek to unify the output space of visual tasks, so that we  can also build a unified model for visual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' To this end,  we demonstrate a single unified model that simultaneously  handles two typical visual tasks of instance segmentation  and depth estimation, which have discrete/fixed-length and  continuous/varied-length outputs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We propose  several new techniques that take into account the particu-  larity of visual tasks: 1) Soft token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We employ soft token  to represent the task output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Unlike hard tokens in the com-  mon VQ-VAE which are assigned one-hot to discrete code-  books/vocabularies, the soft token is assigned softly to the  codebook embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Soft token can improve the accu-  racy of both the next token inference and decoding of the  task output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 2) Mask augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Many visual tasks have  corruption, undefined or invalid values in label annotations,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=', occluded area of depth maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We show that a mask aug-  mentation technique can greatly benefit these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' With  these new techniques and other designs, we show that the  proposed general-purpose task-solver can perform both in-  stance segmentation and depth estimation well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Particu-  larly, we achieve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='279 RMSE on the specific task of NYUv2  depth estimation, setting a new record on this benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  The general-purpose task-solver, dubbed AiT, is available at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='com/SwinTransformer/AiT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Introduction  A unified model for various tasks across modalities is  one of the most ambitious goals of artificial intelligence  (AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' However, this is challenging due to the diversity and  Equal Contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Jia, Chen, and Zigang are interns at MSRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  complexity of real-world tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Despite the difficulties,  large-scale language models in the field of natural language  processing (NLP), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=', GPT-3 [6], have demonstrated im-  pressive capabilities as a general-purpose solver for lan-  guage tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Encouraged by the success of NLP, this paper  attempts to investigate the possibility of universal models  for various computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Most existing related research for universal computer vi-  sion models focuses on the unification of architectures [19,  39] and pre-training [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' A few works aim for develop-  ing one model for multiple visual tasks, yet they are usu-  ally applicable to limited tasks:  [1] handles only tasks  with language as output;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' CLIP models [32] and the follow-  ups [47, 48, 51] tackle only retrieval and image classifica-  tion tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' [9] deals with tasks that have describable and  sequential outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In this paper, we aim for a truly general-  purpose solver for various vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' To this end, we  first notice that a key obstacle was overlooked and under-  explored: unlike language tasks whose inputs and outputs  are represented by language tokens of the same form, the  output forms for different computer vision tasks are more  diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For example, the output of object detection is a set  of coordinates and labels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' the output of semantic segmenta-  tion is a discrete label map;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' the output of depth estimation is  an image with floating-point values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' and the output of flow  estimation is a vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  In this paper, we address this obstacle by unifying the  output space of various visual tasks through a general to-  kenizer to encode the output space into a set of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Specifically, the proposed framework consists of three com-  ponents: tokenizer, detokenizer, and task solver, as illus-  trated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The tokenizer and detokenizer are in-  stantiated by VQ-VAE [37], which encodes the task output  into a set of tokens, which are then reconstructed into the  original output by the decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The task solver for different  vision tasks is instantiated using an auto-regressive encoder-  decoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The encoder-decoder model takes images  as inputs and predicts a token sequence in a causal mode,  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='02229v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='CV]  5 Jan 2023  which is decoded into the original task-specific output form  by the VQ-VAE decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  To improve the effectiveness, we propose several new  techniques that take into account the particularity of visual  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' First, we propose soft token instead of hard ones as  used in language tasks or by the original visual VQ-VAE  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The soft token is represented by probability  vectors, with each value in a vector denoting the probabil-  ity belonging to a codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' When a soft token is fed to  the input of the detokenizer or to the input of the next token  prediction network, its input embedding is computed by the  weighted average of the codebook embeddings based on the  probability values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' This indicates that the soft token embed-  ding has spanned an interpolable continuous space, which  may better represent the visual output, especially when it’s  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The continuous nature of the soft token also al-  lows the introduction of an auxiliary loss which learns the  task output end-to-end, back from the detokenizer output to  the task-solver input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In experiments, we show that the three  usages of the soft token can all lead to better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Second, we propose a mask augmentation technique to  deal with visual tasks which have corrupted, undefined, or  invalid values in annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The depth estimation is a typ-  ical example of such a task, where the occluded area is not  defined [36], as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The undefined regions  make it difficult to the training of VQ-VAE tokenizers and  detokenizers because it does not know what to reconstruct  in the undefined area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' To solve this problem, we randomly  mask several patches of the input depth map when train-  ing VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Unlike undefined areas, manually masked  patches have ground truth annotations, which help train the  VQ-VAE network to be able to recover the ground truth for  undefined areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In experiments, we show this technique  to notably improve the accuracy of depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The  mask augmentation technique is also expected to be use-  ful for other visual problems with similar undefined or cor-  rupted annotations, and therefore strengthens the generality  of our unified task solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Third, we systematically study the impact of architec-  tures in the VQ-VAE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We find that a very lightweight  VQ-VAE with up to 5 convolution layers, 2 residual blocks,  and 128 codebooks can serve as a very powerful tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  The number of parameters and computations can be as small  as 2M and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='06G FLOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' This suggests that the overhead  of the VQ-VAE model is small, which improves the practi-  cality of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  We choose two classical visual tasks with varying out-  put spaces to validate our approach: depth estimation, and  instance segmentation, whose output spaces are in formats  of the floating-point maps and binary masks, and have fixed  size and variable size, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We achieve competitive  accuracy on COCO instance segmentation [26], and state-  of-the-art accuracy for NYUv2 depth estimation [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The  proposed framework and techniques are generic and can be  applied to other visual tasks, which will be our future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Related Works  Unified Frameworks in Computer Vision  Encouraged  by the success of T5 [33]/GPT [6] in NLP, the explo-  ration of a single unified model for various tasks in com-  puter vision has emerged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' However, most existing works  [1, 39, 47, 48, 51] are focused on the training algorithm or  model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' This makes these models either only  available as pre-trained models [39,47,48] or only for VL-  related tasks [1, 51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Perceiver-IO [19] presents a frame-  work that can process different vision tasks, and it adopts  the learned positional encoding or Fourier feature to unify  the output of different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Very recently, Pix2SeqV2 [9] proposed to unify different  vision tasks into tokens and the tokens in Pix2SeqV2 need  to be designed manually for different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For example,  they use the polygon to represent the instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  The most related works with ours are UViM [21] and  Unified-IO [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' They also adopt VQ-GAN/VQ-VAE as a  general tokenizer/detokenizer and an auto-regressive Trans-  former encoder/decoder to solve different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' However,  they are direct applications of these techniques without an  in-depth consideration of the particularity of visual prob-  lems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Our study, while concurrently started, takes more in-  depth consideration of the particularity of visual problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  We propose techniques of soft token and mask augmenta-  tion, which prove beneficial generally for visual tasks or a  part of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We also extensively investigate the architec-  ture of the VQ-VAE, which shows that this part can be made  very light-weight, and thus make the framework more prac-  tical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Vector-Quantization  Discretized token output space is  widely used in generative models, such as DALL-E [34],  VQGAN [14], and VQ-Diffusion [16], to represent high-  dimensional complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Models like VQ-VAE [37],  dVAE [3] define a discrete latent space with the encoder-  decoder architecture and a fixed size of codebook, The in-  put is mapped to the discrete tokens of the codebook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We  adopt the VQ-VAE framework to build our token space, and  our soft token approach that treats the token space as a con-  tinuous one instead of the original discrete one expands the  usage of VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Monocular Depth Estimation  Monocular depth estima-  tion is a fundamental task for 3D perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Deep learning  dominates the depth estimation since Eigen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' [13] intro-  duces it into the depth task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The follow-up works include  proposing powerful network [23, 24, 35], designing novel  augmentation [18, 20], making use of the geometric con-  straints [31,46], exploring pairwise relationship [10,22,52],  combing with conditional random field [27,43,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Some works [29, 35, 38, 45, 50] combine depth estima-  tion with other tasks, such as semantic segmentation, and  edge estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' However, they design different heads and  loss functions for different tasks respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' There are also  some methods [4, 5, 12, 15, 25] discretizing the continuous  depth and cast the depth estimation as a per-pixel classi-  fication task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Our approach represents the depth maps as  a set of tokens, and unifies it with other visual tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  instance segmentation, in a unified network structure and  output space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' More importantly, we show that a general  framework for various visual tasks can achieve state-of-the-  art accuracy on the NYUv2 depth estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Instance Segmentation  Instance segmentation aims to  predict the segments of each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  There are many  works [17,40,44] studying how to represent the masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For  example, MaskRCNN [17] used a binary mask, Dense Rep-  Points [44] adopts a set of deformable points to represent  the segments, and PolarMask [40] models the segments by  polygons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' While their representation is specific to instance  segmentation, we model the instance segmentation by a set  of discrete (soft) tokens, which is more general for visual  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Framework  The goal of this work is to unify the output space of vi-  sual tasks into discrete tokens and to build a single model  that can handle different tasks simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In this sec-  tion, we present the framework to achieve this goal, which  is shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The framework consists of three mod-  ules, a tokenizer that encodes the task output to the discrete  tokens, a detokenizer that decodes tokens to the task output,  and a task-solver that predicts tokens from images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In our  approach, the encoder and decoder of VQ-VAE are used as  the tokenizer and detokenizer, and the task-solver is imple-  mented by an auto-regressive encoder-decoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Dur-  ing training, task annotations are first mapped by the tok-  enizer as discrete tokens and used as supervision to train the  task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In inference, the tokens predicted by the task-  solver are decoded by the detokenizer into task outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Tokenizer and Detokenizer  VQ-VAE is an encoder-decoder model with a set of la-  tent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' It was originally proposed to learn discrete rep-  resentation for natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In this work, we use its en-  coder and decoder as the tokenizer and detokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In train-  ing, the input image is encoded as a set of contiguous em-  beddings, and these embeddings are assigned to their near-  est latent codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In the decoder, the corresponding codes are  used as inputs instead of contiguous embeddings and then  decoded into the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' By minimizing the reconstruction  loss between the input image and decoded images, the en-  coder, decoder and latent codes can be trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Since we adopt discrete tokens as targets in the task-  solver, the accuracy of reconstruction has an upper-bound  on the performance of the whole framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In addition,  both training and inference of task-solver require the tok-  enizer and detokenizer, the fast inference speed is also de-  sired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The original network architecture of VQ-VAE is de-  signed for natural images, which have more complex tex-  tures and colors than the task output, making it not the opti-  mal design for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Therefore, we have exhaustively studied  the effects of VQ-VAE with different design choices in our  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Typical reconstruction losses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' l-1 loss, MSE loss,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=') cannot directly reflect the realistic performance of the  task, we use standard evaluation metrics for different tasks  to measure VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' As shown in Table 2 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We  found that a very lightweight VQ-VAE can achieve promis-  ing results in depth estimation and instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Since the input value ranges for depth estimation and in-  stance segmentation are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We train two VQ-VAE  models for two tasks separately, and they have similar archi-  tecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For depth estimation, the encoder consists of 5 con-  volution layers (kernel size is 3 and stride is 2) and follows  2 residual blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The output feature map has a downsam-  ple ratio of 32, and the channel dimension is progressively  increased from 16 to 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The architecture of the decoder  is symmetrical to the encoder, only replacing the convolu-  tion layers with the deconvolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For instance seg-  mentation, we reduce the convolution and deconvolution of  the encoder and decoder from 5 layers to 4 layers and keep  all others the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Subsequently, the downsample ratio is  changed to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  In addition, compared to the standard VQ-VAE usually  adopts a large codebook size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 8192), our codebook size  can be reduced to 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We note that though larger codebook  size consistently improves VQ-VAE reconstruction ability,  they show no difference when applied to task-solver (see Ta-  ble 1 and Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' There are two speculations on the effec-  tiveness of a small codebook: 1) the output space of depth  and segmentation is simple, without the need for a large  codebook;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 2) the large codebook may increase the learning  difficulty for task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Task-solver  The task-solver is an auto-regressive encoder-decoder  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The encoder is a Swin Transformer with 6 ad-  ditional standard transformer blocks, each block consists of  a self-attention and an FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The decoder has 6 blocks, each  block consists of a self-attention, a cross-attention, and an  FFN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The architecture we used is similar to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Illustration of our unified framework with two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In this framework, various vision task outputs are transferred to discrete  token space by a VQ-VAE tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In this way, discrete or continuous visual tasks can be converted into one discrete classified task,  Given an input image, the encoder is first applied to learn  a generic representation of all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Then, based on the  given task token, the decoder is used to predict a token se-  quence in an auto-regressive manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For different tasks,  we customize their sequence formats, as described in the  following:  Depth Estimation  The task token of depth estimation is  denoted as [DEP].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' It has a straightforward format, which  is a token sequence of length HW  322 , where H and W are the  height and width of input images, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Instance Segmentation  The sequence format of instance  segmentation is more complicated than depth estimation,  which consists of three parts: bounding box coordinates,  class of bounding box, and a binary mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We follow the  practice of Pix2Seq [8] for representing coordinates and the  class of a bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  The coordinates are manually  quantized into 2000 bins, and different classes are repre-  sented by different tokens (including a background class,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' COCO dataset has 81 class tokens in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For the bi-  nary mask, we use 4 × 4 tokens to represent one mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' It  is worth noting that since the computational complexity of  auto-regressive is proportional to the square of the output  sequence length, we have to use very few tokens to repre-  sent the mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Nevertheless, benefitting from our powerful  detokenizer, these tokens can be decoded to a 64×64 mask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Based on these designs, each instance is represented by a  total of 21 tokens (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 4 tokens for coordinates, 1 token for  class, 16 tokens for mask), and we use [INS] as the task  token of instance segmentation, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We  append the meaningless zero mask for the noise box and do  not add loss on those mask tokens during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Illustration of instance segmentation and depth estima-  tion token format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' (a) We organize every object by 21 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For  positive objects, this format includes 4 box, 1 label and 16 mask  tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' While for noise tokens, 4 noise, 1 background and 16 zero  tokens are employed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' (b) For dense tasks like depth estimation, we  treat every patch a token to form a token map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Soft Token  In a typical auto-regressive prediction procedure, the to-  ken with the maximal predicted probability is selected as the  output, and used its embedding as the input to the decoder  for the next prediction step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' This approach is called hard-  inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' since the tokens learned by VQ-VAE  are not completely independent of each other,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' the correla-  967353  VQVAE  83187  Soft Token  9  Encoder  7548318  4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 35,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3  3536754  Backbone  Transformer  Transformer  Encoder  Decoder  Position  Embeddings  Soft  433353  Token  83188  VQVAE  9  7548318  Decoder  3534154  Decoder Loss4x Box  1x Label  16x Mask  4x Box  1x Label  16x Mask  4x Noise  1x BG  16x Zero  Token  Token  Token  Token  Token  Token  Token  Token  Token  33  33  75B  8  6  35  Transformer Decoder  4  333539336424  3534556  TransformerDecoderFigure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' There are some corrupted regions (black regions/pixels)  in the GT depth map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' While we have ignored these regions in  training VQ-VAE as well, the reconstructed regions are still ab-  normal, which is reflected in the shadows in reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  This phenomenon can be alleviated by adding masked augmenta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  tion between the tokens may affect the token prediction ac-  curacy, making the hard inference probably not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' To  leverage the correlation, a soft token technique is presented  in the inference: instead of directly using the embedding  of a single token, the soft token is the weighted averaged  embedding of different tokens by their prediction probabil-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In addition to being applied in task-solver to predict the  next token more accurately, the same idea can also be used  in detokenizor to get better reconstruction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Furthermore, the soft token results in the embedding  space being spanned to an interpolable continuous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Therefore, we can introduce an auxiliary loss that learns  the task-specific output targets in an end-to-end manner by  backing from the detokenzior output to the task-solver in-  put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  We examined the soft token in both instance segmen-  tation and depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' It can consistently improve  the performance without any additional computational cost,  which is a free-lunch technique in inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Mask Augmentation in Depth Estimation  The ground-truth depth maps in the depth estimation  dataset often have some corrupted regions that are not anno-  tated with depth information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In the conventional depth esti-  mation frameworks, these regions are ignored during train-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' However, the same solution cannot be applied to our  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' There are two challenges: First, while we have  ignored these regions in training VQ-VAE as well, the re-  constructed regions are still abnormal (see Figure 3) and  further affect the training of the task-solver and make the  final result also have many artifacts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Second, a token pre-  dicted by VQ-VAE corresponds to a 322 patch, which may  contain both normal and corrupted pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Therefore, it is  hard to deal with this issue by ignoring the tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  As shown in Figure 4, we present to introduce mask  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Illustration of our masked augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In order to  make the tokenizer have a stronger complement capability, we add  random mask noise under a certain mask ratio and patch size to the  original GT depth maps as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' But we still use the GT depth  maps to apply reconstruction loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  augmentation in the training of VQ-VAE to alleviate this  challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Specifically, we randomly mask some regions  in the input depth images and then use their original depth  information as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In this way, the VQ-VAE can  complete/recover some corrupted regions with reasonable  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Figure 3 shows the visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In Table 8, we no-  tice that applying mask augmentation can improve the per-  formance of depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Tasks and Datasets  To examine the generalizability of our framework, we  choose depth estimation and instance segmentation, which  are two tasks with very different output spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Instance Segmentation  The instance segmentation re-  quires predicting the location, the class, and the mask of  each instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The COCO2017 benchmark is one of the  most challenging datasets for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' It consists of 117K  training images, 5K validation images, and 41K test im-  ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  A total of 80 classes annotation are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  In  our experiments, we follow the common setting of previ-  ous works [17, 40, 44] that report the performance on the  validation set for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Depth Estimation  Depth estimation is a fundamental  problem in computer vision, which requires estimating the  depth for each pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Unlike segmentation whose output is  a binary mask, the depth map is a floating point image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In  this work, we use the NYUv2 Depth dataset, which consists  of 24K training images and 654 validation images, and the  RMSE is used as the major metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Implementation Details  We train two separate VQ-VAE for depth estimation and  instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The input image size used in depth  VQVAE  Codebook  Encoder  Decoder  0  1  2  N-1 Nis 4802, with a batch size of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The Adam optimizer is used  with the base learning rate of 1e-3, α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='9, α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  An exponential learning rate schedule is applied with the  learning rate decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='98 and a total of 20 training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  For instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The input image size is 642 with  a batch size of 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The Adam optimizer is used with the  base learning rate of 3e-4, α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='9, α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' An ex-  ponential learning rate schedule is applied with the learning  rate decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='98 and a total of 20 training epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' By  default, the EMA model update technique is used for all  VQ-VAE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  For the task-solver,  we adopt the auto-regressive  encoder-decoder architecture, which is similar to Pix2Seq  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' It consists of a backbone, 6 encoder layers, and 6 de-  coder layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We use the Swin-B as the backbone, which  is pre-trained with SimMIM [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Most experiments in  the ablation study are separately trained in depth estima-  tion and instance segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In depth estimation, we  use the AdamW optimizer with a base learning rate of 2e-4,  α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='9, α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='999, a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='05, and a layer  decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The total training length is 25 epochs and the  batch size is 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The step learning rate schedule is used and  the learning rate dropped to 2e-5 at the 18th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For data  argumentation, the random cropping of 4802 and horizon-  tal flip with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5 are employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We also append  random brightness contrast, random gamma, and hue sat-  uration value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In instance segmentation, the AdamW op-  timizer with a base learning rate of 1e-4, alpha1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='9,  alpha2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='999, a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='05, and a layer decay  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='85, linear decay learning rate scheduler is applied, and  the total training length is 50 epochs with a batch size of 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  For data augmentation, large-scale jittering with the range  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1 to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0 and crop size of 6402 are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation Study  We ablate the key design choices and techniques in this  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' By default, we train the model separately for each  task in the ablation study for better illustration, and Swin-B  is used as the default backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation study on codebook size of VQ-VAE on depth  and instance segmentation in reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Width #tokens Depth Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Mask mAP  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1025  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='94  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0918  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='34  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0902  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='22  Architecture of VQ-VAE  We study how different de-  signs of VQ-VAE affect performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We first evaluate the  reconstruction performance of different codebook sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' To  more accurately and intuitively observe the reconstruction  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation study on codebook size of VQ-VAE on depth  and instance segmentation in task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Width #tokens Depth  Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Box mAP Mask mAP  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3105  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3084  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  256  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3098  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='4  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='4  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation study on the width of VQ-VAE on depth and  instance segmentation in reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Width #tokens Depth Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Mask mAP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1196  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='97  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0918  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='34  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1025  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='92  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation study on the width of VQ-VAE on depth and  instance segmentation in task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Width #tokens Depth  Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Box mAP Mask mAP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3171  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='4  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3084  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0×  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3151  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1  performance, our evaluation is performed on the validation  set of different tasks and adopts mask mAP and RMSE as  metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The results are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We find that  although the large codebook size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 256) benefits the re-  construction performance, the small codebook size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 64)  can also yield sufficiently good reconstruction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Further applying to the task-solver, we found different code-  book size has little effect on final performance (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Using the same evaluation method, we study the effect  of the width of VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Table 3 shows the reconstruction  performance and Table 4 shows the performance of apply-  ing to task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Similar to the observation on codebook  size, we find that the network width has little effect on the  final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  The downsample ratio of VQ-VAE is another key that  may affect network performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We vary the downsam-  ple ratio in [8, 16, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We note that the instance segmen-  tation cannot support a downsample ratio of 8 because it  results in too long sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Table 5 shows the recon-  struction performance, as the downsample ratio increases,  the reconstruction performance gets worse, satisfying the  intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' However, we find that better reconstruction per-  formance does not always lead to better performance when  applying the VQ-VAE in task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In Table 6, the best  performance is achieved at downsample ratio of 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We ex-  plain this phenomenon is that a smaller downsample ratio  facilitates reconstruction, but it also increases the length of  the token sequence, which is detrimental to the task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation study on the downsample ratio of VQ-VAE on  depth and instance segmentation in reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Downsample Ratio Depth Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Mask mAP  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0918  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='91  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0696  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='34  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0515 Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation study on the downsample ratio of VQ-VAE on  depth and instance segmentation in task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Downsample Ratio Depth  Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Box mAP Mask mAP  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3084  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='7  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3  16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3280  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3303 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation on the effectiveness of soft token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Description  Depth  Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE  box mAP  mask mAP  Baseline (hard-inf)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3163  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1  On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' task-solver  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3108  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1  On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' detokenizor  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3113  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3  On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' both  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3084  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' both + aux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3071  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  Soft Token  To leverage the correlation between tokens,  we introduce the soft token techniques, which can be used to  improve the performance in the inference stage for free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We  examined this technique in Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Compared with the hard  inference baseline, applying the soft token in task-solver  and detokenzior alone can bring performance gains, and  further performance improvement is achieved when used in  two stages at the same time: the depth performance is im-  proved by +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='008 RMSE and the instance segmentation is  improved by +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1 mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' On top of it, adding the auxiliary  loss on the output of detokenizor enlarges the gain to +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='009  RMSE and +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1 mAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Affects of mask augmentation in training VQ-VAE on  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The patch size of all models is set to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Mask Ratio VQ-VAE task-solver  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0831  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3138  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0893  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3135  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0918  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3084  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1196  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3194  Mask Augmentation  We evaluate the effectiveness of  mask augmentation in depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The different mask  ratios vary from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='7 are used, and Table 8 shows the  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The best performance is achieved by using the mask  ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5, which is +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='005 better than the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Results of monocular depth estimation task on  NYUv2 [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Both AiT and AiT-P are our methods, where AiT  indicates auto-regressive prediction, and AiT-P indicates parallel  prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Method  RMSE ↓  δ1 ↑  δ2 ↑  δ3 ↑  REL ↓  log10 ↓  DORN [15]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='509  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='828  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='965  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='992  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='051  BTS [23]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
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+page_content=' Preliminary Study on Parallel Prediction  In previous sections, we mainly demonstrate the use of  auto-regressive models to unify various visual tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Note  the bi-directional or even qua-directional natural may en-  courage parallel decoding approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Therefore, we make  some preliminary studies on the parallel decoder instead of  the auto-regressive decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  As parallel decoder has been prevalent in object de-  tection and instance segmentation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' DETR [7] and  Mask2Former [11], we mainly studied the parallel decoder  for the depth estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' As shown in Table 9,  the parallel decoder performs better than the auto-regressive  counterpart by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='01 RMSE (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='279 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='289).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' This indicates  a strong potential than the auto-regressive models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Our techniques such as soft token also benefit the parallel  decoder, as shown in Table 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' This implies the generality of  the proposed techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Comparison with Other Unified Frameworks  and State-of-the-arts in Depth Estimation  UViM and Unified-IO are the most relevant works to  ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  We compare the performance with these methods  on the overlap task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' NYUv2 depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The re-  sults are shown in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Our auto-regressive approach  achieves 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='289 RMSE, which is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='178 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='096 better than  UViM and Unified-IO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Moreover, our parallel approach fur-  ther improves the performance to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='279 RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' This re-  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Visualization on instance segmentation task of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Visualization of depth estimation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' With the well-designed structure and techniques, our model is capable for this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  sult surpasses previous state-of-the-arts by +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='008 RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  While UViM and Unified-IO mainly conceptually propose  unified frameworks for various visual tasks, we push more  solid steps through in-depth study of the general visual task-  solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' One Model for Multiple Tasks  We train the instance segmentation and depth jointly us-  ing a shared task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Table 10 shows that the joint train-  ing with shared model weights has marginal performance  gradation compared to using separate task solvers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Conclusion  In this work, we investigate the unification of output  spaces for various vision tasks by a set of visual tokens, and  Table 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Joint training of depth estimation and instance segmen-  tation using a single task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The performance of joint training  is slightly worse compared to using separate task-solvers for each  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Depth  Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Box mAP Mask mAP  separate training 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3071  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  joint training  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3103  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='1  further develop a unified auto-regressive encoder-decoder  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Two new techniques are proposed which take the  particularity of visual tasks into account to improve the sys-  tem: 1) Soft token can leverage the correlation between  RGB Image  GT Depth Map  OursTable 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation study on increasing the number of residual  blocks on the 16× downsampling setting of depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  #Resblock Tokenizor Task-solver  RMSE  RMSE  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0676  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3229  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0683  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3251  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0695  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3277  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0690  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3293  tokens to improve performance in the inference stage and  enables end-to-end learning for the final visual targets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' 2)  Mask augmentation is used to alleviate the issue of cor-  rupted/undefined areas of visual tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' depth estima-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' With these two techniques, our general method set a  new state-of-the-art on the NYUv2 depth dataset, as well as  competitive accuracy on the COCO 2017 object detection  and instance segmentation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We also conducted  preliminary studies on the parallel decoder which is show  promising results on the depth estimation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Ablation on the Depths of VQ-VAE  In Table 3 and Table 4, we have studied how different  widths of VQ-VAE affect the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' In Table 11,  we further examine the effect of depth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' different num-  ber of residual blocks in VQ-VAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The result shows that  the increasing residual blocks lead to worse performance in  VQ-VAE and task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' VQ-VAE vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Interpolation Tokenizer  We use a lightweight VQ-VAE as the tokenizer to reduce  the resolution and computation for the task-solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' A intu-  itive baseline to perform down-sampling is the interpolation  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For example, in instance segmentation, we can  use a nearest-neighbor interpolation method to up-sample  the binary masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' For depth estimation, we use a bilin-  ear interpolation method and discretize the floating num-  bers between 0 to 10 by 128 bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We select the same in-  put and target resolution as our VQ-VAE tokenizer for a  fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' As shown in Table 12, the tokenizer using  nearest-neighbor interpolation is much worse than our VQ-  VAE method, showing the effectiveness of using VQ-VAE  to represent the task output space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Details for Joint Training  The AdamW optimizer with a base learning rate of 8e-  4, α1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='9, α2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='999, a weight decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='05, and  a layer decay of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='85, linear decay learning rate scheduler  is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We train the model with 300k iterations and  batch size 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' The decoder loss weights of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='0 are  Table 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Comparison of VQ-VAE and interpolation as the tok-  enizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Depth  Instance Seg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  RMSE Box mAP Mask mAP  Ours  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='3084  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='6  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  interpolation 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='7544  43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2  used for instance segmentation and depth estimation respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' A depth loss weight of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' We process these  batch data individually i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' different tasks do task-specific  augmentation and generate their discrete training token se-  quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  References  [1] Jean-Baptiste Alayrac, Jeff Donahue, Pauline Luc, Antoine  Miech, Iain Barr, Yana Hasson, Karel Lenc, Arthur Mensch,  Katie Millican, Malcolm Reynolds, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Flamingo: a vi-  sual language model for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' arXiv preprint  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
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+page_content=' Data2vec: A general framework  for self-supervised learning in speech, vision and language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
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+page_content=' Beit: Bert pre-training  of image transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content='  arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
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+page_content=' In CVPR,  pages 4009–4018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
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+page_content=' Brostow, Moustapha  Ciss´e, Giovanni Maria Farinella, and Tal Hassner, editors,  ECCV, volume 13661 of Lecture Notes in Computer Science,  pages 480–496.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
+page_content=' Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE0T4oBgHgl3EQfTAAm/content/2301.02229v1.pdf'}
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+A Digital Twin Assisted Framework for Interference
+Nulling in Millimeter Wave MIMO Systems
+Yu Zhang, Tawfik Osman, and Ahmed Alkhateeb
+Abstract—Millimeter wave (mmWave) and terahertz MIMO
+systems rely on pre-defined beamforming codebooks for both
+initial access and data transmission. However, most of the
+existing codebooks adopt pre-defined beams that focus mainly
+on improving the gain of their target users, without taking
+interference into account, which could incur critical performance
+degradation in the dense networks. To address this problem, in
+this paper, we propose a sample-efficient digital twin-assisted
+beam pattern design framework that learns how to form the
+beam pattern to reject the signals from the interfering directions.
+The proposed approach does not require any explicit channel
+knowledge or any coordination with the interferers. The adoption
+of the digital twin improves the sample efficiency by better
+leveraging the underlying signal relationship and by incorporat-
+ing a demand-based data acquisition strategy. Simulation results
+show that the developed signal model-based learning framework
+can significantly reduce the actual interaction with the radio
+environment (i.e., number of measurements) compared to the
+model-unaware design, leading to a more practical and efficient
+interference-aware beam design approach.
+I. INTRODUCTION
+Rejecting in-band interference in the RF domain of the
+hybrid or fully analog MIMO systems is possible owing to
+the extra degree of freedom brought by multiple antennas.
+However, realizing its full potential is challenging in practical
+systems due to the lack of channel knowledge as well as
+the additional hardware constraints. As a result, the existing
+approaches normally require large beam learning overhead in
+order to form well-shaped analog beams. Moreover, when
+the codebook design problem is considered, such learning
+overhead gets magnified and increases linearly with respect
+to the codebook size, since the learning experience of one
+beam is not transferable to the others. Such prohibitive over-
+head downplays the potential of the system, and it motivates
+the development of sample efficient interference-aware beam
+codebook design framework, which is the focus of this paper.
+Contributions: In this paper, we develop a model-based
+digital twin-assisted learning framework that achieves higher
+sample efficiency by better leveraging the underlying signal
+model. The improvement on the sample efficiency has the
+potential of reducing the beam learning overhead as well as
+shortening the convergence time of the proposed solution.
+The proposed digital twin-assisted learning framework also
+provides flexibility to the practical deployment in terms of
+enabling data sharing and cooperation (hence better learning)
+Yu Zhang, Tawfik Osman, and Ahmed Alkhateeb are with Arizona State
+University (Email: y.zhang, tmosman, alkhateeb@asu.edu). This work is sup-
+ported in part by the National Science Foundation under Grant No. 1923676
+and by a U. S. Army research program under contract No. W911NF21C0015.
+in complex scenarios. We extensively evaluate the proposed
+interference-aware beam learning framework using numerical
+simulations. This provides a comprehensive assessment of the
+capability of the proposed approach in nulling interference
+without requiring any knowledge about the channel, array
+geometry, or user location. The role of the digital twin model
+is also tested which empirically shows its efficacy in guiding
+the beam learning process and highlights its potential gain of
+reducing the overall learning overhead.
+Prior work: The prior work of interference nulling beam-
+forming design either focuses on MIMO transceivers with
+fully-digital architectures [1]–[3], or assumes some kind of
+channel information, such as covariance, of both the target
+and interferers [4], [5]. Some other approaches that do not
+rely on channel information are also proposed in the past [6],
+[7]. But they, in general, do not leverage underlying signal
+relationship which leads to large beam learning overhead and
+are not robust.
+II. SYSTEM AND CHANNEL MODELS
+We consider the communication system where a mmWave
+MIMO base station (BS), equipped with M antennas, is
+communicating with a single-antenna user equipment (UE) in
+an uplink mode. Moreover, we assume that there exist K (≥ 1)
+non-cooperative interference transmitters1 in the vicinity of the
+BS, operating at the same frequency bands and hence causing
+interference to the BS receiver. Therefore, if the UE transmits
+a symbol x ∈ C to the BS, and the other K interference
+transmitters also transmit symbols xk ∈ C, k = 1, . . . , K at
+the same time and frequency slot, such that all the transmit-
+ted symbols satisfy the same average power constraint, i.e.,
+E[|x|2] = Px and E[|xk|2] = Px, ∀k, the received signal at
+the BS after combining can then be expressed as
+y = wHhx +
+K
+�
+k=1
+wHhkxk + wHn,
+(1)
+where h ∈ CM×1 is the channel between the BS and the
+UE, hk ∈ CM×1 is the channel between the BS and the
+k-th interference transmitter. It is worth pointing out here
+that for clarity, we subsume the factors such as path-loss and
+transmission power into the channels. n ∼ CN(0, σ2I) is the
+receive noise vector at the BS with σ2 being the noise power
+and w ∈ CM×1 is the combining vector used by the BS.
+1For ease of exposition, each interference transmitter is also assumed to
+have a single antenna. This means that the interference signals are being
+transmitted omni-directionally.
+arXiv:2301.13311v1  [cs.IT]  30 Jan 2023
+
+Furthermore, given the high cost and power consumption of
+the mixed-signal components, we consider a practical system
+where the BS has only one radio frequency (RF) chain2 and
+employs analog-only beamforming/combining using a network
+of r-bit quantized phase shifters. Therefore, the combining
+vector at the BS can be written as
+w =
+1
+√
+M
+�
+ejθ1, ejθ2, . . . , ejθM �T ,
+(2)
+where each phase shift θm, ∀m = 1, . . . , M is selected from a
+finite set Ψ with 2r possible discrete values drawn uniformly
+from (−π, π]. The normalization factor M −1/2 is to make sure
+the combiner has unit power, i.e., ∥w∥2
+2 = 1.
+We adopt a geometric channel model for the channel
+between BS and UE, as well as the interference channels be-
+tween BS and any interfering transmitters. Hence, the channel
+between BS and its served UE takes the following form (the
+channel between BS and any interference transmitter takes
+similar form)
+h =
+L
+�
+ℓ=1
+αℓa(φℓ, ϑℓ),
+(3)
+where L is the number of multi-paths. Each path ℓ has a
+complex gain αℓ, which includes the path-loss. The angles φℓ
+and ϑℓ represent the ℓ-th path’s azimuth and elevation angles
+of arrival respectively, and a(φℓ, ϑℓ) is the BS array response
+vector. The exact expression of a(φℓ, ϑℓ) depends on the array
+geometry and possible hardware impairments.
+III. PROBLEM FORMULATION
+In this paper, we investigate the design of the analog com-
+bining/precoding that achieves interference awareness (i.e., at-
+tempts to address the interference) without explicitly knowing
+any channel state information. Given the receive signal (1) at
+the BS, the achievable rate of its target user can be written as
+R = log2
+�
+1 +
+|wHh|2Px
+�K
+k=1 |wHhk|2Px + σ2
+�
+.
+(4)
+The objective is to design the combining vector w such that the
+achievable rate of the target user, i.e., (4), can be maximized.
+Given the monotonicity of the logarithm function, this is
+equivalent to maximize the SINR term in (4). Therefore, the
+problem can be cast as
+w⋆ = arg max
+w
+|wHh|2Px
+�K
+k=1 |wHhk|2Px + σ2 ,
+(5)
+s. t. wm =
+1
+√
+M
+ejθm, ∀m = 1, ..., M,
+(6)
+θm ∈ Ψ, ∀m = 1, ..., M,
+(7)
+2It is very important to note that the RF precoder in a system with
+hybrid architecture is normally constructed using pre-defined codebooks that
+have pre-determined beams. Therefore, the learned beams in this paper can
+be included in such codebooks and be used in the hybrid analog/digital
+architectures as well.
+where wm is the m-th element of the combining vector w 3.
+Solving the interference-aware beam pattern design problem
+formulated in (5) is challenging due to the non-convex and
+discrete hardware constraints (6) and (7), as well as the
+unknown channels in the objective function (5). Therefore,
+it is hard to solve (5) using the conventional optimization
+methods [1]–[3]. An important observation, however, is that
+for a given combining beam w, evaluating the SINR requires
+only the power values (after combining) of the desired and
+interference signals, and does not require explicit knowledge
+about the channel vectors. With this observation, we cast our
+problem as developing an online machine learning approach
+that learns how to design an interference-aware beam pat-
+tern w that optimizes (5), given only the receive power
+measurements for the signal plus interference and noise,
+��wHh
+��2 Px + �K
+k=1
+��wHhk
+��2 Px + σ2, and the interference
+plus noise, �K
+k=1
+��wHhk
+��2 Px + σ2.
+IV. ONLINE LEARNING OF INTERFERENCE AWARE BEAM
+PATTERN DESIGN
+In this section, we present the proposed online reinforce-
+ment learning based interference-aware beam pattern learning
+approach. To solve the problem with reinforcement learning,
+we first fit all the ingredients of problem (5) into a general
+reinforcement learning framework as follows:
+• State: We define the state st as a vector that consists of
+the phases of all the phase shifters at the t-th iteration,
+that is, st = [θ1, θ2, . . . , θM]T .
+• Action: We define the action at as the element-wise
+changes to all the phases in st. Since the phases can
+only take values in Ψ, a change of a phase represents
+the action that a phase shifter selects a value from Ψ.
+Therefore, the action is directly specified as the next state,
+i.e., at = st+1, which can be viewed as a deterministic
+transition in the Markov Decision Process (MDP).
+• Reward: We define a binary reward mechanism, i.e.,
+the reward rt takes values from {+1, −1}. Since the
+objective of (5) is to maximize the SINR performance,
+we compare the SINR achieved by the current combining
+vector, denoted as SINRt, with the previous one, i.e.,
+SINRt−1. The reward is determined according to the
+following rule: rt = +1, if SINRt > SINRt−1; rt = −1,
+otherwise.
+Deep Reinforcement Learning Architecture: Given the re-
+inforcement learning formulation above for the interference-
+aware beam learning problem, we adopt an actor-critic based
+deep reinforcement learning architecture. This follows the
+learning framework that we proposed earlier in [8]. In sum-
+mary, both the actor and critic networks are implemented using
+elegant fully-connected (FC) feed-forward neural networks.
+The input of the actor network is the state and the output
+3It is important to note that the proposed interference-aware beam learning
+approach can be straightforwardly extended to learning a codebook with
+multiple beams by, for example, using the user clustering and assignment
+algorithm proposed in [8].
+
+is the action, while the critic network takes in the state-action
+pair and outputs the predicted Q value.4 Moreover, to respect
+the discrete phase shifter hardware constraint (7), we perform
+an element-wise quantization to make the predicted action a
+valid one. To be more specific, assume that �at is the predicted
+action from the actor network at time t. Then, the action that
+finally gets implemented to the system is given by
+[at]m = arg min
+θ∈Ψ
+|[�at]m − θ| , ∀m = 1, . . . , M.
+(8)
+It is worth emphasizing that such quantization operation is
+only activated when the system is actually implementing the
+predicted action by the actor network to obtain reward. It is
+not involved in the training process of the actor network due
+to its non-differentiability.
+Despite its full compatibility with the considered system,
+the proposed interference-aware beam learning solution still
+has two drawbacks. First, it requires a relatively large number
+of iterations to find a qualified beam pattern, especially when
+the number of antennas is large. As a result, this incurs a large
+beam learning overhead, since these iterations are done over
+the air. Second, as indicated by the objective function of (5),
+the SINR performance of a given beam is determined by two
+factors: (i) The desired beamforming gain and (ii) The effec-
+tiveness of suppressing the undesired interference. However,
+the proposed solution does not fully leverage this information
+as it only focuses on the overall SINR performance. It turns out
+that the decomposition of these two factors, as will be further
+discussed in the next section, makes the data sharing among
+the learning processes of different beams possible, which has
+the potential of improving the convergence behavior of the
+beam/codebook learning algorithm.
+V. DIGITAL TWIN ASSISTED BEAM LEARNING
+FRAMEWORK
+In this section, we describe in detail the proposed dig-
+ital twin assisted interference-aware beam pattern learning
+framework. The motivations of introducing the digital twin
+are mainly two-folds. First, it has the potential of improving
+the sample efficiency (i.e., reducing the number of interac-
+tions with the actual environment) of the learning process
+[9]. Second, it facilitates other more complex tasks (than
+learning a single beamforming vector), such as data sharing
+(which can be very useful in learning interference-aware beam
+codebooks) and cooperative learning5 (among multiple BSs to
+avoid interfering each other).
+A. Digital Twin for Beam Pattern Learning
+In this subsection, we introduce the proposed digital twin
+that assists the learning of interference-aware beams. As men-
+tioned before, in order to acquire the reward signal that is used
+4The detailed architectures and the parameters of the adopted neural
+networks are provided in Section VI-A.
+5For instance, as the system has full knowledge of its simulated environ-
+ment, it can assign accurate reward to each agent. This has the potential of
+mitigating the non-stationary environment problem that exists in most of the
+multi-agent learning tasks.
+for training the RL agent, the system needs to estimate two
+quantities, i.e., the signal power, PS =
+��wHh
+��2 Px, and the in-
+terference plus noise power, PI+N = �K
+k=1
+��wHhk
+��2 Px+σ2.
+Therefore, correspondingly, there are two major components
+in the considered digital twin that provide the agent with
+such information, i.e., an interference predictor and a signal
+predictor, as will be discussed in this subsection.
+1) The key idea of digital twin: The machine learning
+model that virtually interacts with the agent can be considered
+as a digital twin. This model is used to imitate the behavior
+of the actual environment, aiming to reduce the expensive
+(sometimes, even impossible) actual evaluations of the design.
+In this paper, we design the digital twin with a particular
+emphasis on two important aspects. First, a digital twin should
+be able to accurately model the behavior of the actual envi-
+ronment, i.e., having accurate predictions. Second, training
+a digital twin should, in general, require less actual data
+samples than directly interacting with the actual environment,
+which yields a high sample efficiency. With these important
+criterions in mind, we next describe the adopted digital twin.
+As mentioned before, the considered digital twin consists of
+two major components, i.e., an interference prediction model
+and a signal prediction model. Formally, the interference
+prediction model predicts the interference plus noise power
+based on the configuration of the receive combining vector,
+which can be expressed as
+�PI+N = fin(w; Θin),
+(9)
+where w ∈ CM×1 is the input of the model, representing
+the designed receive combining vector, and the output is the
+predicted interference plus noise power, i.e., �PI+N ∈ R. The
+model is parameterized by Θin. Similarly, the signal prediction
+model predicts the signal power of a given receive combining
+vector, which can be written as
+�PS = fs(w; Θs),
+(10)
+where �PS ∈ R is the predicted signal power value and Θs
+denotes the model parameters. It is worth mentioning that the
+architecture of fin and fs is not unique and is a design choice.
+Next, we present two candidates that could be used in the
+considered beam learning task.
+2) Digital twin architecture: In this paper, we study two
+specific designs: (i) A model-based prediction architecture,
+and (ii) a fully-connected neural network based prediction
+architecture.
+Model-based architecture: The model-based architecture, as
+its name suggests, is inspired by the underlying signal model.
+For instance, as can be seen from the expression of the interfer-
+ence plus noise power, i.e., PI+N = �K
+k=1
+��wHhk
+��2 Px + σ2,
+it takes a quadratic form of the receive combining vector w.
+To see this, by defining H = [h1, h2, . . . , hK], PI+N can be
+
+expressed as
+PI+N =
+��HHw
+��2
+2 Px + σ2,
+(11)
+= wH �
+PxHHH + σ2I
+�
+w,
+(12)
+= wHAw,
+(13)
+where A = PxHHH+σ2I. The signal power can be expressed
+in the similar form, i.e., PS = wHPxhhHw. Therefore,
+the interference prediction network is essentially leveraged
+to learn the relationship (13). Inspired by this, we design
+the interference prediction network with a focus on imitating
+the “behavior” of A. Specifically, the interference prediction
+network is chosen to take the following form
+fin(w) = wHQinQH
+inw,
+(14)
+where Qin
+∈ CM×rin with rin being a hyperparameter.
+Therefore, the parameter of the interference prediction network
+is essentially Qin, i.e., Θin = {Qin}. The signal prediction
+network takes the similar form, i.e., fs(w) = wHQsQH
+s w,
+where Qs ∈ CM×rs with rs being a hyperparameter as well,
+which makes Θs = {Qs}.
+Fully-connected neural network based architecture: De-
+spite being lightweight and a better fit to the signal model,
+the model-based architecture, fundamentally, suffers from any
+mismatch between the assumed signal model and the actual
+signal relationship. For instance, there are normally unknown
+non-linearities in the practical hardware that undermine the
+validity of the assumed relationship between the receive com-
+bining vector and the interference plus noise power (similarly
+for the signal power). As a result, the signal model cannot
+always be met and the model-based architecture will show up
+certain level of residual prediction errors that are very hard
+to be eliminated given the less powerful expressive capability
+of its architecture. Motivated by this, we also investigate a
+more general architecture, which is built upon fully-connected
+neural network, given its powerful universal approximation
+capability [10]. Specifically, both fin and fs are modeled with
+feed-forward fully-connected neural networks. The detailed
+network parameters will be provided in Section VI-A.
+3) Training dataset and loss function: We denote the train-
+ing dataset of the interference prediction network as
+Din =
+��
+w(n), P (n)
+I+N
+�Nin
+n=1
+�
+,
+(15)
+where each data sample is comprised of a combining vector
+and its corresponding interference plus noise power value
+obtained from the actual environment, i.e., from the real
+measurement. Nin is the total number of data samples in the
+dataset, i.e., |Din| = Nin. And the dataset used for training
+the signal prediction network can be similarly denoted as
+Ds =
+��
+w(n), P (n)
+S
+�Ns
+n=1
+�
+,
+(16)
+with Ns being its size. Since the target of these two networks
+is to predict the power values, we pose the learning problem
+as a regression problem conducted in a supervised fashion.
+RF
+Chain
+RL
+Agent
+Actual Environment
+RF Frontend
+Beam
+Reward
+Measurement
+Module
+Calculate SINR
+Dataset
+Store
+Measurements
+Interference dataset
+can be shared among agents
+Interference link
+Switching
+Control
+Training
+Control
+Digital Twin
+Fig. 1. An illustration of the proposed digital twin-assisted interference-aware
+beam pattern learning framework.
+Furthermore, we employ mean squared error (MSE) as the
+training loss function. Using the interference prediction net-
+work as an example, for the n-th data sample in Din, the loss
+function is defined as
+L
+�
+P (n)
+I+N, �P (n)
+I+N
+�
+=
+���P (n)
+I+N − �P (n)
+I+N
+���
+2
+,
+(17)
+=
+���P (n)
+I+N − fin(w(n); Θin)
+���
+2
+.
+(18)
+The loss function used for the signal prediction network is
+identical.
+B. Digital Twin Assisted Learning
+In this subsection, we discuss how to integrate the digital
+twin with the proposed RL based beam learning framework.
+Since the digital twin is essentially used to provide the RL
+agent with a simulated environment to interact with, it plays
+the same role as the actual environment. However, in order
+to provide high quality synthetic feedback, it requires training
+process that relies on the authentic data collected from the
+actual environment. Based on the trained digital twin, the
+system can virtually evaluate its designed beams without
+measuring the physical signals. Moreover, the system might
+require constantly switching between the digital twin and the
+actual environment, triggered by the demand for the authentic
+data. Next, we summarize the key components of the proposed
+digital twin assisted beam learning.
+Initial interaction and data acquisition: The system starts
+with the normal interaction between the RL agent and the
+actual environment. To be more specific, upon forming a new
+beam ˜w, the BS estimates the interference plus noise power
+PI+N and the signal power PS. The reward signal used for RL
+agent learning will then be generated. Moreover, these authen-
+tic power measurements together with the beam will be stored
+in the two datasets, i.e., Din and Ds, respectively. During this
+interaction process, two initial datasets are established.
+Digital twin training: Based on the collected initial datasets
+Din and Ds, the two sub-networks of the digital twin, i.e., the
+interference prediction network fin and the signal prediction
+network fs, are trained in a supervised manner. After the
+training process saturates, the digital twin is ready to interact
+with the RL agent.
+
+Environment switching and virtual interaction: The switch-
+ing from the actual environment to the digital twin is triggered
+based on multiple factors, for example, when the interferers are
+not transmitting signal or when the digital twin can provide
+accurate predictions, etc. As a result, after the switching is
+finished, the reward signal required by the RL agent will
+be provided by the trained digital twin instead of the actual
+environment. The agent keeps interacting with the digital twin
+until it does not improve, which marks the saturation of the
+agent learning and the end of the virtual interaction process.
+Demand based switching and active data acquisition: The
+system might require executing the above steps multiple times,
+based on the achieved performance. The motivation of such
+repetition can be summarized as follows. From the model
+training perspective, the quality of the collected datasets,
+i.e., Din and Ds, has significant influence on the prediction
+accuracy of the trained digital twin. To be more specific, during
+the initial interaction process, most of the beams tried out by
+the agent are relatively random and hence have relatively poor
+quality in terms of SINR performance. This means that the
+datasets are, intuitively speaking, biased towards the “poor-
+quality” beams. As a result, the trained digital twin will have
+relatively inaccurate predictions on the beams that actually
+have better performance. The incurred residual prediction error
+will in turn influence the learning of the agent, leading to
+unsatisfactory performance.
+However, as the policy of the RL agent gets improved over
+time, the actions performed by the agent, i.e., the beams,
+are more likely to be in the beam space where the achieved
+SINR is high. Therefore, it is advisable to switching back
+to the actual environment to re-collect data (through agent-
+environment interaction). Such active data acquisition can
+enhance the training datasets with “high-quality” beams. Using
+those better data samples to refine the parameters of the
+digital twin can help achieve higher prediction accuracy in the
+interested beam space, which further helps the learning of the
+agent. By alternatingly performing these steps, the system has
+higher chance to collect data samples that are more useful for
+the agent learning, which has the potential of further enhancing
+both sample efficiency and learning convergence. We show
+such interplay between the RL agent, actual environment and
+the digital twin in Fig. 1.
+VI. SIMULATION RESULTS
+A. Deep Learning Models and Training Procedures
+1) DRL agent architecture: Since the input of the actor
+network is the state and the output is the action, the size of
+both the input and output of the actor network is M, i.e.,
+the number of antennas. The critic network takes in the state-
+action pair and outputs the predicted Q value and hence it has
+an input size of 2M and an output size of 1. Both the actor
+and critic networks have two hidden layers in our proposed
+architecture, with the size of the first hidden layer being 16
+times of the input size and the size of the second hidden layer
+being 16 times of the output size in both networks. All the
+hidden layers are followed by the batch normalization layer
+TABLE I
+HYPER-PARAMETERS FOR DIGITAL TWIN TRAINING
+Parameter
+Model-based
+FC-based
+Batch size
+512
+512
+Number of epochs
+500
+500
+Optimizer
+Adam
+Adam
+Initial learning rate
+1 × 10−1
+1 × 10−2
+Learning rate schedule
+0.1@[50, 300, 400]
+0.1@[100, 300, 400]
+for an efficient training experience and the Rectified Linear
+Unit (ReLU) activation layer. The output layer of the actor
+network is followed by a Tanh activation layer scaled by π
+to make sure that the predicted phases are within (−π, π]
+interval. The output layer of the critic network is a linear
+layer. Moreover, we adopt the same DRL architecture for both
+solutions, regardless of having digital twin or not.
+2) Digital twin architecture: We describe the two different
+architectures of the digital twin studied in this paper. Also,
+as the signal prediction network and the interference pre-
+diction network have identical architecture in both solutions
+(i.e., model-based solution and fully-connected neural network
+based solution), for brevity, we only use the interference
+prediction network as an example.
+Signal model-based prediction network: As mentioned be-
+fore in (14), the interference prediction network is essentially
+devised to take a quadratic form of the combining vector
+determined by a positive semi-definite matrix QinQH
+in, leaving
+the matrix Qin to be the model parameter. Moreover, Qin has
+a shape of M ×rin with M being the number of antennas and
+rin being a hyper-parameter. The choice of rin is empirically
+guided by the following rules: (i) rin should not be too large as
+it will increase the model complexity and hence the required
+amount of training data; (ii) rin should not be too small as
+it will limit the expressive capability of the model, leading to
+unsatisfactory prediction accuracy.
+Fully-connected neural network based prediction network:
+We adopt the fully-connected neural network with two hidden
+layers to be the interference prediction network. The input
+layer of the network has M neurons, which is equal to the
+number of antennas. The output layer of the network has only
+one neuron with linear activation. Both hidden layers have
+M ′ neurons. Similar to rin in the model-based architecture,
+the selection of M ′ needs to strike a balance between model
+complexity and model expressive capability. Moreover, all the
+hidden layers are followed by the batch normalization layer
+and ReLU activation layer.
+3) Training parameters: As mentioned before, the digital
+twin is trained in a supervised fashion, based on the collected
+power datasets, i.e., Din and Ds. Moreover, the interference
+prediction network and the signal prediction network are inde-
+pendently trained. However, for the same type of digital twin,
+i.e., either model-based or fully-connected neural network
+based, we adopt the same training parameters for interference
+and signal prediction networks. We summarize the detailed
+hyper-parameters used for training the digital twins in Table I.
+
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+Number of training samples
+10-3
+10-2
+10-1
+100
+101
+102
+MSE
+FC-based network with 10k training samples (signal)
+Signal model based network (signal)
+FC-based network with 10k training samples (interference)
+Signal model based network (interference)
+(a) Signal and interference prediction (M = 8)
+0
+2000
+4000
+6000
+8000
+10000
+Number of iteration
+15
+10
+5
+0
+5
+10
+15
+20
+25
+SIR (dB)
+Actual environment based
+(b) Interaction with the actual environment
+0
+2000
+4000
+6000
+8000
+10000
+Number of iteration
+15
+10
+5
+0
+5
+10
+15
+20
+25
+SIR (dB)
+Digital twin based
+(c) Interaction with the digital twin
+Fig. 2. The performance of the proposed digital twin-assisted beam learning framework. In (a), we compare the proposed signal model-based digital twin
+design with the FC-based design to show the significant reduction on the required real measurements. In (b) and (c), we show the learning experience of the
+DRL agent when interacting with the actual environment and the trained digital twin, to highlight the efficacy of such “virtual” environment.
+B. Numerical Results
+In this subsection, we provide the simulation results of
+the proposed digital twin-assisted interference-aware beam
+learning solutions. We first evaluate the prediction accuracy
+of the two proposed prediction network architectures, which
+provides insight on how much data samples are required
+in order to have a reasonable performance as well as the
+practicality of the solutions. We show the prediction accuracy
+of both the signal power and the interference power. As can be
+seen, the signal model-based architecture requires much less
+data samples to achieve higher prediction accuracy than the
+FC-based architecture trained with much more data samples.
+For instance, as indicated in Fig. 2(a), with only 50 samples,
+the signal model-based prediction architecture can achieve
+even more accurate interference prediction than the FC-based
+architecture trained with 10, 000 samples. This saves almost
+99.5% of the measurements, yielding a more sample-
+efficient solution for the practical system deployment.
+Moreover, as there are more data samples, the prediction
+accuracy of the signal model-based architecture also gets
+improved quite significantly. Such performance is achieved by
+better leveraging the underlying signal relationships and hence
+the model parameters are essentially searched over a much
+smaller space. The trained digital twin is utilized to interact
+with the DRL agent, aiming to reduce the expensive actual
+measurements conducted by the hardware. In Fig. 2, we show
+the performance of the DRL agent when interacting with the
+actual environment as well as the digital twin. The training
+of the DRL agent is repeated for 100 times and the average
+performance as well as the standard deviation are reported in
+Fig. 2. We test the performance of a system with 8 antennas,
+and the digital twin is trained using 1, 000 data samples, i.e.,
+|Din| = |Ds| = 1000. As can be seen, the learning experience
+based on the digital twin is quite similar to that of the one
+based on the actual environment. This empirically shows the
+effectiveness of using the digital twin in training the DRL
+agent. As a result, although the DRL agent requires almost
+a total number of 5, 000 interactions with the environment to
+converge, in the digital twin assisted learning framework,
+all these interactions are with the digital twin and hence
+the expensive evaluations on the real hardware are avoided.
+VII. CONCLUSION
+In this paper, we developed a sample-efficient digital twin-
+assisted online interference-aware beam design framework.
+The proposed solution learns how to design beam patterns that
+can effectively manage interference, relying only on the power
+measurements and without any channel knowledge. The design
+of the digital twin leverages the underlying signal relationship,
+leading to a significant reduction on the required interactions
+with the actual environment. Moreover, it also facilitates other
+tasks such as interference-aware codebook learning, where the
+data sharing among different beam learning agents/engines
+becomes possible. The results highlight the efficacy of the
+trained digital twin in guiding the beam learning process.
+REFERENCES
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+
diff --git a/PNFQT4oBgHgl3EQfYTZl/content/tmp_files/load_file.txt b/PNFQT4oBgHgl3EQfYTZl/content/tmp_files/load_file.txt
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@@ -0,0 +1,436 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf,len=435
+page_content='A Digital Twin Assisted Framework for Interference  Nulling in Millimeter Wave MIMO Systems  Yu Zhang, Tawfik Osman, and Ahmed Alkhateeb  Abstract—Millimeter wave (mmWave) and terahertz MIMO  systems rely on pre-defined beamforming codebooks for both  initial access and data transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' However, most of the  existing codebooks adopt pre-defined beams that focus mainly  on improving the gain of their target users, without taking  interference into account, which could incur critical performance  degradation in the dense networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' To address this problem, in  this paper, we propose a sample-efficient digital twin-assisted  beam pattern design framework that learns how to form the  beam pattern to reject the signals from the interfering directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  The proposed approach does not require any explicit channel  knowledge or any coordination with the interferers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The adoption  of the digital twin improves the sample efficiency by better  leveraging the underlying signal relationship and by incorporat-  ing a demand-based data acquisition strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Simulation results  show that the developed signal model-based learning framework  can significantly reduce the actual interaction with the radio  environment (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', number of measurements) compared to the  model-unaware design, leading to a more practical and efficient  interference-aware beam design approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' INTRODUCTION  Rejecting in-band interference in the RF domain of the  hybrid or fully analog MIMO systems is possible owing to  the extra degree of freedom brought by multiple antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  However, realizing its full potential is challenging in practical  systems due to the lack of channel knowledge as well as  the additional hardware constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As a result, the existing  approaches normally require large beam learning overhead in  order to form well-shaped analog beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, when  the codebook design problem is considered, such learning  overhead gets magnified and increases linearly with respect  to the codebook size, since the learning experience of one  beam is not transferable to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Such prohibitive over-  head downplays the potential of the system, and it motivates  the development of sample efficient interference-aware beam  codebook design framework, which is the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Contributions: In this paper, we develop a model-based  digital twin-assisted learning framework that achieves higher  sample efficiency by better leveraging the underlying signal  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The improvement on the sample efficiency has the  potential of reducing the beam learning overhead as well as  shortening the convergence time of the proposed solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  The proposed digital twin-assisted learning framework also  provides flexibility to the practical deployment in terms of  enabling data sharing and cooperation (hence better learning)  Yu Zhang, Tawfik Osman, and Ahmed Alkhateeb are with Arizona State  University (Email: y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='zhang, tmosman, alkhateeb@asu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This work is sup-  ported in part by the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' 1923676  and by a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Army research program under contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' W911NF21C0015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  in complex scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' We extensively evaluate the proposed  interference-aware beam learning framework using numerical  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This provides a comprehensive assessment of the  capability of the proposed approach in nulling interference  without requiring any knowledge about the channel, array  geometry, or user location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The role of the digital twin model  is also tested which empirically shows its efficacy in guiding  the beam learning process and highlights its potential gain of  reducing the overall learning overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Prior work: The prior work of interference nulling beam-  forming design either focuses on MIMO transceivers with  fully-digital architectures [1]–[3], or assumes some kind of  channel information, such as covariance, of both the target  and interferers [4], [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Some other approaches that do not  rely on channel information are also proposed in the past [6],  [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' But they, in general, do not leverage underlying signal  relationship which leads to large beam learning overhead and  are not robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' SYSTEM AND CHANNEL MODELS  We consider the communication system where a mmWave  MIMO base station (BS), equipped with M antennas, is  communicating with a single-antenna user equipment (UE) in  an uplink mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, we assume that there exist K (≥ 1)  non-cooperative interference transmitters1 in the vicinity of the  BS, operating at the same frequency bands and hence causing  interference to the BS receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Therefore, if the UE transmits  a symbol x ∈ C to the BS, and the other K interference  transmitters also transmit symbols xk ∈ C, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' , K at  the same time and frequency slot, such that all the transmit-  ted symbols satisfy the same average power constraint, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=',  E[|x|2] = Px and E[|xk|2] = Px, ∀k, the received signal at  the BS after combining can then be expressed as  y = wHhx +  K  �  k=1  wHhkxk + wHn,  (1)  where h ∈ CM×1 is the channel between the BS and the  UE, hk ∈ CM×1 is the channel between the BS and the  k-th interference transmitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' It is worth pointing out here  that for clarity, we subsume the factors such as path-loss and  transmission power into the channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' n ∼ CN(0, σ2I) is the  receive noise vector at the BS with σ2 being the noise power  and w ∈ CM×1 is the combining vector used by the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  1For ease of exposition, each interference transmitter is also assumed to  have a single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This means that the interference signals are being  transmitted omni-directionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='13311v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='IT]  30 Jan 2023  Furthermore, given the high cost and power consumption of  the mixed-signal components, we consider a practical system  where the BS has only one radio frequency (RF) chain2 and  employs analog-only beamforming/combining using a network  of r-bit quantized phase shifters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Therefore, the combining  vector at the BS can be written as  w =  1  √  M  �  ejθ1, ejθ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' , ejθM �T ,  (2)  where each phase shift θm, ∀m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' , M is selected from a  finite set Ψ with 2r possible discrete values drawn uniformly  from (−π, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The normalization factor M −1/2 is to make sure  the combiner has unit power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', ∥w∥2  2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  We adopt a geometric channel model for the channel  between BS and UE, as well as the interference channels be-  tween BS and any interfering transmitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Hence, the channel  between BS and its served UE takes the following form (the  channel between BS and any interference transmitter takes  similar form)  h =  L  �  ℓ=1  αℓa(φℓ, ϑℓ),  (3)  where L is the number of multi-paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Each path ℓ has a  complex gain αℓ, which includes the path-loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The angles φℓ  and ϑℓ represent the ℓ-th path’s azimuth and elevation angles  of arrival respectively, and a(φℓ, ϑℓ) is the BS array response  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The exact expression of a(φℓ, ϑℓ) depends on the array  geometry and possible hardware impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' PROBLEM FORMULATION  In this paper, we investigate the design of the analog com-  bining/precoding that achieves interference awareness (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', at-  tempts to address the interference) without explicitly knowing  any channel state information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Given the receive signal (1) at  the BS, the achievable rate of its target user can be written as  R = log2  �  1 +  |wHh|2Px  �K  k=1 |wHhk|2Px + σ2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  (4)  The objective is to design the combining vector w such that the  achievable rate of the target user, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', (4), can be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Given the monotonicity of the logarithm function, this is  equivalent to maximize the SINR term in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Therefore, the  problem can be cast as  w⋆ = arg max  w  |wHh|2Px  �K  k=1 |wHhk|2Px + σ2 ,  (5)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' wm =  1  √  M  ejθm, ∀m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', M,  (6)  θm ∈ Ψ, ∀m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', M,  (7)  2It is very important to note that the RF precoder in a system with  hybrid architecture is normally constructed using pre-defined codebooks that  have pre-determined beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Therefore, the learned beams in this paper can  be included in such codebooks and be used in the hybrid analog/digital  architectures as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  where wm is the m-th element of the combining vector w 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Solving the interference-aware beam pattern design problem  formulated in (5) is challenging due to the non-convex and  discrete hardware constraints (6) and (7), as well as the  unknown channels in the objective function (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Therefore,  it is hard to solve (5) using the conventional optimization  methods [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' An important observation, however, is that  for a given combining beam w, evaluating the SINR requires  only the power values (after combining) of the desired and  interference signals, and does not require explicit knowledge  about the channel vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' With this observation, we cast our  problem as developing an online machine learning approach  that learns how to design an interference-aware beam pat-  tern w that optimizes (5), given only the receive power  measurements for the signal plus interference and noise,  ��wHh  ��2 Px + �K  k=1  ��wHhk  ��2 Px + σ2, and the interference  plus noise, �K  k=1  ��wHhk  ��2 Px + σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' ONLINE LEARNING OF INTERFERENCE AWARE BEAM  PATTERN DESIGN  In this section, we present the proposed online reinforce-  ment learning based interference-aware beam pattern learning  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' To solve the problem with reinforcement learning,  we first fit all the ingredients of problem (5) into a general  reinforcement learning framework as follows:  State: We define the state st as a vector that consists of  the phases of all the phase shifters at the t-th iteration,  that is, st = [θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' , θM]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Action: We define the action at as the element-wise  changes to all the phases in st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Since the phases can  only take values in Ψ, a change of a phase represents  the action that a phase shifter selects a value from Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Therefore, the action is directly specified as the next state,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', at = st+1, which can be viewed as a deterministic  transition in the Markov Decision Process (MDP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Reward: We define a binary reward mechanism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=',  the reward rt takes values from {+1, −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Since the  objective of (5) is to maximize the SINR performance,  we compare the SINR achieved by the current combining  vector, denoted as SINRt, with the previous one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=',  SINRt−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The reward is determined according to the  following rule: rt = +1, if SINRt > SINRt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' rt = −1,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Deep Reinforcement Learning Architecture: Given the re-  inforcement learning formulation above for the interference-  aware beam learning problem, we adopt an actor-critic based  deep reinforcement learning architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This follows the  learning framework that we proposed earlier in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' In sum-  mary, both the actor and critic networks are implemented using  elegant fully-connected (FC) feed-forward neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  The input of the actor network is the state and the output  3It is important to note that the proposed interference-aware beam learning  approach can be straightforwardly extended to learning a codebook with  multiple beams by, for example, using the user clustering and assignment  algorithm proposed in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  is the action, while the critic network takes in the state-action  pair and outputs the predicted Q value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='4 Moreover, to respect  the discrete phase shifter hardware constraint (7), we perform  an element-wise quantization to make the predicted action a  valid one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' To be more specific, assume that �at is the predicted  action from the actor network at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Then, the action that  finally gets implemented to the system is given by  [at]m = arg min  θ∈Ψ  |[�at]m − θ| , ∀m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  (8)  It is worth emphasizing that such quantization operation is  only activated when the system is actually implementing the  predicted action by the actor network to obtain reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' It is  not involved in the training process of the actor network due  to its non-differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Despite its full compatibility with the considered system,  the proposed interference-aware beam learning solution still  has two drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' First, it requires a relatively large number  of iterations to find a qualified beam pattern, especially when  the number of antennas is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As a result, this incurs a large  beam learning overhead, since these iterations are done over  the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Second, as indicated by the objective function of (5),  the SINR performance of a given beam is determined by two  factors: (i) The desired beamforming gain and (ii) The effec-  tiveness of suppressing the undesired interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' However,  the proposed solution does not fully leverage this information  as it only focuses on the overall SINR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' It turns out  that the decomposition of these two factors, as will be further  discussed in the next section, makes the data sharing among  the learning processes of different beams possible, which has  the potential of improving the convergence behavior of the  beam/codebook learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' DIGITAL TWIN ASSISTED BEAM LEARNING  FRAMEWORK  In this section, we describe in detail the proposed dig-  ital twin assisted interference-aware beam pattern learning  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The motivations of introducing the digital twin  are mainly two-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' First, it has the potential of improving  the sample efficiency (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', reducing the number of interac-  tions with the actual environment) of the learning process  [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Second, it facilitates other more complex tasks (than  learning a single beamforming vector), such as data sharing  (which can be very useful in learning interference-aware beam  codebooks) and cooperative learning5 (among multiple BSs to  avoid interfering each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Digital Twin for Beam Pattern Learning  In this subsection, we introduce the proposed digital twin  that assists the learning of interference-aware beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As men-  tioned before, in order to acquire the reward signal that is used  4The detailed architectures and the parameters of the adopted neural  networks are provided in Section VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  5For instance, as the system has full knowledge of its simulated environ-  ment, it can assign accurate reward to each agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This has the potential of  mitigating the non-stationary environment problem that exists in most of the  multi-agent learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  for training the RL agent, the system needs to estimate two  quantities, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', the signal power, PS =  ��wHh  ��2 Px, and the in-  terference plus noise power, PI+N = �K  k=1  ��wHhk  ��2 Px+σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Therefore, correspondingly, there are two major components  in the considered digital twin that provide the agent with  such information, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', an interference predictor and a signal  predictor, as will be discussed in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  1) The key idea of digital twin: The machine learning  model that virtually interacts with the agent can be considered  as a digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This model is used to imitate the behavior  of the actual environment, aiming to reduce the expensive  (sometimes, even impossible) actual evaluations of the design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  In this paper, we design the digital twin with a particular  emphasis on two important aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' First, a digital twin should  be able to accurately model the behavior of the actual envi-  ronment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', having accurate predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Second, training  a digital twin should, in general, require less actual data  samples than directly interacting with the actual environment,  which yields a high sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' With these important  criterions in mind, we next describe the adopted digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  As mentioned before, the considered digital twin consists of  two major components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', an interference prediction model  and a signal prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Formally, the interference  prediction model predicts the interference plus noise power  based on the configuration of the receive combining vector,  which can be expressed as  �PI+N = fin(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Θin),  (9)  where w ∈ CM×1 is the input of the model, representing  the designed receive combining vector, and the output is the  predicted interference plus noise power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', �PI+N ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The  model is parameterized by Θin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Similarly, the signal prediction  model predicts the signal power of a given receive combining  vector, which can be written as  �PS = fs(w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Θs),  (10)  where �PS ∈ R is the predicted signal power value and Θs  denotes the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' It is worth mentioning that the  architecture of fin and fs is not unique and is a design choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Next, we present two candidates that could be used in the  considered beam learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  2) Digital twin architecture: In this paper, we study two  specific designs: (i) A model-based prediction architecture,  and (ii) a fully-connected neural network based prediction  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Model-based architecture: The model-based architecture, as  its name suggests, is inspired by the underlying signal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  For instance, as can be seen from the expression of the interfer-  ence plus noise power, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', PI+N = �K  k=1  ��wHhk  ��2 Px + σ2,  it takes a quadratic form of the receive combining vector w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  To see this, by defining H = [h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' , hK], PI+N can be  expressed as  PI+N =  ��HHw  ��2  2 Px + σ2,  (11)  = wH �  PxHHH + σ2I  �  w,  (12)  = wHAw,  (13)  where A = PxHHH+σ2I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The signal power can be expressed  in the similar form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', PS = wHPxhhHw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Therefore,  the interference prediction network is essentially leveraged  to learn the relationship (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Inspired by this, we design  the interference prediction network with a focus on imitating  the “behavior” of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Specifically, the interference prediction  network is chosen to take the following form  fin(w) = wHQinQH  inw,  (14)  where Qin  ∈ CM×rin with rin being a hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Therefore, the parameter of the interference prediction network  is essentially Qin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', Θin = {Qin}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The signal prediction  network takes the similar form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', fs(w) = wHQsQH  s w,  where Qs ∈ CM×rs with rs being a hyperparameter as well,  which makes Θs = {Qs}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Fully-connected neural network based architecture: De-  spite being lightweight and a better fit to the signal model,  the model-based architecture, fundamentally, suffers from any  mismatch between the assumed signal model and the actual  signal relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' For instance, there are normally unknown  non-linearities in the practical hardware that undermine the  validity of the assumed relationship between the receive com-  bining vector and the interference plus noise power (similarly  for the signal power).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As a result, the signal model cannot  always be met and the model-based architecture will show up  certain level of residual prediction errors that are very hard  to be eliminated given the less powerful expressive capability  of its architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Motivated by this, we also investigate a  more general architecture, which is built upon fully-connected  neural network, given its powerful universal approximation  capability [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Specifically, both fin and fs are modeled with  feed-forward fully-connected neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The detailed  network parameters will be provided in Section VI-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  3) Training dataset and loss function: We denote the train-  ing dataset of the interference prediction network as  Din =  ��  w(n), P (n)  I+N  �Nin  n=1  �  ,  (15)  where each data sample is comprised of a combining vector  and its corresponding interference plus noise power value  obtained from the actual environment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', from the real  measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Nin is the total number of data samples in the  dataset, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', |Din| = Nin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' And the dataset used for training  the signal prediction network can be similarly denoted as  Ds =  ��  w(n), P (n)  S  �Ns  n=1  �  ,  (16)  with Ns being its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Since the target of these two networks  is to predict the power values, we pose the learning problem  as a regression problem conducted in a supervised fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  RF  Chain  RL  Agent  Actual Environment  RF Frontend  Beam  Reward  Measurement  Module  Calculate SINR  Dataset  Store  Measurements  Interference dataset  can be shared among agents  Interference link  Switching  Control  Training  Control  Digital Twin  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' An illustration of the proposed digital twin-assisted interference-aware  beam pattern learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Furthermore, we employ mean squared error (MSE) as the  training loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Using the interference prediction net-  work as an example, for the n-th data sample in Din, the loss  function is defined as  L  �  P (n)  I+N, �P (n)  I+N  �  =  ���P (n)  I+N − �P (n)  I+N  ���  2  ,  (17)  =  ���P (n)  I+N − fin(w(n);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Θin)  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  (18)  The loss function used for the signal prediction network is  identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Digital Twin Assisted Learning  In this subsection, we discuss how to integrate the digital  twin with the proposed RL based beam learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Since the digital twin is essentially used to provide the RL  agent with a simulated environment to interact with, it plays  the same role as the actual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' However, in order  to provide high quality synthetic feedback, it requires training  process that relies on the authentic data collected from the  actual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Based on the trained digital twin, the  system can virtually evaluate its designed beams without  measuring the physical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, the system might  require constantly switching between the digital twin and the  actual environment, triggered by the demand for the authentic  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Next, we summarize the key components of the proposed  digital twin assisted beam learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Initial interaction and data acquisition: The system starts  with the normal interaction between the RL agent and the  actual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' To be more specific, upon forming a new  beam ˜w, the BS estimates the interference plus noise power  PI+N and the signal power PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The reward signal used for RL  agent learning will then be generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, these authen-  tic power measurements together with the beam will be stored  in the two datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', Din and Ds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' During this  interaction process, two initial datasets are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Digital twin training: Based on the collected initial datasets  Din and Ds, the two sub-networks of the digital twin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', the  interference prediction network fin and the signal prediction  network fs, are trained in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' After the  training process saturates, the digital twin is ready to interact  with the RL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Environment switching and virtual interaction: The switch-  ing from the actual environment to the digital twin is triggered  based on multiple factors, for example, when the interferers are  not transmitting signal or when the digital twin can provide  accurate predictions, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As a result, after the switching is  finished, the reward signal required by the RL agent will  be provided by the trained digital twin instead of the actual  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The agent keeps interacting with the digital twin  until it does not improve, which marks the saturation of the  agent learning and the end of the virtual interaction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Demand based switching and active data acquisition: The  system might require executing the above steps multiple times,  based on the achieved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The motivation of such  repetition can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' From the model  training perspective, the quality of the collected datasets,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', Din and Ds, has significant influence on the prediction  accuracy of the trained digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' To be more specific, during  the initial interaction process, most of the beams tried out by  the agent are relatively random and hence have relatively poor  quality in terms of SINR performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This means that the  datasets are, intuitively speaking, biased towards the “poor-  quality” beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As a result, the trained digital twin will have  relatively inaccurate predictions on the beams that actually  have better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The incurred residual prediction error  will in turn influence the learning of the agent, leading to  unsatisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  However, as the policy of the RL agent gets improved over  time, the actions performed by the agent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', the beams,  are more likely to be in the beam space where the achieved  SINR is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Therefore, it is advisable to switching back  to the actual environment to re-collect data (through agent-  environment interaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Such active data acquisition can  enhance the training datasets with “high-quality” beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Using  those better data samples to refine the parameters of the  digital twin can help achieve higher prediction accuracy in the  interested beam space, which further helps the learning of the  agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' By alternatingly performing these steps, the system has  higher chance to collect data samples that are more useful for  the agent learning, which has the potential of further enhancing  both sample efficiency and learning convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' We show  such interplay between the RL agent, actual environment and  the digital twin in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' SIMULATION RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Deep Learning Models and Training Procedures  1) DRL agent architecture: Since the input of the actor  network is the state and the output is the action, the size of  both the input and output of the actor network is M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=',  the number of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The critic network takes in the state-  action pair and outputs the predicted Q value and hence it has  an input size of 2M and an output size of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Both the actor  and critic networks have two hidden layers in our proposed  architecture, with the size of the first hidden layer being 16  times of the input size and the size of the second hidden layer  being 16 times of the output size in both networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' All the  hidden layers are followed by the batch normalization layer  TABLE I  HYPER-PARAMETERS FOR DIGITAL TWIN TRAINING  Parameter  Model-based  FC-based  Batch size  512  512  Number of epochs  500  500  Optimizer  Adam  Adam  Initial learning rate  1 × 10−1  1 × 10−2  Learning rate schedule  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='1@[50, 300, 400]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='1@[100, 300, 400]  for an efficient training experience and the Rectified Linear  Unit (ReLU) activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The output layer of the actor  network is followed by a Tanh activation layer scaled by π  to make sure that the predicted phases are within (−π, π]  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The output layer of the critic network is a linear  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, we adopt the same DRL architecture for both  solutions, regardless of having digital twin or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  2) Digital twin architecture: We describe the two different  architectures of the digital twin studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Also,  as the signal prediction network and the interference pre-  diction network have identical architecture in both solutions  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', model-based solution and fully-connected neural network  based solution), for brevity, we only use the interference  prediction network as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Signal model-based prediction network: As mentioned be-  fore in (14), the interference prediction network is essentially  devised to take a quadratic form of the combining vector  determined by a positive semi-definite matrix QinQH  in, leaving  the matrix Qin to be the model parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, Qin has  a shape of M ×rin with M being the number of antennas and  rin being a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The choice of rin is empirically  guided by the following rules: (i) rin should not be too large as  it will increase the model complexity and hence the required  amount of training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' (ii) rin should not be too small as  it will limit the expressive capability of the model, leading to  unsatisfactory prediction accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Fully-connected neural network based prediction network:  We adopt the fully-connected neural network with two hidden  layers to be the interference prediction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The input  layer of the network has M neurons, which is equal to the  number of antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The output layer of the network has only  one neuron with linear activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Both hidden layers have  M ′ neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Similar to rin in the model-based architecture,  the selection of M ′ needs to strike a balance between model  complexity and model expressive capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, all the  hidden layers are followed by the batch normalization layer  and ReLU activation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  3) Training parameters: As mentioned before, the digital  twin is trained in a supervised fashion, based on the collected  power datasets, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', Din and Ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, the interference  prediction network and the signal prediction network are inde-  pendently trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' However, for the same type of digital twin,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=', either model-based or fully-connected neural network  based, we adopt the same training parameters for interference  and signal prediction networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' We summarize the detailed  hyper-parameters used for training the digital twins in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
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+page_content='10-3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
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+page_content='MSE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='FC-based network with 10k training samples (signal)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='Signal model based network (signal)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='FC-based network with 10k training samples (interference)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='Signal model based network (interference)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='(a) Signal and interference prediction (M = 8)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
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+page_content='SIR (dB)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='Actual environment based  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='(b) Interaction with the actual environment  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
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+page_content='Number of iteration  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
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+page_content='SIR (dB)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='Digital twin based  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='(c) Interaction with the digital twin  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The performance of the proposed digital twin-assisted beam learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' In (a), we compare the proposed signal model-based digital twin  design with the FC-based design to show the significant reduction on the required real measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' In (b) and (c), we show the learning experience of the  DRL agent when interacting with the actual environment and the trained digital twin, to highlight the efficacy of such “virtual” environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Numerical Results  In this subsection, we provide the simulation results of  the proposed digital twin-assisted interference-aware beam  learning solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' We first evaluate the prediction accuracy  of the two proposed prediction network architectures, which  provides insight on how much data samples are required  in order to have a reasonable performance as well as the  practicality of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' We show the prediction accuracy  of both the signal power and the interference power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As can be  seen, the signal model-based architecture requires much less  data samples to achieve higher prediction accuracy than the  FC-based architecture trained with much more data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  For instance, as indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' 2(a), with only 50 samples,  the signal model-based prediction architecture can achieve  even more accurate interference prediction than the FC-based  architecture trained with 10, 000 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This saves almost  99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='5% of the measurements, yielding a more sample-  efficient solution for the practical system deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  Moreover, as there are more data samples, the prediction  accuracy of the signal model-based architecture also gets  improved quite significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Such performance is achieved by  better leveraging the underlying signal relationships and hence  the model parameters are essentially searched over a much  smaller space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The trained digital twin is utilized to interact  with the DRL agent, aiming to reduce the expensive actual  measurements conducted by the hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' 2, we show  the performance of the DRL agent when interacting with the  actual environment as well as the digital twin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The training  of the DRL agent is repeated for 100 times and the average  performance as well as the standard deviation are reported in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' We test the performance of a system with 8 antennas,  and the digital twin is trained using 1, 000 data samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=',  |Din| = |Ds| = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As can be seen, the learning experience  based on the digital twin is quite similar to that of the one  based on the actual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' This empirically shows the  effectiveness of using the digital twin in training the DRL  agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' As a result, although the DRL agent requires almost  a total number of 5, 000 interactions with the environment to  converge, in the digital twin assisted learning framework,  all these interactions are with the digital twin and hence  the expensive evaluations on the real hardware are avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' CONCLUSION  In this paper, we developed a sample-efficient digital twin-  assisted online interference-aware beam design framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  The proposed solution learns how to design beam patterns that  can effectively manage interference, relying only on the power  measurements and without any channel knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The design  of the digital twin leverages the underlying signal relationship,  leading to a significant reduction on the required interactions  with the actual environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' Moreover, it also facilitates other  tasks such as interference-aware codebook learning, where the  data sharing among different beam learning agents/engines  becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content=' The results highlight the efficacy of the  trained digital twin in guiding the beam learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
+page_content='  REFERENCES  [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNFQT4oBgHgl3EQfYTZl/content/2301.13311v1.pdf'}
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+Probing Out-of-Distribution Robustness of Language Models with
+Parameter-Efficient Transfer Learning Methods
+Hyunsoo Cho†, Choonghyun Park†, Junyeop Kim†, Hyuhng Joon Kim†,
+Kang Min Yoo†‡, and Sang-goo Lee†
+† Seoul National University, ‡ NAVER
+{johyunsoo,pch330,juny116,heyjoonkim,sglee}@europa.snu.ac.kr
+{kangmin.yoo}@navercorp.com
+Abstract
+As the size of the pre-trained language
+model (PLM) continues to increase, numer-
+ous parameter-efficient transfer learning meth-
+ods have been proposed recently to compen-
+sate for the tremendous cost of fine-tuning.
+Despite the impressive results achieved by
+large pre-trained language models (PLMs) and
+various parameter-efficient transfer learning
+(PETL) methods on sundry benchmarks, it re-
+mains unclear if they can handle inputs that
+have been distributionally shifted effectively.
+In this study, we systematically explore how
+the ability to detect out-of-distribution (OOD)
+changes as the size of the PLM grows or the
+transfer methods are altered. Specifically, we
+evaluated various PETL techniques, including
+fine-tuning, Adapter, LoRA, and prefix-tuning,
+on three different intention classification tasks,
+each utilizing various language models with
+different scales.
+1
+Introduction
+Pre-trained language models (PLM), which are pre-
+trained on large-scale corpora using transformer-
+based architectures (Vaswani et al., 2017), have
+achieved groundbreaking success on sundry bench-
+marks (Wang et al., 2019b; Rajpurkar et al., 2016;
+Wang et al., 2019a), establishing themselves as the
+standard neural model in countless applications.
+Moreover, language models pre-trained with larger
+parameters on a rich volume of corpora tend to ex-
+hibit more intriguing potentials, such as the ability
+to capture world knowledge (Petroni et al., 2019),
+generate codes (Poesia et al., 2022), and even solve
+mathematical problems (Henighan et al., 2020), on
+top of understanding linguistic knowledge (e.g.,
+semantic or syntactic). To explore the apex of pre-
+trained language models (PLMs), the size of PLMs
+is growing exponentially and has reached billions
+to a trillion (Brown et al., 2020; Chowdhery et al.,
+2022; Fedus et al., 2022; Hoffmann et al., 2022).
+Under these circumstances, the conventional
+method for transferring PLMs to a target task
+(i.e., fine-tuning) is now infeasible as it entails
+prohibitive costs to train and store the entire pa-
+rameters of large PLMs for every desired task.
+To mitigate this issue, several recent parameter-
+efficient transfer learning (PETL) methods have
+been proposed to improve task scalability. For in-
+stance, adapter-based (Houlsby et al., 2019; Hu
+et al., 2022) approaches insert small neural mod-
+ules into each layer of the PLM and update those
+lightweight modules in the training phase. Inspired
+by the recent success of textual prompts (Brown
+et al., 2020), prompt-based methods (Li and Liang,
+2021; Lester et al., 2021; Shin et al., 2020) concate-
+nate extra tunable tokens to the front of the input or
+hidden layers and update prepended soft prompts
+in the training phase.
+Despite these breakthroughs in NLP, even very
+recent anomaly detection studies (Cho et al., 2022;
+Shen et al., 2021) are still limited to relatively small
+bi-directional PLMs (e.g., BERT, RoBERTa). Thus,
+how large-scale PLMs or auto-regressive PLMs
+cope with outliers is uncharted territory, naturally
+begging the following questions:
+• Q1: Does increasing model size improve OOD
+detection performance without model parameters?
+• Q2: If so, does scaling the size of PLM makes
+the model robust enough to utilize them without
+any additional process?
+• Q3: Do fine-tuning and various PETL method-
+ologies display differences in OOD detection per-
+formance according to the size of PLMs?
+• Q4: Can the OOD detection methods from previ-
+ous works (usually for the bi-directional PLMs) be
+transferred to auto-regressive PLMs (GPT)?
+To resolve these questions, this paper investi-
+gates the capability of large PLMs as outlier de-
+tectors from various perspectives. Specifically, we
+compare the robustness to outliers with various
+transfer learning techniques on several OOD bench-
+arXiv:2301.11660v1  [cs.CL]  27 Jan 2023
+
+marks: Full fine-tuning, LoRA (Hu et al., 2022),
+Adapter (Houlsby et al., 2019), and prefix-tuning
+(Li and Liang, 2021) on various auto-regressive
+PLMs with different sizes, i.e., GPT2-S, M, L, XL
+(Radford et al., 2019), GPT-Neo (Black et al., 2021)
+and GPT-J (Wang and Komatsuzaki, 2021). From
+in-depth investigations, we share several intriguing
+observations: (1) As the size of the PLM increases,
+the performance improves without any update of
+model parameters. However, it is still challeng-
+ing to use it without supervision since their perfor-
+mances still lag far behind compared to the fine-
+tuned small PLM (i.e., BERT-base). (2) PETLs out-
+perform fine-tuning with sufficiently large PLMs
+in both IND and OOD metrics. (3) Lastly, lever-
+aging the information of the last hidden represen-
+tation, which is the most prevailing method for
+bi-directional PLM in recent OOD detection, does
+not transfer well in auto-regressive PLM, requiring
+a novel representation extracting technique. We
+believe that these findings will help future anomaly
+detection studies.
+2
+Related Work
+Parameter-Efficient Transfer Learning is draw-
+ing considerable attention lately, emerging as an
+alternative strategy to fine-tuning. Compared to
+fine-tuning, parameter-efficient transfer methods
+show superiority in the number of trainable param-
+eter usage while achieving performance analogous
+to fine-tuning. Depending on the characteristics of
+the methods, parameter-efficient transfer methods
+can be categorized into Adapter-based and Prompt-
+Engineering approaches.
+Adapter (Houlsby et al., 2019; Pfeiffer et al.,
+2021) refers to a lightweight neural module in-
+jected within each layer of PLM. The structure of
+the adapter generally consists of a bottleneck layer
+(down-projection and up-projection), a nonlinear
+function, a normalization layer, and a residual con-
+nection. The adapter has many different variants
+due to numerous design choices, such as the order
+or specifics of each component (e.g., which nor-
+malization technique will be used) and where the
+adapter will be attached. For example, LoRA (Hu
+et al., 2022) inserts low-rank decomposition matri-
+ces in each weight in self-attention (Vaswani et al.,
+2017) (i.e., query, key, and value).
+Another line of work, prompt engineering, casts
+the existing task as a text generation problem to
+fully leverage the capability of PLMs to predict
+the appropriate word in the given sentence. This
+approach requires an empirical endeavor of opti-
+mizing the prompt to maximize a PLM’s perfor-
+mance. Earlier works exploit handcrafted manual
+prompts (Schick and Schütze, 2021; Jiang et al.,
+2020) or by providing demonstrations to PLM 1
+(Brown et al., 2020; Raffel et al., 2020; Gao et al.,
+2021; Zhao et al., 2021). More recent work re-
+places the manual prompt with a soft prompt (Li
+and Liang, 2021; Lester et al., 2021; Shin et al.,
+2020; Liu et al., 2021), a machine trainable contin-
+uous vector. The soft prompt is a more modular
+and versatile method that evades additional latency
+in the inference phase because it detaches the ad-
+ditionally trained parameters and solely employs
+the final output of the trained parameters as the
+prompt.
+While former parameter-efficient transfer meth-
+ods showed noticeable achievements, their evalu-
+ations generally assume the train and test distribu-
+tions are identical (i.e., i.i.d. assumption); however,
+this condition is rarely satisfied in real-world sce-
+narios due to the diversity and volatility of user
+input. Consequently, if the model can not correctly
+handle distribution-shifted malicious input and mis-
+conceives it as an in-distribution (IND) example, it
+may lead to fatal accidents.
+Despite its practical importance, how large
+PLMs or parameter-efficient transfer learning cope
+with unknown input is poorly understood. This
+work aims to understand language models’ capabil-
+ities to detect outliers through parameter-efficient
+transfer learning methods.
+3
+Probing OOD Robustness
+3.1
+Backbones and Models
+To investigate the trend of OOD performance under
+varying scales of PLM, we consider three factors
+during backbone selection. They should be (1)
+publicly available, (2) reasonably large, and (3)
+share identical structures to eliminate factors other
+than size. Since recent large PLMs utilize auto-
+regressive objectives due to their computational
+complexity, we adopt six auto-regressive PLMs
+as the backbone of our experiments accordingly:
+GPT2 (S,M,L,XL), GPT-Neo, and GPT-J.
+For the parameter-efficient transfer methods,
+we selected two methods: two adapter-based and
+one prompt engineering-based. Namely, Adapter
+(Houlsby et al., 2019), LoRA (Hu et al., 2022), and
+1also termed as in-context learning.
+
+Prefix-tuning (Li and Liang, 2021) are selected
+for the adapter approach, which is compatible with
+classification tasks, for the prompt approach. We
+also report the performance of linear evaluation,
+i.e., single layer perceptron (SLP) on top of PLMs,
+and fine-tuning, which act like a lower-bound and
+upper-bound, respectively.
+3.2
+Dataset and Metrics
+Dataset. We evaluate our model on two datasets,
+CLINC150 and Banking77, widely used in OOD
+detection. CLINC150 dataset (Larson et al., 2019)
+contains 150 class labels (15 intents for 10 do-
+mains), while Banking77 dataset (Casanueva et al.,
+2020) consists of fine-grained 77 bank-related in-
+tents. Following the experimental settings from pre-
+vious works (Cho et al., 2022; Zhang et al., 2022;
+Shu et al., 2017; Fei and Liu, 2016; Lin and Xu,
+2019), we validate our models in two different sce-
+narios: far-OOD setting and close-OOD setting.
+For CLINC dataset, we train our model with the
+whole training dataset and test with an indepen-
+dent OOD test split from CLINC dataset, which
+does not overlap with 150 classes in the training
+dataset. Outliers in CLINC OOD split are distribu-
+tionally far from the training distribution (Zhang
+et al., 2022), so it is relatively easy to discern. For
+Banking77, we partition the dataset into 2 disjoint
+datasets (i.e., IND / OOD dataset) based on the
+class label. Since both IND and OOD datasets orig-
+inated from the equivalent dataset, they share sim-
+ilar distributions and properties, making the task
+more demanding. Thus, we refer to a CLINC OOD
+setting as far-OOD and split settings in Banking as
+close-OOD settings, respectively.
+Metrics. To evaluate IND performance, we mea-
+sured the classification accuracy. And for OOD per-
+formance, we adopt two metrics commonly used
+in recent OOD detection literature:
+• FPR@95. The false-positive rate at the true-
+positive rate of 95% (FPR@95) measures the prob-
+ability of classifying OOD input as IND input when
+the true-positive rate is 95%.
+• AUROC. The area under the receiver operating
+characteristic curve (AUROC) is a threshold-free
+metric that indicates the ability of the model to
+discriminate outliers from IND samples.
+3.3
+OOD Evaluation Methods
+Evaluation in OOD detection is done via a scoring
+function, which outputs the appropriateness of the
+input into a single scalar value (p). Then we com-
+pare p with the pre-set threshold δ to determine
+whether the input is an outlier or not:
+Iδ(x) =
+� IND
+p(x) ≥ δ
+OOD
+p(x) < δ,
+(1)
+In this paper, we evaluate the performance of
+our method in 4 different evaluation methods,
+which can be categorized into 2 higher branches:
+representation-based and logit-based.
+Logit-based approaches exploit the PLM’s predic-
+tion result extracted from the classification layer as
+their primary information to discern outliers. Logit-
+based approaches are simple and have their own
+dominance in computational cost since it pursues
+OOD detection and general classification nigh si-
+multaneously.
+• MSP is a baseline method in this branch that
+employs the maximum softmax probability to score
+the appropriateness of the given input, based on the
+idea that the model will output more certain output
+(higher probability) to a normal sample (Hendrycks
+and Gimpel, 2017):
+p(x) =
+efi(x)
+ΣN
+j=1efj(x) ,
+(2)
+where fi(x) refer to as max value from the classifi-
+cation layer (max logit value).
+• Energy is a variant of MSP, which calibrates logit
+value based on energy function (Liu et al., 2020):
+p(x) = −E(x; f) = T · log ΣN
+i ef(x)/T .
+(3)
+Representation-based approaches, on the other
+hand, employ the hidden representation from PLM
+as their primary source. Since the size of the hidden
+representation is larger and inheres more copious
+information, they generally yield a more precise
+decision than logit-based approaches. However,
+they require more inference time to derive a final
+score. We employed Mahalanobis distance-based
+and cosine similarity-based methods in this branch.
+• Mahalanobis distance refers to the distance be-
+tween the specific distribution and the input. In
+OOD detection, we estimate the gaussian distribu-
+tion of the training dataset and utilize the minimum
+Mahalanobis distance to score the input suitability
+(Lee et al., 2018):
+p(x) = (h − µk)⊤Σ−1(h − µk),
+(4)
+
+(a) Performance on far-OOD setting.
+(b) Performance on close-OOD setting.
+Figure 1: OOD detection performance of PLMs without updating the model parameters.
+Setting
+Backbone
+Evaluation Method
+MSP
+Energy
+Mahal.
+Cosine
+CLINC
+Setting
+GPT2-S
+93.22
+95.79
+77.63
+76.34
+GPT2-M
+95.41
+97.63
+82.42
+79.82
+GPT2-L
+96.21
+97.77
+96.93
+97.57
+GPT2-XL
+96.48
+97.99
+97.28
+97.66
+GPT-Neo
+96.04
+97.72
+96.59
+97.64
+GPT-J
+97.34
+98.50
+97.91
+98.20
+Banking
+Split 25%
+GPT2-S
+90.12
+91.32
+75.32
+73.11
+GPT2-M
+91.74
+92.78
+78.03
+76.56
+GPT2-L
+93.02
+93.45
+92.44
+93.41
+GPT2-XL
+94.29
+94.95
+93.24
+94.10
+GPT-Neo
+93.83
+94.85
+92.79
+93.88
+GPT-J
+94.11
+95.10
+93.66
+94.80
+Table 1: AUROC of each PLMs trained with LoRA. Energey function consistently outperforms other methods .
+where training distribution is (N(µi, Σ) for i ∈
+i = {1, 2, · · · , |C|}), and k refers to a index of
+minimum mahalanobis distance.
+• Cosine Similarity method utilizes the cosine dis-
+tance between the representation of the given input
+(z(x)) and the nearest neighbor z(xnn) (Tack et al.,
+2020):
+p(x) = sim(z(x), z(xnn))
+(5)
+4
+Analysis
+In this section, we share several intriguing findings
+and insights from various settings.
+4.1
+OOD Robustness of PLMs without
+Supervision.
+In this experiment, we investigate the OOD detec-
+tion capability of PLMs without parameter tuning.
+Precisely, we extract the final layer representation
+from each frozen PLM and evaluate their perfor-
+mance via representation-based evaluation meth-
+ods. (Logit-based evaluation methods are not used
+as they require additional training of the classifi-
+cation layer.) Figure 1 summarizes the results in
+two scenarios (i.e., far-OOD and close-OOD). We
+verified the correlation between the size of PLMs
+and their OOD detection ability, but utilizing them
+without parameter supervision is roughly impossi-
+ble since they still lag far behind the small super-
+vised methods (i.e., BERT-base with Mahalanobis
+evaluation) in a barebone setting. Moreover, perfor-
+mance improvement from the scaling saturates in a
+more harsh setting (i.e., close-OOD), displaying an
+unbridgeable gap with the fine-tuned model.
+4.2
+Evaluation methods for auto-regressive
+PLMs.
+Many recent OOD works (Zhou et al., 2021; Shen
+et al., 2021) leverage hidden representation-based
+
+98
+91
+84
+77
+70
+GPT2-S
+GPT2-M
+GPT2-L
+GPT2-XLGPT-NeO
+GPT-J
+(117M)
+(345M)
+(762M)
+(1.5B)
+(2.7B)
+(6B)
+Mahalanobis Distance
+Cosine Similarity
+BERT-base (finetuned)94
+83
+72
+61
+50
+GPT2-S
+GPT2-M
+GPT2-L
+GPT2-XLGPT-Neo
+GPT-J
+(117M)
+(345M)
+(762M)
+(1.5B)
+(2.7B)
+(6B)
+Mahalanobis Distance
+Cosine Similarity
+BERT-base (finetuned)Setting
+Method
+# Params.
+Backbone
+GPT2
+(S)
+GPT2
+(M)
+GPT2
+(L)
+GPT2
+(XL)
+GPT
+Neo
+GPT-J
+CLINC
+(far-ood)
+Linear (SLP)
+0%
+83.03
+87.39
+88.47
+89.55
+89.44
+91.94
+Fine-tuning
+100%
+96.84
+97.71
+98.24
+98.33
+98.01
+98.41
+LoRA
+0.1%
+95.00
+96.54
+97.66
+97.72
+98.14
+97.79
+0.5%
+96.41
+96.04
+97.52
+97.45
+98.12
+97.89
+1%
+96.13
+95.89
+97.61
+97.40
+98.11
+98.50
+Adapter
+0.1%
+96.62
+97.52
+97.74
+97.71
+97.81
+96.80
+0.5%
+95.64
+97.07
+97.86
+96.94
+97.98
+98.37
+1%
+95.79
+97.63
+97.77
+97.99
+98.12
+98.50
+Prefix
+0.1%
+95.53
+96.93
+96.38
+97.88
+90.25
+98.55
+0.5%
+96.91
+96.96
+97.78
+97.88
+89.81
+97.92
+1%
+96.97
+97.50
+97.69
+97.81
+88.98
+98.62
+Banking
+split 25%
+(close-ood)
+Linear (SLP)
+0%
+72.97
+75.17
+80.46
+77.59
+86.55
+89.12
+Fine-tuning
+100%
+90.06
+92.06
+93.14
+93.23
+92.54
+93.73
+LoRA
+0.1%
+91.18
+91.74
+94.65
+94.58
+94.29
+95.82
+0.5%
+91.16
+92.98
+94.54
+94.04
+94.55
+94.65
+1%
+91.39
+92.39
+93.45
+93.59
+94.81
+95.29
+Adapter
+0.1%
+91.97
+93.24
+94.90
+94.69
+93.26
+95.59
+0.5%
+92.90
+92.63
+95.18
+95.24
+93.61
+95.83
+1%
+91.32
+92.78
+95.41
+94.95
+94.41
+95.37
+Prefix
+0.1%
+91.22
+91.92
+93.96
+93.48
+81.9
+94.93
+0.5%
+91.85
+92.55
+93.84
+93.34
+80.82
+93.99
+1%
+92.09
+92.65
+94.38
+93.74
+89.66
+94.39
+Table 2: AUROC of various PETL methods with various number of parameters evaluated by the energy function.
+evaluation, as they generally surpass logit-based
+evaluations (Podolskiy et al., 2021). The reason-
+able conjecture behind their success is that hid-
+den representations have more copious information
+than the logit value. However, in auto-regressive
+PLMs, logit-based evaluations (i.e., MSP and En-
+ergy) outperform representation-based methods
+(i.e., Mahalanobis distance and cosine similarity),
+as shown in Table 1. The reasonable conjecture
+for this phenomenon is due to the characteristic
+of the language model. Unlike bi-directional mod-
+els (e.g., BERT, RoBERTa, DeBERTa), decoder
+models (e.g., GPT and its variants) do not have
+[CLS] embedding, which assembles the token em-
+beddings to capture holistic information (Devlin
+et al., 2019; Kim et al., 2021). Therefore, auto-
+regressive PLMs generally utilize the last token
+embedding as a final feature embedding replacing
+[CLS] embedding of encoder-based models. While
+the last token of GPT is befitted for predicting the
+next token, however, it cannot extract the holistic
+semantics of the sentence suitably, unlike [CLS]
+embedding. We believe extracting a better repre-
+sentation through various pooling (Wang and Kuo,
+2020) methods might be a possible avenue for auto-
+regressive models to improve the OOD robustness
+further.
+4.3
+PETLs VS. Fine-tuning
+In this experiment, we investigate the performance
+gap between various PETL methods (i.e., Adapter,
+LoRA, prefix-tuning) and model fine-tuning. To
+compare the performance of each method under
+similar circumstances, we set every PETL method
+to utilize a similar number of parameters sufficient
+enough to reach maximum accuracy. Moreover,
+we utilized the energy function to evaluate each
+method as they displayed the best performance
+among other evaluation methods, i.e., cosine, Ma-
+halanobis, and MSP, in the previous experiments.
+Table 2 summarizes the results.
+From this experiment, we observed that PETL
+methods are more robust than fine-tuning with
+reasonably large PLMs (i.e., GPT-J). Specifically,
+most PELT methods on GPT-J outperform fine-
+tuning with proper tunable parameters. Neverthe-
+less, size is not the ultimate answer. While it is
+clear that the scale of a model is an essential factor
+in OOD robustness, larger models are still vulnera-
+ble to close-OOD inputs. The capability to detect
+far-OOD inputs (far from the training distribution)
+
+95
+96
+97
+98
+99
+GPT2-S GPT2-M GPT2-L GPT2-XLGPT-Neo GPT-J
+Fine-tuned
+LoRA
+Adapter
+Prefix
+(a) Performance on far-OOD setting.
+89
+90
+91
+92
+93
+94
+95
+96
+GPT2-S
+GPT2-M
+GPT2-L
+GPT2-XL
+GPT-Neo
+GPT-J
+Fine-tuned
+LoRA
+Adapter
+Prefix
+(b) Performance on close-OOD setting.
+Figure 2: OOD detection performance of PLMs without updating the model parameters.
+improves proportionally as the size grows, while
+the ability to identify close-OOD input improves
+rather trivially. PLM’s vulnerability to close-OOD
+has already been reported in other studies (Zhang
+et al., 2022), and this may be related to shortcut
+learning (Geirhos et al., 2020) that predicts with
+high probability by looking at specific words. Gen-
+erating OOD data with particular keywords or uti-
+lizing another pretext task, such as (Moon et al.,
+2021), can be worthy approaches to alleviate such
+phenomena. A suitable OOD approach is neces-
+sary to alleviate the aforementioned issue, as it
+can further boost the robustness. We conduct addi-
+tional experiments with PETLs on three different
+numbers of tunable parameters: 0.1%, 0.5%, and
+1% of the PLM parameters. Figure 2 summarizes
+the results. With sufficient parameters to reach
+maximum performance, there is no meaningful dif-
+ference or improvement within each methodology.
+Also, empirically, we confirmed that LoRA is the
+most stable during learning and that prefix-tuning
+fluctuates severely according to learning.
+5
+Conclusion and Future Work
+In this study, we showed that the scale of the lan-
+guage model is an important factor in OOD ro-
+bustness. Moreover, we also showed that various
+methodologies outperform fine-tuning when ap-
+plied to sufficiently large PLM. Our follow-up work
+seeks to create a methodology that allows large
+PLMs to be more robust to OOD input. The per-
+formance improvement that can be achieved by the
+size of PLM and OOD technique is orthogonal. In
+line with the growing size of PLM, the OOD tech-
+nique needs to be developed in a more parameter-
+efficient way. As such, developing a proper OOD
+technique compatible with the parameter-efficient
+transfer methods is our proper goal.
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+
+Appendix
+A
+Parameter-Efficient Transfer
+Learning
+Adapter The adapter approach inserts small train-
+able adapter modules between transformer layers
+while the parameters of the original network remain
+fixed. The adapter module uses a bottleneck archi-
+tecture which projects the input dimension h to a
+lower-dimensional space specified by bottleneck
+dimension r, followed by a nonlinear activation
+function, and a up-projection to initial dimension
+h. In this work, we attach adapter modules in two
+places, i.e., after the projection following multi-
+head attention and after the two feed-forward lay-
+ers, following original implementation in (Houlsby
+et al., 2019). Also, we use relu as a nonlinear func-
+tion and layer normalization (Ba et al., 2016).
+LoRA LoRA injects trainable low-rank matrices
+into transformer layers to approximate the weight
+updates. For a pre-trained weight matrix W ∈
+Rh×k, LoRA decompose ∆W
+= WdownWup
+where Wdown ∈ Rh×r, Wup ∈ Rr×k are trainable
+parameters. Specifically we attach LoRA in weight
+matrices in the self attention module. Specifically
+we attached LoRA to query and key vector follow-
+ing the original implementation.
+Prefix-Tuning Prefix tuning prepends l tunable
+prefix vectors to the keys and values of the multi-
+head attention at every layer. Following the original
+implementation, we reparametrize the prefix matrix
+of dimension h by a smaller matrix of dimension r
+composed with a large feedforward neural network
+with tanh as a nonlinear function.
+B
+Expanded Configuration Details
+B.1
+Common Environment
+For the experiments, 4 Tesla V100 SXM2 32GB
+GPUs are used. The batch size is 8 per GPU. When
+the GPU is too small for the batch size, we set
+batch size to 4 and the number of gradient accu-
+mulation steps to 2. We implemented our model
+based on Transformers (Wolf et al., 2020) library
+by Huggingface. Additionally, we used deepspeed
+(Rajbhandari et al., 2020) to train models. Specifi-
+cally, we used ZeRO2 with cpu offload on a 240GB
+RAM CPU. In this setting, fine-tuning GPT-J on
+CLINC150 full dataset takes about 7.1 GPU hours
+per epoch. We used AdamW (Loshchilov and Hut-
+ter, 2019) optimizer with epsilon 1e-6 and weight
+dataset
+#domain
+#intent
+#data (train/val/test/ood)
+CLINC
+10
+15
+15000/3000/4500/1000
+Banking
+1
+77
+7812 / 1520 / 3040
+Table 3: Dataset statistics.
+BERT-base
+CLINC150 Full
+ACC ↑
+FPR-95 ↓
+AUROC ↑
+Shu et al. (2017)
+94.51±0.45
+23.33±1.27
+95.92±0.05
+Li et al. (2021)
+96.1±0.37
+10.6±0.26
+97.72±0.03
+Zeng et al. (2021)
+94.19±0.28
+23.4±1.97
+95.75±0.2
+Zhou et al. (2021)
+95.79±0.13
+10.7±0.95
+97.6±0.11
+Shen et al. (2021)
+96.66
+10.88
+97.43
+Cho et al. (2022)
+96.96±0.39
+6.67 ±0.51
+98.27 ±0.16
+Table 4:
+Results of each model trained on the
+CLINC150 dataset. The best performance in each met-
+ric is indicated in bold.
+decay 0.1. Furthermore, we apply the cosine an-
+nealing scheduler. For GPT-neo, the minimum
+learning rate is 0. For GPT-J, the minimum learn-
+ing rate is the one fifth of maximum learning rate.
+B.2
+Number of Trainable-Parameter
+For each method, a feed-forward layer is added at
+the end of the model. In this section, we will calcu-
+late the number of additional trainable parameters
+of each training methods discussed in this paper.
+Biases are omitted for better readability.
+Adapter Adapter method adds four feed-forward
+layers per transformer layer in the model. Two of
+them are down-projection layers, and the others are
+up-projection layers. When the original embedding
+size of the model is h, the bottleneck dimension is
+r, and the number of transformer layers is L, the
+number of the trainable parameters of these layers
+is calculated as 4Lhr, excluding the bias of the
+added layers.
+LoRA Similar to adapter, LoRA also adds feed-
+forward layers per transformer layer. Therefore,
+the number of the trainable parameters of 4Lhr.
+However, the number of parameters are less than
+adapter if h and r is the same, since LoRA does
+not use bias of the feed-forward layers.
+Prefix-Tuning There are two trainable elements
+in prefix tuning. The first one is the prefix em-
+beddings. When the number of prefixes is l, and
+the embedding size is h, lh parameters are used
+by the prefixes. Second, the reparametrization ma-
+trix is also trained. The down-projection matrix
+has hr parameters, when the reduced dimension
+for reparametrization is r. The up-projection ma-
+trix has 2Lhr parameters. As a result, there are
+
+Method
+Parameters
+Values
+LoRA
+Learning rate
+2e-4 (GPT-Neo), 5e-5 (GPT-J)
+Bottleneck dim
+8 (GPT-Neo / 0.1%), 80 (GPT-Neo / 1%), 12 (GPT-J /0.1%), 128 (GPT-J / 1%)
+Location
+query, value
+Adapter
+Learning rate
+8e-5 (GPT-Neo / 0.1%), 1e-4 (GPT-Neo / 1%), 5e-5 (GPT-J), 5e-4 (GPT-J / 0.1%), 1e-4 (GPT-J / 1%)
+Bottleneck dim
+6 (GPT-Neo / 0.1%), 80 (GPT-Neo / 1%), 11 (GPT-J /0.1%), 128 (GPT-J / 1%)
+Location
+after Multi-head, after Feed-forward,
+Prefix-tuning
+Learning rate
+2E-4 (GPT-Neo), 5E-5 (GPT-J)
+Bottleneck dim
+12 (GPT-Neo / 0.1%), 160 (GPT-Neo / 1%), 20 (GPT-J /0.1%), 256 (GPT-J / 1%)
+Prefix length
+5, 10, 20
+Table 5: Hyper-parameter search for each model.
+h(2Lr + l) trainable parameters on prefix tuning
+approach.
+B.3
+Hyper-parameter Search
+Tab 5 summarizes hyper parameters for each
+model.
+C
+Selecting SOTA OOD Method.
+The Tab.4 summarizes the results with recently
+proposed OOD approaches on BERT-base with
+CLINC dataset. The best performing model (Cho
+et al., 2022) is selected as the baseline.
+
diff --git a/RNFJT4oBgHgl3EQf3S0N/content/tmp_files/load_file.txt b/RNFJT4oBgHgl3EQf3S0N/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,785 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf,len=784
+page_content='Probing Out-of-Distribution Robustness of Language Models with  Parameter-Efficient Transfer Learning Methods  Hyunsoo Cho†, Choonghyun Park†, Junyeop Kim†, Hyuhng Joon Kim†,  Kang Min Yoo†‡, and Sang-goo Lee†  † Seoul National University, ‡ NAVER  {johyunsoo,pch330,juny116,heyjoonkim,sglee}@europa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='snu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='kr  {kangmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='yoo}@navercorp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='com  Abstract  As the size of the pre-trained language  model (PLM) continues to increase, numer-  ous parameter-efficient transfer learning meth-  ods have been proposed recently to compen-  sate for the tremendous cost of fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Despite the impressive results achieved by  large pre-trained language models (PLMs) and  various parameter-efficient transfer learning  (PETL) methods on sundry benchmarks, it re-  mains unclear if they can handle inputs that  have been distributionally shifted effectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In this study, we systematically explore how  the ability to detect out-of-distribution (OOD)  changes as the size of the PLM grows or the  transfer methods are altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Specifically, we  evaluated various PETL techniques, including  fine-tuning, Adapter, LoRA, and prefix-tuning,  on three different intention classification tasks,  each utilizing various language models with  different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  1  Introduction  Pre-trained language models (PLM), which are pre-  trained on large-scale corpora using transformer-  based architectures (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2017), have  achieved groundbreaking success on sundry bench-  marks (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019a), establishing themselves as the  standard neural model in countless applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Moreover, language models pre-trained with larger  parameters on a rich volume of corpora tend to ex-  hibit more intriguing potentials, such as the ability  to capture world knowledge (Petroni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019),  generate codes (Poesia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022), and even solve  mathematical problems (Henighan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020), on  top of understanding linguistic knowledge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  semantic or syntactic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' To explore the apex of pre-  trained language models (PLMs), the size of PLMs  is growing exponentially and has reached billions  to a trillion (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Chowdhery et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Fedus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Hoffmann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Under these circumstances, the conventional  method for transferring PLMs to a target task  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', fine-tuning) is now infeasible as it entails  prohibitive costs to train and store the entire pa-  rameters of large PLMs for every desired task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  To mitigate this issue, several recent parameter-  efficient transfer learning (PETL) methods have  been proposed to improve task scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' For in-  stance, adapter-based (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Hu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022) approaches insert small neural mod-  ules into each layer of the PLM and update those  lightweight modules in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Inspired  by the recent success of textual prompts (Brown  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020), prompt-based methods (Li and Liang,  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020) concate-  nate extra tunable tokens to the front of the input or  hidden layers and update prepended soft prompts  in the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Despite these breakthroughs in NLP, even very  recent anomaly detection studies (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021) are still limited to relatively small  bi-directional PLMs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', BERT, RoBERTa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Thus,  how large-scale PLMs or auto-regressive PLMs  cope with outliers is uncharted territory, naturally  begging the following questions:  Q1: Does increasing model size improve OOD  detection performance without model parameters?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Q2: If so, does scaling the size of PLM makes  the model robust enough to utilize them without  any additional process?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Q3: Do fine-tuning and various PETL method-  ologies display differences in OOD detection per-  formance according to the size of PLMs?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Q4: Can the OOD detection methods from previ-  ous works (usually for the bi-directional PLMs) be  transferred to auto-regressive PLMs (GPT)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  To resolve these questions, this paper investi-  gates the capability of large PLMs as outlier de-  tectors from various perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Specifically, we  compare the robustness to outliers with various  transfer learning techniques on several OOD bench-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='11660v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='CL]  27 Jan 2023  marks: Full fine-tuning, LoRA (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022),  Adapter (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019), and prefix-tuning  (Li and Liang, 2021) on various auto-regressive  PLMs with different sizes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', GPT2-S, M, L, XL  (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019), GPT-Neo (Black et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021)  and GPT-J (Wang and Komatsuzaki, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' From  in-depth investigations, we share several intriguing  observations: (1) As the size of the PLM increases,  the performance improves without any update of  model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' However, it is still challeng-  ing to use it without supervision since their perfor-  mances still lag far behind compared to the fine-  tuned small PLM (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', BERT-base).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (2) PETLs out-  perform fine-tuning with sufficiently large PLMs  in both IND and OOD metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (3) Lastly, lever-  aging the information of the last hidden represen-  tation, which is the most prevailing method for  bi-directional PLM in recent OOD detection, does  not transfer well in auto-regressive PLM, requiring  a novel representation extracting technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We  believe that these findings will help future anomaly  detection studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  2  Related Work  Parameter-Efficient Transfer Learning is draw-  ing considerable attention lately, emerging as an  alternative strategy to fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Compared to  fine-tuning, parameter-efficient transfer methods  show superiority in the number of trainable param-  eter usage while achieving performance analogous  to fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Depending on the characteristics of  the methods, parameter-efficient transfer methods  can be categorized into Adapter-based and Prompt-  Engineering approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Adapter (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Pfeiffer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2021) refers to a lightweight neural module in-  jected within each layer of PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The structure of  the adapter generally consists of a bottleneck layer  (down-projection and up-projection), a nonlinear  function, a normalization layer, and a residual con-  nection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The adapter has many different variants  due to numerous design choices, such as the order  or specifics of each component (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', which nor-  malization technique will be used) and where the  adapter will be attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' For example, LoRA (Hu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022) inserts low-rank decomposition matri-  ces in each weight in self-attention (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2017) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', query, key, and value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Another line of work, prompt engineering, casts  the existing task as a text generation problem to  fully leverage the capability of PLMs to predict  the appropriate word in the given sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' This  approach requires an empirical endeavor of opti-  mizing the prompt to maximize a PLM’s perfor-  mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Earlier works exploit handcrafted manual  prompts (Schick and Schütze, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Jiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2020) or by providing demonstrations to PLM 1  (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' More recent work re-  places the manual prompt with a soft prompt (Li  and Liang, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Lester et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021), a machine trainable contin-  uous vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The soft prompt is a more modular  and versatile method that evades additional latency  in the inference phase because it detaches the ad-  ditionally trained parameters and solely employs  the final output of the trained parameters as the  prompt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  While former parameter-efficient transfer meth-  ods showed noticeable achievements, their evalu-  ations generally assume the train and test distribu-  tions are identical (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' assumption);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' however,  this condition is rarely satisfied in real-world sce-  narios due to the diversity and volatility of user  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Consequently, if the model can not correctly  handle distribution-shifted malicious input and mis-  conceives it as an in-distribution (IND) example, it  may lead to fatal accidents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Despite its practical importance, how large  PLMs or parameter-efficient transfer learning cope  with unknown input is poorly understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' This  work aims to understand language models’ capabil-  ities to detect outliers through parameter-efficient  transfer learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  3  Probing OOD Robustness  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1  Backbones and Models  To investigate the trend of OOD performance under  varying scales of PLM, we consider three factors  during backbone selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' They should be (1)  publicly available, (2) reasonably large, and (3)  share identical structures to eliminate factors other  than size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Since recent large PLMs utilize auto-  regressive objectives due to their computational  complexity, we adopt six auto-regressive PLMs  as the backbone of our experiments accordingly:  GPT2 (S,M,L,XL), GPT-Neo, and GPT-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  For the parameter-efficient transfer methods,  we selected two methods: two adapter-based and  one prompt engineering-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Namely, Adapter  (Houlsby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019), LoRA (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022), and  1also termed as in-context learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Prefix-tuning (Li and Liang, 2021) are selected  for the adapter approach, which is compatible with  classification tasks, for the prompt approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We  also report the performance of linear evaluation,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', single layer perceptron (SLP) on top of PLMs,  and fine-tuning, which act like a lower-bound and  upper-bound, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='2  Dataset and Metrics  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We evaluate our model on two datasets,  CLINC150 and Banking77, widely used in OOD  detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' CLINC150 dataset (Larson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019)  contains 150 class labels (15 intents for 10 do-  mains), while Banking77 dataset (Casanueva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2020) consists of fine-grained 77 bank-related in-  tents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Following the experimental settings from pre-  vious works (Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Fei and Liu, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Lin and Xu,  2019), we validate our models in two different sce-  narios: far-OOD setting and close-OOD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  For CLINC dataset, we train our model with the  whole training dataset and test with an indepen-  dent OOD test split from CLINC dataset, which  does not overlap with 150 classes in the training  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Outliers in CLINC OOD split are distribu-  tionally far from the training distribution (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022), so it is relatively easy to discern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' For  Banking77, we partition the dataset into 2 disjoint  datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', IND / OOD dataset) based on the  class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Since both IND and OOD datasets orig-  inated from the equivalent dataset, they share sim-  ilar distributions and properties, making the task  more demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Thus, we refer to a CLINC OOD  setting as far-OOD and split settings in Banking as  close-OOD settings, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' To evaluate IND performance, we mea-  sured the classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' And for OOD per-  formance, we adopt two metrics commonly used  in recent OOD detection literature:  FPR@95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The false-positive rate at the true-  positive rate of 95% (FPR@95) measures the prob-  ability of classifying OOD input as IND input when  the true-positive rate is 95%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  AUROC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The area under the receiver operating  characteristic curve (AUROC) is a threshold-free  metric that indicates the ability of the model to  discriminate outliers from IND samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='3  OOD Evaluation Methods  Evaluation in OOD detection is done via a scoring  function, which outputs the appropriateness of the  input into a single scalar value (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Then we com-  pare p with the pre-set threshold δ to determine  whether the input is an outlier or not:  Iδ(x) =  � IND  p(x) ≥ δ  OOD  p(x) < δ,  (1)  In this paper, we evaluate the performance of  our method in 4 different evaluation methods,  which can be categorized into 2 higher branches:  representation-based and logit-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Logit-based approaches exploit the PLM’s predic-  tion result extracted from the classification layer as  their primary information to discern outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Logit-  based approaches are simple and have their own  dominance in computational cost since it pursues  OOD detection and general classification nigh si-  multaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  MSP is a baseline method in this branch that  employs the maximum softmax probability to score  the appropriateness of the given input, based on the  idea that the model will output more certain output  (higher probability) to a normal sample (Hendrycks  and Gimpel, 2017):  p(x) =  efi(x)  ΣN  j=1efj(x) ,  (2)  where fi(x) refer to as max value from the classifi-  cation layer (max logit value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Energy is a variant of MSP, which calibrates logit  value based on energy function (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020):  p(x) = −E(x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' f) = T · log ΣN  i ef(x)/T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  (3)  Representation-based approaches, on the other  hand, employ the hidden representation from PLM  as their primary source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Since the size of the hidden  representation is larger and inheres more copious  information, they generally yield a more precise  decision than logit-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' However,  they require more inference time to derive a final  score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We employed Mahalanobis distance-based  and cosine similarity-based methods in this branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Mahalanobis distance refers to the distance be-  tween the specific distribution and the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In  OOD detection, we estimate the gaussian distribu-  tion of the training dataset and utilize the minimum  Mahalanobis distance to score the input suitability  (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2018):  p(x) = (h − µk)⊤Σ−1(h − µk),  (4)  (a) Performance on far-OOD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  (b) Performance on close-OOD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Figure 1: OOD detection performance of PLMs without updating the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Setting  Backbone  Evaluation Method  MSP  Energy  Mahal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Cosine  CLINC  Setting  GPT2-S  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='79  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='63  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='34  GPT2-M  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='41  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='63  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='42  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='82  GPT2-L  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='21  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='57  GPT2-XL  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='48  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='99  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='66  GPT-Neo  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='34  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='20  Banking  Split 25%  GPT2-S  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='32  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='11  GPT2-M  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='74  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='78  78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='02  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='29  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='83  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='88  GPT-J  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='11  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='10  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='66  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='80  Table 1: AUROC of each PLMs trained with LoRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Energey function consistently outperforms other methods .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  where training distribution is (N(µi, Σ) for i ∈  i = {1, 2, · · · , |C|}), and k refers to a index of  minimum mahalanobis distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Cosine Similarity method utilizes the cosine dis-  tance between the representation of the given input  (z(x)) and the nearest neighbor z(xnn) (Tack et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2020):  p(x) = sim(z(x), z(xnn))  (5)  4  Analysis  In this section, we share several intriguing findings  and insights from various settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1  OOD Robustness of PLMs without  Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In this experiment, we investigate the OOD detec-  tion capability of PLMs without parameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Precisely, we extract the final layer representation  from each frozen PLM and evaluate their perfor-  mance via representation-based evaluation meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (Logit-based evaluation methods are not used  as they require additional training of the classifi-  cation layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=') Figure 1 summarizes the results in  two scenarios (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', far-OOD and close-OOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We  verified the correlation between the size of PLMs  and their OOD detection ability, but utilizing them  without parameter supervision is roughly impossi-  ble since they still lag far behind the small super-  vised methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', BERT-base with Mahalanobis  evaluation) in a barebone setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Moreover, perfor-  mance improvement from the scaling saturates in a  more harsh setting (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', close-OOD), displaying an  unbridgeable gap with the fine-tuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='2  Evaluation methods for auto-regressive  PLMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Many recent OOD works (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Shen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021) leverage hidden representation-based  98  91  84  77  70  GPT2-S  GPT2-M  GPT2-L  GPT2-XLGPT-NeO  GPT-J  (117M)  (345M)  (762M)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='5B)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='7B)  (6B)  Mahalanobis Distance  Cosine Similarity  BERT-base (finetuned)94  83  72  61  50  GPT2-S  GPT2-M  GPT2-L  GPT2-XLGPT-Neo  GPT-J  (117M)  (345M)  (762M)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='5B)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='7B)  (6B)  Mahalanobis Distance  Cosine Similarity  BERT-base (finetuned)Setting  Method  # Params.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Backbone  GPT2  (S)  GPT2  (M)  GPT2  (L)  GPT2  (XL)  GPT  Neo  GPT-J  CLINC  (far-ood)  Linear (SLP)  0%  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='94  Fine-tuning  100%  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', MSP and En-  ergy) outperform representation-based methods  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', Mahalanobis distance and cosine similarity),  as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The reasonable conjecture  for this phenomenon is due to the characteristic  of the language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Unlike bi-directional mod-  els (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', BERT, RoBERTa, DeBERTa), decoder  models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', GPT and its variants) do not have  [CLS] embedding, which assembles the token em-  beddings to capture holistic information (Devlin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Therefore, auto-  regressive PLMs generally utilize the last token  embedding as a final feature embedding replacing  [CLS] embedding of encoder-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' While  the last token of GPT is befitted for predicting the  next token, however, it cannot extract the holistic  semantics of the sentence suitably, unlike [CLS]  embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We believe extracting a better repre-  sentation through various pooling (Wang and Kuo,  2020) methods might be a possible avenue for auto-  regressive models to improve the OOD robustness  further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='3  PETLs VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Fine-tuning  In this experiment, we investigate the performance  gap between various PETL methods (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', Adapter,  LoRA, prefix-tuning) and model fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' To  compare the performance of each method under  similar circumstances, we set every PETL method  to utilize a similar number of parameters sufficient  enough to reach maximum accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Moreover,  we utilized the energy function to evaluate each  method as they displayed the best performance  among other evaluation methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', cosine, Ma-  halanobis, and MSP, in the previous experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Table 2 summarizes the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  From this experiment, we observed that PETL  methods are more robust than fine-tuning with  reasonably large PLMs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', GPT-J).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Specifically,  most PELT methods on GPT-J outperform fine-  tuning with proper tunable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Neverthe-  less, size is not the ultimate answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' While it is  clear that the scale of a model is an essential factor  in OOD robustness, larger models are still vulnera-  ble to close-OOD inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The capability to detect  far-OOD inputs (far from the training distribution)  95  96  97  98  99  GPT2-S GPT2-M GPT2-L GPT2-XLGPT-Neo GPT-J  Fine-tuned  LoRA  Adapter  Prefix  (a) Performance on far-OOD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  89  90  91  92  93  94  95  96  GPT2-S  GPT2-M  GPT2-L  GPT2-XL  GPT-Neo  GPT-J  Fine-tuned  LoRA  Adapter  Prefix  (b) Performance on close-OOD setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Figure 2: OOD detection performance of PLMs without updating the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  improves proportionally as the size grows, while  the ability to identify close-OOD input improves  rather trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' PLM’s vulnerability to close-OOD  has already been reported in other studies (Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022), and this may be related to shortcut  learning (Geirhos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020) that predicts with  high probability by looking at specific words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Gen-  erating OOD data with particular keywords or uti-  lizing another pretext task, such as (Moon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=',  2021), can be worthy approaches to alleviate such  phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' A suitable OOD approach is neces-  sary to alleviate the aforementioned issue, as it  can further boost the robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We conduct addi-  tional experiments with PETLs on three different  numbers of tunable parameters: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='5%, and  1% of the PLM parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Figure 2 summarizes  the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' With sufficient parameters to reach  maximum performance, there is no meaningful dif-  ference or improvement within each methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Also, empirically, we confirmed that LoRA is the  most stable during learning and that prefix-tuning  fluctuates severely according to learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  5  Conclusion and Future Work  In this study, we showed that the scale of the lan-  guage model is an important factor in OOD ro-  bustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Moreover, we also showed that various  methodologies outperform fine-tuning when ap-  plied to sufficiently large PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Our follow-up work  seeks to create a methodology that allows large  PLMs to be more robust to OOD input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The per-  formance improvement that can be achieved by the  size of PLM and OOD technique is orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In  line with the growing size of PLM, the OOD tech-  nique needs to be developed in a more parameter-  efficient way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' As such, developing a proper OOD  technique compatible with the parameter-efficient  transfer methods is our proper goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  References  Jimmy Lei Ba, Jamie Ryan Kiros, and Geoffrey E Hin-  ton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' In Pro-  ceedings of the 2nd Workshop on Natural Language  Processing for Conversational AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Hyunsoo Cho, Choonghyun Park, Jaewook Kang,  Kang Min Yoo, Tae-uk Kim, and Sang-goo Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='  Enhancing out-of-distribution detection in  natural language understanding via implicit layer en-  semble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Findings of the Association for Computa-  tional Linguistics: EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Aakanksha Chowdhery, Sharan Narang, Jacob Devlin,  Maarten Bosma, Gaurav Mishra, Adam Roberts,  Paul Barham, Hyung Won Chung, Charles Sutton,  Sebastian Gehrmann, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' arXiv preprint  arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='02311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Jacob Devlin, Ming-Wei Chang, Kenton Lee, and  Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  BERT: pre-training of  deep bidirectional transformers for language under-  standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 2019 Conference of  the North American Chapter of the Association for  Computational Linguistics, NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  William Fedus, Barret Zoph, and Noam Shazeer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Switch transformers: Scaling to trillion parameter  models with simple and efficient sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Journal  of Machine Learning Research, 23(120):1–39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Geli Fei and Bing Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Breaking the closed  world assumption in text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceed-  ings of the 2016 Conference of the North American  Chapter of the Association for Computational Lin-  guistics, NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Tianyu Gao, Adam Fisch, and Danqi Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Making pre-trained language models better few-shot  learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 59th Annual Meet-  ing of the Association for Computational Linguistics,  ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Robert  Geirhos,  Jörn-Henrik  Jacobsen,  Claudio  Michaelis,  Richard  Zemel,  Wieland  Brendel,  Matthias Bethge, and Felix A Wichmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Shortcut learning in deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Nature  Machine Intelligence, 2(11):665–673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Dan Hendrycks and Kevin Gimpel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' A baseline  for detecting misclassified and out-of-distribution  examples in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In 5th International  Conference on Learning Representations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Tom Henighan, Jared Kaplan, Mor Katz, Mark Chen,  Christopher Hesse, Jacob Jackson, Heewoo Jun,  Tom B Brown, Prafulla Dhariwal, Scott Gray, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Scaling laws for autoregressive generative  modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' arXiv preprint arXiv:2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='  Jordan Hoffmann, Sebastian Borgeaud, Arthur Men-  sch, Elena Buchatskaya, Trevor Cai, Eliza Ruther-  ford, Diego de Las Casas, Lisa Anne Hendricks,  Johannes Welbl, Aidan Clark, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Train-  ing compute-optimal large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' arXiv  preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='15556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski,  Bruna Morrone, Quentin De Laroussilhe, Andrea  Gesmundo, Mona Attariyan, and Sylvain Gelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Parameter-efficient transfer learning for nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Edward J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Hu, Yelong Shen, Phillip Wallis, Zeyuan  Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and  Weizhu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Lora: Low-rank adaptation of  large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In The Tenth International  Conference on Learning Representations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Zhengbao Jiang, Frank F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Xu, Jun Araki, and Graham  Neubig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  How can we know what language  models know.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Transactions of the Association for  Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Taeuk Kim, Kang Min Yoo, and Sang-goo Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Self-guided contrastive learning for BERT sentence  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 59th Annual  Meeting of the Association for Computational Lin-  guistics ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Stefan Larson, Anish Mahendran, Joseph J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Peper,  Christopher Clarke,  Andrew Lee,  Parker Hill,  Jonathan K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Kummerfeld, Kevin Leach, Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Laurenzano, Lingjia Tang, and Jason Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  An evaluation dataset for intent classification and  out-of-scope prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 2019  Conference on Empirical Methods in Natural Lan-  guage Processing EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Kimin Lee, Kibok Lee, Honglak Lee, and Jinwoo Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' A simple unified framework for detecting out-  of-distribution samples and adversarial attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In  Advances in Neural Information Processing Systems,  pages 7167–7177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  The power of scale for parameter-efficient prompt  tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 2021 Conference on  Empirical Methods in Natural Language Processing,  EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Tian Li, Xiang Chen, Shanghang Zhang, Zhen Dong,  and Kurt Keutzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Cross-domain senti-  ment classification with contrastive learning and mu-  tual information maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Xiang Lisa Li and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' Prefix-tuning:  Optimizing continuous prompts for generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In  Proceedings of the 59th Annual Meeting of the Asso-  ciation for Computational Linguistics, ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='  Deep unknown in-  tent detection with margin loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of  the 57th Conference of the Association for Computa-  tional Linguistics, ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Weitang Liu, Xiaoyun Wang, John Owens, and Yix-  uan Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Energy-based out-of-distribution de-  tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Advances in Neural Information Processing  Systems, 33:21464–21475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Xiao Liu, Yanan Zheng, Zhengxiao Du, Ming Ding,  Yujie Qian, Zhilin Yang, and Jie Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Gpt  understands, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='  Ilya Loshchilov and Frank Hutter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Decoupled  weight decay regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In 7th International  Conference on Learning Representations, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Seung Jun Moon, Sangwoo Mo, Kimin Lee, Jaeho Lee,  and Jinwoo Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  MASKER: masked key-  word regularization for reliable text classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In Thirty-Fifth AAAI Conference on Artificial Intel-  ligence, AAAI 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Fabio Petroni, Tim Rocktäschel, Sebastian Riedel,  Patrick S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Language models as  knowledge bases?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 2019 Con-  ference on Empirical Methods in Natural Language  Processing, EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Jonas Pfeiffer, Aishwarya Kamath, Andreas Rücklé,  Kyunghyun  Cho,  and  Iryna  Gurevych.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Adapterfusion:  Non-destructive task composition  for transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In Proceedings of the 16th  Conference of the European Chapter of the Associ-  ation for Computational Linguistics, EACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Alexander Podolskiy, Dmitry Lipin, Andrey Bout, Eka-  terina Artemova, and Irina Piontkovskaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Re-  visiting mahalanobis distance for transformer-based  out-of-domain detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Thirty-Fifth AAAI Con-  ference on Artificial Intelligence, AAAI 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Gabriel Poesia, Alex Polozov, Vu Le, Ashish Tiwari,  Gustavo Soares, Christopher Meek, and Sumit Gul-  wani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Synchromesh: Reliable code genera-  tion from pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In The Tenth  International Conference on Learning Representa-  tions, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Alec Radford, Jeffrey Wu, Rewon Child, David Luan,  Dario Amodei, Ilya Sutskever, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Lan-  guage models are unsupervised multitask learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  OpenAI blog, 1(8):9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Colin Raffel, Noam Shazeer, Adam Roberts, Katherine  Lee, Sharan Narang, Michael Matena, Yanqi Zhou,  Wei Li, and Peter J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Exploring the limits  of transfer learning with a unified text-to-text trans-  former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Journal of Machine Learning Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Samyam Rajbhandari, Jeff Rasley, Olatunji Ruwase,  and Yuxiong He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Zero: Memory optimiza-  tions toward training trillion parameter models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In  SC20: International Conference for High Perfor-  mance Computing, Networking, Storage and Anal-  ysis, pages 1–16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and  Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' Squad: 100, 000+ questions for  machine comprehension of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of  the 2016 Conference on Empirical Methods in Natu-  ral Language Processing, EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Timo Schick and Hinrich Schütze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Exploiting  cloze-questions for few-shot text classification and  natural language inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the  16th Conference of the European Chapter of the As-  sociation for Computational Linguistics, EACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Yilin Shen, Yen-Chang Hsu, Avik Ray, and Hongxia  Jin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Enhancing the generalization for intent  classification and out-of-domain detection in SLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In Proceedings of the 59th Annual Meeting of the  Association for Computational Linguistics ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Taylor Shin, Yasaman Razeghi, Robert L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Logan IV,  Eric Wallace, and Sameer Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Autoprompt:  Eliciting knowledge from language models with au-  tomatically generated prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of  the 2020 Conference on Empirical Methods in Natu-  ral Language Processing, EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Lei Shu, Hu Xu, and Bing Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' DOC: deep open  classification of text documents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of  the 2017 Conference on Empirical Methods in Natu-  ral Language Processing, EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Jihoon Tack, Sangwoo Mo, Jongheon Jeong, and Jin-  woo Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  CSI: novelty detection via con-  trastive learning on distributionally shifted instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In Advances in Neural Information Processing Sys-  tems 33, NeurIPS 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob  Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz  Kaiser, and Illia Polosukhin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Attention is all  you need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Advances in neural information pro-  cessing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Alex Wang,  Yada Pruksachatkun,  Nikita Nangia,  Amanpreet Singh, Julian Michael, Felix Hill, Omer  Levy, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Superglue: A  stickier benchmark for general-purpose language un-  derstanding systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Advances in neural informa-  tion processing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Alex Wang, Amanpreet Singh, Julian Michael, Felix  Hill, Omer Levy, and Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  GLUE: A multi-task benchmark and analysis plat-  form for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In 7th  International Conference on Learning Representa-  tions, ICLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Ben Wang and Aran Komatsuzaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content='  GPT-  J-6B:  A  6  Billion  Parameter  Autoregressive  Language Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='com/  kingoflolz/mesh-transformer-jax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Bin Wang and C-C Jay Kuo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+page_content=' Sbert-wk: A sen-  tence embedding method by dissecting bert-based  word models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  IEEE/ACM Transactions on Audio,  Speech, and Language Processing, 28:2146–2157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Thomas Wolf, Lysandre Debut, Victor Sanh, Julien  Chaumond, Clement Delangue, Anthony Moi, Pier-  ric Cistac, Tim Rault, Remi Louf, Morgan Funtow-  icz, Joe Davison, Sam Shleifer, Patrick von Platen,  Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu,  Teven Le Scao, Sylvain Gugger, Mariama Drame,  Quentin Lhoest, and Alexander Rush.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Trans-  formers: State-of-the-art natural language process-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 2020 Conference on Em-  pirical Methods in Natural Language Processing:  System Demonstrations, pages 38–45, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Asso-  ciation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Zhiyuan Zeng, Keqing He, Yuanmeng Yan, Zijun Liu,  Yanan Wu, Hong Xu, Huixing Jiang, and Weiran Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Modeling discriminative representations for  out-of-domain detection with supervised contrastive  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 59th Annual Meet-  ing of the Association for Computational Linguistics,  ACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Jianguo Zhang, Kazuma Hashimoto, Yao Wan, Zhiwei  Liu, Ye Liu, Caiming Xiong, and Philip Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Are pre-trained transformers robust in intent classi-  fication?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' a missing ingredient in evaluation of out-  of-scope intent detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In Proceedings of the 4th  Workshop on NLP for Conversational AI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Zihao Zhao, Eric Wallace, Shi Feng, Dan Klein, and  Sameer Singh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Calibrate before use: Improv-  ing few-shot performance of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In  Proceedings of the 38th International Conference on  Machine Learning, ICßML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Wenxuan Zhou, Fangyu Liu, and Muhao Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Contrastive out-of-distribution detection for pre-  trained transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  In Proceedings of the 2021  Conference on Empirical Methods in Natural Lan-  guage Processing, EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Appendix  A  Parameter-Efficient Transfer  Learning  Adapter The adapter approach inserts small train-  able adapter modules between transformer layers  while the parameters of the original network remain  fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The adapter module uses a bottleneck archi-  tecture which projects the input dimension h to a  lower-dimensional space specified by bottleneck  dimension r, followed by a nonlinear activation  function, and a up-projection to initial dimension  h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In this work, we attach adapter modules in two  places, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', after the projection following multi-  head attention and after the two feed-forward lay-  ers, following original implementation in (Houlsby  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Also, we use relu as a nonlinear func-  tion and layer normalization (Ba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  LoRA LoRA injects trainable low-rank matrices  into transformer layers to approximate the weight  updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' For a pre-trained weight matrix W ∈  Rh×k, LoRA decompose ∆W  = WdownWup  where Wdown ∈ Rh×r, Wup ∈ Rr×k are trainable  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Specifically we attach LoRA in weight  matrices in the self attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Specifically  we attached LoRA to query and key vector follow-  ing the original implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Prefix-Tuning Prefix tuning prepends l tunable  prefix vectors to the keys and values of the multi-  head attention at every layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Following the original  implementation, we reparametrize the prefix matrix  of dimension h by a smaller matrix of dimension r  composed with a large feedforward neural network  with tanh as a nonlinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  B  Expanded Configuration Details  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1  Common Environment  For the experiments, 4 Tesla V100 SXM2 32GB  GPUs are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The batch size is 8 per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' When  the GPU is too small for the batch size, we set  batch size to 4 and the number of gradient accu-  mulation steps to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We implemented our model  based on Transformers (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020) library  by Huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Additionally, we used deepspeed  (Rajbhandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2020) to train models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Specifi-  cally, we used ZeRO2 with cpu offload on a 240GB  RAM CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In this setting, fine-tuning GPT-J on  CLINC150 full dataset takes about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1 GPU hours  per epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' We used AdamW (Loshchilov and Hut-  ter, 2019) optimizer with epsilon 1e-6 and weight  dataset  #domain  #intent  #data (train/val/test/ood)  CLINC  10  15  15000/3000/4500/1000  Banking  1  77  7812 / 1520 / 3040  Table 3: Dataset statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  BERT-base  CLINC150 Full  ACC ↑  FPR-95 ↓  AUROC ↑  Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (2017)  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='51±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='45  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='33±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='27  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='92±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='05  Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (2021)  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='37  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='26  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='72±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='03  Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (2021)  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='19±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='28  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='4±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='97  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='75±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='2  Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (2021)  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='79±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='13  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='95  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='6±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='11  Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (2021)  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='66  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='88  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='43  Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' (2022)  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='96±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='39  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='67 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='51  98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='27 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='16  Table 4:  Results of each model trained on the  CLINC150 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The best performance in each met-  ric is indicated in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Furthermore, we apply the cosine an-  nealing scheduler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' For GPT-neo, the minimum  learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' For GPT-J, the minimum learn-  ing rate is the one fifth of maximum learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='2  Number of Trainable-Parameter  For each method, a feed-forward layer is added at  the end of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' In this section, we will calcu-  late the number of additional trainable parameters  of each training methods discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Biases are omitted for better readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Adapter Adapter method adds four feed-forward  layers per transformer layer in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Two of  them are down-projection layers, and the others are  up-projection layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' When the original embedding  size of the model is h, the bottleneck dimension is  r, and the number of transformer layers is L, the  number of the trainable parameters of these layers  is calculated as 4Lhr, excluding the bias of the  added layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  LoRA Similar to adapter, LoRA also adds feed-  forward layers per transformer layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Therefore,  the number of the trainable parameters of 4Lhr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  However, the number of parameters are less than  adapter if h and r is the same, since LoRA does  not use bias of the feed-forward layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  Prefix-Tuning There are two trainable elements  in prefix tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The first one is the prefix em-  beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' When the number of prefixes is l, and  the embedding size is h, lh parameters are used  by the prefixes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' Second, the reparametrization ma-  trix is also trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The down-projection matrix  has hr parameters, when the reduced dimension  for reparametrization is r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The up-projection ma-  trix has 2Lhr parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' As a result, there are  Method  Parameters  Values  LoRA  Learning rate  2e-4 (GPT-Neo), 5e-5 (GPT-J)  Bottleneck dim  8 (GPT-Neo / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 80 (GPT-Neo / 1%), 12 (GPT-J /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 128 (GPT-J / 1%)  Location  query, value  Adapter  Learning rate  8e-5 (GPT-Neo / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 1e-4 (GPT-Neo / 1%), 5e-5 (GPT-J), 5e-4 (GPT-J / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 1e-4 (GPT-J / 1%)  Bottleneck dim  6 (GPT-Neo / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 80 (GPT-Neo / 1%), 11 (GPT-J /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 128 (GPT-J / 1%)  Location  after Multi-head, after Feed-forward,  Prefix-tuning  Learning rate  2E-4 (GPT-Neo), 5E-5 (GPT-J)  Bottleneck dim  12 (GPT-Neo / 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 160 (GPT-Neo / 1%), 20 (GPT-J /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='1%), 256 (GPT-J / 1%)  Prefix length  5, 10, 20  Table 5: Hyper-parameter search for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  h(2Lr + l) trainable parameters on prefix tuning  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='3  Hyper-parameter Search  Tab 5 summarizes hyper parameters for each  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  C  Selecting SOTA OOD Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='  The Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content='4 summarizes the results with recently  proposed OOD approaches on BERT-base with  CLINC dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=' The best performing model (Cho  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
+page_content=', 2022) is selected as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RNFJT4oBgHgl3EQf3S0N/content/2301.11660v1.pdf'}
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+arXiv:2301.13147v1  [math.CV]  30 Jan 2023
+NEARGEODESICS IN GROMOV HYPERBOLIC JOHN DOMAINS
+IN INFINITE DIMENSIONAL BANACH SPACES
+VASUDEVARAO ALLU AND ABHISHEK PANDEY
+Abstract. Rasila, Talponen and Zhang in [Proc. Amer. Math. Soc. 146 (2018),
+3863–3873] have proposed a general question that which Banach space properties
+can be characterized by the corresponding quasihyperbolic metric? A domain D in
+a Banach space E is said to be Gromov hyperbolic if it is δ ≥ 0-hyperbolic with
+respect to quasihyperbolic metric. In this paper we show that every neargeodesic
+in a Gromov hyperbolic John domain in infinite dimensional Banach space is a
+cone arc. Thus, our result gives an answer to the above problem in the sense
+that it gives a necessary condition for a John domain to be a Gromov hyperbolic
+space in infinite dimensional Banach space. Moreover, our main result gives an
+improvement of a result of Li [Theorem 1, Int. J. Math. 25 (2014)].
+1. Introduction
+Let Ω be an open ball or half space in Rn, we denote hΩ be the hyperbolic metric
+in Ω with constant curvature −1. If Ω is an upper half plane then hΩ is defined in
+Ω by the means of the density
+ρ(z) =
+1
+Im z =
+1
+d(z, ∂Ω).
+This density can be used to introduce an analogue of the hyperbolic metric in an
+arbitrary subdomain D in Rn. The quasihyperbolic metric in Rn is a generalization
+of hyperbolic metric, which was introduced by Gehring and Palka [3] and they
+have studied the characterization of domains D in the one point compactification
+of Rn which are quasiconformally equivalent to ball in Rn. Infact such domains are
+homogeneous via quasiconformal mappings. Gehring and Palka [3] have proved that
+the maximal dilatation of such quasiconformal mappings can be estimated in terms
+of quasihyperbolic metric and using these estimates they obtained useful results
+related to the homogeneity of domains with respect to quasiconformal family. Thus,
+the importance of the quasihyperbolic metric is quite clear as it is well-behaved with
+the quasiconformal mappings. The quasihyperbolic metric can be naturally defined
+on a wide range of metric spaces, including infinite dimensional Banach spaces. The
+study of this concept is known as free quasiconformality which has been introduced
+and developed by Väisälä (see [11, 12]).
+In this paper, we mainly focus on the geometry of neargeodesics in John domains
+in infinite dimensional Banach spaces.
+File: QHG-P-1.tex, printed: 2023-1-31, 1.52
+2010 Mathematics Subject Classification. Primary 30C65, 30L10, 30F45. Secondary 30C20.
+Key words and phrases. Quasihyperbolic metric, Gromov hyperbolic spaces, John domain, near-
+geodesics, quasiconformal mappings, quasihyperbolic geodesic, cone arc.
+1
+
+2
+Vasudevarao Allu and Abhishek Pandey
+2. Preliminaries
+In this section we mention some basic as well as advanced concepts and related
+results in the literature, which are useful to state and prove our main result.
+2.1. Metric Geometry. Let (X, d) be a metric space. A curve is a continuous
+function γ : [a, b] → X. If a curve γ : [a, b] → X is an embedding of [a, b], then it is
+called an arc. Let P denote set of all partitions a = t0 < t1 < t2 < · · · < tn = b of
+the interval [a, b]. The length of the curve γ in the metric space (X, d) is
+ld(γ) = sup
+P
+n−1
+�
+k=0
+d(γ(tk), γ(tk+1)).
+A curve is said to be rectifiable if ld(γ) < ∞.
+A metric space X is said to be
+rectifiably connected if every pair of points x, y ∈ X can be joined by a rectifi-
+able curve. For a rectifiable curve γ we define arc length s : [a, b] → [0, ld(γ)] by
+s(t) = ld(γ|[a,t]). The arc length function is of bounded variation. For any rectifiable
+curve γ : [a, b] → X, there is a unique map γs : [0, ld(γ)] → X such that γ = γs ◦ s,
+and such a curve γs is called the arclength parametrization of γ.
+For x, y ∈ X, the inner length metric λX(x, y) is defined by
+λX(x, y) = inf{ld(γ) : γ is a rectifiable arc joining x and y}.
+Let ρ : X → [0, ∞] be a Borel function. The ρ-length of a rectifiable curve γ is
+�
+γ
+ρ ds =
+� b
+a
+ρ(γ(t)) ds(t) =
+� ld(γ)
+0
+ρ ◦ γs(t) dt.
+If X is rectifiably connected then ρ induces a distance function which is defined
+by
+dρ(x, y) = inf
+�
+γ
+ρ ds,
+where infimum is taken over all rectifiable curves joining x and y in X. We note
+that, in general, dρ need not to be a metric however dρ is a metric if ρ is positive and
+continuous. If dρ is a metric, we say (X, dρ) is a conformal deformation of (X, d) by
+the conformal factor ρ.
+A curve γ : [a, b] → X is said to be geodesic if for all t, t′ ∈ [a, b],
+d(γ(t), γ(t′)) = |t − t′|.
+A metric space X is said to be a geodesic metric space if every pair of points x, y ∈ X
+can be joined by a geodesic, and is said to be proper if every closed ball is compact.
+A metric space (X, d) is said to be intrinsic or path metric space/ or length metric
+space if for all x, y ∈ X, we have
+d(x, y) = λX(x, y).
+Definition 2.1. A metric space (X, d) is said to be minimally nice if it is locally
+compact, rectifiably connected and non-complete.
+
+Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces
+3
+2.2. Gromov hyperbolicity. Gromov hyperbolicity was introduced by Gromov in
+the 1980s. It is often assumed that the under lying space is geodesic metric space
+and usually it is a proper metric space. As we will see later that infinite dimensional
+Banach space with quasihyperbolic metric need not be geodesic space, therefore we
+adapt the definition of Gromov hyperbolicity which works well with the non geodesic
+spaces. For this, we refer to the work of Väisälä [16] where the concept of Gromov
+product has been used.
+Definition 2.2. Let (X, d) be a metric space and let p ∈ X. The Gromov Product
+(x|y)p of x, y ∈ X with respect to p is defined by
+(x|y)p = 1
+2
+�
+d(x, p) + d(y, p) − d(x, y)
+�
+.
+Therefore, (x|y)p measures to what extent the triangle inequality for the triangle
+∆xyp is far from being an equality.
+Definition 2.3. Let δ ≥ 0. A metric space (X, d) is said to be δ-hyperbolic if for
+every x, y, z, p ∈ X, the following inequality holds
+(x|z)p ≥ min
+�
+(x|y)p, (y|z)p
+�
+− δ.
+A metric space (X, d) is said to be Gromov hyperbolic if it is δ-hyperbolic for some
+δ ≥ 0.
+2.3. Examples. Clearly, R is a 0-hyperbolic. Any tree T is 0-hyperbolic. The unit
+disk with hyperbolic metric is Gromov hyperbolic with δ = log 3. The complex
+plane is not Gromov hyperbolic.
+We shall mention an important result here due to Väisälä [15, 3.7], which is known
+as stability theorem. Infact, stability theorem says that in an intrinsic hyperbolic
+space any two quasi-isometric paths with same initial and terminal points x, y runs
+close to each other even if |x − y| is large. To mention this theorem we first need to
+mention the concept of quasi-isometry and Hausdorff distance.
+Definition 2.4. Let X and Y be two non-empty subsets of a metric space (M, d).
+We define their Hausdorff distance dH(X, Y ) by
+dH(X, Y ) = max
+�
+sup
+x∈X
+d(x, Y ), sup
+y∈Y
+d(X, y)
+�
+,
+where d(a, B) = infb∈B d(a, b) denotes the distance from a point a ∈ M to a subset
+B ⊂ M.
+Definition 2.5. Let λ ≥ 1 and µ ≥ 0 and (X, d) and (Y, d′) be metric spaces. We
+say that a map f : X → Y is a (λ, µ)-quasi-isometry if
+λ−1d(x, y) − µ ≤ d′(f(x), f(y)) ≤ λd(x, y) + µ
+for all x, y ∈ X. In the case where f : I → Y is a map of an real interval I, we say
+that such a map (λ, µ)-quasi-isometric path.
+Remark 2.1. An arc γ joining x and y in a metric space X is said to be λ-
+quasigeodesic, λ ≥ 1, if
+l(γ[u, v]) ≤ λd(u, v)
+
+4
+Vasudevarao Allu and Abhishek Pandey
+for all u, v ∈ γ. In such a case the arc length parametrization γs : [0, l(γ)] → γ
+satisfies the inequality
+λ−1|t − t′| ≤ d(γs(t), γs(t′)) ≤ λ|t − t′|.
+Thus γ is (λ, 0)-quasi-isometry.
+The following theorem (see [15, 3.7]) is vital to prove our main result.
+Theorem A. Suppose that X is an intrinsic δ-hyperbolic space and that γ and α
+are (λ, µ)-quasi-isometric path with same initial and terminal points. Then there
+exists a constant M such that
+dH(γ, α) ≤ M,
+where M depends only on δ, λ, µ and dH denote the Hausdorff distance between γ
+and α.
+Next we mention an important result due to Väisälä (see [16, Theorem 3.18])
+which says that quasi-isometries preserves hyperbolicity.
+Theorem 2.1. Let X and Y be intrinsic spaces and let f : X → Y be a (λ, µ)-
+quasi-isometry. If Y is δ-hyperbolic, then X is δ′-hyperbolic with δ′ = δ′(δ, λ, µ).
+2.4. Quasihyperbolic metric. In view of our frame work, we consider the quasi-
+hyperbolic metric in the setting of Banach spaces.
+Definition 2.6. Let E be a real Banach space of dimension at least 2 and let D ⊊ E
+be a domain (open, connected nonempty set). Let the metric induced by norm on
+D is |.|. Define k : D → (0, ∞) by
+k(z) =
+1
+∆(z) =
+1
+d(z, ∂D).
+Define the quasihyperbolic length of a rectifiable curve γ in D by
+lk(γ) =
+�
+γ
+ρ(z) ds =
+�
+γ
+ds
+d(z, ∂D).
+Let kD denote the quasihyperbolic metric in D which is defined by
+(2.1)
+kD(x, y) = inf
+γ lk(γ)
+where the infimum is taken over all rectifiable curves γ in D joining x to y.
+Observe that (D, kD) is always an intrinsic space. For further geometric properties
+of hyperbolic metric, we refer to [4] and [21]. Quasihyperbolic metric satisfies the
+following properties.
+(i) kD = hD when D is a half space in Rn.
+(ii) kD ≤ hD ≤ 2kD when D is a ball in Rn.
+Remark 2.2. [2, Proposition 2.8] If (X, d) is a locally compact, incomplete and
+rectifiably connected metric space then (X, kX) is a complete, proper and geodesic
+metric space.
+
+Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces
+5
+Finally, we recall some estimates for the quasihyperbolic metric which have been
+first introduced by Gehring and Palka [3, 2.1] in Rn. Later, Väisälä [11, Lemma
+2.2] have proved these inequalities in the case of Banach spaces. Let D ⊊ E be a
+domain and x, y ∈ D and let γ be a rectifiable curve joining x and y. Then we have
+the following:
+(2.2)
+k(x, y) ≥ log
+�
+1 +
+|x − y|
+min{∆(x), ∆(y)}
+�
+≥ log ∆(y)
+∆(x)
+and
+(2.3)
+lk(γ) ≥ log
+�
+1 +
+l(γ)
+min{∆(x), ∆(y)}
+�
+.
+We are going to use frequently (2.2) and (2.3) to prove our main results.
+2.5. Quasi-hyperbolic geodesic. A rectifiable curve γ from a to b in D is said to
+be quasihyperbolic geodesic if kD(a, b) = lk(γ). Obviously each subarc of a quasi-
+hyperbolic geodesic is again a geodesic. Observe that quasi-hyperbolic geodesic is
+curve γ for which the infimum in (2.1) is attained.
+At this juncture, the natural question is does quasi-hyperbolic geodesic always
+exists? A result of Gehring and Osgood [4] shows that for a proper subdomain D
+in Rn, the answer to this question is affirmative. For important results on quasihy-
+perbolic geodesics in domain in Rn, we refer to G. Martin [9]. Moreover, Martin [9]
+has proved that quasihyperbolic geodesics in Rn are C1 smooth. It is well known
+that a quasihyperbolic geodesic between any two points exists if dim (E) is finite
+(see [4, Lemma 1]). Note that the quasihyperbolic geodesic may not exists in an
+infinite-dimensional Banach spaces (see [11, Example 2.9] and [13, 3.5]). However,
+Väisälä [15, Theorem 2.1] has proved the existence of quasihyperbolic geodesic when
+E is reflexive Banach space and D is a convex domain. Furthermore, in 2011, Martio
+and Väisälä [10, Theorem 2.11] proved the existence as well as uniqueness of quasi-
+hyperbolic geodesic when E is uniformly convex Banach space and D is a convex
+domain.
+2.6. Some special domains.
+2.7. Uniform Domains.
+Definition 2.7. Let c ≥ 1. A domain D in E is said to be c-uniform in the norm
+metric if for each pair of points x, y ∈ D, there exists a rectifiable arc γ in D joining
+x to y such that
+(i) min{l(γ[x, z]), l(γ[z, y])} ≤ c d(z, ∂D), for all z ∈ γ, and
+(ii) l(γ) ≤ c|x − y|,
+where l(γ) denotes the arc length of γ in (D, |. |), where |. | is metric induced by
+norm, and γ[x, z] denotes the part of γ between x and z. Also we say that curve γ
+is a double c-cone arc.
+
+6
+Vasudevarao Allu and Abhishek Pandey
+2.8. Inner uniform domains.
+Definition 2.8. Let c ≥ 1. A domain D is said to be c-inner uniform in the norm
+metric if for each pair of points x, y ∈ D, there exists a rectifiable arc γ in D joining
+x to y such that
+(i) min{l(γ[x, z]), l(γ[z, y])} ≤ c d(z, ∂D), for all z ∈ γ,
+(ii) l(γ) ≤ cλD(x, y).
+2.9. John Domain.
+Definition 2.9. Let c ≥ 1. A domain D is said to be c-John in the norm metric if
+for each pair of points x, y ∈ D there exists a rectifiable curve γ in D joining x to y
+such that for all z ∈ γ
+(2.4)
+min{l(γ[x, z]), l(γ[z, y])} ≤ c d(z, ∂D)
+and we say that such a curve γ is a c-cone arc.
+Remark 2.3. We mention some of the remarks with regards to the connection
+between the aforesaid domains.
+(i) c-uniform implies inner c-uniform implies c-John
+(ii) R2 \ {(x, 0) : x ≥ 0} is inner uniform but not unifrom.
+(iii) R2 \ {(n, 0) : n ∈ N} is a John domain but not inner uniform.
+2.10. Gromov hyperbolicity of domains.
+Definition 2.10. Let δ ≥ 0.
+A domain D in a Banach space E is said to be
+δ-hyperbolic if (D, kD) is δ-hyperbolic.
+Definition 2.11. A domain D in a Banach space E is said to be Gromov hyperbolic
+if (D, kD) is δ-hyperbolic for some δ ≥ 0.
+All c- uniform and inner c- uniform domains are Gromov δ = δ(c) hyperbolic.
+R2 \ {(n, 0) : n ∈ N} is a John domain which is not hyperbolic. The broken tube
+(see [11, 2.12] and for a detailed treatment we refer to [14]) in an infinite dimensional
+seperable Hilbert space is a classical example of a domain which is hyperbolic but
+not John and hence not uniform.
+2.11. Quasihyperbolic geodesic in Uniform Domains in Rn. Gehring and
+Osgood [4] have proved that each quasihyperbolic geodesic in a c-uniform domain
+D ∈ Rn is a double b-cone arc, where the constant b depends only on c.
+2.12. Quasihyperbolic Geodesic in John Domain in Rn. Since John domains
+can be thought of one sided uniform domains, it is natural to ask whether the result
+also holds for John domains or not? In fact, this problem has been proposed by
+Gehring et. al. [5] in the following form.
+Problem 2.5. Suppose D ⊂ Rn is a c− John domain and that γ is a quasihyperbolic
+geodesic in D. Is γ a b-cone arc for some b = b(c)?
+In 1989, Gehring et. al. [5] proved the following result.
+Theorem B. [5, Theorem 4.1] If D ⊂ R2 is a simply connected c-John domain
+then every quasihyperbolic or hyperbolic geodesic in D is a b-cone arc, where b only
+depends on c.
+
+Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces
+7
+Infact, Gehring et. al. [5] have constructed several examples which shows that
+a quasihyperbolic geodesic in a c- John domain need not to be a b-cone arc with
+b = b(c) unless n = 2 and D is simply connected. Thus, this suggest us that we
+need some extra restriction on c-John domain so that the Problem 2.5 holds good.
+This raises the following natural question.
+Problem 2.6. Determine necessary and/or sufficient conditions for quasihyperbolic
+geodesics in a John space to be cone arcs.
+In 1989, Heinonen [6, Question 2] posed the following problem.
+Problem 2.7. Suppose D ⊂ R2 is a c-John domain which is quasiconformally
+equivalent to the unit ball Bn in Rn and γ is a quasihyperbolic geodesic in D. Is γ
+a b = b(c)-cone arc for some constant b?
+In 2001, Bonk, Heinonen and Koskela [2, Proposition 7.12] gave an affirmative
+answer to Probelm 2.7. In particular, they have proved the following result.
+Theorem C. [2, Proposition 7.12] If D ⊂ Rn is a bounded a-John domain which
+is homeomorphic to inner c-uniform domain via K-quasiconformal map, then each
+quasihyperbolic geodesic in D is a b-cone arc with b = b(a, c, K, n)
+Since every ball is inner uniform, Theorem C gives an affirmative answer to Prob-
+lem 2.7.
+3. Metric Space setting
+In 2022, Zhou, Li and Rasila [20] consider the Problem 2.6 further and proved the
+following.
+Theorem 3.1. [20, Theorem 1.1] Let (D, d) be a locally compact, rectifiablly con-
+nected and non complete metric space, and k be the quasihyperbolic metric of (D, d).
+If (D, d) is a-John and if (D, k) is K-roughly starlike and Gromov δ-hyperbolic.
+Then every quasihyperbolic geodesic in D is a b-cone arc, where b depends only on
+a, K and δ.
+Remark 3.1. Observe that any proper subdomain D in Rn is locally compact,
+rectifiablly connected and noncomplete with respect to the usual metric. Theorefore,
+Theorem 3.1 gives an improvement of Theorem C.
+The next result of Zhou and Ponnusamy [18, 19] shows that the condition of rough
+starlikeness is not required in Theorem 3.1. Thus the following results of Zhou and
+Ponnusamy [19] give an improvement of Theorem 3.1.
+Theorem 3.2. [19, Theorem 1.4] Let (D, d) be a locally compact, rectifiablly con-
+nected and non complete metric space, and k be the quasihyperbolic metric of (D, d).
+If (D, d) is a-John and if D is Gromov hyperbolic i.e., (D, kD) is δ-hyperbolic. Then
+every quasihyperbolic geodesic in D is a b-cone arc with b = b(a, δ).
+4. Banach space setting
+The main aim of this paper is to study Problem 2.6 by replacing quasihyperbolic
+geodesic with neargeodesics in Banach space settings precisely in infinite dimansional
+
+8
+Vasudevarao Allu and Abhishek Pandey
+Banach spaces. Note that the quasihyperbolic geodesic may not exists in the infinite-
+dimensional Banach spaces (see [11, Example 2.9] and [13, Remark 3.5]). However,
+these examples of domains are in Hilbert spaces such that the complement of these
+domains are uncountable. For example of a domain that is not geodesic with respect
+to quasihyperbplic metric such that the complement of domain is countable we refer
+to [7, Example 4.1]. In order to overcome this shortage, Väisälä [12] introduced the
+concept of neargeodesic. We recall the definition of neargeodesic.
+Definition 4.1. Let D ⊊ E be a domain and c ≥ 1. An arc γ in D is said to be
+c-neargeodesic if for all x, y ∈ γ, we have
+lk(γ[x, y]) ≤ ckD(x, y).
+Thus an arc γ is a quasihyperbolic geodesic if, and only if, it is a 1-neargeodesic.
+In [12] Väisälä proved the following existence theorem for neargeodesics.
+Theorem 4.1. [12, Theorem 3.3] Let c > 1. Then for every pair of points z1, z2 ∈ D
+there exists a c-neargeodesic joining z1 and z2.
+Observe that the constant b in Theorem B depends on n, so it is natural to ask
+the following question.
+Problem 4.1. Does there exists a dimension free anologue of Theorem B. In other
+words does it holds in the setting of Banach spaces or not if we replace quasihyper-
+bolic geodesic with neargeodesics?
+Li [8, Theorem 1] gave an affirmative answer to Problem 4.1 and proved the
+following
+Theorem 4.2. If D ⊂ E is an a-John domain which is homeomorphic to c-inner
+uniform domain via an (M, C) − CQH. Let z1, z2 ∈ D and γ is a c0-neargeodesic
+joining z1 and z2 in D. Then γ is a b-cone arc with b = b(a, c, c0, M, C).
+5. Main Result
+In this section we state our main result of this paper. We observe that in Theorem
+3.1 and 3.2, authors have considered that metric space (X, D) which is minimally
+nice. Once you have such metric space, Remark 2.2 immediately tells us that (X, kX)
+is a geodesic metric space i.e., for any papir of points quasihyperbolic geodesic exists.
+Note that if we consider a domain D in an infinite dimensional Banach space
+then D with the metric induced by norm satisfies all the properties of minimally
+nice space except locally compact. This motivates us to ask does Theorem 3.1 and
+3.2 type results still holds if we consider John domain D in infinite dimensional
+Banach space? In such a case we know that (D, kD) need not be geodesic and hence
+the natural choice is to consider the neargeodesics at the place of quasihyperbolic
+geodesics.
+Theorem 5.1. Let E be an infinite dimensional Banach space and D ⊊ E is a
+c-John domain and suppose that D is Gromov hyperbolic i.e. (D, kD) is δ-hyperbolic
+for some δ ≥ 0. Then every c0-neargeodesic in D is a b-cone arc, where b depends
+only on c, c0 and δ.
+
+Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces
+9
+Remark 5.1.
+(1) Observe that Theorem 5.1 is different from Theorem 3.1 and
+Theorem 3.2. Note that in Theorem 5.1 D is a domain in infinite dimansional
+Banach spce therefore D is not locally compact with metric induced by norm.
+Hence Remark 2.2 is also no more valid and we can’t use the results of
+Gromov hyperbolicity for geodesic metric space. Further, in this paper, we
+are considering the neargeodesics rather than quasihyperbolic geodesic.
+(2) To prove our main result we need the results on Gromov hyperbolicity for
+general metric spaces that are need not to be geodesic. Thanks to the work
+of Väisälä [16] where he has established the theory of Gromov hyperbolicity
+for domains in Banach spaces which obviously need not to be geodesic (in
+quasihyperbolic metric) when the Banach space is infinite dimensional.
+(2) In view of Theorem 2.1 and the following two facts:
+(i) all inner c-uniform domains are Gromov hyperbolic with respect to
+quasihyperbolic metric (see [15, Remark 2.16])
+(ii) every (M, C) − CQH map is (M, C)-quasi-isometry in quasihyperbolic
+metric
+we conclude that Theorem 5.1 is an improvement of Theorem 4.2 of Li [8,
+Theorem 1].
+6. Proof of Main Result
+We shall devide the proof of Theorem 5.1 into two theorems. Theorem 6.1 below
+gives a sufficient condition for a neargeodesic to be a cone arc and Theorem 6.2
+below says that in Gromov hyperbolic John domains, every neargeodesic satisfies
+this sufficient condition.
+Therefore, combining these two theorem we obtain the
+proof of our main result.
+Theorem 6.1. Let γ be a c0- neargeodesic joining x, y such that the following holds:
+If z1, z2 ∈ γ such that each w ∈ γ[z1, z2] satisfies ∆(w) ≤ 2 min{∆(z1), ∆(z2)}, then
+there exists a constant A ≥ 1 such that
+kD(z1, z2) ≤ A.
+Then γ is a C-cone arc, where C depends only on A and co.
+Proof. Suppose γ is a c0- neargeodesic joining x, y and if z1 and z2 are points on
+γ satisfying for each w ∈ γ[z1, z2], ∆(w) ≤ 2 min{∆(z1), ∆(z2)}, then there exists a
+constant A ≥ 1 such that
+kD(z1, z2) ≤ A.
+Since ∆ is continous and γ is compact, choose a point z0 ∈ γ with
+∆(z0) = max
+z∈γ ∆(z)
+Then there exists unique non-negative integer n such that
+2n∆(x) ≤ ∆(z0) ≤ 2n+1∆(x).
+Let x = x0. For each i = 1, 2, . . . , n, let xi be the first point on γ[x, z0] such that
+∆(xi) = 2i∆(x0).
+Similarly, there exists m ≥ 0 such that
+2m∆(y) ≤ ∆(z0) ≤ 2m+1∆(y).
+
+10
+Vasudevarao Allu and Abhishek Pandey
+Let y = y0. For each j = 1, 2, . . . , m, let yj be the first point on γ[z0, y] such that
+∆(yj) = 2j∆(y0).
+In this manner we have devided the arc γ into (n + m + 1) subarcs
+γ[x0, x1], . . . , γ[xn−1, xn], γ[xn, ym], γ[ym, ym+1], . . . , γ[y1, y0].
+It is easy to observe that all these subarcs are also c0-neargeodesic between their
+respective end points. Let u and v be any two adjacent points along γ. That is u
+and v are xi, xi+1 or xn, ym or yj+1, yj, 0 ≤ i ≤ n − 1 and 0 ≤ j ≤ m − 1. By the
+construction, it is clear that for each z ∈ γ[u, v], we have
+∆(z) = 2 min{∆(u), ∆(v)}.
+By assumption, there exists a constant A ≥ 1 such that
+kD(u, v) ≤ A.
+To show that γ is a C-cone arc for some C, we need to show that
+(6.1)
+min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)
+for all w ∈ γ.
+In order to prove (6.1), it is sufficient to prove either l(γ[x, w] ≤ C∆(w) or l(γ[w, y]) ≤
+C∆(w). For this we need to consider three cases depending upon the location of w
+on γ.
+Case 1. Suppose w ∈ γ[xi, xi+1] for some 0 ≤ i ≤ n − 1. Then in view of (2.2), we
+have
+log∆(xi)
+∆(w) ≤ kD(w, xi) = kD(xi, w) ≤ kD(xi, xi+1) ≤ A.
+Therefore, we have
+(6.2)
+∆(xi) ≤ eA∆(w).
+Moreover, we have
+(6.3)
+l(γ[x, w]) ≤
+i
+�
+t=0
+l(γ[xs, xs+1])
+In view of (2.3), we have
+log
+�
+1 +
+l(γ[xs, xs+1])
+min{∆(xs), ∆(xs+1)}
+�
+≤
+lk(γ[xs, xs+1])
+≤
+c0 kD(xs, xs+1)
+≤
+c0 A
+Therefore, this gives us that
+logl(γ[xs, xs+1])
+∆(xs)
+≤ log
+�
+1 + l(γ[xs, xs+1])
+∆(xs)
+�
+≤ c0 A
+and hence
+(6.4)
+l(γ[xs, xs+1]) ≤ ec0 A∆(xs)
+
+Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces
+11
+Using (6.4) in (6.3), we obtain
+l(γ[x, w])
+≤
+i
+�
+t=0
+ec0 A∆(xt)
+(6.5)
+≤
+2 ec0 A ∆(xi)
+≤
+2 eA(c0+1) ∆(w)
+and hence we have
+l(γ[x, w]) ≤ C ∆(w),
+where C = 2eA(c0+1).
+Theorefore, we have
+(6.6)
+min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)
+for all w ∈ γ[xi, xi+1].
+Case 2. Suppose w ∈ γ[xn, ym]. Then it is easy to see that
+log∆(xn)
+∆(w) ≤ kD(w, xn) = kD(xn, w) ≤ kD(xn, ym) ≤ A,
+which implies
+∆(xn) ≤ eA ∆(w).
+An easy computation shows that
+l(γ[x, w])
+≤
+n−1
+�
+i=0
+l(γ[xi, xi+1]) + l(γ[xn, ym])
+≤
+m−1
+�
+i=0
+ec0 A∆(xi) + ec0 A∆(xn)
+≤
+3ec0 A∆(xn)
+≤
+3 eA(c0+1) ∆(w).
+That is,
+l(γ[x, w]) ≤ C ∆(w),
+where C = 3 eA(c0+1)
+and hence we have
+(6.7)
+min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)
+for all w ∈ γ[xn, ym].
+Case 3. Suppose w ∈ γ[yj, yj+1] for some 0 ≤ j ≤ m − 1. Using the same argument
+as in earlier, we obtain
+l(γ[w, y]) ≤ C ∆(w),
+where C = 2 eA(c0+1)
+and hence
+(6.8)
+min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)
+for all w ∈ γ[yj, yj+1].
+Thus, combining (6.6), (6.7) and (6.8), we conclude that γ is a C- cone arc, where
+C = 3 eA(c0+1).
+□
+
+12
+Vasudevarao Allu and Abhishek Pandey
+Theorem 6.2. Let D be a c- John domain in an infinite dimensional Banach space
+E such that (D, kD) is Gromov hyperbolic space. Fix x, y ∈ D and let γ be a c0
+neargeodesic joining x, y ∈ D. Then there exists a constant A ≥ 1 depending only c,
+c0 and δ such that the following holds: If there are u, v ∈ γ such that each w ∈ γ[u, v]
+satisfies ∆(w) ≤ 2 min{∆(u), ∆(v)}, then
+kD(u, v) ≤ A.
+Proof. Let D be a c-John domain in a Banach space E such that (D, kD) is Gromov
+δ-hyperbolic. By definition (D, kD) is an intrinsic space. Fix x, y ∈ D and c0 > 1. In
+view of Theorem 4.1, let γ be a c0-neargeodesic joining x and y. In view of Remark
+2.1, it is clear that a c-neargeodesic γ is (c, 0)- quasi-isometry. Since D is a c-John,
+there is a c-cone arc α joining x and y. To use Theorem A, we first need to show
+that every c-cone arc α is (λ, µ)-quasi-isometry for some λ and µ depending only on
+c. Which we will prove in the following lemma
+Lemma 6.9. Let c ≥ 1 and α be a c-cone arc. Then its arc length parametrization
+αs satisfies the following inequality
+1
+3c|t − t′| − 3c log(3c) ≤ kD(αs(t), αs(t′)) ≤ 3c|t − t′| + 3c log(3c)
+and hence α is (3c, 3c log(3c))-quasi-isometry.
+Proof. Let α be a c-cone arc joining x and y in D and z1, z2 ∈ α. Without loss
+of generality we can assume that z1 ∈ α[x, z2]. Since α is a c-cone arc, for each
+w ∈ α[z1, z2], we have
+l(α[w, z2]) ≤ min{l(α[x, w]), l(α[w, y])} ≤ c∆(w)
+and hence
+(6.10)
+l(α[w, z2]) ≤ c∆(w).
+Next, our aim is to show that ∆(z2) ≤ 2c∆(w). To prove this, we consider two
+cases:
+Case (i) If d(w, z2) < ∆(y)/2, then
+∆(w) ≥ ∆(z2) − d(w, z2) ≥ ∆(z2)/2
+and hence, ∆(z2) ≤ 2∆(w).
+Case (ii) Since α is a c-cone arc we obtain
+c∆(w) ≥ l(α[z2, w]) ≥ d(z2, w) ≥ ∆(z2)/2
+and hence
+(6.11)
+∆(z2) ≤ 2c∆(w)
+By adding (6.10) and (6.11), we obtain
+l(α[w, z2]) + ∆(z2) ≤ 3c∆(w)
+which gives,
+(6.12)
+1
+∆(w) ≤
+3c
+l(α[w, z2]) + ∆(z2)
+
+Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces
+13
+A simple computation shows that
+kD(z1, z2)
+≤
+lk(α[z1, z2]) =
+�
+α[z1,z2]
+ds
+∆(w)
+(6.13)
+≤
+� l(α[z1,z2])
+0
+3c
+t + ∆(z2) dt.
+lk(α[z1, z2])
+≤
+3c(log(l(α[z1, z2]) + ∆(z2)) − log(∆(z2))
+(6.14)
+≤
+3c log
+�
+1 + l(α[z1, z2])
+∆(z2)
+�
+≤
+3c log
+�
+1 + c∆(z1)
+∆(z2)
+�
+(6.15)
+≤
+3c
+�
+log
+�∆(z1)
+∆(z2)
+�
++ log3c
+�
+.
+lk(α[z1, z2])
+≤
+3ckD(z1, z2) + 3c log 3c
+(6.16)
+Theorefore, from (6.16), it is clear that the arc length parametrization of α satisfies
+the inequality
+1
+3c|t − t′| − 3c log(3c) ≤ kD(αs(t), αs(t′)) ≤ 3c|t − t′| + 3c log(3c)
+and hence α is (3c, 3c log(3c))-quasi-isometry. This completes the proof of Lemma
+6.9.
+□
+Since γ is (c0, 0)-quasi-isometry and α is (3c, 3c log(3c))-quasi-isometry in view of
+Theorem A, we have that quasihyperbolic Hausdorff distance satisfies the following
+inequality
+kDH(γ, α) ≤ R = R(c, c0, δ).
+Let u, v ∈ γ such that each w ∈ γ[u, v] satisfies ∆(w) ≤ 2 min{∆(u), ∆(v)}. Our
+aim is to show that there exists a constant A ≥ 1 such that kD(u, v) ≤ A. For this
+we need to estimate kD(u, v). Since kDH(γ, α) ≤ R i.e.,
+max
+�
+sup
+x∈γ kD(x, α), sup
+y∈α kD(γ, y)
+�
+≤ R
+Theorefore,
+sup
+x∈γ kD(x, α) ≤ R
+In particular,
+M1 = kD(u, α) = inf
+b∈α kD(u, b) ≤ R
+and
+M2 = kD(v, α) = inf
+b∈α kD(v, b) ≤ R
+There exists a sequence bi and bj such that kD(u, bi) converges to M1 and kD(u, bj)
+converges to M2. Since α is compact, bi and bj have convergent subsequence which
+
+14
+Vasudevarao Allu and Abhishek Pandey
+converges to say Zu and Zv respectively such that kD(u, Zu) ≤ R and kD(v, Zv) ≤ R.
+Therefore,
+log ∆(u)
+∆(Zu) ≤ kD(u, Zu) ≤ R
+implies
+(6.17)
+∆(Zu) ≤ eR∆(u)
+and
+log ∆(v)
+∆(Zv) ≤ kD(v, Zv) ≤ R
+implies
+(6.18)
+e−R∆(v) ≤ ∆(Zv).
+Since each w ∈ γ[u, v] satisfies
+(6.19)
+∆(w) ≤ 2 min{∆(u), ∆(v)}
+by taking w = v in (6.19), we obtain
+(6.20)
+∆(v) = 2∆(u)
+and by taking w = u in (6.19), we obtain
+(6.21)
+∆(u) ≤ 2∆(v).
+Using (6.21) in (6.17), we obtain
+(6.22)
+∆(Zu) ≤ eR∆(u) ≤ 2eR∆(v).
+Using (6.18) in (6.22), we obtain
+∆(Zu) ≤ 2e2R∆(Zv)
+which implies that
+(6.23)
+∆(Zu)
+∆(Zv) ≤ 2e2R.
+In view of (6.15), we have
+(6.24)
+kD(Zu, Zv) ≤ 3c log
+�
+1 + c∆(Zu)
+∆(Zv)
+�
+.
+Using (6.23) in (6.24), we obtain
+(6.25)
+kD(Zu, Zv) ≤ 3c log
+�
+1 + 2ce2R�
+.
+Therefore, we have
+kD(u, v)
+≤
+kD(u, Zu) + kD(Zu, Zv) + kD(Zv, v)
+≤
+2R + kD(Zu, Zv)
+≤
+2R + 3c log(1 + 2ce2R)
+(by (6.25)
+This completes the proof of Theorem 6.2 by taking A = 2R + 3c log(1 + 2ce2R).
+□
+Proof of Theorem 5.1. Let D be a c- John domain such that (D, kD) is δ-
+hyperbolic for some δ and let γ be a c0- neargeodesic in D.
+Then from Theo-
+rem 6.2 we have that if there are u, v ∈ γ such that each w ∈ γ[u, v] satisfies
+
+Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces
+15
+∆(w) ≤ 2 min{∆(u), ∆(v)}, then kD(u, v) ≤ A. Therefore, by Theorem 6.1, we con-
+clude that γ is a b-cone arc, where b depends only on c, δ and c0. This completes
+the proof of Theorem 5.1.
+Acknowledgement: The first named author thanks SERB-CRG, and the second
+named author thanks PMRF-MHRD (Id: 1200297), Govt. of India for their support.
+References
+[1] G. D. Anderson, M. K. Vamanamurthy and M. Vuorinen, Dimension-free quasiconfor-
+mal deformation in n-space, Trans. Amer. Math. Soc. 297 (1986), 687–706.
+[2] M. Bonk, J. Heinonen and P. Koskela, Uniformizing Gromov hyperbolic domains, As-
+terisque 270 (2001), 1–99
+[3] F. W. Gehring and B. P. Palka, Quasiconformally Homogeneous Domains, J. Anal. Math.
+30 (1976), 172–199.
+[4] F. W. Gehring and B. G. Osgood, Uniform domains and the Quasi-Hyperbolic Metric, J.
+Anal. Math. 36 (1979), 50–74.
+[5] F. W. Gehring, K. Hag and O. Martio, Quasihyperbolic geodesics in John domains, Math.
+Scand. 65 (1989), 75–92.
+[6] J. Heinonen, Quasiconformal mappings onto John domains, Rev. Math. Iber. 5 (1989), 97–
+123.
+[7] R. Klél, A. Rasila and J. Talponen, On the smoothness of quasihyperbolic balls, Ann.
+Acad. Sci. Fenn. Math. 42 (2017), 439–452.
+[8] Y. Li, Neargeodesics in John domains in Banach spaces, Int. J. Math. 25, 1450041 (2014),
+https://doi.org/10.1142/S0129167X14500414.
+[9] G. J. Martin, Quasiconformal and bi-Lipschtiz homeomorphisms, uniform domains and the
+quasihyperbolic metric, Trans. Amer. Math. Soc. 292, 1985, 169–191.
+[10] Olli Martio and J. Väisälä, Quasihyperbolic geodesic in convex domains II, Pure Appl.
+Math. Q. 7 (2011), 395–409.
+[11] J. Väisälä, Free quasiconformality in Banach spaces I, Ann. Acad. Sci. Fenn. Ser. A I Math.
+15 (1990), 355–379.
+[12] J. Väisälä, Free quasiconformality in Banach spaces II, Ann. Acad. Sci. Fenn. Ser. A I Math.
+16 (1991), 255–310.
+[13] J. Väisälä, The Free quasiworld, Quasiconformal and related maps in Banach spaces, Banach
+Center Publ. 48 (1999), 55–118.
+[14] J. Väisälä, Broken tube in Hilbert spaces Analysis 24 (2004), 227–238.
+[15] J. Väisälä, Quasihyperbolic geodesic in convex domains, Results Math. 48 (2005), 184–195.
+[16] J. Väisälä, Gromov hyperbolic spaces, Expo. Math. 23 (2005), 187–231.
+[17] J. Väisälä, Hyperbolic and Uniform domains in Banach spaces, Ann. Acad. Sci. Fenn. 30
+(2005), 261–302.
+[18] Q. Zhou and S. Ponnusamy, Gromov hyperbolic John is Quasihyperbolic John I,
+arXiv:2111.14457v1
+[19] Q. Zhou and S. Ponnusamy, Gromov hyperbolic John is Quasihyperbolic John II,
+arXiv:2111.14457v1.
+[20] Q. Zhou, Yaxiang Li and Antti Rasila, Gromov hyperbolicity, John Spaces, and Quasihy-
+perbolic Geodesics, J. Geom. Anal. 32, 228 (2022) https://doi.org/10.1007/s12220-022-00968-
+2.
+[21] M. Vourinen, Conformal geometry and Quasiregular mappings, Lecture notes in mathemat-
+ics, Vol. 1319 (Springer, 1988).
+
+16
+Vasudevarao Allu and Abhishek Pandey
+Vasudevarao Allu, Discipline of Mathematics, School of Basic Sciences, Indian
+Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha
+(State), India.
+Email address: avrao@iitbbs.ac.in
+Abhishek Pandey, Discipline of Mathematics, School of Basic Sciences, Indian
+Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha
+(State), India.
+Email address: ap57@iitbbs.ac.in
+
diff --git a/XNFPT4oBgHgl3EQfsDWM/content/tmp_files/load_file.txt b/XNFPT4oBgHgl3EQfsDWM/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf,len=637
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='13147v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='CV]  30 Jan 2023  NEARGEODESICS IN GROMOV HYPERBOLIC JOHN DOMAINS  IN INFINITE DIMENSIONAL BANACH SPACES  VASUDEVARAO ALLU AND ABHISHEK PANDEY  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Rasila, Talponen and Zhang in [Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' 146 (2018),  3863–3873] have proposed a general question that which Banach space properties  can be characterized by the corresponding quasihyperbolic metric?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A domain D in  a Banach space E is said to be Gromov hyperbolic if it is δ ≥ 0-hyperbolic with  respect to quasihyperbolic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In this paper we show that every neargeodesic  in a Gromov hyperbolic John domain in infinite dimensional Banach space is a  cone arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Thus, our result gives an answer to the above problem in the sense  that it gives a necessary condition for a John domain to be a Gromov hyperbolic  space in infinite dimensional Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Moreover, our main result gives an  improvement of a result of Li [Theorem 1, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' 25 (2014)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Introduction  Let Ω be an open ball or half space in Rn, we denote hΩ be the hyperbolic metric  in Ω with constant curvature −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' If Ω is an upper half plane then hΩ is defined in  Ω by the means of the density  ρ(z) =  1  Im z =  1  d(z, ∂Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  This density can be used to introduce an analogue of the hyperbolic metric in an  arbitrary subdomain D in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The quasihyperbolic metric in Rn is a generalization  of hyperbolic metric, which was introduced by Gehring and Palka [3] and they  have studied the characterization of domains D in the one point compactification  of Rn which are quasiconformally equivalent to ball in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Infact such domains are  homogeneous via quasiconformal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Gehring and Palka [3] have proved that  the maximal dilatation of such quasiconformal mappings can be estimated in terms  of quasihyperbolic metric and using these estimates they obtained useful results  related to the homogeneity of domains with respect to quasiconformal family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Thus,  the importance of the quasihyperbolic metric is quite clear as it is well-behaved with  the quasiconformal mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The quasihyperbolic metric can be naturally defined  on a wide range of metric spaces, including infinite dimensional Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The  study of this concept is known as free quasiconformality which has been introduced  and developed by Väisälä (see [11, 12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In this paper, we mainly focus on the geometry of neargeodesics in John domains  in infinite dimensional Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  File: QHG-P-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='tex, printed: 2023-1-31, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='52  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Primary 30C65, 30L10, 30F45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Secondary 30C20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Quasihyperbolic metric, Gromov hyperbolic spaces, John domain, near-  geodesics, quasiconformal mappings, quasihyperbolic geodesic, cone arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  1  2  Vasudevarao Allu and Abhishek Pandey  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Preliminaries  In this section we mention some basic as well as advanced concepts and related  results in the literature, which are useful to state and prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Metric Geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A curve is a continuous  function γ : [a, b] → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' If a curve γ : [a, b] → X is an embedding of [a, b], then it is  called an arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let P denote set of all partitions a = t0 < t1 < t2 < · · · < tn = b of  the interval [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The length of the curve γ in the metric space (X, d) is  ld(γ) = sup  P  n−1  �  k=0  d(γ(tk), γ(tk+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  A curve is said to be rectifiable if ld(γ) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  A metric space X is said to be  rectifiably connected if every pair of points x, y ∈ X can be joined by a rectifi-  able curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For a rectifiable curve γ we define arc length s : [a, b] → [0, ld(γ)] by  s(t) = ld(γ|[a,t]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The arc length function is of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For any rectifiable  curve γ : [a, b] → X, there is a unique map γs : [0, ld(γ)] → X such that γ = γs ◦ s,  and such a curve γs is called the arclength parametrization of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  For x, y ∈ X, the inner length metric λX(x, y) is defined by  λX(x, y) = inf{ld(γ) : γ is a rectifiable arc joining x and y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Let ρ : X → [0, ∞] be a Borel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The ρ-length of a rectifiable curve γ is  �  γ  ρ ds =  � b  a  ρ(γ(t)) ds(t) =  � ld(γ)  0  ρ ◦ γs(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  If X is rectifiably connected then ρ induces a distance function which is defined  by  dρ(x, y) = inf  �  γ  ρ ds,  where infimum is taken over all rectifiable curves joining x and y in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' We note  that, in general, dρ need not to be a metric however dρ is a metric if ρ is positive and  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' If dρ is a metric, we say (X, dρ) is a conformal deformation of (X, d) by  the conformal factor ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  A curve γ : [a, b] → X is said to be geodesic if for all t, t′ ∈ [a, b],  d(γ(t), γ(t′)) = |t − t′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  A metric space X is said to be a geodesic metric space if every pair of points x, y ∈ X  can be joined by a geodesic, and is said to be proper if every closed ball is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  A metric space (X, d) is said to be intrinsic or path metric space/ or length metric  space if for all x, y ∈ X, we have  d(x, y) = λX(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A metric space (X, d) is said to be minimally nice if it is locally  compact, rectifiably connected and non-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Gromov hyperbolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Gromov hyperbolicity was introduced by Gromov in  the 1980s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' It is often assumed that the under lying space is geodesic metric space  and usually it is a proper metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' As we will see later that infinite dimensional  Banach space with quasihyperbolic metric need not be geodesic space, therefore we  adapt the definition of Gromov hyperbolicity which works well with the non geodesic  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For this, we refer to the work of Väisälä [16] where the concept of Gromov  product has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let (X, d) be a metric space and let p ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The Gromov Product  (x|y)p of x, y ∈ X with respect to p is defined by  (x|y)p = 1  2  �  d(x, p) + d(y, p) − d(x, y)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Therefore, (x|y)p measures to what extent the triangle inequality for the triangle  ∆xyp is far from being an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A metric space (X, d) is said to be δ-hyperbolic if for  every x, y, z, p ∈ X, the following inequality holds  (x|z)p ≥ min  �  (x|y)p, (y|z)p  �  − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  A metric space (X, d) is said to be Gromov hyperbolic if it is δ-hyperbolic for some  δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Clearly, R is a 0-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Any tree T is 0-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The unit  disk with hyperbolic metric is Gromov hyperbolic with δ = log 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The complex  plane is not Gromov hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  We shall mention an important result here due to Väisälä [15, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7], which is known  as stability theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Infact, stability theorem says that in an intrinsic hyperbolic  space any two quasi-isometric paths with same initial and terminal points x, y runs  close to each other even if |x − y| is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' To mention this theorem we first need to  mention the concept of quasi-isometry and Hausdorff distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let X and Y be two non-empty subsets of a metric space (M, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  We define their Hausdorff distance dH(X, Y ) by  dH(X, Y ) = max  �  sup  x∈X  d(x, Y ), sup  y∈Y  d(X, y)  �  ,  where d(a, B) = infb∈B d(a, b) denotes the distance from a point a ∈ M to a subset  B ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let λ ≥ 1 and µ ≥ 0 and (X, d) and (Y, d′) be metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' We  say that a map f : X → Y is a (λ, µ)-quasi-isometry if  λ−1d(x, y) − µ ≤ d′(f(x), f(y)) ≤ λd(x, y) + µ  for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In the case where f : I → Y is a map of an real interval I, we say  that such a map (λ, µ)-quasi-isometric path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' An arc γ joining x and y in a metric space X is said to be λ-  quasigeodesic, λ ≥ 1, if  l(γ[u, v]) ≤ λd(u, v)  4  Vasudevarao Allu and Abhishek Pandey  for all u, v ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In such a case the arc length parametrization γs : [0, l(γ)] → γ  satisfies the inequality  λ−1|t − t′| ≤ d(γs(t), γs(t′)) ≤ λ|t − t′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Thus γ is (λ, 0)-quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  The following theorem (see [15, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7]) is vital to prove our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Suppose that X is an intrinsic δ-hyperbolic space and that γ and α  are (λ, µ)-quasi-isometric path with same initial and terminal points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then there  exists a constant M such that  dH(γ, α) ≤ M,  where M depends only on δ, λ, µ and dH denote the Hausdorff distance between γ  and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Next we mention an important result due to Väisälä (see [16, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='18])  which says that quasi-isometries preserves hyperbolicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let X and Y be intrinsic spaces and let f : X → Y be a (λ, µ)-  quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' If Y is δ-hyperbolic, then X is δ′-hyperbolic with δ′ = δ′(δ, λ, µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Quasihyperbolic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In view of our frame work, we consider the quasi-  hyperbolic metric in the setting of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let E be a real Banach space of dimension at least 2 and let D ⊊ E  be a domain (open, connected nonempty set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let the metric induced by norm on  D is |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Define k : D → (0, ∞) by  k(z) =  1  ∆(z) =  1  d(z, ∂D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Define the quasihyperbolic length of a rectifiable curve γ in D by  lk(γ) =  �  γ  ρ(z) ds =  �  γ  ds  d(z, ∂D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Let kD denote the quasihyperbolic metric in D which is defined by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1)  kD(x, y) = inf  γ lk(γ)  where the infimum is taken over all rectifiable curves γ in D joining x to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Observe that (D, kD) is always an intrinsic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For further geometric properties  of hyperbolic metric, we refer to [4] and [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Quasihyperbolic metric satisfies the  following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  (i) kD = hD when D is a half space in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  (ii) kD ≤ hD ≤ 2kD when D is a ball in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [2, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='8] If (X, d) is a locally compact, incomplete and  rectifiably connected metric space then (X, kX) is a complete, proper and geodesic  metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces  5  Finally, we recall some estimates for the quasihyperbolic metric which have been  first introduced by Gehring and Palka [3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1] in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Later, Väisälä [11, Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2] have proved these inequalities in the case of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let D ⊊ E be a  domain and x, y ∈ D and let γ be a rectifiable curve joining x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then we have  the following:  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2)  k(x, y) ≥ log  �  1 +  |x − y|  min{∆(x), ∆(y)}  �  ≥ log ∆(y)  ∆(x)  and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3)  lk(γ) ≥ log  �  1 +  l(γ)  min{∆(x), ∆(y)}  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  We are going to use frequently (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3) to prove our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Quasi-hyperbolic geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A rectifiable curve γ from a to b in D is said to  be quasihyperbolic geodesic if kD(a, b) = lk(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Obviously each subarc of a quasi-  hyperbolic geodesic is again a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Observe that quasi-hyperbolic geodesic is  curve γ for which the infimum in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1) is attained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  At this juncture, the natural question is does quasi-hyperbolic geodesic always  exists?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A result of Gehring and Osgood [4] shows that for a proper subdomain D  in Rn, the answer to this question is affirmative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For important results on quasihy-  perbolic geodesics in domain in Rn, we refer to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Martin [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Moreover, Martin [9]  has proved that quasihyperbolic geodesics in Rn are C1 smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' It is well known  that a quasihyperbolic geodesic between any two points exists if dim (E) is finite  (see [4, Lemma 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Note that the quasihyperbolic geodesic may not exists in an  infinite-dimensional Banach spaces (see [11, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='9] and [13, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' However,  Väisälä [15, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1] has proved the existence of quasihyperbolic geodesic when  E is reflexive Banach space and D is a convex domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Furthermore, in 2011, Martio  and Väisälä [10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='11] proved the existence as well as uniqueness of quasi-  hyperbolic geodesic when E is uniformly convex Banach space and D is a convex  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Some special domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Uniform Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let c ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A domain D in E is said to be c-uniform in the norm  metric if for each pair of points x, y ∈ D, there exists a rectifiable arc γ in D joining  x to y such that  (i) min{l(γ[x, z]), l(γ[z, y])} ≤ c d(z, ∂D), for all z ∈ γ, and  (ii) l(γ) ≤ c|x − y|,  where l(γ) denotes the arc length of γ in (D, |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' |), where |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' | is metric induced by  norm, and γ[x, z] denotes the part of γ between x and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Also we say that curve γ  is a double c-cone arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  6  Vasudevarao Allu and Abhishek Pandey  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Inner uniform domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let c ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A domain D is said to be c-inner uniform in the norm  metric if for each pair of points x, y ∈ D, there exists a rectifiable arc γ in D joining  x to y such that  (i) min{l(γ[x, z]), l(γ[z, y])} ≤ c d(z, ∂D), for all z ∈ γ,  (ii) l(γ) ≤ cλD(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' John Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let c ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A domain D is said to be c-John in the norm metric if  for each pair of points x, y ∈ D there exists a rectifiable curve γ in D joining x to y  such that for all z ∈ γ  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='4)  min{l(γ[x, z]), l(γ[z, y])} ≤ c d(z, ∂D)  and we say that such a curve γ is a c-cone arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' We mention some of the remarks with regards to the connection  between the aforesaid domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  (i) c-uniform implies inner c-uniform implies c-John  (ii) R2 \\ {(x, 0) : x ≥ 0} is inner uniform but not unifrom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  (iii) R2 \\ {(n, 0) : n ∈ N} is a John domain but not inner uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Gromov hyperbolicity of domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  A domain D in a Banach space E is said to be  δ-hyperbolic if (D, kD) is δ-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' A domain D in a Banach space E is said to be Gromov hyperbolic  if (D, kD) is δ-hyperbolic for some δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  All c- uniform and inner c- uniform domains are Gromov δ = δ(c) hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  R2 \\ {(n, 0) : n ∈ N} is a John domain which is not hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' The broken tube  (see [11, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='12] and for a detailed treatment we refer to [14]) in an infinite dimensional  seperable Hilbert space is a classical example of a domain which is hyperbolic but  not John and hence not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Quasihyperbolic geodesic in Uniform Domains in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Gehring and  Osgood [4] have proved that each quasihyperbolic geodesic in a c-uniform domain  D ∈ Rn is a double b-cone arc, where the constant b depends only on c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Quasihyperbolic Geodesic in John Domain in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Since John domains  can be thought of one sided uniform domains, it is natural to ask whether the result  also holds for John domains or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In fact, this problem has been proposed by  Gehring et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [5] in the following form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Suppose D ⊂ Rn is a c− John domain and that γ is a quasihyperbolic  geodesic in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Is γ a b-cone arc for some b = b(c)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In 1989, Gehring et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [5] proved the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1] If D ⊂ R2 is a simply connected c-John domain  then every quasihyperbolic or hyperbolic geodesic in D is a b-cone arc, where b only  depends on c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces  7  Infact, Gehring et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [5] have constructed several examples which shows that  a quasihyperbolic geodesic in a c- John domain need not to be a b-cone arc with  b = b(c) unless n = 2 and D is simply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Thus, this suggest us that we  need some extra restriction on c-John domain so that the Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='5 holds good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  This raises the following natural question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Determine necessary and/or sufficient conditions for quasihyperbolic  geodesics in a John space to be cone arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In 1989, Heinonen [6, Question 2] posed the following problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Suppose D ⊂ R2 is a c-John domain which is quasiconformally  equivalent to the unit ball Bn in Rn and γ is a quasihyperbolic geodesic in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Is γ  a b = b(c)-cone arc for some constant b?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In 2001, Bonk, Heinonen and Koskela [2, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='12] gave an affirmative  answer to Probelm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In particular, they have proved the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [2, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='12] If D ⊂ Rn is a bounded a-John domain which  is homeomorphic to inner c-uniform domain via K-quasiconformal map, then each  quasihyperbolic geodesic in D is a b-cone arc with b = b(a, c, K, n)  Since every ball is inner uniform, Theorem C gives an affirmative answer to Prob-  lem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Metric Space setting  In 2022, Zhou, Li and Rasila [20] consider the Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='6 further and proved the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [20, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1] Let (D, d) be a locally compact, rectifiablly con-  nected and non complete metric space, and k be the quasihyperbolic metric of (D, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  If (D, d) is a-John and if (D, k) is K-roughly starlike and Gromov δ-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Then every quasihyperbolic geodesic in D is a b-cone arc, where b depends only on  a, K and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Observe that any proper subdomain D in Rn is locally compact,  rectifiablly connected and noncomplete with respect to the usual metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Theorefore,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 gives an improvement of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  The next result of Zhou and Ponnusamy [18, 19] shows that the condition of rough  starlikeness is not required in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Thus the following results of Zhou and  Ponnusamy [19] give an improvement of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [19, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='4] Let (D, d) be a locally compact, rectifiablly con-  nected and non complete metric space, and k be the quasihyperbolic metric of (D, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  If (D, d) is a-John and if D is Gromov hyperbolic i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=', (D, kD) is δ-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then  every quasihyperbolic geodesic in D is a b-cone arc with b = b(a, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Banach space setting  The main aim of this paper is to study Problem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='6 by replacing quasihyperbolic  geodesic with neargeodesics in Banach space settings precisely in infinite dimansional  8  Vasudevarao Allu and Abhishek Pandey  Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Note that the quasihyperbolic geodesic may not exists in the infinite-  dimensional Banach spaces (see [11, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='9] and [13, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' However,  these examples of domains are in Hilbert spaces such that the complement of these  domains are uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For example of a domain that is not geodesic with respect  to quasihyperbplic metric such that the complement of domain is countable we refer  to [7, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In order to overcome this shortage, Väisälä [12] introduced the  concept of neargeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' We recall the definition of neargeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let D ⊊ E be a domain and c ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' An arc γ in D is said to be  c-neargeodesic if for all x, y ∈ γ, we have  lk(γ[x, y]) ≤ ckD(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Thus an arc γ is a quasihyperbolic geodesic if, and only if, it is a 1-neargeodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In [12] Väisälä proved the following existence theorem for neargeodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' [12, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3] Let c > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then for every pair of points z1, z2 ∈ D  there exists a c-neargeodesic joining z1 and z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Observe that the constant b in Theorem B depends on n, so it is natural to ask  the following question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Problem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Does there exists a dimension free anologue of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In other  words does it holds in the setting of Banach spaces or not if we replace quasihyper-  bolic geodesic with neargeodesics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Li [8, Theorem 1] gave an affirmative answer to Problem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 and proved the  following  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' If D ⊂ E is an a-John domain which is homeomorphic to c-inner  uniform domain via an (M, C) − CQH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let z1, z2 ∈ D and γ is a c0-neargeodesic  joining z1 and z2 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then γ is a b-cone arc with b = b(a, c, c0, M, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Main Result  In this section we state our main result of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' We observe that in Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2, authors have considered that metric space (X, D) which is minimally  nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Once you have such metric space, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2 immediately tells us that (X, kX)  is a geodesic metric space i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=', for any papir of points quasihyperbolic geodesic exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Note that if we consider a domain D in an infinite dimensional Banach space  then D with the metric induced by norm satisfies all the properties of minimally  nice space except locally compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' This motivates us to ask does Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2 type results still holds if we consider John domain D in infinite dimensional  Banach space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In such a case we know that (D, kD) need not be geodesic and hence  the natural choice is to consider the neargeodesics at the place of quasihyperbolic  geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let E be an infinite dimensional Banach space and D ⊊ E is a  c-John domain and suppose that D is Gromov hyperbolic i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' (D, kD) is δ-hyperbolic  for some δ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then every c0-neargeodesic in D is a b-cone arc, where b depends  only on c, c0 and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces  9  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  (1) Observe that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 is different from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 and  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Note that in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 D is a domain in infinite dimansional  Banach spce therefore D is not locally compact with metric induced by norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Hence Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2 is also no more valid and we can’t use the results of  Gromov hyperbolicity for geodesic metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Further, in this paper, we  are considering the neargeodesics rather than quasihyperbolic geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  (2) To prove our main result we need the results on Gromov hyperbolicity for  general metric spaces that are need not to be geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Thanks to the work  of Väisälä [16] where he has established the theory of Gromov hyperbolicity  for domains in Banach spaces which obviously need not to be geodesic (in  quasihyperbolic metric) when the Banach space is infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  (2) In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 and the following two facts:  (i) all inner c-uniform domains are Gromov hyperbolic with respect to  quasihyperbolic metric (see [15, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='16])  (ii) every (M, C) − CQH map is (M, C)-quasi-isometry in quasihyperbolic  metric  we conclude that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 is an improvement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2 of Li [8,  Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Proof of Main Result  We shall devide the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 into two theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1 below  gives a sufficient condition for a neargeodesic to be a cone arc and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2  below says that in Gromov hyperbolic John domains, every neargeodesic satisfies  this sufficient condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Therefore, combining these two theorem we obtain the  proof of our main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let γ be a c0- neargeodesic joining x, y such that the following holds:  If z1, z2 ∈ γ such that each w ∈ γ[z1, z2] satisfies ∆(w) ≤ 2 min{∆(z1), ∆(z2)}, then  there exists a constant A ≥ 1 such that  kD(z1, z2) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Then γ is a C-cone arc, where C depends only on A and co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Suppose γ is a c0- neargeodesic joining x, y and if z1 and z2 are points on  γ satisfying for each w ∈ γ[z1, z2], ∆(w) ≤ 2 min{∆(z1), ∆(z2)}, then there exists a  constant A ≥ 1 such that  kD(z1, z2) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Since ∆ is continous and γ is compact, choose a point z0 ∈ γ with  ∆(z0) = max  z∈γ ∆(z)  Then there exists unique non-negative integer n such that  2n∆(x) ≤ ∆(z0) ≤ 2n+1∆(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Let x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For each i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' , n, let xi be the first point on γ[x, z0] such that  ∆(xi) = 2i∆(x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Similarly, there exists m ≥ 0 such that  2m∆(y) ≤ ∆(z0) ≤ 2m+1∆(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  10  Vasudevarao Allu and Abhishek Pandey  Let y = y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For each j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' , m, let yj be the first point on γ[z0, y] such that  ∆(yj) = 2j∆(y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In this manner we have devided the arc γ into (n + m + 1) subarcs  γ[x0, x1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' , γ[xn−1, xn], γ[xn, ym], γ[ym, ym+1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' , γ[y1, y0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  It is easy to observe that all these subarcs are also c0-neargeodesic between their  respective end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let u and v be any two adjacent points along γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' That is u  and v are xi, xi+1 or xn, ym or yj+1, yj, 0 ≤ i ≤ n − 1 and 0 ≤ j ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' By the  construction, it is clear that for each z ∈ γ[u, v], we have  ∆(z) = 2 min{∆(u), ∆(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  By assumption, there exists a constant A ≥ 1 such that  kD(u, v) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  To show that γ is a C-cone arc for some C, we need to show that  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1)  min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)  for all w ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In order to prove (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1), it is sufficient to prove either l(γ[x, w] ≤ C∆(w) or l(γ[w, y]) ≤  C∆(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For this we need to consider three cases depending upon the location of w  on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Case 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Suppose w ∈ γ[xi, xi+1] for some 0 ≤ i ≤ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2), we  have  log∆(xi)  ∆(w) ≤ kD(w, xi) = kD(xi, w) ≤ kD(xi, xi+1) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Therefore, we have  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2)  ∆(xi) ≤ eA∆(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Moreover, we have  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3)  l(γ[x, w]) ≤  i  �  t=0  l(γ[xs, xs+1])  In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3), we have  log  �  1 +  l(γ[xs, xs+1])  min{∆(xs), ∆(xs+1)}  �  ≤  lk(γ[xs, xs+1])  ≤  c0 kD(xs, xs+1)  ≤  c0 A  Therefore, this gives us that  logl(γ[xs, xs+1])  ∆(xs)  ≤ log  �  1 + l(γ[xs, xs+1])  ∆(xs)  �  ≤ c0 A  and hence  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='4)  l(γ[xs, xs+1]) ≤ ec0 A∆(xs)  Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces  11  Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='4) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='3), we obtain  l(γ[x, w])  ≤  i  �  t=0  ec0 A∆(xt)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='5)  ≤  2 ec0 A ∆(xi)  ≤  2 eA(c0+1) ∆(w)  and hence we have  l(γ[x, w]) ≤ C ∆(w),  where C = 2eA(c0+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Theorefore, we have  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='6)  min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)  for all w ∈ γ[xi, xi+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Case 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Suppose w ∈ γ[xn, ym].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then it is easy to see that  log∆(xn)  ∆(w) ≤ kD(w, xn) = kD(xn, w) ≤ kD(xn, ym) ≤ A,  which implies  ∆(xn) ≤ eA ∆(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  An easy computation shows that  l(γ[x, w])  ≤  n−1  �  i=0  l(γ[xi, xi+1]) + l(γ[xn, ym])  ≤  m−1  �  i=0  ec0 A∆(xi) + ec0 A∆(xn)  ≤  3ec0 A∆(xn)  ≤  3 eA(c0+1) ∆(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  That is,  l(γ[x, w]) ≤ C ∆(w),  where C = 3 eA(c0+1)  and hence we have  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7)  min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)  for all w ∈ γ[xn, ym].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Case 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Suppose w ∈ γ[yj, yj+1] for some 0 ≤ j ≤ m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Using the same argument  as in earlier, we obtain  l(γ[w, y]) ≤ C ∆(w),  where C = 2 eA(c0+1)  and hence  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='8)  min{l(γ[x, w], l(γ[w, y]))} ≤ C∆(w)  for all w ∈ γ[yj, yj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Thus, combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='6), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='7) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='8), we conclude that γ is a C- cone arc, where  C = 3 eA(c0+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  □  12  Vasudevarao Allu and Abhishek Pandey  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let D be a c- John domain in an infinite dimensional Banach space  E such that (D, kD) is Gromov hyperbolic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Fix x, y ∈ D and let γ be a c0  neargeodesic joining x, y ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then there exists a constant A ≥ 1 depending only c,  c0 and δ such that the following holds: If there are u, v ∈ γ such that each w ∈ γ[u, v]  satisfies ∆(w) ≤ 2 min{∆(u), ∆(v)}, then  kD(u, v) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let D be a c-John domain in a Banach space E such that (D, kD) is Gromov  δ-hyperbolic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' By definition (D, kD) is an intrinsic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Fix x, y ∈ D and c0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In  view of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1, let γ be a c0-neargeodesic joining x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' In view of Remark  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1, it is clear that a c-neargeodesic γ is (c, 0)- quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Since D is a c-John,  there is a c-cone arc α joining x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' To use Theorem A, we first need to show  that every c-cone arc α is (λ, µ)-quasi-isometry for some λ and µ depending only on  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Which we will prove in the following lemma  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let c ≥ 1 and α be a c-cone arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Then its arc length parametrization  αs satisfies the following inequality  1  3c|t − t′| − 3c log(3c) ≤ kD(αs(t), αs(t′)) ≤ 3c|t − t′| + 3c log(3c)  and hence α is (3c, 3c log(3c))-quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let α be a c-cone arc joining x and y in D and z1, z2 ∈ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Without loss  of generality we can assume that z1 ∈ α[x, z2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Since α is a c-cone arc, for each  w ∈ α[z1, z2], we have  l(α[w, z2]) ≤ min{l(α[x, w]), l(α[w, y])} ≤ c∆(w)  and hence  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='10)  l(α[w, z2]) ≤ c∆(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Next, our aim is to show that ∆(z2) ≤ 2c∆(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' To prove this, we consider two  cases:  Case (i) If d(w, z2) < ∆(y)/2, then  ∆(w) ≥ ∆(z2) − d(w, z2) ≥ ∆(z2)/2  and hence, ∆(z2) ≤ 2∆(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Case (ii) Since α is a c-cone arc we obtain  c∆(w) ≥ l(α[z2, w]) ≥ d(z2, w) ≥ ∆(z2)/2  and hence  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='11)  ∆(z2) ≤ 2c∆(w)  By adding (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='10) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='11), we obtain  l(α[w, z2]) + ∆(z2) ≤ 3c∆(w)  which gives,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='12)  1  ∆(w) ≤  3c  l(α[w, z2]) + ∆(z2)  Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces  13  A simple computation shows that  kD(z1, z2)  ≤  lk(α[z1, z2]) =  �  α[z1,z2]  ds  ∆(w)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='13)  ≤  � l(α[z1,z2])  0  3c  t + ∆(z2) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  lk(α[z1, z2])  ≤  3c(log(l(α[z1, z2]) + ∆(z2)) − log(∆(z2))  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='14)  ≤  3c log  �  1 + l(α[z1, z2])  ∆(z2)  �  ≤  3c log  �  1 + c∆(z1)  ∆(z2)  �  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='15)  ≤  3c  �  log  �∆(z1)  ∆(z2)  �  + log3c  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  lk(α[z1, z2])  ≤  3ckD(z1, z2) + 3c log 3c  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='16)  Theorefore, from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='16), it is clear that the arc length parametrization of α satisfies  the inequality  1  3c|t − t′| − 3c log(3c) ≤ kD(αs(t), αs(t′)) ≤ 3c|t − t′| + 3c log(3c)  and hence α is (3c, 3c log(3c))-quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' This completes the proof of Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  □  Since γ is (c0, 0)-quasi-isometry and α is (3c, 3c log(3c))-quasi-isometry in view of  Theorem A, we have that quasihyperbolic Hausdorff distance satisfies the following  inequality  kDH(γ, α) ≤ R = R(c, c0, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Let u, v ∈ γ such that each w ∈ γ[u, v] satisfies ∆(w) ≤ 2 min{∆(u), ∆(v)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Our  aim is to show that there exists a constant A ≥ 1 such that kD(u, v) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' For this  we need to estimate kD(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Since kDH(γ, α) ≤ R i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=',  max  �  sup  x∈γ kD(x, α), sup  y∈α kD(γ, y)  �  ≤ R  Theorefore,  sup  x∈γ kD(x, α) ≤ R  In particular,  M1 = kD(u, α) = inf  b∈α kD(u, b) ≤ R  and  M2 = kD(v, α) = inf  b∈α kD(v, b) ≤ R  There exists a sequence bi and bj such that kD(u, bi) converges to M1 and kD(u, bj)  converges to M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Since α is compact, bi and bj have convergent subsequence which  14  Vasudevarao Allu and Abhishek Pandey  converges to say Zu and Zv respectively such that kD(u, Zu) ≤ R and kD(v, Zv) ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Therefore,  log ∆(u)  ∆(Zu) ≤ kD(u, Zu) ≤ R  implies  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='17)  ∆(Zu) ≤ eR∆(u)  and  log ∆(v)  ∆(Zv) ≤ kD(v, Zv) ≤ R  implies  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='18)  e−R∆(v) ≤ ∆(Zv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Since each w ∈ γ[u, v] satisfies  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='19)  ∆(w) ≤ 2 min{∆(u), ∆(v)}  by taking w = v in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='19), we obtain  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='20)  ∆(v) = 2∆(u)  and by taking w = u in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='19), we obtain  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='21)  ∆(u) ≤ 2∆(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='21) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='17), we obtain  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='22)  ∆(Zu) ≤ eR∆(u) ≤ 2eR∆(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='18) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='22), we obtain  ∆(Zu) ≤ 2e2R∆(Zv)  which implies that  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='23)  ∆(Zu)  ∆(Zv) ≤ 2e2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  In view of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='15), we have  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='24)  kD(Zu, Zv) ≤ 3c log  �  1 + c∆(Zu)  ∆(Zv)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='23) in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='24), we obtain  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='25)  kD(Zu, Zv) ≤ 3c log  �  1 + 2ce2R�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Therefore, we have  kD(u, v)  ≤  kD(u, Zu) + kD(Zu, Zv) + kD(Zv, v)  ≤  2R + kD(Zu, Zv)  ≤  2R + 3c log(1 + 2ce2R)  (by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='25)  This completes the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2 by taking A = 2R + 3c log(1 + 2ce2R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  □  Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Let D be a c- John domain such that (D, kD) is δ-  hyperbolic for some δ and let γ be a c0- neargeodesic in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Then from Theo-  rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='2 we have that if there are u, v ∈ γ such that each w ∈ γ[u, v] satisfies  Neargeodesics in Gromov hyperbolic John domains in infinite dimensional Banach spaces  15  ∆(w) ≤ 2 min{∆(u), ∆(v)}, then kD(u, v) ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Therefore, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1, we con-  clude that γ is a b-cone arc, where b depends only on c, δ and c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' This completes  the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Acknowledgement: The first named author thanks SERB-CRG, and the second  named author thanks PMRF-MHRD (Id: 1200297), Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' of India for their support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
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+page_content=' Hag and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Martio, Quasihyperbolic geodesics in John domains, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Scand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
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+page_content=' Rasila and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Talponen, On the smoothness of quasihyperbolic balls, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Fenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' 42 (2017), 439–452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content=' Li, Neargeodesics in John domains in Banach spaces, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
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+page_content='  16  Vasudevarao Allu and Abhishek Pandey  Vasudevarao Allu, Discipline of Mathematics, School of Basic Sciences, Indian  Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha  (State), India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Email address: avrao@iitbbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='in  Abhishek Pandey, Discipline of Mathematics, School of Basic Sciences, Indian  Institute of Technology Bhubaneswar, Argul, Bhubaneswar, PIN-752050, Odisha  (State), India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='  Email address: ap57@iitbbs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/XNFPT4oBgHgl3EQfsDWM/content/2301.13147v1.pdf'}
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+arXiv:2301.05178v1  [hep-th]  12 Jan 2023
+A guide to frames, 2π’s, scales and corrections
+in string compactifications
+B. V. Bentoa, D. Chakrabortyb, S. Parameswarana, I. Zavalac
+a Department of Mathematical Sciences,
+University of Liverpool, Liverpool L69 7ZL
+b Department of Physics, Ashoka University,
+Plot 2, Rajiv Gandhi Education City, P.O. Rai, Sonipat 131029, Haryana, India
+c Department of Physics, Swansea University,
+Singleton Park, Swansea, SA2 8PP, UK
+Abstract
+This note is intended to serve as a reference for conventions used in the literature on
+string compactifications, and how to move between them, collected in a single and easy-
+to-find place, using type IIB as an illustrative example. We hope it may be useful to
+beginners in the field and busy experts. E.g. string constructions proposed to address
+the moduli stabilisation problem are generically in regions of parameter space at the
+boundaries of control, so that consistent use of 2π’s and frame conventions can be
+pivotal when computing their potentially dangerous corrections.
+E-mail: Bruno.Bento@liv.ac.uk, dibya.chakraborty@ashoka.edu.in, susha@liv.ac.uk,
+e.i.zavalacarrasco@swansea.ac.uk
+
+1
+Introduction
+The idea that this pedagogical note could be a useful contribution to the community came
+about after several discussions with colleagues on the robustness of various candidate string
+constructions for moduli stabilisation, towards particle physics and cosmology, against po-
+tentially dangerous corrections. To scrutinise these constructions, it becomes necessary to
+use other people’s conventions (or indeed one’s own in one’s past worldline), and although
+there is nothing deep in changing conventions, and a change of frames is simply a field re-
+definition1, it is tedious, and possibly tricky unless starting from scratch. We thus present
+the various choices most commonly used for 10d and 4d string and Einstein frames in type
+IIB compactifications, and 2π’s, together with the map between them. We emphasise how
+physical quantities such as mass ratios, which determine the size of leading corrections to
+explicit string compactifications, are of course convention-independent. We hope that this
+may help both beginners in the field and busy experts to save some time.
+In Section 2, we present the 10d type IIB supergravity action in the string frame and the
+Einstein frame, using the two main choices of conventions for change of frames, and including
+the SL(2, R) manifest action. In Section 3, we dimensionally reduce to 4d using a general
+warped compactification, and present the 4d Einstein frame in different conventions which,
+however, always allow to recover the unwarped limit from the warped case in an intuitive
+way. We identify the (warped) KK scale, presenting a single expression that covers the var-
+ious conventions considered. In Section 4 we similarly work out the flux superpotential and
+gravitino mass, and show how the mass ratio
+m3/2
+mKK – which not only determines whether we
+have a consistent supergravity description in 4d, but also controls the higher F-term correc-
+tions to KKLT/LVS type compactifications – is of course convention-independent. In Section
+5 we give the leading perturbative corrections to the Kähler potential and non-perturbative
+corrections to the superpotential in the most common conventions, again showing how the
+convention-independence can be made manifest.
+1See e.g. [1, 2, 3, 4, 5] and references therein for some interesting discussions on the equivalence of the
+Einstein and Jordan frames in cosmology.
+1
+
+2
+Type IIB supergravity
+Our starting point is the type IIB low-energy supergravity action in string frame, which is
+given by2
+SS
+IIB = 1
+2κ2
+10
+�
+d10x
+�
+−GS
+�
+e−2Φ
+�
+R + 4∂µΦ∂µΦ − 1
+2|H3|2
+�
+−
+�1
+2|F1|2 + 1
+2| ˜F3|2 + 1
+4| ˜F5|2
+� �
+− 1
+4κ2
+10
+�
+C4 ∧ H3 ∧ F3 ,
+(1)
+where R is the Ricci scalar, Φ is the dilaton, H3 is the field-strength of the NS 2-form B2
+and Fp is the field-strength of the RR (p − 1)-forms Cp−1, and ˜F3, ˜F5 are defined as below:
+H3 = dB2 ,
+˜F3 = F3 − C0H3 ,
+Fp = dCp−1 ,
+˜F5 = F5 − 1
+2C2 ∧ H3 + 1
+2B2 ∧ F3 ,
+|Fp|2 = 1
+p!Fµ1...µpF µ1...µp .
+Moreover, the type IIB action must be supplemented with the self-duality condition3
+˜F5 = ⋆ ˜F5 .
+The relation between the string scale α′ and the 10d gravitational coupling in string
+frame κ10 is
+2κ2
+10 = (2π)7α′4 .
+(2)
+A common convention for the string length4 ls, which we use below, is
+(2π)2α′ = l2
+s ,
+(3)
+although sometimes α′ = l2
+s is used instead.
+2The action can be found on p.90 of [6], p.314 of [7], p.114 of [8], p.79 of [9] and p.625 of [10], along with
+the necessary definitions. There is a different definition of F5, for example in [11], which also contains the
+equations of motion. See also [12] for the dilaton dependence of the RR sector.
+3Notice the factor of 1
+4 rather than 1
+2 in the kinetic term, which accounts for the fact that only half the
+degrees of freedom should be present.
+4The string scale (corresponding to the mass of the tower of string states) is Ms =
+1
+√
+α′ = 2π
+ls for this
+choice of conventions. Sometimes the notation ms =
+1
+ls is used with this convention, with the relation
+Ms = 2πms.
+2
+
+2.1
+Einstein frame
+In the string frame, the gravitational part of the action is not in the canonical Einstein-
+Hilbert form. In order to obtain the latter, we perform a conformal transformation of the
+metric in 10d, GS → GE = e2ΩGS.
+The frame in which the gravitational part of the action takes the canonical Einstein-
+Hilbert form – i.e. the Ricci scalar does not couple to anything other than
+√
+−GE – is the
+Einstein frame. This choice fixes the required conformal transformation up to a constant5
+Ω = −Φ − Φ0
+4
+,
+(4)
+where the constant Φ0 is a choice of convention, and the two metrics are related by
+GE
+MN = e− Φ−Φ0
+2
+GS
+MN .
+(5)
+The first term in the action (1), namely the Ricci scalar, in the Einstein frame becomes
+SE
+grav = 1
+2κ2
+�
+d10x
+�
+−GE
+�
+RE − 9
+2(GE)µν(∂µΦ)(∂νΦ)
+�
+,
+(6)
+where κ ≡ eΦ0κ10 is the rescaled coupling.
+Including the contribution from the kinetic
+term of Φ, which also transforms under this conformal transformation, the Einstein frame
+gravitational plus dilaton action becomes
+SE
+grav+Φ =
+1
+2κ2
+�
+d10x
+�
+−GE
+�
+RE − 1
+2(∂µΦ)(∂µΦ)
+�
+.
+(7)
+Note that the dilaton is canonically normalised in Einstein frame.
+The kinetic terms of the NS and RR form fields include an implicit metric associated to
+the index contraction of the forms. For a generic p-form η we have
+|η|2
+S = 1
+p!(GS)µ1ν1...(GS)µpνpηµ1...µpην1...νp
+= 1
+p!(e2Ω)p(GE)µ1ν1...(GE)µpνpηµ1...µpην1...νp = e2Ω·p |η|2
+E .
+(8)
+Putting everything together, with the appropriate choice (4), the action (1) in Einstein frame
+5Note that a constant multiplying RE is a simple rescaling of the coupling constant κ, so that one still
+obtains the Einstein frame. The constant is a matter of convention.
+3
+
+becomes
+SE
+IIB =
+1
+2κ2
+� �
+d10x
+�
+−GE
+�
+RE − 1
+2(∂µΦ)(∂µΦ) − eΦ0
+2 e−Φ|H3|2
+E
+�
+(9)
+−
+�
+d10x
+�
+−GE
+�e2Φ
+2 |F1|2
+E + eΦ0
+2 eΦ| ˜F3|2
+E + e2Φ0
+4 | ˜F5|2
+E
+�
+− e2Φ0
+2
+�
+C4 ∧ H3 ∧ F3
+�
+.
+Note that the Chern-Simons term in the action does not transform, apart from via the
+constant relating κ and κ10, as it is a topological term, independent of the metric.
+A common choice of Φ0 is such that the metric in the string frame and the metric in
+the Einstein frame are the same at the vacuum, i.e. Φ0 = ⟨Φ⟩ – this allows us to discuss
+quantities in a frame-independent way at the vacuum. For that choice the action in Einstein
+frame reads
+SE
+IIB = 1
+2κ2
+� �
+d10x
+√
+−G
+�
+R − 1
+2(∂µΦ)(∂µΦ) − gs
+2 e−Φ|H3|2
+�
+−
+�
+d10x
+√
+−G
+�e2Φ
+2 |F1|2 + gs
+2 eΦ| ˜F3|2 + g2
+s
+4 | ˜F5|2
+�
+− g2
+s
+2
+�
+C4 ∧ H3 ∧ F3
+�
+,
+(10)
+where we dropped the E, as all metrics are in Einstein frame. With this choice, the gravita-
+tional coupling is related to the string scale as
+2κ2 = 2κ2
+10g2
+s = (2π)7g2
+sα′4
+or
+2κ2 = g2
+s l8
+s
+2π
+.
+(11)
+Another common choice of convention is Φ0 = 0.
+In this case, volumes are frame-
+dependent in the vacuum (see eq. (46) below) and one needs to be careful in using the right
+frame, e.g. when checking whether the α′-expansion is under control for a certain vacuum,
+which should be done using the string frame volume.
+For this choice the gravitational
+coupling is related to the string scale as
+2κ2 = 2κ2
+10 = (2π)7α′4
+or
+2κ2 = l8
+s
+2π .
+(12)
+2.2
+SL(2, R) manifest action
+We now express the Einstein frame action (10) in terms of the fields G3 and τ, such that
+the underlying SL(2, R) symmetry becomes manifest, which is sometimes useful when doing
+4
+
+calculations and it is commonly used in the literature. We define the fields
+τ = C0 + ie−Φ ,
+(13)
+G3 = ˜F3 − ie−ΦH3 = F3 − τH3 ,
+(14)
+where τ is known as the axio-dilaton.
+In terms of these fields, the action takes the form
+SE
+IIB =
+1
+2κ2
+�
+d10x
+√
+−G
+�
+R − (∂µτ)(∂µ¯τ)
+2(Im τ)2
+−
+eΦ0
+2(Im τ)|G3|2 − e2Φ0
+4 | ˜F5|2
+�
+−
+1
+2κ2
+ie2Φ0
+4
+�
+1
+(Im τ)C4 ∧ G3 ∧ G3 ,
+(15)
+where we recall κ = eΦ0κ10. We can also write the action in differential form language,
+SE
+IIB = 1
+2κ2
+� �
+R ⋆ 1 − dτ ∧ ⋆dτ
+2(Im τ)2 −
+eΦ0
+2(Im τ)G3 ∧ ⋆G3 − e2Φ0
+4
+˜F5 ∧ ⋆ ˜F5
+−
+ie2Φ0
+4(Im τ)C4 ∧ G3 ∧ G3
+�
+.
+(16)
+Written in this form, the SL(2, R) symmetry of the type IIB action becomes manifest — it
+leaves the metric and 4-form invariant, and acts on the remaining fields as
+τ → aτ + b
+cτ + d ,
+�
+C2
+B2
+�
+=
+�
+a
+b
+c
+d
+� �
+C2
+B2
+�
+,
+with
+�
+a
+b
+c
+d
+�
+∈ SL(2, R) ,
+(17)
+that is, ad − bc = 1.
+Another occasionally used convention (see e.g. [13, 14, 15, 16]) is to redefine the RR
+forms in Einstein frame as CE
+p = eΦ0CS
+p ; the action then becomes
+SE
+IIB =
+1
+2κ2
+� �
+R ⋆ 1 − dτ ∧ ⋆dτ
+2(Im τ)2 − G3 ∧ ⋆G3
+2(Im τ) − 1
+4
+˜F5 ∧ ⋆ ˜F5 −
+i
+4(Im τ)C4 ∧ G3 ∧ G3
+�
+, (18)
+where the axio-dilaton was also redefined as τ E = eΦ0τ S = eΦ0CS
+0 +ie−ϕ, with e−ϕ = e−(Φ−Φ0),
+and GE
+3 = eΦ0GS
+3 = F E
+3 − τ EH3. Note that in terms of τ E we have ⟨Im τ E⟩ = 1. With this
+field redefinition the action looks the same regardless of the choice of convention, apart from
+having a different gravitational coupling κ.
+5
+
+3
+Dimensional Reduction
+In order to obtain a 4d EFT at low energies, we consider a compactification (or dimensional
+reduction) of the 10d theory down to 4 dimensions. The 4d theory describes perturbations
+around a 10d vacuum solution and is valid for energies much lower than the compactification
+scale6. We consider a vacuum solution which corresponds to a warped product spacetime
+M10 = R1,3 ×w X6, where R1,3 is a 4d Lorentzian spacetime and X6 is a 6d compact space.
+The Einstein frame metric takes the form7
+ds2
+10 = H−1/2(y) e2ω(x)gµνdxµdxν + H1/2(y) V1/3gmndymdyn ,
+(19)
+where xµ (µ = 0, ..., 3) are 4d coordinates and ym (m = 4, ..., 9) are 6d coordinates on the
+compact space X6. The metric gmn = (g6)mn is the 6d metric of a Calabi-Yau (Ricci flat)
+manifold normalised such that
+�
+d6y√g6 ≡ l6
+s,
+with V = VE(x) keeping track of the physical size of the compact space. We define the warp
+factor H as
+H(y) ≡ 1 + e−4A0(y)
+V2/3
+,
+(20)
+which is motivated as follows. First, the background warp factor – commonly written as
+e−4A(y) – that solves the 10d Einstein equations in the presence of fluxes is only fixed up to
+a constant shift, e−4A(y) = e−4A0(y) + c, which becomes a modulus in the 4d EFT [17]. The
+fact that gmn → λgmn together with e2A → λe2A is a gauge redundancy of the metric [18, 19]
+allows us to choose λ = c1/2 and rewrite e−4A(y) = 1 + e−4A0(y)
+c
+, which naturally recovers the
+unwarped case in the c → ∞ limit – this relates c = V2/3 with the unwarped volume of the
+compact space. The factor e2ω(x) is introduced to Weyl rescale to the 4d Einstein frame,
+with metric gµν, as we now describe.
+Dimensionally reducing the 10d Einstein-Hilbert term (in Einstein frame)
+SE
+IIB =
+1
+2κ2
+�
+d10x
+√
+−G R10
+(21)
+down to 4d using the ansatz (19) gives, among other contributions, the term
+S4d ⊃
+1
+2κ2
+�
+d4x√−g4 · e2ω(x)�
+V
+�
+d6y√g6 · H(y)
+�
+R4 .
+(22)
+6Depending on the details of the compactification, this scale could correspond to e.g. mKK or mw
+KK.
+7Since we start with Einstein frame metric (19) and action (21), the volumes V and Vw are Einstein frame
+volumes.
+6
+
+Any choice of e2ω(x) that leaves a non-canonical coupling of the volume modulus V to R4 is
+said to be in the Jordan frame. Requiring a canonical form for the Einstein-Hilbert term
+instead – which defines the 4d Einstein frame – fixes the Weyl rescaling e2ω(x), up to a
+constant factor e2ω0, as
+e2ω(x) =
+e2ω0 · l6
+s
+V
+�
+d6y√g6 · H(y) ≡ e2ω0 · l6
+s
+Vw
+= e2ω0
+Vw
+,
+(23)
+where we defined the warped volume Vw = Vw · l6
+s as8
+Vw ≡ V
+�
+d6y√g6 · H(y)
+�
+��
+�
+⟨H⟩av· l6s
+.
+(24)
+This definition of Vw only differs from V · l6
+s by the factor ⟨H⟩av, the average of the warp
+factor over the compact space. If the integral is dominated by the unwarped bulk, then
+⟨H⟩av ≈ 1 and Vw ≈ V · l6
+s.
+Note the similarities with the conformal transformation in 10d to go from string frame
+to Einstein frame, where we also had some freedom in the form of a constant. There a
+convenient choice was the one for which the two metrics matched at the vacuum. Here we
+are going from the Jordan frame, in which some scalars couple to the Ricci scalar in the
+action, to the 4d Einstein frame, in which we recover the canonical Einstein-Hilbert term.
+The two metrics will match at the vacuum if we choose e2ω0 = ⟨Vw⟩. The action in Einstein
+frame for general ω0 becomes
+SE
+4d ⊃ e2ω0 · l6
+s
+2κ2
+�
+d4x√−g4 · R4 ≡ M2
+Pl
+2
+�
+d4x√−g4 · R4 ,
+(25)
+which defines the relation between the string scale9 (ms = 1/ls) and the Planck scale as
+ms =
+eΦ0
+√
+4πe2ω0 MPl .
+(26)
+8Note that this differs from the volume of the 6d compact space in the ansatz (19), which is
+V
+�
+d6y√g6 H3/2(y) .
+9See footnote 4.
+7
+
+For the convenient choice eΦ0 = gs and e2ω0 = ⟨Vw⟩, this relation becomes
+ms =
+gs
+√4πVw
+MPl .
+(27)
+Note also that in the unwarped limit the warped volume tends to the volume modulus of the
+compactification, Vw → V, and – with these choices of convention for the Weyl rescalings –
+we recover the common expression for the ratio ms/MPl. If instead we choose conventions
+Φ0 = 0 = ω0, then ms = MPl/
+√
+4π. Note that the convention dependence of MPl with
+respect to ms makes sense, as MPl measures the coupling strength of the Einstein frame
+gravitational field, whose definition depends on the convention chosen.
+At the same time, the warped string scale is given by
+mw
+s ≡ H−1/4(y0) ms ,
+(28)
+and corresponds to the scale perceived by a 4d observer living at some fixed position y0 along
+the warping direction of the compact space.
+We now determine the Kaluza-Klein (KK) scale at which the towers of massive states
+associated with the compact dimensions appear. Considering the simple case of a 10d scalar
+field ρ,
+S =
+�
+d10x
+√
+−G
+�
+−1
+2GMN(∂Mρ)(∂Nρ)
+�
+(29)
+=
+�
+d4x
+�
+d6y · H−1(y)e2ω(x)√−g4 · H3/2(y) V√g6
+�
+−1
+2H1/2(y)e−2ω(x)gµν(∂µρ)(∂νρ) − 1
+2H−1/2(y)V−1/3gmn(∂mρ)(∂nρ)
+�
+(30)
+=
+�
+d4x√−g4 V
+�
+d6y√g6
+�
+−1
+2H(y)gµν(∂µρ)(∂νρ) − 1
+2
+e2ω(x)
+V1/3 gmn(∂mρ)(∂nρ)
+�
+(31)
+=
+�
+d4x√−g4 V
+�
+d6y√g6
+�
+−1
+2H(y)gµν(∂µρ)(∂νρ) + 1
+2
+e2ω(x)
+V1/3 H(y)(∆6ρ) · ρ
+�
+,
+(32)
+where in the last step we integrated the second term by parts and defined the internal space
+Laplacian operator
+∆6ρ ≡ H−1(y)
+√g6
+∂m(√g6 gmn∂nρ) .
+(33)
+Decomposing the field ρ(x, y) in a basis of eigenfunctions of ∆6 (i.e. ∆6ξk = −λ2
+kξk, with no
+8
+
+sum over k),
+ρ(x, y) =
+�
+k
+̺k(x)ξk(y) ,
+(34)
+the action for the 10d scalar ρ becomes an action for infinitely many 4d scalars ̺k(x),
+S =
+�
+d4x√−g4 V
+�
+d6y√g6
+�
+k,l
+�
+− 1
+2H(y)gµν(∂µ̺k)(∂ν̺l)(ξkξl)
+− 1
+2
+e2ω(x)
+V1/3 H(y)λ2
+k ̺k̺l (ξkξl)
+�
+(35)
+=
+�
+d4x√−g4 V
+�
+k,l
+�
+− 1
+2gµν(∂µ̺k)(∂ν̺l) ·
+�
+d6y√g6 · H(y) ξkξl
+− 1
+2
+e2ω(x)
+V1/3 λ2
+k̺k̺l ·
+�
+d6y√g6 · H(y) ξkξl
+�
+.
+(36)
+Since the eigenmodes ξk satisfy the orthogonality condition
+�
+d6y√g6 · H(y) ξkξl = c1δkl ,
+(37)
+the KK modes decouple and the action reduces to
+S =
+�
+k
+�
+d4x√−g4
+�
+− 1
+2gµν(∂µ̺k)(∂ν̺k)(c1V) + 1
+2
+e2ω(x)
+V1/3 λ2
+k̺2
+k(c1V)
+�
+(38)
+=
+�
+k
+�
+d4x√−g4
+�
+− 1
+2gµν(∂µ̺c
+k)(∂ν̺c
+k) + 1
+2 · λ2
+k
+V1/3
+e2ω0
+Vw
+· (̺c
+k)2
+�
+,
+(39)
+where in the second line we canonically normalise the fields ̺k. Therefore, the mass of ̺c
+k is
+mk = λk
+V1/6
+�e2ω0
+Vw
+�1/2
+=⇒
+mKK = λ1
+V1/6
+�e2ω0
+Vw
+�1/2
+,
+(40)
+where we identify the KK scale, mKK, with the mass of the lightest mode m1.
+To determine λk (and the eigenfunctions10) one must solve the eigenvalue equation
+1
+√g6
+∂m(√g6 gmn∂nξk) + H(y) · λ2
+k · ξk = 0 ,
+(41)
+10These are commonly referred to as the wavefunctions of the modes ̺k.
+9
+
+together with appropriate boundary conditions. It is therefore not possible to give a fully
+generic expression for λk, and thus mKK, as it depends on the details of the compactification.
+If we consider the case of a torus with a single common radius as the prototypical example
+of an isotropic compact space11 with characteristic scale ls and constant warp factor12, the
+eigenvalue is λ1 = H−1/2
+0
+· (2π) · ms. Then the KK scale becomes
+mKK =
+�e2ω0
+Vw
+�1/2
+H−1/2
+0
+· 2π
+V1/6ms = H−1/2
+0
+· 2π
+V1/6 ·
+eΦ0
+√
+4πV1/2
+w
+MPl .
+(42)
+We should note that generally there are two distinct volumes appearing in mKK.
+One
+usually takes Vw ≈ V, i.e. one assumes that the unwarped region of the compact space
+dominates the volume integral, in which case we recover the well-known volume suppression
+mKK ∼ MPl/V2/3. For the convenient choice eΦ0 = gs and e2ω0 = ⟨Vw⟩, and approximating
+Vw ≈ V, we have
+mKK = H−1/2
+0
+· 2π
+V1/6 ms = H−1/2
+0
+·
+2πgs
+√
+4πV2/3MPl ,
+(43)
+while for the common alternative choice Φ0 = 0, the factor of gs will be absent.
+One also often finds in the literature another scale
+mw
+KK ≡ H−1/4(y0) mloc
+KK ,
+(44)
+where mloc
+KK corresponds to a KK scale associated with modes localised on a subspace at
+some fixed y = y0 (e.g. the tower of states associated with fields living on the world-volume
+of a brane wrapping an internal cycle at the tip of some warped throat).
+It is worth emphasising that volumes measured using a string frame metric may differ from
+the ones measured using an Einstein frame metric, depending on the convention used, i.e. on
+the choice of Φ0. To see this, recall that the two metrics are related by GE
+MN = e− Φ−Φ0
+2
+GS
+MN.
+11More generically one could consider different scales in different directions, which would result in different
+KK scales.
+12This is consistent with our normalisation for the coordinates ym, such that
+�
+d6y√g6 = l6
+s – it corresponds
+to an identification of the normalised coordinates ym ∼ ym + 1 (with ds2
+6 = l2
+s dymdym), rather than
+ym ∼ ym + 2π.
+Moreover, with a constant warp factor, H(y) ≡ H0, the eigenfunctions respecting the
+(periodic) boundary conditions on the torus would be ξ⃗k ∝ e2πi ⃗k·⃗y, with a vector of integers ⃗k labeling the
+modes, and hence ∆6ξ⃗k = −H−1
+0
+· (2π)2k2 · ms · ξ⃗k, giving λ⃗k = H−1/2
+0
+· (2π|⃗k|) · ms. When the warp factor
+is not trivial, its functional form will qualitatively change the eigenvalue equation (41) and the solutions λk
+will depend, in particular, on the balance between warped and unwarped regions of the compact space.
+10
+
+Therefore, a generic d-dimensional volume can be written as
+V E
+d =
+�
+ddy
+�
+gE
+d f(y) =
+�
+ddy
+�
+gS
+d e− d
+4 (Φ−Φ0)f(y) ,
+(45)
+where we allow for some function f(y), such as H(y) in the definition of Vw (24). Therefore,
+the volumes in the two frames (assuming Φ is stabilised) are related as V S
+d =
+�
+e⟨Φ⟩−Φ0� d
+4 V E
+d ⇔
+VS = e
+3
+2(⟨Φ⟩−Φ0)VE .
+(46)
+Hence, it is important to note the convention being used for the Einstein frame metric and
+how it relates to quantities that are obtained in either string frame or Einstein frame (e.g.
+perturbative and non-perturbative corrections).
+Finally, it is interesting to note that the ratio mKK/ms is manifestly independent of the
+choice for eΦ0, i.e. on the convention used in changing from string frame to Einstein frame,
+whereas the ratio mKK/MPl is manifestly independent of the choice for eω0, i.e.
+on the
+convention used in going to 4d Einstein frame. After taking into account the dependence
+of the Einstein frame volume on the frame convention, the ratio mKK/MPl is also actually
+independent of the choice for eΦ0, as it must be. Approximating Vw ≈ V and expressing the
+ratio in terms of the string-frame volume, the convention-dependent factors fall out:
+mKK
+MPl
+= H−1/2
+0
+·
+2πgs
+√
+4πV2/3
+S
+.
+(47)
+4
+Flux scalar potential
+The scalar potential for the moduli fields and the dilaton comes from the terms R, |G3|2 and
+| ˜F5|2 in the action (9), after dimensional reduction to 4d. The contributions from the R and
+˜F5 terms can be shown to give (see section 5.3 of [20])
+1
+2κ2
+� �
+R ⋆ 1 − e2Φ0
+4
+˜F5 ∧ ⋆ ˜F5
+�
+= eΦ0
+2κ2
+�
+d4x√−g4 · e4ω(x)
+�
+H−1G3 ∧ iG3
+2(Imτ) ,
+(48)
+which we can put together with the G3 ∧ ⋆G3 term to give in total
+SE
+IIB ⊃ eΦ0
+2κ2
+�
+d4x√−g4 · e4ω(x)
+�
+H−1
+2(Imτ)G3 ∧ (iG3 + ⋆6G3)
+(49)
+= eΦ0
+2κ2
+�
+d4x√−g4 · e4ω(x)
+�
+H−1
+(Imτ)G+
+3 ∧ ⋆6G
++
+3 ,
+(50)
+11
+
+with G+
+3 = 1
+2(G3 + i ⋆6 G3) such that ⋆6G+
+3 = −iG+
+3 [20]. Using the metric (19) we can
+rewrite this action in terms of gmn,
+SE
+IIB ⊃
+�
+d4x√−g4
+�
+eΦ0
+2κ2 e4ω(x)
+�
+H−1 G+
+3 ∧ ⋆g6G
++
+3
+(Imτ)
+�
+≡ −
+�
+d4x√−g4 V ,
+(51)
+which defines the 4d scalar potential V as
+V = −i eΦ0
+2κ2 e4ω(x)
+�
+H (H−1G+
+3 ) ∧ (H−1G
++
+3 )
+(Imτ)
+.
+(52)
+It is now possible to rewrite this potential in an N = 1 supergravity form, by defining
+W = 1
+a
+�
+G3 ∧ Ω ,
+(53)
+where a is a normalisation constant to be determined below, and using that
+�
+H (H−1G+
+3 ) ∧ (H−1G
++
+3 ) =
+a2
+�
+H Ω ∧ ΩGα¯β(DαW)(D ¯βW)
+(54)
+where α, β run over the complex structure moduli and the axio-dilaton. Using this, the
+scalar potential becomes
+V = −i eΦ0
+2κ2
+e4ω(x)
+(Im τ)
+a2
+�
+H Ω ∧ Ω
+�
+Gi¯(DiW)(D¯W) − 3|W|2�
+= eΦ0
+2κ2
+�e2ω0 · l6
+s
+Vw
+�2
+1
+(Im τ)
+a2
+l6
+s
+l6
+s
+i
+�
+H Ω ∧ Ω
+�
+Gi¯(DiW)(D¯W) − 3|W|2�
+=
+2π
+eΦ0 · l8s
+�e2Φ0M2
+Pl · l2
+s · l6
+s
+4πVw
+�2
+1
+(Im τ)
+a2
+l6s
+l6
+s
+i
+�
+H Ω ∧ Ω
+�
+Gi¯(DiW)(D¯W) − 3|W|2�
+= e3Φ0
+4π · l4s
+M4
+Pl
+a2
+l6s
+·
+� l6
+s
+Vw
+�2
+1
+2(Im τ)
+l6
+s
+i
+�
+H Ω ∧ Ω · M2
+Pl
+� Gi¯
+M2
+Pl
+(DiW)(D¯W) −
+3
+M2
+Pl
+|W|2
+�
+=
+�
+e3Φ0
+4π · l10
+s
+M6
+Pl
+�
+eK/M2
+Pl
+�
+Ki¯(DiW)(D¯W) −
+3
+M2
+Pl
+|W|2
+�
+,
+(55)
+where now i, j run over complex structure moduli, Kähler moduli and the axio-dilaton, the
+Kähler potential K is given by
+12
+
+K/M2
+Pl = −2 log Vw − log(−i(τ − ¯τ)) − log
+� i
+l6s
+�
+H Ω ∧ Ω
+�
+(56)
+and Ki¯ is the inverse field space metric that follows from Ki¯ = ∂i∂¯K.
+Note that the volume term in K includes not only the overall volume modulus V, but
+also the other Kähler moduli. This scalar potential leads to the normalisation
+W/M3
+Pl =
+e
+3
+2 Φ0
+√
+4π · l5s
+�
+G3 ∧ Ω .
+(57)
+We can see that the normalisation constant, a, is convention-dependent through the choice
+of eΦ0.
+This gives for the gravitino mass
+m3/2 = e
+K
+2M2
+Pl |W|
+M2
+Pl
+=
+e
+1
+2⟨Φ⟩
+Vw ||Ω||w
+e
+3
+2 Φ0W0
+√
+8π
+MPl ,
+(58)
+where e
+1
+2⟨Φ⟩ comes from ⟨Im τ⟩, ||Ω||2
+w · l6
+s = i
+�
+H Ω ∧ Ω and we define
+W0/M3
+Pl ≡
+� 1
+l5
+s
+�
+G3 ∧ Ω
+�
+.
+(59)
+It follows from (58) and (42) that the important13 ratio (assuming the bulk dominates
+all the integrals, so that Vw ≈ V and ||Ω||w ≈ ||Ω||)
+m3/2
+mKK
+= H1/2
+0
+e
+1
+2(⟨Φ⟩+Φ0)
+V1/3
+E
+W0
+√
+2(2π)||Ω|| ,
+(60)
+where we highlight the fact that the volume being used is the Einstein frame volume, VE.
+Note that this mass ratio, as written in terms of the Einstein frame volume, seems to depend
+on the convention used for the 10d change of frames, i.e. the choice of Φ0. It is however
+convention-independent, as it must be, since the Einstein frame volume also depends on the
+choice of Φ0. If we express the mass ratio in terms of the string frame volume instead, which
+13Not only is this ratio important because a consistent 4d supergravity description requires that the
+gravitino remains in the theory, i.e. its mass is not above the EFT cutoff – typically mKK – and therefore
+integrated out, but it was shown that it also serves as a control parameter for certain corrections to the
+scalar potential, e.g. from higher F-terms [21].
+13
+
+corresponds to the volume perceived by the string itself, using (46) we find
+m3/2
+mKK
+= H1/2
+0
+e⟨Φ⟩
+V1/3
+S
+W0
+√
+2(2π)||Ω|| ,
+(61)
+which is manifestly independent of conventions14.
+5
+Corrections to the scalar potential
+Since the flux superpotential leaves all Kähler moduli unstabilised, either leaving them as
+flat directions or generating runaways, one must resort to higher-order corrections to the
+EFT in order to stabilise them. Both perturbative and non-perturbative corrections have
+been considered in the literature — while the former are computed at the level of the 10d
+EFT and in string frame, the latter are obtained directly at the level of the 4d EFT and are
+computed in Einstein frame. One must therefore be careful with the conventions being used
+to change frames, i.e. the choice of Φ0, in order to remain consistent.
+5.1
+Perturbative corrections
+In [22], it was shown that α′-corrections to the Type IIB effective action (9) manifest as
+corrections to the 4d volume modulus Kähler potential and spoil the no-scale structure
+of its scalar potential. These corrections arise from higher-derivative terms at order (α′)3
+appearing in the type IIB effective action,
+SS
+IIB =
+1
+2κ2
+10
+�
+d10x
+�
+−GS e−2Φ�
+RS + 4(∂Φ)2
+S + (α′)3 · ζ(3)
+3 · 211 · J0
+�
+,
+(62)
+where the higher-order term is schematically given by
+J0 ∼ (RMNP Q)4 .
+(63)
+One must also add a term
+δSS
+Φ ∼
+�
+d10x
+�
+−GSe−2Φ(α′)3(∇2Φ) Q ,
+(64)
+14Note that we give mass ratios for canonically normalised fields defined in the Einstein frame. Whilst
+these mass ratios must be invariant under change of conventions, a change in frame would come with field
+redefinitions, and new masses and couplings. In a setup in which all couplings, including the gravitational
+coupling, are constant (e.g. assuming that the dilaton and volume modulus are stabilised and integrated
+out), the change of frames becomes a change of convention from one Einstein frame to another Einstein
+frame, and the mass ratios would be invariant.
+14
+
+where Q ∼ (RMNP Q)3 is a generalisation of the 6d Euler integrand
+�
+X6 d6y√g6 Q = χ, with
+χ the Euler characteristic of X6 [22]. This term corrects the 10d solution to the equation
+of motion for Φ, such that Φ = Φ10 + ζ(3)
+16 Q. It is then shown in [22] that this leads to a
+correction to the Kähler potential of the form
+K = −2 log
+�
+VS + ξ
+2
+�
+= −2 log
+�
+VE e
+3
+2(Φ−Φ0) + ξ
+2
+�
+(65)
+= −2 log
+�
+VE + ξ
+2e− 3
+2(Φ−Φ0)�
++ ... ,
+(66)
+where we have used (46) with d = 6, keeping Φ0 unspecified, and ξ is defined as15
+ξ = − ζ(3)χ
+2(2π)3 .
+(67)
+Note in particular that the correction expressed in Einstein frame depends on the convention
+one chooses for Φ0,
+Φ0 = 0 =⇒ K = −2 log
+�
+V +
+ξ
+2g3/2
+s
+�
+,
+(68)
+Φ0 = ⟨Φ⟩ =⇒ K = −2 log
+�
+V + ξ
+2
+�
+,
+(69)
+where we have assumed as usual that the dilaton has been stabilised by fluxes.
+5.2
+Non-perturbative corrections
+Although the superpotential W does not receive perturbative corrections, it may receive non-
+perturbative corrections from either instantons arising from Euclidean D3-branes wrapping
+4-cycles or gaugino condensation on the world-volume theory of D7-branes wrapped around
+internal 4-cycles. Let us consider the latter case in some detail. In what follows, Tp =
+2π
+lp+1
+s
+is the brane tension. The DBI action for a Dp-brane, in the string frame, is given by [6, 7]
+SDBI
+Dp = −Tp
+�
+dp+1σ e−Φ
+�
+− det
+�
+gS + B + l2
+s
+2πF
+�
+,
+(70)
+15In [22], we find the definition ξ = − ζ(3)χ(X6)
+2
+. The missing factor of (2π)3 comes from their conventions
+for the volume, V[22] = V6/(2πα′)3, whereas we are using V = V6/l6
+s = (2π)−3 V6/(2πα′)3, with the convention
+(2π)2α′ = l2
+s. There are also instances in the literature where the factor of 1/2 is absorbed into the definition
+of ξ.
+15
+
+where gS and B refer to the pull-back of the string frame metric (GS)MN and 2-form BMN
+onto the world-volume of the brane and F to the field-strength Fab of the brane gauge fields.
+Rewriting the action in terms of the Einstein frame metric,
+SDBI
+Dp = −Tp
+�
+dp+1σ e−Φ�
+− det gS
+�
+det
+�
+1 + (gS)−1
+�
+B + l2
+s
+2πF
+��
+(71)
+⊃ −Tp
+�
+dp+1σ e−Φ�
+− det gS 1
+4
+� l2
+s
+2π
+�2
+(gS)ac(gS)bdFabFcd
+(72)
+= −Tp
+4
+l4
+s
+(2π)2
+�
+dp+1σ
+�
+− det gE e
+p−3
+4 (Φ−Φ0)e−ΦFabF ab ,
+(73)
+where the indices in FabF ab are contracted with Einstein frame metrics. This is the kinetic
+term for the brane gauge bosons (see Appendix A.2. of [23]) and tells us the gauge coupling
+of the corresponding theory, which is a key parameter for gaugino condensation. If the brane
+is wrapping a (p − 3)–cycle Σp−3, we find the corresponding 4d term (assuming that Φ is
+constant over the cycle)
+S4d
+Dp ⊃ −
+1
+8πlp−3
+s
+�
+d4x
+�
+− det gE
+4 e
+p−3
+4 (Φ−Φ0)e−Φ
+��
+dp−3σ
+�
+gE
+p−3
+�
+�
+��
+�
+τ E
+Σp−3lp−3
+s
+FabF ab
+(74)
+= −
+�
+d4x
+�
+− det gE
+4
+�
+τ E
+Σp−3
+8πeΦ e
+p−3
+4 (Φ−Φ0)
+�
+FabF ab ,
+(75)
+and we can read off the gauge coupling gc,
+1
+g2
+c
+=
+τ E
+Σp−3
+4πe⟨Φ⟩ e
+p−3
+4 (⟨Φ⟩−Φ0) ,
+(76)
+where we have assumed as usual that the dilaton has been stabilised by fluxes at some
+higher scale. Gaugino condensation on the world-volume theory of D7-branes will then give
+a non-perturbative contribution to the superpotential [24],16
+Wnp ∼ e
+− 8π2
+g2c
+1
+N = e− 2π
+N
+τE
+gs e⟨Φ⟩−Φ0 ,
+(77)
+16Here N is the number of branes stacked on top of each other, responsible for the gauge group. It appears
+through the beta-function coefficient [24].
+16
+
+where we used e⟨Φ⟩ = gs. Holomorphicity of W then leads to the general contribution
+Wnp =
+�
+i
+Aiei ai
+gs e⟨Φ⟩−Φ0T E
+i
+,
+(78)
+where the sum is over the contributing cycles, ai = 2π
+Ni and the fields Ti = bi + iτi are the
+complexified Kähler moduli. Hence, we can compare the two most common conventions for
+Φ0,
+Φ0 = 0 =⇒ Wnp =
+�
+i
+AieiaiT E
+i ,
+(79)
+Φ0 = ⟨Φ⟩ =⇒ Wnp =
+�
+i
+Aiei ai
+gs T E
+i .
+(80)
+As for the mass ratio m3/2/mKK, the superpotential as written, in terms of Einstein frame
+4-cycle volumes, appears to depend on the choice of convention for Φ0, but one should recall
+that the 4-cycle volumes also depend on this choice of convention. If we express the 4-cycle
+volumes in terms of the string frame metric (46),
+τ S
+i = e⟨Φ⟩−Φ0τ E
+i ,
+(81)
+the convention-independence becomes manifest.
+Acknowledgements
+IZ is partially supported by STFC, grant ST/P00055X/1.
+References
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+19
+
diff --git a/Z9E4T4oBgHgl3EQfnw0i/content/tmp_files/load_file.txt b/Z9E4T4oBgHgl3EQfnw0i/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..d7bd3b69316da2cb26df5e06fa63adf43d17eef4
--- /dev/null
+++ b/Z9E4T4oBgHgl3EQfnw0i/content/tmp_files/load_file.txt
@@ -0,0 +1,492 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf,len=491
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='05178v1  [hep-th]  12 Jan 2023  A guide to frames, 2π’s, scales and corrections  in string compactifications  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Bentoa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Chakrabortyb, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Parameswarana, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Zavalac  a Department of Mathematical Sciences,  University of Liverpool, Liverpool L69 7ZL  b Department of Physics, Ashoka University,  Plot 2, Rajiv Gandhi Education City, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Rai, Sonipat 131029, Haryana, India  c Department of Physics, Swansea University,  Singleton Park, Swansea, SA2 8PP, UK  Abstract  This note is intended to serve as a reference for conventions used in the literature on  string compactifications, and how to move between them, collected in a single and easy-  to-find place, using type IIB as an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We hope it may be useful to  beginners in the field and busy experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' string constructions proposed to address  the moduli stabilisation problem are generically in regions of parameter space at the  boundaries of control, so that consistent use of 2π’s and frame conventions can be  pivotal when computing their potentially dangerous corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  E-mail: Bruno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Bento@liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='uk, dibya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='chakraborty@ashoka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='in, susha@liv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='uk,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='zavalacarrasco@swansea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='uk  1  Introduction  The idea that this pedagogical note could be a useful contribution to the community came  about after several discussions with colleagues on the robustness of various candidate string  constructions for moduli stabilisation, towards particle physics and cosmology, against po-  tentially dangerous corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' To scrutinise these constructions, it becomes necessary to  use other people’s conventions (or indeed one’s own in one’s past worldline), and although  there is nothing deep in changing conventions, and a change of frames is simply a field re-  definition1, it is tedious, and possibly tricky unless starting from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We thus present  the various choices most commonly used for 10d and 4d string and Einstein frames in type  IIB compactifications, and 2π’s, together with the map between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We emphasise how  physical quantities such as mass ratios, which determine the size of leading corrections to  explicit string compactifications, are of course convention-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We hope that this  may help both beginners in the field and busy experts to save some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  In Section 2, we present the 10d type IIB supergravity action in the string frame and the  Einstein frame, using the two main choices of conventions for change of frames, and including  the SL(2, R) manifest action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' In Section 3, we dimensionally reduce to 4d using a general  warped compactification, and present the 4d Einstein frame in different conventions which,  however, always allow to recover the unwarped limit from the warped case in an intuitive  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We identify the (warped) KK scale, presenting a single expression that covers the var-  ious conventions considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' In Section 4 we similarly work out the flux superpotential and  gravitino mass, and show how the mass ratio  m3/2  mKK – which not only determines whether we  have a consistent supergravity description in 4d, but also controls the higher F-term correc-  tions to KKLT/LVS type compactifications – is of course convention-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' In Section  5 we give the leading perturbative corrections to the Kähler potential and non-perturbative  corrections to the superpotential in the most common conventions, again showing how the  convention-independence can be made manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  1See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' [1, 2, 3, 4, 5] and references therein for some interesting discussions on the equivalence of the  Einstein and Jordan frames in cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  1  2  Type IIB supergravity  Our starting point is the type IIB low-energy supergravity action in string frame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' which is  given by2  SS  IIB = 1  2κ2  10  �  d10x  �  −GS  �  e−2Φ  �  R + 4∂µΦ∂µΦ − 1  2|H3|2  �  −  �1  2|F1|2 + 1  2| ˜F3|2 + 1  4| ˜F5|2  � �  − 1  4κ2  10  �  C4 ∧ H3 ∧ F3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (1)  where R is the Ricci scalar,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Φ is the dilaton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' H3 is the field-strength of the NS 2-form B2  and Fp is the field-strength of the RR (p − 1)-forms Cp−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' and ˜F3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' ˜F5 are defined as below:  H3 = dB2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  ˜F3 = F3 − C0H3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Fp = dCp−1 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  ˜F5 = F5 − 1  2C2 ∧ H3 + 1  2B2 ∧ F3 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  |Fp|2 = 1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Fµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='µpF µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='µp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Moreover, the type IIB action must be supplemented with the self-duality condition3  ˜F5 = ⋆ ˜F5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  The relation between the string scale α′ and the 10d gravitational coupling in string  frame κ10 is  2κ2  10 = (2π)7α′4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (2)  A common convention for the string length4 ls, which we use below, is  (2π)2α′ = l2  s ,  (3)  although sometimes α′ = l2  s is used instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  2The action can be found on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='90 of [6], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='314 of [7], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='114 of [8], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='79 of [9] and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='625 of [10], along with  the necessary definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' There is a different definition of F5, for example in [11], which also contains the  equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' See also [12] for the dilaton dependence of the RR sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  3Notice the factor of 1  4 rather than 1  2 in the kinetic term, which accounts for the fact that only half the  degrees of freedom should be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  4The string scale (corresponding to the mass of the tower of string states) is Ms =  1  √  α′ = 2π  ls for this  choice of conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Sometimes the notation ms =  1  ls is used with this convention, with the relation  Ms = 2πms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='1  Einstein frame  In the string frame, the gravitational part of the action is not in the canonical Einstein-  Hilbert form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' In order to obtain the latter, we perform a conformal transformation of the  metric in 10d, GS → GE = e2ΩGS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  The frame in which the gravitational part of the action takes the canonical Einstein-  Hilbert form – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' the Ricci scalar does not couple to anything other than  √  −GE – is the  Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' This choice fixes the required conformal transformation up to a constant5  Ω = −Φ − Φ0  4  ,  (4)  where the constant Φ0 is a choice of convention, and the two metrics are related by  GE  MN = e− Φ−Φ0  2  GS  MN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (5)  The first term in the action (1), namely the Ricci scalar, in the Einstein frame becomes  SE  grav = 1  2κ2  �  d10x  �  −GE  �  RE − 9  2(GE)µν(∂µΦ)(∂νΦ)  �  ,  (6)  where κ ≡ eΦ0κ10 is the rescaled coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Including the contribution from the kinetic  term of Φ, which also transforms under this conformal transformation, the Einstein frame  gravitational plus dilaton action becomes  SE  grav+Φ =  1  2κ2  �  d10x  �  −GE  �  RE − 1  2(∂µΦ)(∂µΦ)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (7)  Note that the dilaton is canonically normalised in Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  The kinetic terms of the NS and RR form fields include an implicit metric associated to  the index contraction of the forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' For a generic p-form η we have  |η|2  S = 1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='(GS)µ1ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' (GS)µpνpηµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='µpην1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='νp  = 1  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='(e2Ω)p(GE)µ1ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' (GE)µpνpηµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='µpην1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='νp = e2Ω·p |η|2  E .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (8)  Putting everything together, with the appropriate choice (4), the action (1) in Einstein frame  5Note that a constant multiplying RE is a simple rescaling of the coupling constant κ, so that one still  obtains the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The constant is a matter of convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  3  becomes  SE  IIB =  1  2κ2  � �  d10x  �  −GE  �  RE − 1  2(∂µΦ)(∂µΦ) − eΦ0  2 e−Φ|H3|2  E  �  (9)  −  �  d10x  �  −GE  �e2Φ  2 |F1|2  E + eΦ0  2 eΦ| ˜F3|2  E + e2Φ0  4 | ˜F5|2  E  �  − e2Φ0  2  �  C4 ∧ H3 ∧ F3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Note that the Chern-Simons term in the action does not transform, apart from via the  constant relating κ and κ10, as it is a topological term, independent of the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  A common choice of Φ0 is such that the metric in the string frame and the metric in  the Einstein frame are the same at the vacuum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Φ0 = ⟨Φ⟩ – this allows us to discuss  quantities in a frame-independent way at the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' For that choice the action in Einstein  frame reads  SE  IIB = 1  2κ2  � �  d10x  √  −G  �  R − 1  2(∂µΦ)(∂µΦ) − gs  2 e−Φ|H3|2  �  −  �  d10x  √  −G  �e2Φ  2 |F1|2 + gs  2 eΦ| ˜F3|2 + g2  s  4 | ˜F5|2  �  − g2  s  2  �  C4 ∧ H3 ∧ F3  �  ,  (10)  where we dropped the E, as all metrics are in Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' With this choice, the gravita-  tional coupling is related to the string scale as  2κ2 = 2κ2  10g2  s = (2π)7g2  sα′4  or  2κ2 = g2  s l8  s  2π  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (11)  Another common choice of convention is Φ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  In this case, volumes are frame-  dependent in the vacuum (see eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' (46) below) and one needs to be careful in using the right  frame, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' when checking whether the α′-expansion is under control for a certain vacuum,  which should be done using the string frame volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  For this choice the gravitational  coupling is related to the string scale as  2κ2 = 2κ2  10 = (2π)7α′4  or  2κ2 = l8  s  2π .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (12)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='2  SL(2, R) manifest action  We now express the Einstein frame action (10) in terms of the fields G3 and τ, such that  the underlying SL(2, R) symmetry becomes manifest, which is sometimes useful when doing  4  calculations and it is commonly used in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We define the fields  τ = C0 + ie−Φ ,  (13)  G3 = ˜F3 − ie−ΦH3 = F3 − τH3 ,  (14)  where τ is known as the axio-dilaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  In terms of these fields, the action takes the form  SE  IIB =  1  2κ2  �  d10x  √  −G  �  R − (∂µτ)(∂µ¯τ)  2(Im τ)2  −  eΦ0  2(Im τ)|G3|2 − e2Φ0  4 | ˜F5|2  �  −  1  2κ2  ie2Φ0  4  �  1  (Im τ)C4 ∧ G3 ∧ G3 ,  (15)  where we recall κ = eΦ0κ10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We can also write the action in differential form language,  SE  IIB = 1  2κ2  � �  R ⋆ 1 − dτ ∧ ⋆dτ  2(Im τ)2 −  eΦ0  2(Im τ)G3 ∧ ⋆G3 − e2Φ0  4  ˜F5 ∧ ⋆ ˜F5  −  ie2Φ0  4(Im τ)C4 ∧ G3 ∧ G3  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (16)  Written in this form, the SL(2, R) symmetry of the type IIB action becomes manifest — it  leaves the metric and 4-form invariant, and acts on the remaining fields as  τ → aτ + b  cτ + d ,  �  C2  B2  �  =  �  a  b  c  d  � �  C2  B2  �  ,  with  �  a  b  c  d  �  ∈ SL(2, R) ,  (17)  that is, ad − bc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Another occasionally used convention (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' [13, 14, 15, 16]) is to redefine the RR  forms in Einstein frame as CE  p = eΦ0CS  p ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' the action then becomes  SE  IIB =  1  2κ2  � �  R ⋆ 1 − dτ ∧ ⋆dτ  2(Im τ)2 − G3 ∧ ⋆G3  2(Im τ) − 1  4  ˜F5 ∧ ⋆ ˜F5 −  i  4(Im τ)C4 ∧ G3 ∧ G3  �  , (18)  where the axio-dilaton was also redefined as τ E = eΦ0τ S = eΦ0CS  0 +ie−ϕ, with e−ϕ = e−(Φ−Φ0),  and GE  3 = eΦ0GS  3 = F E  3 − τ EH3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Note that in terms of τ E we have ⟨Im τ E⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' With this  field redefinition the action looks the same regardless of the choice of convention, apart from  having a different gravitational coupling κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  5  3  Dimensional Reduction  In order to obtain a 4d EFT at low energies, we consider a compactification (or dimensional  reduction) of the 10d theory down to 4 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The 4d theory describes perturbations  around a 10d vacuum solution and is valid for energies much lower than the compactification  scale6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We consider a vacuum solution which corresponds to a warped product spacetime  M10 = R1,3 ×w X6, where R1,3 is a 4d Lorentzian spacetime and X6 is a 6d compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  The Einstein frame metric takes the form7  ds2  10 = H−1/2(y) e2ω(x)gµνdxµdxν + H1/2(y) V1/3gmndymdyn ,  (19)  where xµ (µ = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=', 3) are 4d coordinates and ym (m = 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=', 9) are 6d coordinates on the  compact space X6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The metric gmn = (g6)mn is the 6d metric of a Calabi-Yau (Ricci flat)  manifold normalised such that  �  d6y√g6 ≡ l6  s,  with V = VE(x) keeping track of the physical size of the compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' We define the warp  factor H as  H(y) ≡ 1 + e−4A0(y)  V2/3  ,  (20)  which is motivated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' First, the background warp factor – commonly written as  e−4A(y) – that solves the 10d Einstein equations in the presence of fluxes is only fixed up to  a constant shift, e−4A(y) = e−4A0(y) + c, which becomes a modulus in the 4d EFT [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The  fact that gmn → λgmn together with e2A → λe2A is a gauge redundancy of the metric [18, 19]  allows us to choose λ = c1/2 and rewrite e−4A(y) = 1 + e−4A0(y)  c  , which naturally recovers the  unwarped case in the c → ∞ limit – this relates c = V2/3 with the unwarped volume of the  compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The factor e2ω(x) is introduced to Weyl rescale to the 4d Einstein frame,  with metric gµν, as we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Dimensionally reducing the 10d Einstein-Hilbert term (in Einstein frame)  SE  IIB =  1  2κ2  �  d10x  √  −G R10  (21)  down to 4d using the ansatz (19) gives, among other contributions, the term  S4d ⊃  1  2κ2  �  d4x√−g4 · e2ω(x)�  V  �  d6y√g6 · H(y)  �  R4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (22)  6Depending on the details of the compactification, this scale could correspond to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' mKK or mw  KK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  7Since we start with Einstein frame metric (19) and action (21), the volumes V and Vw are Einstein frame  volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  6  Any choice of e2ω(x) that leaves a non-canonical coupling of the volume modulus V to R4 is  said to be in the Jordan frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Requiring a canonical form for the Einstein-Hilbert term  instead – which defines the 4d Einstein frame – fixes the Weyl rescaling e2ω(x), up to a  constant factor e2ω0, as  e2ω(x) =  e2ω0 · l6  s  V  �  d6y√g6 · H(y) ≡ e2ω0 · l6  s  Vw  = e2ω0  Vw  ,  (23)  where we defined the warped volume Vw = Vw · l6  s as8  Vw ≡ V  �  d6y√g6 · H(y)  �  ��  �  ⟨H⟩av· l6s  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (24)  This definition of Vw only differs from V · l6  s by the factor ⟨H⟩av, the average of the warp  factor over the compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' If the integral is dominated by the unwarped bulk, then  ⟨H⟩av ≈ 1 and Vw ≈ V · l6  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Note the similarities with the conformal transformation in 10d to go from string frame  to Einstein frame, where we also had some freedom in the form of a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' There a  convenient choice was the one for which the two metrics matched at the vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Here we  are going from the Jordan frame, in which some scalars couple to the Ricci scalar in the  action, to the 4d Einstein frame, in which we recover the canonical Einstein-Hilbert term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  The two metrics will match at the vacuum if we choose e2ω0 = ⟨Vw⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The action in Einstein  frame for general ω0 becomes  SE  4d ⊃ e2ω0 · l6  s  2κ2  �  d4x√−g4 · R4 ≡ M2  Pl  2  �  d4x√−g4 · R4 ,  (25)  which defines the relation between the string scale9 (ms = 1/ls) and the Planck scale as  ms =  eΦ0  √  4πe2ω0 MPl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (26)  8Note that this differs from the volume of the 6d compact space in the ansatz (19), which is  V  �  d6y√g6 H3/2(y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  9See footnote 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  7  For the convenient choice eΦ0 = gs and e2ω0 = ⟨Vw⟩, this relation becomes  ms =  gs  √4πVw  MPl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (27)  Note also that in the unwarped limit the warped volume tends to the volume modulus of the  compactification, Vw → V, and – with these choices of convention for the Weyl rescalings –  we recover the common expression for the ratio ms/MPl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' If instead we choose conventions  Φ0 = 0 = ω0, then ms = MPl/  √  4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Note that the convention dependence of MPl with  respect to ms makes sense, as MPl measures the coupling strength of the Einstein frame  gravitational field, whose definition depends on the convention chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  At the same time, the warped string scale is given by  mw  s ≡ H−1/4(y0) ms ,  (28)  and corresponds to the scale perceived by a 4d observer living at some fixed position y0 along  the warping direction of the compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  We now determine the Kaluza-Klein (KK) scale at which the towers of massive states  associated with the compact dimensions appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Considering the simple case of a 10d scalar  field ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  S =  �  d10x  √  −G  �  −1  2GMN(∂Mρ)(∂Nρ)  �  (29)  =  �  d4x  �  d6y · H−1(y)e2ω(x)√−g4 · H3/2(y) V√g6  �  −1  2H1/2(y)e−2ω(x)gµν(∂µρ)(∂νρ) − 1  2H−1/2(y)V−1/3gmn(∂mρ)(∂nρ)  �  (30)  =  �  d4x√−g4 V  �  d6y√g6  �  −1  2H(y)gµν(∂µρ)(∂νρ) − 1  2  e2ω(x)  V1/3 gmn(∂mρ)(∂nρ)  �  (31)  =  �  d4x√−g4 V  �  d6y√g6  �  −1  2H(y)gµν(∂µρ)(∂νρ) + 1  2  e2ω(x)  V1/3 H(y)(∆6ρ) · ρ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (32)  where in the last step we integrated the second term by parts and defined the internal space  Laplacian operator  ∆6ρ ≡ H−1(y)  √g6  ∂m(√g6 gmn∂nρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (33)  Decomposing the field ρ(x, y) in a basis of eigenfunctions of ∆6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' ∆6ξk = −λ2  kξk, with no  8  sum over k),  ρ(x, y) =  �  k  ̺k(x)ξk(y) ,  (34)  the action for the 10d scalar ρ becomes an action for infinitely many 4d scalars ̺k(x),  S =  �  d4x√−g4 V  �  d6y√g6  �  k,l  �  − 1  2H(y)gµν(∂µ̺k)(∂ν̺l)(ξkξl)  − 1  2  e2ω(x)  V1/3 H(y)λ2  k ̺k̺l (ξkξl)  �  (35)  =  �  d4x√−g4 V  �  k,l  �  − 1  2gµν(∂µ̺k)(∂ν̺l) ·  �  d6y√g6 · H(y) ξkξl  − 1  2  e2ω(x)  V1/3 λ2  k̺k̺l ·  �  d6y√g6 · H(y) ξkξl  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (36)  Since the eigenmodes ξk satisfy the orthogonality condition  �  d6y√g6 · H(y) ξkξl = c1δkl ,  (37)  the KK modes decouple and the action reduces to  S =  �  k  �  d4x√−g4  �  − 1  2gµν(∂µ̺k)(∂ν̺k)(c1V) + 1  2  e2ω(x)  V1/3 λ2  k̺2  k(c1V)  �  (38)  =  �  k  �  d4x√−g4  �  − 1  2gµν(∂µ̺c  k)(∂ν̺c  k) + 1  2 · λ2  k  V1/3  e2ω0  Vw  (̺c  k)2  �  ,  (39)  where in the second line we canonically normalise the fields ̺k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Therefore, the mass of ̺c  k is  mk = λk  V1/6  �e2ω0  Vw  �1/2  =⇒  mKK = λ1  V1/6  �e2ω0  Vw  �1/2  ,  (40)  where we identify the KK scale, mKK, with the mass of the lightest mode m1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  To determine λk (and the eigenfunctions10) one must solve the eigenvalue equation  1  √g6  ∂m(√g6 gmn∂nξk) + H(y) · λ2  k · ξk = 0 ,  (41)  10These are commonly referred to as the wavefunctions of the modes ̺k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  9  together with appropriate boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' It is therefore not possible to give a fully  generic expression for λk, and thus mKK, as it depends on the details of the compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  If we consider the case of a torus with a single common radius as the prototypical example  of an isotropic compact space11 with characteristic scale ls and constant warp factor12, the  eigenvalue is λ1 = H−1/2  0  (2π) · ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Then the KK scale becomes  mKK =  �e2ω0  Vw  �1/2  H−1/2  0  2π  V1/6ms = H−1/2  0  2π  V1/6 ·  eΦ0  √  4πV1/2  w  MPl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (42)  We should note that generally there are two distinct volumes appearing in mKK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  One  usually takes Vw ≈ V, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' one assumes that the unwarped region of the compact space  dominates the volume integral, in which case we recover the well-known volume suppression  mKK ∼ MPl/V2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' For the convenient choice eΦ0 = gs and e2ω0 = ⟨Vw⟩, and approximating  Vw ≈ V, we have  mKK = H−1/2  0  2π  V1/6 ms = H−1/2  0 2πgs  √  4πV2/3MPl ,  (43)  while for the common alternative choice Φ0 = 0, the factor of gs will be absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  One also often finds in the literature another scale  mw  KK ≡ H−1/4(y0) mloc  KK ,  (44)  where mloc  KK corresponds to a KK scale associated with modes localised on a subspace at  some fixed y = y0 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' the tower of states associated with fields living on the world-volume  of a brane wrapping an internal cycle at the tip of some warped throat).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  It is worth emphasising that volumes measured using a string frame metric may differ from  the ones measured using an Einstein frame metric, depending on the convention used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' on  the choice of Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' To see this, recall that the two metrics are related by GE  MN = e− Φ−Φ0  2  GS  MN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  11More generically one could consider different scales in different directions, which would result in different  KK scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  12This is consistent with our normalisation for the coordinates ym, such that  �  d6y√g6 = l6  s – it corresponds  to an identification of the normalised coordinates ym ∼ ym + 1 (with ds2  6 = l2  s dymdym), rather than  ym ∼ ym + 2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Moreover, with a constant warp factor, H(y) ≡ H0, the eigenfunctions respecting the  (periodic) boundary conditions on the torus would be ξ⃗k ∝ e2πi ⃗k·⃗y, with a vector of integers ⃗k labeling the  modes, and hence ∆6ξ⃗k = −H−1  0  (2π)2k2 · ms · ξ⃗k, giving λ⃗k = H−1/2  0  (2π|⃗k|) · ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' When the warp factor  is not trivial, its functional form will qualitatively change the eigenvalue equation (41) and the solutions λk  will depend, in particular, on the balance between warped and unwarped regions of the compact space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  10  Therefore, a generic d-dimensional volume can be written as  V E  d =  �  ddy  �  gE  d f(y) =  �  ddy  �  gS  d e− d  4 (Φ−Φ0)f(y) ,  (45)  where we allow for some function f(y), such as H(y) in the definition of Vw (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Therefore,  the volumes in the two frames (assuming Φ is stabilised) are related as V S  d =  �  e⟨Φ⟩−Φ0� d  4 V E  d ⇔  VS = e  3  2(⟨Φ⟩−Φ0)VE .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (46)  Hence, it is important to note the convention being used for the Einstein frame metric and  how it relates to quantities that are obtained in either string frame or Einstein frame (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  perturbative and non-perturbative corrections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Finally, it is interesting to note that the ratio mKK/ms is manifestly independent of the  choice for eΦ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' on the convention used in changing from string frame to Einstein frame,  whereas the ratio mKK/MPl is manifestly independent of the choice for eω0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  on the  convention used in going to 4d Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' After taking into account the dependence  of the Einstein frame volume on the frame convention, the ratio mKK/MPl is also actually  independent of the choice for eΦ0, as it must be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Approximating Vw ≈ V and expressing the  ratio in terms of the string-frame volume, the convention-dependent factors fall out:  mKK  MPl  = H−1/2  0 2πgs  √  4πV2/3  S  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (47)  4  Flux scalar potential  The scalar potential for the moduli fields and the dilaton comes from the terms R, |G3|2 and  | ˜F5|2 in the action (9), after dimensional reduction to 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The contributions from the R and  ˜F5 terms can be shown to give (see section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='3 of [20])  1  2κ2  � �  R ⋆ 1 − e2Φ0  4  ˜F5 ∧ ⋆ ˜F5  �  = eΦ0  2κ2  �  d4x√−g4 · e4ω(x)  �  H−1G3 ∧ iG3  2(Imτ) ,  (48)  which we can put together with the G3 ∧ ⋆G3 term to give in total  SE  IIB ⊃ eΦ0  2κ2  �  d4x√−g4 · e4ω(x)  �  H−1  2(Imτ)G3 ∧ (iG3 + ⋆6G3)  (49)  = eΦ0  2κ2  �  d4x√−g4 · e4ω(x)  �  H−1  (Imτ)G+  3 ∧ ⋆6G  +  3 ,  (50)  11  with G+  3 = 1  2(G3 + i ⋆6 G3) such that ⋆6G+  3 = −iG+  3 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Using the metric (19) we can  rewrite this action in terms of gmn,  SE  IIB ⊃  �  d4x√−g4  �  eΦ0  2κ2 e4ω(x)  �  H−1 G+  3 ∧ ⋆g6G  +  3  (Imτ)  �  ≡ −  �  d4x√−g4 V ,  (51)  which defines the 4d scalar potential V as  V = −i eΦ0  2κ2 e4ω(x)  �  H (H−1G+  3 ) ∧ (H−1G  +  3 )  (Imτ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (52)  It is now possible to rewrite this potential in an N = 1 supergravity form, by defining  W = 1  a  �  G3 ∧ Ω ,  (53)  where a is a normalisation constant to be determined below, and using that  �  H (H−1G+  3 ) ∧ (H−1G  +  3 ) =  a2  �  H Ω ∧ ΩGα¯β(DαW)(D ¯βW)  (54)  where α, β run over the complex structure moduli and the axio-dilaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Using this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' the  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='scalar potential becomes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='V = −i eΦ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='2κ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e4ω(x)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='(Im τ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='a2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='H Ω ∧ Ω  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Gi¯\uf6be(DiW)(D¯\uf6beW) − 3|W|2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='= eΦ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='2κ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�e2ω0 · l6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Vw  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='(Im τ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='a2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='H Ω ∧ Ω  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Gi¯\uf6be(DiW)(D¯\uf6beW) − 3|W|2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='2π  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='eΦ0 · l8s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�e2Φ0M2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Pl · l2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='4πVw  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='(Im τ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='a2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='H Ω ∧ Ω  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Gi¯\uf6be(DiW)(D¯\uf6beW) − 3|W|2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='= e3Φ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='4π · l4s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='M4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='a2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='l6s � l6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Vw  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='2(Im τ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='i  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='H Ω ∧ Ω · M2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='� Gi¯\uf6be  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='M2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='(DiW)(D¯\uf6beW) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='|W|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e3Φ0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='eK/M2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Ki¯\uf6be(DiW)(D¯\uf6beW) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='M2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='Pl  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='|W|2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (55)  where now i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' j run over complex structure moduli,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Kähler moduli and the axio-dilaton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' the  Kähler potential K is given by  12  K/M2  Pl = −2 log Vw − log(−i(τ − ¯τ)) − log  � i  l6s  �  H Ω ∧ Ω  �  (56)  and Ki¯\uf6be is the inverse field space metric that follows from Ki¯\uf6be = ∂i∂¯\uf6beK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Note that the volume term in K includes not only the overall volume modulus V, but  also the other Kähler moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' This scalar potential leads to the normalisation  W/M3  Pl =  e  3  2 Φ0  √  4π · l5s  �  G3 ∧ Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (57)  We can see that the normalisation constant, a, is convention-dependent through the choice  of eΦ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  This gives for the gravitino mass  m3/2 = e  K  2M2  Pl |W|  M2  Pl  =  e  1  2⟨Φ⟩  Vw ||Ω||w  e  3  2 Φ0W0  √  8π  MPl ,  (58)  where e  1  2⟨Φ⟩ comes from ⟨Im τ⟩, ||Ω||2  w · l6  s = i  �  H Ω ∧ Ω and we define  W0/M3  Pl ≡  � 1  l5  s  �  G3 ∧ Ω  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (59)  It follows from (58) and (42) that the important13 ratio (assuming the bulk dominates  all the integrals, so that Vw ≈ V and ||Ω||w ≈ ||Ω||)  m3/2  mKK  = H1/2  0  e  1  2(⟨Φ⟩+Φ0)  V1/3  E  W0  √  2(2π)||Ω|| ,  (60)  where we highlight the fact that the volume being used is the Einstein frame volume, VE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Note that this mass ratio, as written in terms of the Einstein frame volume, seems to depend  on the convention used for the 10d change of frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' the choice of Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' It is however  convention-independent, as it must be, since the Einstein frame volume also depends on the  choice of Φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' If we express the mass ratio in terms of the string frame volume instead, which  13Not only is this ratio important because a consistent 4d supergravity description requires that the  gravitino remains in the theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' its mass is not above the EFT cutoff – typically mKK – and therefore  integrated out, but it was shown that it also serves as a control parameter for certain corrections to the  scalar potential, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' from higher F-terms [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  13  corresponds to the volume perceived by the string itself, using (46) we find  m3/2  mKK  = H1/2  0  e⟨Φ⟩  V1/3  S  W0  √  2(2π)||Ω|| ,  (61)  which is manifestly independent of conventions14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  5  Corrections to the scalar potential  Since the flux superpotential leaves all Kähler moduli unstabilised, either leaving them as  flat directions or generating runaways, one must resort to higher-order corrections to the  EFT in order to stabilise them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Both perturbative and non-perturbative corrections have  been considered in the literature — while the former are computed at the level of the 10d  EFT and in string frame, the latter are obtained directly at the level of the 4d EFT and are  computed in Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' One must therefore be careful with the conventions being used  to change frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' the choice of Φ0, in order to remain consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='1  Perturbative corrections  In [22], it was shown that α′-corrections to the Type IIB effective action (9) manifest as  corrections to the 4d volume modulus Kähler potential and spoil the no-scale structure  of its scalar potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' These corrections arise from higher-derivative terms at order (α′)3  appearing in the type IIB effective action,  SS  IIB =  1  2κ2  10  �  d10x  �  −GS e−2Φ�  RS + 4(∂Φ)2  S + (α′)3 · ζ(3)  3 · 211 · J0  �  ,  (62)  where the higher-order term is schematically given by  J0 ∼ (RMNP Q)4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (63)  One must also add a term  δSS  Φ ∼  �  d10x  �  −GSe−2Φ(α′)3(∇2Φ) Q ,  (64)  14Note that we give mass ratios for canonically normalised fields defined in the Einstein frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Whilst  these mass ratios must be invariant under change of conventions, a change in frame would come with field  redefinitions, and new masses and couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' In a setup in which all couplings, including the gravitational  coupling, are constant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' assuming that the dilaton and volume modulus are stabilised and integrated  out), the change of frames becomes a change of convention from one Einstein frame to another Einstein  frame, and the mass ratios would be invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  14  where Q ∼ (RMNP Q)3 is a generalisation of the 6d Euler integrand  �  X6 d6y√g6 Q = χ, with  χ the Euler characteristic of X6 [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' This term corrects the 10d solution to the equation  of motion for Φ, such that Φ = Φ10 + ζ(3)  16 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' It is then shown in [22] that this leads to a  correction to the Kähler potential of the form  K = −2 log  �  VS + ξ  2  �  = −2 log  �  VE e  3  2(Φ−Φ0) + ξ  2  �  (65)  = −2 log  �  VE + ξ  2e− 3  2(Φ−Φ0)�  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' ,  (66)  where we have used (46) with d = 6, keeping Φ0 unspecified, and ξ is defined as15  ξ = − ζ(3)χ  2(2π)3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (67)  Note in particular that the correction expressed in Einstein frame depends on the convention  one chooses for Φ0,  Φ0 = 0 =⇒ K = −2 log  �  V +  ξ  2g3/2  s  �  ,  (68)  Φ0 = ⟨Φ⟩ =⇒ K = −2 log  �  V + ξ  2  �  ,  (69)  where we have assumed as usual that the dilaton has been stabilised by fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='2  Non-perturbative corrections  Although the superpotential W does not receive perturbative corrections, it may receive non-  perturbative corrections from either instantons arising from Euclidean D3-branes wrapping  4-cycles or gaugino condensation on the world-volume theory of D7-branes wrapped around  internal 4-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Let us consider the latter case in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' In what follows, Tp =  2π  lp+1  s  is the brane tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The DBI action for a Dp-brane, in the string frame, is given by [6, 7]  SDBI  Dp = −Tp  �  dp+1σ e−Φ  �  − det  �  gS + B + l2  s  2πF  �  ,  (70)  15In [22], we find the definition ξ = − ζ(3)χ(X6)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' The missing factor of (2π)3 comes from their conventions  for the volume, V[22] = V6/(2πα′)3, whereas we are using V = V6/l6  s = (2π)−3 V6/(2πα′)3, with the convention  (2π)2α′ = l2  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' There are also instances in the literature where the factor of 1/2 is absorbed into the definition  of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  15  where gS and B refer to the pull-back of the string frame metric (GS)MN and 2-form BMN  onto the world-volume of the brane and F to the field-strength Fab of the brane gauge fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Rewriting the action in terms of the Einstein frame metric,  SDBI  Dp = −Tp  �  dp+1σ e−Φ�  − det gS  �  det  �  1 + (gS)−1  �  B + l2  s  2πF  ��  (71)  ⊃ −Tp  �  dp+1σ e−Φ�  − det gS 1  4  � l2  s  2π  �2  (gS)ac(gS)bdFabFcd  (72)  = −Tp  4  l4  s  (2π)2  �  dp+1σ  �  − det gE e  p−3  4 (Φ−Φ0)e−ΦFabF ab ,  (73)  where the indices in FabF ab are contracted with Einstein frame metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' This is the kinetic  term for the brane gauge bosons (see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' of [23]) and tells us the gauge coupling  of the corresponding theory, which is a key parameter for gaugino condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' If the brane  is wrapping a (p − 3)–cycle Σp−3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' we find the corresponding 4d term (assuming that Φ is  constant over the cycle)  S4d  Dp ⊃ −  1  8πlp−3  s  �  d4x  �  − det gE  4 e  p−3  4 (Φ−Φ0)e−Φ  ��  dp−3σ  �  gE  p−3  �  �  ��  �  τ E  Σp−3lp−3  s  FabF ab  (74)  = −  �  d4x  �  − det gE  4  �  τ E  Σp−3  8πeΦ e  p−3  4 (Φ−Φ0)  �  FabF ab ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (75)  and we can read off the gauge coupling gc,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  1  g2  c  =  τ E  Σp−3  4πe⟨Φ⟩ e  p−3  4 (⟨Φ⟩−Φ0) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (76)  where we have assumed as usual that the dilaton has been stabilised by fluxes at some  higher scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Gaugino condensation on the world-volume theory of D7-branes will then give  a non-perturbative contribution to the superpotential [24],16  Wnp ∼ e  − 8π2  g2c  1  N = e− 2π  N  τE  gs e⟨Φ⟩−Φ0 ,  (77)  16Here N is the number of branes stacked on top of each other, responsible for the gauge group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' It appears  through the beta-function coefficient [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  16  where we used e⟨Φ⟩ = gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Holomorphicity of W then leads to the general contribution  Wnp =  �  i  Aiei ai  gs e⟨Φ⟩−Φ0T E  i  ,  (78)  where the sum is over the contributing cycles, ai = 2π  Ni and the fields Ti = bi + iτi are the  complexified Kähler moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Hence, we can compare the two most common conventions for  Φ0,  Φ0 = 0 =⇒ Wnp =  �  i  AieiaiT E  i ,  (79)  Φ0 = ⟨Φ⟩ =⇒ Wnp =  �  i  Aiei ai  gs T E  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  (80)  As for the mass ratio m3/2/mKK, the superpotential as written, in terms of Einstein frame  4-cycle volumes, appears to depend on the choice of convention for Φ0, but one should recall  that the 4-cycle volumes also depend on this choice of convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' If we express the 4-cycle  volumes in terms of the string frame metric (46),  τ S  i = e⟨Φ⟩−Φ0τ E  i ,  (81)  the convention-independence becomes manifest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  Acknowledgements  IZ is partially supported by STFC, grant ST/P00055X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content='  References  [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Faraoni and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Gunzig, Einstein frame or Jordan frame?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=', Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Z9E4T4oBgHgl3EQfnw0i/content/2301.05178v1.pdf'}
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+Astronomy & Astrophysics manuscript no. main
+©ESO 2023
+January 30, 2023
+Magnetic reconnection plasmoid model for Sagittarius A* flares
+N. Aimar1, A. Dmytriiev2, F. H. Vincent1, I. El Mellah3, T. Paumard1, G. Perrin1, and A. Zech4
+1 LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, 5 place Jules Janssen, 92195
+Meudon, France
+e-mail: nicolas.aimar@obspm.fr
+2 Centre for Space Research, North-West University, Potchefstroom, 2531, South Africa
+3 Univ. Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
+4 LUTH, Observatoire de Paris, CNRS, Université Paris Diderot, 5 place Jules Janssen, 92190 Meudon, France
+Received September 09, 2022; accepted January 27, 2023
+ABSTRACT
+Context. Sagittarius A*, the supermassive black hole at the center of our galaxy, exhibits episodic near-infrared flares. The recent
+monitoring of three such events by the GRAVITY instrument has shown that some flares are associated with orbital motions in the
+close environment of the black hole. The GRAVITY data analysis points at super-Keplerian azimuthal velocity, while (sub-)Keplerian
+velocity is expected for the hot flow surrounding the black hole.
+Aims. We develop a semi-analytic model of Sagittarius A* flares based on an ejected large plasmoid, inspired by recent particle-in-
+cell global simulations of black hole magnetospheres. We model the infrared astrometric and photometric signatures associated to this
+model.
+Methods. We consider a spherical macroscopic hot plasma region, that we call a large plasmoid. This structure is ejected along a
+conical orbit in the vicinity of the black hole. This plasmoid is assumed to be formed by successive mergers of smaller plasmoids
+produced through magnetic reconnection that we do not model. Non-thermal electrons are injected in the plasmoid. We compute the
+evolution of the electron-distribution function under the influence of synchrotron cooling. We solve the radiative transfer problem
+associated to this scenario and transport the radiation along null geodesics of the Schwarzschild spacetime. We also take into account
+the quiescent radiation of the accretion flow, on top of which the flare evolves.
+Results. For the first time, we successfully account for the astrometric and flux variations of the GRAVITY data with a flare model
+that incorporates an explicit modeling of the emission mechanism. We find good agreement between the prediction of our model and
+the recent data. In particular, the azimuthal velocity of the plasmoid is set by the magnetic field line it belongs to, which is anchored in
+the inner parts of the accretion flow, hence the super-Keplerian motion. The astrometric track is also shifted with respect to the center
+of mass due to the quiescent radiation, in agreement with the difference measured with the GRAVITY data.
+Conclusions. These results support the picture of magnetic reconnection in a black hole magnetosphere as a viable model for Sagit-
+tarius A* infrared flares.
+Key words. Accretion, accretion disk - Magnetic reconnection - Black hole physics - Relativistic processes - Radiative transfer -
+Radiation mechanisms: non-thermal
+1. Introduction
+The Galactic Center hosts the compact radio source Sagittar-
+ius A* (Sgr A*) with an estimated mass of 4.297 million solar
+masses at a distance of only 8.277 kpc (GRAVITY Collaboration
+et al. 2022). This makes the compact object associated to Sgr A*
+the closest supermassive black hole (SMBH) candidate to Earth.
+Sgr A* is a low-luminosity accretion flow with an accretion rate
+of (5.2 − 9.5) × 10−9M⊙ yr−1 and a bolometric luminosity of
+(6.8 − 9.2) × 1035 erg s−1 (Bower et al. 2019; Event Horizon
+Telescope Collaboration et al. 2022b) and thus is accreting at a
+very sub-Eddington rate. It has been the subject of numerous ob-
+serving campaigns over the past two decades in order to test the
+massive black hole (MBH) paradigm (see Gravity Collaboration
+et al. (2020b)) and study the physics of radiatively inefficient ac-
+cretion flows (RIAF) around SMBH.
+Sgr A* shows a slow and low amplitude variability in ra-
+dio (Lo et al. 1975; Backer 1978; Krichbaum et al. 1998; Falcke
+1999; Bower et al. 2006; Michail et al. 2021b), in millimetre
+and submillimetre (Mauerhan et al. 2005; Macquart et al. 2006;
+Yusef-Zadeh et al. 2006; Marrone et al. 2008; Brinkerink et al.
+2015; Wielgus et al. 2022a), but also large amplitude and rapid
+variability in near infrared (NIR; Genzel et al. 2003; Ghez et al.
+2004; Hornstein et al. 2007; Hora et al. 2014) and in X-rays
+(Baganoff et al. 2001; Nowak et al. 2012; Neilsen et al. 2013;
+Barrière et al. 2014; Ponti et al. 2015). The flux distribution in
+the NIR of Sgr A* has been the subject of numerous studies.
+Some claim a single state modeled by rednoise (Witzel et al.
+2018; Do et al. 2019) for the variability of Sgr A* while others
+claim that there are two states for Sgr A* (Genzel et al. 2003;
+Dodds-Eden et al. 2011; Gravity Collaboration et al. 2020a;
+Witzel et al. 2021): a continuously low amplitude variable state
+called "quiescent state" and the "flare state" described by short
+and bright flux with a typical timescale of 30 minutes to 1 hour
+with a rate of ∼ 4 a day. Multi-wavelength studies show that
+when an X-ray flare is observed, there is a counterpart in NIR
+suggesting a common origin but the reverse is not true (Fazio
+et al. 2018). Moreover, the flare can also be observed in sub-
+mm but with a time lag of several minutes (Eckart et al. 2008,
+2009; Dodds-Eden et al. 2009; Michail et al. 2021a; Witzel et al.
+Article number, page 1 of 20
+arXiv:2301.11874v1  [astro-ph.HE]  27 Jan 2023
+
+A&A proofs: manuscript no. main
+2021) following a dimming (Wielgus et al. 2022a; Ripperda et al.
+2022).
+Recently, the GRAVITY instrument (Gravity Collaboration
+et al. 2017; Eisenhauer et al. 2008, 2011; Paumard et al. 2008)
+was able to resolve the motion of the NIR centroid during three
+bright flare events, showing a clockwise, continuous rotation at
+low inclination close to face-on (i ∼ 20◦) consistent with a re-
+gion of emission located at a few gravitational radii rg = GM/c2
+from the central black hole (Gravity Collaboration et al. 2018).
+These flares are thus powered very close to the event horizon
+of the black hole. The exploration of a relativistic accretion re-
+gion as close to the event horizon with high-precision astrometry
+and imaging techniques like GRAVITY and the Event Horizon
+Telescope (EHT) (Event Horizon Telescope Collaboration et al.
+2022a) promises important information for physics and astron-
+omy, including new tests of the MBH paradigm.
+Significant efforts have been made to explain the flares of
+Sgr A*: rednoise (Do et al. 2009), hot spot (Hamaus et al. 2009;
+Genzel et al. 2003; Broderick & Loeb 2006), ejected blob (Vin-
+cent et al. 2014), star-disk interaction (Nayakshin et al. 2004)
+and disk instability (Tagger & Melia 2006). The GRAVITY
+observations in 2018 (Gravity Collaboration et al. 2018) sup-
+port the hot spot model. However, the physical origin of such
+hot spots remains an open question. Instabilities in black hole
+accretion disks are a candidate, for instance the triggering of
+Rossby Waves Instabilities (RWI; Tagger & Melia 2006; Vin-
+cent et al. 2014). Alternatively, it could originate from the dis-
+sipation of electromagnetic energy through magnetic reconnec-
+tion. This modification of the magnetic field topology results
+from the inversion of the magnetic field orientation across a cur-
+rent sheet which eventually breaks into magnetic islands called
+plasmoids (Komissarov 2004, 2005; Komissarov & McKinney
+2007; Loureiro et al. 2007; Sironi & Spitkovsky 2014; Parfrey
+et al. 2019; Ripperda et al. 2020; Porth et al. 2021). In the past
+years, numerical simulations have repeatedly highlighted the
+ubiquity of magnetic reconnection in BH magnetospheres, what-
+ever the physical point of view adopted: global particle-in-cell
+(PIC) simulations in Kerr metrics (El Mellah et al. 2022; ?), re-
+sistive general-relativistic magneto-hydrodynamics (GRMHD)
+simulations (Ripperda et al. 2020; Dexter et al. 2020a,b) or resis-
+tive force-free simulations (Parfrey et al. 2015). PIC simulations
+show that magnetic reconnection in the collisionless corona of
+spinning BHs can accelerate leptons up to relativistic Lorentz
+factors of γ ∼ 103...7 (El Mellah et al. 2022), sufficiently high
+to generate the variable IR (and X-ray) emission (Rowan et al.
+2017; Werner et al. 2018; Ball et al. 2018; Zhang et al. 2021;
+Scepi et al. 2022).
+The GRMHD and PIC frameworks each have different lim-
+itations. GRMHD simulations describe the evolution of the ac-
+cretion flow over long time scales, typically of the order of sev-
+eral 100,000 rg/c, but they rely on a fluid representation. Conse-
+quently, they cannot self-consistently capture the kinetic effects
+which are important to constrain dissipation, particle accelera-
+tion and subsequent non-thermal radiation. On the other hand,
+PIC simulations provide an accurate description of the micro-
+physics but at the cost of simulations which can only span a few
+100rg/c in time and with limited scale separation between global
+scales and plasma scales.
+We develop a semi-analytical model, fed by the knowledge
+accumulated by recent GRMHD and GRPIC simulations. The
+aim is to condense into a reasonably small set of simple param-
+eters the complex physics of GRMHD and GRPIC models, and
+thus allow to probe a large parameter space within a reasonable
+computing time. We also want to remain as agnostic as possi-
+ble regarding the initial conditions of the flow. In this context,
+we discuss the interpretation of the Gravity Collaboration et al.
+(2018) flare data paying particular attention to the following di-
+agnostics:
+– the marginally detected shift between the astrometric data
+and the center-of-mass location;
+– the tension between the data and the hot spot model used
+by Gravity Collaboration et al. (2018), which assumes a Ke-
+plerian orbit;
+– the physical origin of the rising and decaying phases of the
+flare light curve in the context of magnetic reconnection.
+The first point can be discussed in the context of a very sim-
+ple hot-spot model and is the main topic of Sect. 2. Sect. 3 is
+the core of our study and focuses on the second and third points
+above. It presents a semi-analytical large plasmoid model due to
+magnetic reconnection. It highlights in particular the impact of
+considering a self-consistent evolution of the electron distribu-
+tion function through kinetic modeling. This section shows that
+our plasmoid model is able to reasonably account for the Grav-
+ity Collaboration et al. (2018) flare data. The limitations of our
+plasmoid model are discussed in Sect. 4. The conclusions and
+perspectives are given in Sect. 5.
+2. Quiescent flow impact on astrometry: shifting
+and rotating the orbit
+Gravity Collaboration et al. (2018) used a hot spot model in an
+equatorial circular orbit to fit the astrometry of three bright flares.
+They considered a constant radiation flux from the emitting re-
+gion orbiting the black hole to fit the orbital motion. The effect
+of out-of-plane motion and orbital shear have also been stud-
+ied by Gravity Collaboration et al. (2020c) to model the flares.
+However, the impact of the quiescent radiation surrounding the
+hot spot was not taken into account. The aim of this section is to
+show that taking into account the quiescent radiation can lead to
+shifting and rotating the orbit on sky. We note a 1-σ difference
+between the center of the orbit of the hot spot and the center of
+mass derived from the orbit of S2 in Gravity Collaboration et al.
+(2018) which makes this shift marginal.
+In this section, we will use a simplified hot spot model that
+is sufficient to highlight the main effects of the quiescent radia-
+tion. This simple model will also allow us to introduce the most
+important relativistic effects at play, that were already studied in
+many previous works (Broderick & Loeb 2006; Hamaus et al.
+2009). These reminders will be helpful when we turn to a more
+complex hot spot model in the Sect. 3, which is the main aim of
+this paper.
+2.1. Simple hot spot + quiescent model for the flaring Sgr A*
+The quiescent radiation of Sgr A* is modeled by means of the
+torus-jet model as derived in Vincent et al. (2019), to which we
+refer for all details. Figure 1 resumes the main features of the
+model. The torus emits thermal synchrotron radiation, while the
+flux emitted by the jet follows a κ distribution (i.e. a thermal
+core with a power-law tail). The multi-wavelength spectrum of
+the quiescent Sgr A* is well fit with this model. The κ distribu-
+tion emission from the jets dominates at most wavelength except
+at the sub-mm bump where the flux comes mostly from the ther-
+mal disk. We summarize the best-fit parameters in Table 1, and
+the resulting best-fit quiescent spectrum is given in Fig. 2. More
+details on the fitting procedure are given in Appendix A. With
+Article number, page 2 of 20
+
+N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+Fig. 1. Scheme of the torus-jet model for the quiescent state in blue and flares in red. Two trajectories are considered for the flare, which can either
+rotate in the torus (hot spot model) or be ejected along the jet sheath (plasmoid model). The jet is parametrized by the angles θ1 and θ2 that describe
+the angular opening of the radiation-emitting sheath, by the base height zb, the constant Lorentz factor Γ j, and the temperature power-law index
+sT. The jet is symmetrical with respect to the equatorial plane, and axisymmetric.
+these parameters, the flux of the torus-jet model at 2.2 µm is 1.1
+mJy. It is in perfect agreement with the median quiescent dered-
+dened flux provided by Gravity Collaboration et al. (2020a) of
+1.1 ± 0.3 mJy. At this wavelength, the torus is optically thin and
+its emission is negligible compared to the jet. In the remainder
+of this paper, where we focus only on the infrared band, we will
+thus neglect the torus and consider a pure jet quiescent model,
+unless otherwise noted.
+The only relevant features of our quiescent model for the
+rest of this paper are the location of its infrared centroid and
+its NIR flux. As depicted in the right panel of Fig. 2, the centroid
+of our jet-dominated model lies very close to the mass center.
+We have checked that considering a disk-dominated model only
+very marginally changes the position of the quiescent centroid
+at low inclination (see the blue and green dots in the left panel
+of Fig. 3). Our conclusions are thus not biased by our particular
+choice of a jet-dominated quiescent model.
+The hot spot model is composed of a plasma sphere of radius
+1 rg (fixed) with a uniform but time-dependent κ-distribution for
+the electrons. The emissivity jν and absorptivity αν coefficients
+depend on the density, temperature, and magnetic field which we
+considered uniform. We use the fitting formula of Pandya et al.
+(2016) to compute these coefficients. The typical light curve of
+a flare is characterized by a phase with increasing flux and one
+with decreasing flux. We model this behavior by a Gaussian time
+modulation on the density and temperature as follows
+ne(t) = nhs
+e exp
+�������−0.5 ×
+�t − tre f
+tσ
+�2������� ,
+(1)
+Te(t) = T hs
+e exp
+�������−0.5 ×
+�t − tre f
+tσ
+�2�������
+(2)
+where tσ is the typical duration of the flare. As ne varies over
+time (Eq. 1), the magnetic field strength also varies since we set
+a constant magnetization σ = B2/4πmpc2ne.
+Contrary to Gravity Collaboration et al. (2020c), we keep the
+circular equatorial orbit of Gravity Collaboration et al. (2018) as
+we assume that the hot spot is formed in the equatorial plane and
+we do not take into account any shearing effect and assume a
+constant spherical geometry of the hot spot. We summarize all
+the input parameters of the hot spot in Table 2.
+2.2. Shifting the orbit on sky
+Figure 3 shows the impact of taking into account the quiescent
+radiation on the astrometry of the flare, considering the trivial
+Article number, page 3 of 20
+
+20
+Jet
+sheath
+[j
+Te(z) α (Z/z)
+01
+S-
+10
+EjectedPlasmoid
+(aun w)
+Torus
+N
+Orbiting Hot spot
+-10
+-20
+0
+5
+10
+15
+20
+X (M unit)A&A proofs: manuscript no. main
+1010
+1012
+1014
+1016
+1018
+ [Hz]
+1032
+1033
+1034
+1035
+L  [erg/s]
+150
+100
+50
+0
+50
+100
+150
+X [ as]
+150
+100
+50
+0
+50
+100
+150
+Y [ as]
+10
+27
+10
+26
+10
+25
+10
+24
+10
+23
+I  [erg cm
+2 s
+1 sr
+1 Hz
+1]
+Fig. 2. Left: Spectrum associated to the best-fit of the torus-jet model (see Table 1) for the quiescent state of Sgr A* (χ2
+red = 0.91 with ndof=27).
+The data are taken from Bower et al. (2015) for ν < 50 GHz, Brinkerink et al. (2015) for the 2 points around 100 GHz, Liu et al. (2016) for the
+492 GHz point, Marrone et al. (2006) for the 690 GHz point, von Fellenberg et al. (2018) for the far infrared upper limits, Witzel et al. (2018) for
+the mid infrared data, and Baganoff et al. (2001) for the X-ray bow-tie. We note that as in Vincent et al. (2019), the X-ray data are not fitted as we
+do not take into account bremsstrahlung nor Comptonized emission. Right: Best-fit image at 2.2 µm of the torus-jet model with a field of view
+of 150 µas seen with an inclination of 20◦ and a Position Angle of the Line of Nodes (PALN) of π rad. The color bar gives the values of specific
+intensity in cgs units in log-scale. The outer region emission comes from the backward jet’s part while the emission close to the center comes from
+the forward part of the jet. The centroid of the jet is represented by the blue dot at ∼(0, −2.2).
+Parameter
+Symbol
+Value
+Black Hole
+mass [M⊙]
+M
+4.297 × 106
+distance [kpc]
+d
+8.277
+spin
+a
+0
+inclination [◦]
+i
+20
+Torus
+angular momentum [rg/c]
+l
+4
+inner radius [rg]
+rin
+8
+polytropic index
+k
+5/3
+central density [cm−3]
+nT
+e
+1.2 × 109
+central temperature [K]
+T T
+e
+7 × 109
+magnetization parameter
+σT
+0.002
+Jet
+inner opening angle [◦]
+θ1
+20
+outer opening angle [◦]
+θ2
+θ1 + 3.5
+jet base height [rg]
+zb
+2
+bulk Lorentz factor
+Γ j
+1.15
+base number density [cm−3]
+nJ
+e
+3.5 × 106
+base temperature [K]
+T J
+e
+3 × 1010
+temperature slope
+sT
+0.21
+κ index
+κJ
+5.5
+magnetization parameter
+σJ
+(fixed) 1
+Table 1. Best fit parameters of the torus+jet quiescent model. We keep
+the same geometrical parameters, bulk Lorentz factor and κ-index as
+Vincent et al. (2019) and we fit the base number density, base temper-
+ature and temperature slope of the jet considering the correction (see
+bellow) and the new value of the jet magnetization parameter. The pa-
+rameters of the torus are unchanged.
+case of a constant-emission hot spot, as well as the varying-
+emission hot spot introduced in section 2.1.
+Obviously, whether or not the hot spot intrinsic emission
+varies, the first effect of adding a quiescent radiation is to shrink
+the orbit’s size, because the overall centroid is moved towards
+the quiescent radiation’s centroid, which always lies close to the
+mass center.
+A slightly less obvious effect is that, when the hot spot emis-
+sion varies in time, the orbit can shift in the plane of sky, and no
+longer be centered at the center-of-mass location. This is clearly
+apparent on the solid-red orbit of the left panel of Fig. 3. This is
+simply due to the time variation of the intensity ratio between the
+quiescent and the hot spot radiation. At early and late times, the
+hot spot has a weaker emission than the quiescent component,
+and the overall centroid coincides with the quiescent centroid.
+As the hot spot emission increases and dominates, the overall
+centroid will be driven towards it. Such a shift between the as-
+trometric data and the center-of-mass position is visible at 1-σ
+significance in the the Gravity Collaboration et al. (2018) data.
+We note another non-trivial effect appearing in the varying-
+emission hot-spot orbit without any quiescent radiation (red-
+dotted orbit in Fig. 3). The orbit is not closing, due to the time
+delay between the primary and secondary images. Indeed, at the
+end of the simulation, the flux from the secondary image is in-
+trinsically higher than the primary (the emission times of the pri-
+mary and secondary are different), and is amplified by the beam-
+ing effect. When the centroid is computed, the secondary image
+has a larger impact at this time than before, resulting in a closer
+centroid position relative to the black hole. This astrometric im-
+pact of the secondary image was already discussed by Hamaus
+et al. (2009).
+2.3. Rotating the orbit on sky
+It is not an easy task to disentangle the intrinsic time variability
+of the hot spot from the variability due to the relativistic beam-
+ing effect. Figure 4 illustrates the impact on astrometry and light
+curve of playing with the relative influence between the intrin-
+sic and beaming-related variability. Here, we simply change the
+initial azimuthal coordinate ϕ0 of the hot spot along its orbit,
+in order to change the dephasing between the time of the max-
+imum intrinsic emission (t = tref) which is fixed, and the time
+Article number, page 4 of 20
+
+N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+100
+75
+50
+25
+0
+25
+50
+75
+100
+X ( as)
+100
+75
+50
+25
+0
+25
+50
+75
+100
+Y ( as)
+Astrometry
+constant emission
+Gaussian emission
+no quiescent
+with quiescent
+0
+10
+20
+30
+40
+50
+60
+t (min)
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+F/F(S2)
+Light Curve
+Fig. 3. Astrometry (left) and light curves (right) of the hot spot - jet model with two values for the quiescent state corresponding to no quiescent
+(dashed lines) and the with quiescent state (full lines). In shade of blue, the hot spot has a nearly constant emission (tσ >> torbit). The effect of
+beaming is reflected in the light curves. In shade of red, the hot spot has a Gaussian time emission with tσ = 30 min. The parameters of the hot
+spot are listed in Table 2. We synchronise the maximum of beaming and the intrinsic maximum of the Gaussian modulation. The black, blue and
+green dots in the left panels represent the position of Sgr A*, the jet’s centroid and the disk’s centroid respectively.
+Parameter
+Symbol
+Value
+Hot spot
+number density max [cm−3]
+nhs
+e
+1.05 × 107
+temperature max [K]
+T hs
+e
+9.03 × 1010
+time Gaussian sigma [min]
+tσ
+30
+magnetization parameter
+σhs
+0.01
+κ-distribution index
+κhs
+5
+orbital radius [rg]
+Rhs
+9
+initial azimuth angle [◦]
+ϕhs
+0
+90
+Position Angle of the Line of Nodes [◦]
+Ω
+160
+Table 2. Summary of parameters of the hot spot model. We note that we
+used the maximum number density and temperature of the jet best-fit in
+Table 1 as reference and scale them for the hot spot by a factor 3.01.
+of the maximum constructive beaming effect (when the hot spot
+moves towards the observer). The orbit rotates around the quies-
+cent centroid following the variation of ϕ0 (left panel of Fig. 4).
+The light curve is also strongly affected, reaching much brighter
+levels when the intrinsic emission maximum is in phase with the
+constructive beaming effect.
+Here we show that the quiescent state of Sgr A* can have
+significant impact on the observed astrometry by shrinking the
+apparent orbit, creating a shift between the center of the latter
+and the position of the mass center. One should have these effects
+in mind for the comparison to the flare data at the end of the
+following section.
+3. Plasmoid model from magnetic reconnection
+In this section, we develop a semi-analytical hot-spot-like model
+in order to interpret the rise and decay of Sgr A* flares, thus
+going one step further with respect to the model we used in sec-
+tion 2, where a Gaussian modulation of the emission is enforced
+without physical motivation.
+Black hole magnetospheres naturally lead to the develop-
+ment of equatorial current sheets corresponding to a strong spa-
+tial gradient of the magnetic field which changes sign at the
+equator (Komissarov 2004; Komissarov & McKinney 2007; Par-
+frey et al. 2019; Ripperda et al. 2020). Such a configuration
+results in magnetic reconnection, i.e. a change of the topology
+of the field lines forming X points (Komissarov 2005; Loureiro
+et al. 2007; Sironi & Spitkovsky 2014). This process is intrin-
+sically non-ideal and thus can only be captured either by re-
+sistive MHD or kinetic simulations. For suitable values of the
+magnetic diffusivity, the reconnecting current sheet can break
+into chains of plasmoids, i.e. magnetic islands separated by X
+points (Loureiro et al. 2007; Parfrey et al. 2019; Ripperda et al.
+2020; Porth et al. 2021).
+The reconnection rate (i.e. the typical rate at which magnetic
+energy is dissipated into particle kinetic energy) is equal to the
+ratio vrec/vout with vrec the velocity of matter injected into the
+reconnection region, and vout the bulk outflow velocity of parti-
+cles accelerated by the reconnection event. The outflow velocity
+is of the order of the Alfven speed, vout ≈ vA, which is itself
+of the order of the speed of light, vA ≈ c, for strongly magne-
+tized environments. The reconnection rate has been shown to be
+rather independent of the details of the chosen parameters. For
+PIC simulations, it lies around 10%, i.e. vrec,PIC ≈ 0.1vA ≈ 0.1c,
+for magnetized collisionless plasmas (Sironi & Spitkovsky 2014;
+Werner et al. 2018; Guo et al. 2015), which are the typical
+conditions in the inner flow surrounding Sgr A*1. GRRMHD
+simulations point towards a slower rate of around 1%, so that
+vrec,MHD ≈ 0.01c (see the discussion in Ripperda et al. 2022),
+1 It is likely that the accretion flow surrounding Sgr A* is in a Magnet-
+ically Arrested Disk (MAD, see Narayan et al. 2003) regime, i.e. with
+strong poloidal magnetic fields in the inner regions (Gravity Collabora-
+tion et al. 2018; Dexter et al. 2020b).
+Article number, page 5 of 20
+
+A&A proofs: manuscript no. main
+100
+75
+50
+25
+0
+25
+50
+75
+100
+X ( as)
+100
+75
+50
+25
+0
+25
+50
+75
+100
+Y ( as)
+Astrometry
+orbit
+0
+5
+10
+15
+20
+25
+30
+t (min)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+F/F(S2)
+Light Curve
+0=0.0
+0=90.0
+0=180.0
+0=270.0
+intrinsic
+Fig. 4. Astrometry (left) and light curves (right) of the hot spot - jet model for four initial azimuthal angle ϕ0 of 0◦ in blue, 90◦ in orange, 180◦
+in green and 270◦ in red. The dashed black line shows the primary image centroid track with no quiescent jet (clock-wise). The jet dominates the
+beginning and the end of the flares. The observed centroids thus start and end close to the one of the jet. The apparent orbits rotate around the latter
+with ϕ0 as the maximum of emission occurs at different ϕ. The Gaussian modulation which has a typical duration of tσ = 15 min (grey dashed line;
+which is the same for the four the curves) is affected by relativistic effects. For negative X (right part of the astrometry), the beaming, combined
+with relativistic Doppler effect, amplify the flux from the hot spot while in the positive X (left part of the figure), they lower it. The black dot in
+the left panels represents the position of Sgr A*.
+but this applies to collisional environments, thus less similar to
+Sgr A* vicinity.
+Fresh plasma flows into the current sheet at the reconnec-
+tion rate vrec, is accelerated by the electric field generated in
+the current sheet, usually giving rise to power-law energy dis-
+tributions of electrons (Sironi & Spitkovsky 2014; Werner et al.
+2018). Inside the current sheet, the particles get trapped in the
+plasmoids which act as particle reservoirs (Sironi & Spitkovsky
+2014) which can merge in a macroscopic magnetic island, that
+is, a large plasmoid. In Ripperda et al. (2022), magnetic flux dis-
+sipation through reconnection last for ∼ 100rg/c ∼ 30 min and
+the resulting hot spot orbits for ∼ 500rg/c ∼ 150 min before it
+disappears by losing its coherence through interaction with the
+surrounding flow.
+In the global PIC simulation of El Mellah et al. (2022), the
+authors study magnetic reconnection in the sheath of relativis-
+tic jet working with magnetic field loops coupling the BH to the
+accretion disk. The resulting plasmoids evolve off-plane, propa-
+gate away from the BH and are prone to merge with each other to
+form macroscopic plasmoids susceptible to radiate high amounts
+of energy under the form of non-thermal radiation. The under-
+lying mechanism, first described by Uzdensky (2005) and de
+Gouveia dal Pino & Lazarian (2005), relies on the accretion of
+poloidal magnetic field loops onto a spinning BH. Once the in-
+ner footpoint of the loop reaches the BH ergosphere, the mag-
+netic field line experiences strong torques due to the frame drag-
+ging effect while its other footpoint on the disk rotates at the
+local Keplerian speed. Thus, the toroidal component of the mag-
+netic field quickly grows in the innermost regions, propagates
+upstream along the field line and leads to the opening of the
+magnetic loop above a certain magnetic loop size. On the out-
+ermost closed magnetic field line (called the separatrix), a Y-
+point appears where plasmoids form and flow away along an
+inclined current sheet above the disk (Fig. 5). In the PIC sim-
+ulations of El Mellah et al. (2022), a cone-shaped reconnecting
+current sheet forms where vivid particle acceleration takes place.
+Electrons and positrons pile up into outflowing plasmoids where
+they cool through synchrotron radiation. This topological con-
+figuration where some magnetic field lines anchored in the disk
+close within the event horizon is coherent with what is seen in
+resistive GRMHD simulations during the short episodes of flux
+repulsion which separates different accretion regimes. During
+approximately 100 rg/c, these simulations show an essentially
+force-free funnel surrounded by merging plasmoids formed in
+the jet sheath and in the equatorial plane Ripperda et al. (2022);
+Chashkina et al. (2021).
+3.1. Plasmoid model from magnetic reconnection
+The aim of this section is to develop a semi-analytic large plas-
+moid model (which will be named plasmoid model), inspired
+from the reconnection literature reviewed above. The interest of
+such a model, compared to state-of-the-art GRMHD or GRPIC
+modeling, is twofold:
+– it allows to remain as agnostic as possible regarding the
+physical conditions close to Sgr A* and encapsulate into a
+single model a large parameter space;
+– it allows to perform simulations within a limited computing
+time, allowing to explore the large parameter space and com-
+pare to astrometric and photometric data.
+Our hope is that such a model can not only be fed with the re-
+sults of more elaborated simulations, but also bring constraints to
+these simulations by determining what features of the modeling
+are important in order to explain the data.
+Article number, page 6 of 20
+
+N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+The main features of our model are illustrated in Fig. 5. in-
+spired by the recent GRPIC results of El Mellah et al. (2022).
+Here, we consider a single plasmoid, which is modeled as a
+sphere of hot plasma with a constant radius. This macroscopic
+plasmoid is understood as the end product of a sequence of mi-
+croscopic plasmoid mergers. The spherical geometry is chosen
+only for simplicity, given that current data are certainly unable
+to make a difference between various geometries.
+3.1.1. Plasmoid motion
+We consider that the magnetic reconnection event occurs close
+to the black hole, and the resulting plasmoid is ejected along the
+jet sheath (Ripperda et al. 2020; El Mellah et al. 2022). Thus, we
+define a conical motion (as in Ball et al. (2021)) defined by a con-
+stant polar angle θ = θ0 and the initial conditions r0, θ0, ϕ0, vr0,
+and vϕ0 . The subscript 0 reflects the initial value of a given pa-
+rameter in Boyer-Lindquist coordinate. As in Ball et al. (2021),
+we set a constant radial velocity vr = vr0 and the azimuthal veloc-
+ity is defined through the conservation of the Newtonian angular
+momentum
+vϕ(t) = vϕ0
+r2
+0
+r(t)2 .
+(3)
+The azimuthal angle is obtained by integrating the previous
+Eq. 3
+ϕ(t) = ϕ0 + r2
+0
+vϕ0
+vr
+( 1
+r0
+− 1
+r(t)).
+(4)
+In the GRPIC simulations performed by El Mellah et al.
+(2022), the plasmoids are formed in the vicinity of the black hole
+at the Y-point and are ejected into the black hole magnetosphere,
+we thus restrict our study to vr > 0.
+An important feature of our model is the fact that the initial
+azimuthal velocity of the plasmoid is naturally super-Keplerian.
+Indeed, the Y-point, from which the plasmoid is generated, is an-
+chored to the equatorial plane of the accretion flow through the
+separatrix field line. So it will typically rotate at the Keplerian
+speed corresponding to the foot point of the line, thus at a veloc-
+ity higher than the Keplerian velocity corresponding to the initial
+cylindrical radius of the plasmoid.
+3.1.2. Growth and cooling phases
+We consider two phases in the lifetime of the plasmoid that aim
+at modeling the ascending and descending phases of the ob-
+served flare light curves:
+– during the growth phase, which lasts a total time tgrowth, the
+plasmoid continuously receives fresh accelerated particles at
+a constant rate resulting from the merging of microscopic
+plasmoids from reconnection into our large plasmoid which
+mix with "old" electrons cooled by synchrotron radiation.
+The growth time tgrowth corresponds to the lifetime of the
+reconnection engine, i.e. the duration of magnetic flux dis-
+sipation;
+– after t = tgrowth, the plasmoid enters the cooling phase: we as-
+sume that magnetic reconnection is quenched and plasmoids
+no longer merge so injection of fresh plasma stops and the
+plasmoid cools by emitting synchrotron radiation. We ne-
+glect particle escape and adiabatic losses.
+The duration of the growth phase is set both by the recon-
+nection rate and the speed at which magnetic flux is advected by
+the accretion flow into the current sheet. In Parfrey et al. (2015),
+the accretion of successive magnetic loops of opposite polarity
+activates this process, with typical duration of transition of the
+order of 100rg/c. This duration is representative of the dissipa-
+tion of the magnetic flux of one magnetic loop which is set by
+both the size of the loop and the accretion speed, that the authors
+fix to 2rg and c/200 respectively, and the reconnection rate, fixed
+by the prescribed resistivity. Resistive GRMHD simulations of
+magnetically arrested disks (Narayan et al. 2003) bring support
+to these values (e.g. Ripperda et al. 2020) but fail at reaching re-
+connection rates realistically high (Bransgrove et al. 2021a). In
+the more ab initio PIC simulations of El Mellah et al. (2022), the
+reconnection rate is more realistic (∼ 10%, Sironi & Spitkovsky
+2014; Werner et al. 2018) but due to the high computational cost
+of the simulations, they did not work over duration long enough
+to model the inward drift of the magnetic footpoints on the disk.
+As a consequence, the reconnection rate is accurate but the fuel-
+ing magnetic flux is artificially steady and act as an infinite reser-
+voir over the ∼200rg/c covered by the simulation. A coupling
+between GRMHD, force-free and PIC simulations to jointly de-
+scribe the disk, the corona and the current sheet respectively is
+still missing. In this context, we considered duration tgrowth of the
+growth phase of the order of 100rg/c, corresponding to a typical
+episode of magnetic flux dissipation set by the two rates at which
+magnetic flux is advected into the current sheet and dissipated by
+magnetic reconnection.
+3.1.3. Evolution of the electron distribution
+Next, we prescribe the emission/absorption mechanism in the
+plasmoid. Most studies use chosen electron distributions, with
+analytical prescriptions for their evolution at best. Ball et al.
+(2021) use a fixed thermal distribution with a linear increase of
+the number density for the rising part of the light curve and a
+decrease of the temperature following Eq. D.7 for the cooling.
+Scepi et al. (2022) use a kappa distribution with an exponential
+cutoff and a synchrotron cooling break for the plasma emission
+generating X-ray flares. While their evolution of the plasma pa-
+rameters (number density, temperature, magnetic field) is more
+elaborate than in our model, their approximation is valid only
+while injection and cooling are balanced. When the injection
+stops, the shape of the electron distribution changes rapidly (see
+Fig. 6). Here, we choose a different approach by evolving the
+electron distribution in the plasmoid by solving the kinetic equa-
+tion
+∂Ne(γ, t)
+∂t
+= ∂
+∂γ
+�
+−˙γsyn Ne(γ, t)
+�
++ Qinj(γ),
+(5)
+where γ is the Lorentz factor of the electrons, Qinj is the injection
+rate and Ne = dne/dγ is the electron number density distribution,
+using the EMBLEM code (Dmytriiev et al. 2021). The term
+−˙γsynNe = 4σTUB
+3mec (γ2 − 1)Ne
+(6)
+of the right hand side describes the synchrotron cooling of the
+plasmoid particles, with UB = B2/(8π). In our approach, we do
+not model the details of the magnetic reconnection process but
+instead describe the supply of freshly accelerated particles to the
+plasmoid by magnetic reconnection. Therefore, for the injection
+rate Qinj(γ) in Eq. 5, we use the following expression, assuming
+Article number, page 7 of 20
+
+A&A proofs: manuscript no. main
+Fig. 5. Sketch of magnetic reconnection in the black hole magneto-
+sphere as shown by El Mellah et al. (2022) on which our plasmoid
+model is built. There are three types of magnetic field lines: the ones
+threading the event horizon which goes to infinity, the ones anchored
+in the disk which also go to infinity, and the separatrix which link the
+disk and the black hole event horizon. The latter form a Y-point and a
+current sheet where chain of plasmoids are formed. Here we model a
+single plasmoid as the result of multiple mergers.
+a constant injection rate:
+Qinj(γ) =
+���������
+4πNκ
+e(γ)
+tgrowth
+in the growth phase,
+0
+in the cooling phase,
+(7)
+where Nκ
+e(γ) is the distribution of the injected particles which
+follows the kappa distribution i.e. a thermal core with a power-
+law tail following:
+Nκ
+e(γ) = N
+4πγ(γ2 − 1)1/2
+�
+1 + γ − 1
+κΘe
+�−(κ+1)
+(8)
+with a normalization factor N = 1/2ne(κ−2)(κ−1)κ−2Θ−3
+e , where
+ne and Θe are the density and dimensionless temperature of the
+injected plasma. The index κ is defined as
+κ = p + 1 = Ap + Bp tanh (Cp βb) + 1
+(9)
+where
+Ap = 1.8 + 0.7/ √σb, Bp = 3.7 σ−0.19
+b
+,Cp = 23.4 σ0.26
+b
+,
+(10)
+following Ball et al. (2018); Werner et al. (2018), where p is
+the powerlaw index of the non-thermal part of the distribution,
+σb
+≫
+1 is the plasma magnetization of the accelerating site
+and βb ≪ 1 is the ratio of proton thermal pressure to magnetic
+pressure of the accelerating site. If the magnetization at the ac-
+celerating site satisfies σb ≥ 100 (Crinquand et al. 2021), then κ
+is in the range of [2.8, 4.4] depending on βb. This implies that the
+spectral index α (νFν ∝ να) is between −0.5 and 0.5 which is in
+perfect agreement with the measured indices for flares (Fig. 32
+in Genzel et al. 2010). We note that realistic values for the mag-
+netization in the funnel region of Sgr A* can be orders of mag-
+nitude higher than 100 (see Ripperda et al. 2022) which result in
+a smaller parameter space for κ, closer to the low boundary.
+The bounds of the electron Lorentz factor are chosen to sat-
+isfy γmin = 1 and γmax = 106 (Eq.3 of Ripperda et al. 2022).
+When solving the kinetic Eq. 5, we assume that the density of
+the plasmoid particles follows
+ne(t) =
+�
+nmax
+e
+× t/tgrowth
+in the growth phase,
+nmax
+e
+in the cooling phase.
+(11)
+Such high maximum Lorentz factor is needed to also power X-
+ray flares with only synchrotron. However, Ripperda et al. (2022)
+suggest a lower maximum Lorentz factor γmax ∼ 104 in the plas-
+moid as electrons cool during their travel time between the accel-
+eration site and the later. Such lower value results in a marginally
+lower flux in NIR, as most of the emission at this wavelength
+comes from lower energy electrons, which can be compensated
+with a slightly higher maximum number density.
+The temperature of the injected particles remains fixed in the
+growth phase, and we define a uniform and time-independent
+tangled magnetic field in the plasmoid. This is also a simplifying
+assumption, and we intend to consider in future work the impact
+of the magnetic field geometry on the polarized observables.
+The EMBLEM code does not only solve for the evolu-
+tion of the electron distribution, it also provides the associated
+synchrotron emission and absorption coefficients of the plas-
+moid particles. We can thus compute an image of our plas-
+moid scenario by backwards integrating null geodesics in the
+Schwarzschild spacetime from a distant observer screen, and in-
+tegrate the radiative transfer equation through the plasmoid by
+reading the tabulated emission and absorption provided by EM-
+BLEM. This step is performed by means of the GYOTO2 ray-
+tracing code (see Appendix B for details; Vincent et al. 2011;
+Paumard et al. 2019). The input parameters that we used for the
+code are listed in Table 3. With these values of density and mag-
+netic field strength, we obtain a magnetization inside the plas-
+moid of σp ∼ 10−2 from the end of the growth phase since nei-
+ther the density nor the magnetic field evolve.
+3.2. Importance of evolving the electron distribution
+One of the most important aspects of our model is the self-
+consistent evolution of the electron distribution function, and
+corresponding radiative transfer, in the plasmoid. Here, we il-
+lustrate the importance of taking into account the evolution of
+the electron distribution by comparing our model with another
+reconnection plasmoid model inspired by Ball et al. (2021).
+We show in the top-left panel of Fig. 6 the evolution of the
+electron distribution in our plasmoid model, for the parameters
+listed in Tables 3 and 4 (see Sect. 3.3 for details) and the associ-
+ated spectral energy density (SED) in Fig. 7. During the growth
+phase (tobs < 10 min), for γ > 103 the distribution is stationary
+as the injection is balanced by the cooling. After the end of the
+growth phase, the shape of the distribution changes rapidly as
+there is only cooling left. We show in the right panel of Fig. 6
+the light curve obtained with our model (in red) and with a model
+inspired by Ball et al. (2021) who do not take into account the
+2 https://gyoto.obspm.fr
+Article number, page 8 of 20
+
+Current
+sheet
+Synchrotron
+cooling
+Plasmoid
+v(rcyl)
+yr
+Injection of
+O
+accelerated
+electrons
+Merging
+Y point
+magnetic
+islands
+B lines
+L
+rcyl
+rcyl
+footpoint
+plasmoid
+(depends
+on spin)N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+non-thermal electrons, i.e. using a thermal distribution, with a
+linear increase of the number density with a fixed temperature
+during the growth phase and an analytical prescription for the
+temperature decrease during the cooling phase using Eq. D.7 and
+assuming Θe = γ/3. While this model gives a similar intrinsic
+light curve as our model, the dimensionless temperature required
+is twice as high as ours (Θe = 109) with a magnetic field of
+B = 20 G to cool faster lower energy electrons. The evolution of
+the distribution with this model is shown in the bottom left panel
+of Fig. 6. We do not need such high a temperature as most of
+the emission comes from high-energy electrons which are non-
+thermal in our model as suggested by PIC simulations (Rowan
+et al. 2017; Werner et al. 2018; Ball et al. 2018; Zhang et al.
+2021). Our temperature could be even lower with a harder (i.e.
+lower) κ-index. The cooling of the electron distribution through
+synchrotron radiation is difficult to model properly and needs a
+kinetic approach as we do in our plasmoid model.
+3.3. Comparing GRAVITY 2018 flare data with our plasmoid
+model
+This paper aims at checking whether we can reproduce with our
+plasmoid model the general features of the observed light curve
+and astrometry of the July 22, 2018 flare reported by Gravity
+Collaboration et al. (2018). We show in Fig. 8 a comparison be-
+tween the July 22, 2018 flare data observed by GRAVITY (in
+black) and our plasmoid model (red line) with the parameters
+listed in Tables 3 and 4. For comparison, we show the intrinsic
+light curve (dashed line) obtained by removing all the relativistic
+effects (Doppler effect, beaming, secondary image).
+This comparison is not the result of a fit and was obtained
+by estimating the relevant parameters using simple physical ar-
+guments:
+– The rise time and slope of the light curve are mainly moni-
+tored by (i) the growth time, (ii) our choice of linear evolu-
+tion of the electron density (which enters the injection func-
+tion), (iii) the relativistic beaming effect, and thus (iv) the
+initial azimuthal position of the plasmoid, ϕ0, which has a
+strong impact on beaming as illustrated in the right panel of
+Fig. 4.
+– The decaying part of the light curve is monitored by the syn-
+chrotron cooling time, thus by the magnetic field strength,
+and by the beaming effect.
+– The maximum of the light curve can be estimated by means
+of an analytical formula that we derive in Appendix C.2.
+This maximum depends mainly on the maximum number
+density ne,max, as well as on the temperature and κ-index of
+the distribution. These parameters are degenerate and thus
+not constrained with only the NIR flare data. Nevertheless,
+GRMHD (Dexter et al. 2020b; Ripperda et al. 2022; Scepi
+et al. 2022) and GRPIC simulations (El Mellah et al. 2022)
+of magnetic reconnection suggest that the density in the plas-
+moid is higher than its close environment in the current sheet,
+of the order of the density at the base of the jet, close to the
+event horizon, but lower than in the disk. Still, the two re-
+maining parameters (Θe, κ) which describe the shape of the
+distribution are fully degenerate.
+– The initial position and velocity of the plasmoid have a
+strong impact on the astrometric trace on sky. We guess the
+initial azimuthal velocity based on the following reasoning.
+The Keplerian velocity of the plasmoid at its initial cylindri-
+cal radius is vKep ∼ 0.31c (for our choice of initial cylin-
+drical radius given in Table 4, rcyl = 10.6 rg). However, as
+Parameter
+Symbol
+Value
+Plasmoid
+magnetic field [G]
+Bp
+15
+plasmoid radius [rg]
+Rp
+1
+minimal Lorentz factor
+γmin
+1
+maximum Lorentz factor
+γmax
+106
+kappa distribution index
+κ
+4.0
+kappa distribution temperature
+Θe
+50
+maximum electron number density [cm−3]
+ne,max
+5 × 106
+growth timescale [rg/c]
+tgrowth
+120
+Table 3. Input parameters of the EMBLEM code for the simulation of the
+electron distribution evolution. These parameters are used for the July
+22 flare of Gravity Collaboration et al. (2018).
+Parameter
+Symbol
+July 22
+Plasmoid
+time in EMBLEM at zero observing time [min]
+temblem
+obs,0
+29.6
+initial cylindrical radius [rg]
+rcyl,0
+10.6
+polar angle [◦]
+θ
+135
+initial azimuthal angle [◦]
+ϕ0
+280
+initial radial velocity [c]
+vr,0
+0.01
+initial azimuthal velocity [c]
+vϕ,0
+0.45
+X position of Sgr A* [µas]
+x0
+0
+Y position of Sgr A* [µas]
+y0
+0
+PALN [◦]
+Ω
+160
+Table 4. Orbital parameters of the plasmoid model following a conical
+motion used for the comparison of the July 22, 2018 flares observed by
+Gravity Collaboration et al. (2018).
+discussed in Sect. 3.1, our model naturally leads to a super-
+Keplerian initial velocity to the plasmoid. Indeed, the plas-
+moid initial azimuthal velocity is that of the footpoint of the
+separatrix (see Fig. 5). Based on Fig. 8 of El Mellah et al.
+(2022), we can determine the radius of the footpoint, rf p, of
+a separatrix giving rise to a Y point located at a cylindrical
+radius of ≈ 10 rg. We find r f p = (4.7 ± 0.5) rg, which trans-
+lates in an orbital velocity vϕ,0 between 0.41c and 0.45c. The
+upper bound of this interval compares well with the July 22
+flare data. We note that this estimate of the initial azimuthal
+velocity is anchored in the model of El Mellah et al. (2022)
+and thus depends on their choice of initial condition, in par-
+ticular on the initial profile of their magnetic field.
+We note that the fiducial values proposed in Tables 3 and
+4 represent a set of parameters with values that are inspired by
+numerical simulations of reconnection which reproduce the key
+observational features of the July 22 flare data. This setup is not
+unique and is not the result of a fit. We let the exploration of the
+full parameters space (freeing some fixed/constrained parame-
+ters like maximum number density, growth time, inclination) to
+a future work. Nevertheless, our model disfavor low growth time
+(tgrowth < 50rg/c) for this particular flare.
+Overall, our plasmoid model jointly describes the astrom-
+etry and the flux variation of the 22 July 2018 flare measured
+by (Gravity Collaboration et al. 2018) for the first time, consid-
+ering a model with a specific emission prescription. Magnetic
+reconnection is thus a viable scenario to explain Sgr A* flares.
+Article number, page 9 of 20
+
+A&A proofs: manuscript no. main
+100
+101
+102
+103
+104
+105
+106
+10
+9
+10
+6
+10
+3
+100
+103
+ne [cm
+3]
+kappa
+Distribution
+Thermal
+PL (p=3)
+100
+101
+102
+103
+104
+105
+106
+10
+9
+10
+6
+10
+3
+100
+103
+ne [cm
+3]
+tobs=0.71 min
+tobs=3.53 min
+tobs=6.70 min
+tobs=9.17 min
+tobs=14.46 min
+tobs=16.93 min
+tobs=20.46 min
+tobs=22.93 min
+tobs=26.81 min
+tobs=29.28 min
+30
+20
+10
+0
+10
+20
+30
+40
+50
+tobs [min]
+0
+10
+20
+30
+40
+50
+60
+70
+80
+tintrinsic [min]
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+F  [erg.cm
+2.s
+1]
+1e
+11
+Observing time
+tgrowth
+growth phase
+cooling phase
+EMBLEM intrinsic LC
+Thermal model intrinsic LC
+Fig. 6. (Top-left) Evolution of the electron distribution in our model with EMBLEM at each observing time of the July 22, 2018 flare. The black
+dotted line correspond to the injected κ electron distribution composed of a thermal core with a power law tail. The parameters used are listed in
+Table 3. (Bottom-left) Evolution of the electron distribution in the Thermal model inspired by Ball et al. (2021) at each observing time of the July
+22, 2018 flare. The parameters used for this distribution are the same as in our model (listed in Table 3) but with the dimensionless temperature of
+Θe = 109 and the magnetic field of B = 20 G. (Right) Full intrinsic light curves of the two models. Note that in this panel we plot the light curve
+from the beginning of the growth phase while in the left panels we plot the distribution at the observed time of Fig. 8 (tobs = tintrinsic − 29.6 min).
+1010
+1012
+1014
+1016
+1018
+1020
+1022
+ [Hz]
+10
+19
+10
+17
+10
+15
+10
+13
+10
+11
+F  [erg.cm
+2.s
+1]
+tobs=0.71 min
+tobs=3.53 min
+tobs=6.70 min
+tobs=9.17 min
+tobs=14.46 min
+tobs=16.93 min
+tobs=20.46 min
+tobs=22.93 min
+tobs=26.81 min
+tobs=29.28 min
+Fig. 7. Intrinsic Spectral Energy Density (SED) evolution from radio to X-rays of our plasmoid model with the parameters listed in Table 3. Color
+code the time as in Fig. 6. Grey lines are the SED out of the observing time. The SED peak occurs in NIR with this set of parameters, but with a
+lower κ-index, i.e. a harder power law tail, the peak could reach X-rays. Here, X-rays flux drops quickly compare to the typical timescale of X-ray
+flares as we consider only synchrotron cooling and not Synchrotron-Self Compton (SSC). We note that the flux emitted by our plasmoid at 230
+GHz is ∼ 0.1 Jy which is the good order of magnitude of the variability associated to flares in sub-mm (Wielgus et al. 2022a).
+Article number, page 10 of 20
+
+N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+150
+100
+50
+0
+50
+100
+150
+X ( as)
+150
+100
+50
+0
+50
+100
+150
+Y ( as)
+Astrometry
+0
+5
+10
+15
+20
+25
+30
+t (min)
+0.20
+0.25
+0.30
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+0.65
+F/F(S2)
+Light Curve
+observed
+intrinsic
+data
+Fig. 8. Data and plasmoid models of the flares from July 22, 2018. The left panels shows the astrometry of the flare while the right panel shows
+the observed (full line) and intrinsic (dashed line) light curves. The parameters of the model are listed in Tables 3 and 4. Note that this is not the
+result of a fit. The black dot in the left panels represents the position of Sgr A* in GYOTO and the orange cross represent the position of Sgr A*
+measured through the orbit of S2.
+4. Limitations of our plasmoid model
+Our plasmoid model is vastly simplified with respect to the com-
+plexity of realistic magnetic reconnection events in the environ-
+ment of black holes. We review here its main limitations:
+i. We consider a single plasmoid while the instability of thin
+current sheets gives rise to a dynamic flow of merging
+magnetic islands. Our argument for this simplification is
+that the merging process is certainly very dependent on the
+unknown initial conditions, and that the final, bigger and
+brighter product of the cascade is likely to dominate the
+observed signal;
+ii. The initial condition on the plasmoid’s velocity is simply
+imposed for the radial motion, and based on a particular
+GRPIC model as regards the azimuthal motion;
+iii. The evolution of the plasma parameters (density, temper-
+ature, magnetic field) are chosen to be either constant or
+linear, so very simplified compared to a realistic scenario.
+However, we consider that these evolutions are very likely
+to be strongly dependent on the initial conditions of the flow,
+so that they are weakly constrained;
+iv. The values of almost all the parameters except mass and
+distance of Sgr A* are poorly constrained. We choose a set
+of values which are reasonable according to simulations.
+Future work is needed to investigate the details of the
+parameter space.
+v. We model the plasmoid by a homogeneous sphere for
+simplicity from the circle plasmoid seen in 2D GRMHD
+(Nathanail et al. 2020; Ripperda et al. 2020; Porth et al.
+2021) and PIC simulations (Rowan et al. 2017; Ball et al.
+2018; Werner et al. 2018). The 3D aspect of such plasmoid
+is cylindrical (flux ropes) both in GRMHD (Bransgrove
+et al. 2021b; Nathanail et al. 2022; Ripperda et al. 2022)
+and PIC (Nättilä & Beloborodov 2021; Zhang et al. 2021)
+simulations. Thus, a realistic geometry of the flare source
+is likely more complex than in our model. We note that the
+exact geometry of the flare is not relevant as we only track
+the centroid position, as much the flare source is not too
+extended, and we consider tangled magnetic field. However,
+the coherence time of the structure might be shorter in 3D
+and might have an impact on the rise time of the light curve.
+Further 3D simulations studies are needed to better model
+the shape of the flux ropes and their evolution;
+vi. We neglect any shearing of the plasmoid and consider
+that it remains identical to itself throughout the simulation.
+Differential rotation is however likely to stretch the plasmoid
+over its orbit and destroy its coherence (Hamaus et al. 2009;
+Gravity Collaboration et al. 2020c);
+vii. We consider a tangled magnetic field in the plasmoid
+and thus do not consider the impact of the magnetic field
+geometry on the observable. The magnetic field geometry
+of the quiescent flow is likely to be ordered and vertical if
+Sgr A* is strongly magnetized. The magnetic field in the
+plasmoid, which is our interest here, could be either helical
+(plasmoids) or vertical for large flux tubes (Ripperda et al.
+2022).
+viii. During a flare, the quiescent state can change in a non
+axisymmetrical way (Ripperda et al. 2022). This will push
+the centroid position of the quiescent further away from
+the center-of-mass location which will affect the offset. We
+choose to use a static and axisymmetric quiescent model
+during flare to avoid adding more degrees of freedom
+which would lead to higher degeneracies, rather than clearer
+constraints.
+Article number, page 11 of 20
+
+A&A proofs: manuscript no. main
+ix. We choose a high maximum Lorentz factor γmax = 106 to
+be able to power X-ray flares (but without any constrain
+for this study). However, high energy photons lead to pair-
+production and thus increase the number density in the plas-
+moid which we do not take into account.
+Despite these many limitations, we consider that our model
+is very interesting for fitting flare data, because it allows to cover
+a much broader set of physical scenarios than more elaborate
+simulations that strongly depend on their assumptions regarding
+the relevant physics and the initial conditions.
+5. Conclusion and perspectives
+This paper is mainly focused at developing a new plasmoid
+model for Sgr A* flares, inspired by magnetic reconnection in
+black hole environments. Our semi-analytic model allows to
+study a broad parameter space within a reasonable computing
+time, thus being well suited for data analysis.
+Our model considers non-thermal electrons accelerated by
+magnetic reconnection and injected into a spherical large plas-
+moid. We evolve the electron distribution through a kinetic equa-
+tion taking into account synchrotron cooling and particle injec-
+tion at a constant rate. We show in Appendix C.1 (Fig. C.3) the
+importance of taking into account the cooling of the electrons al-
+ready in plasmoid during the growth phase. Our model also natu-
+rally accounts for a super-Keplerian velocity of the flare source,
+through the dynamical coupling between the plasmoid and the
+inner regions of the accretion flow through magnetic field lines.
+One of the main results of this paper is that for the first time
+we model the astrometry and lightcurve of the flares measured
+by (Gravity Collaboration et al. 2018) by explicit modeling of
+the emission zone.
+Our conclusions regarding the three main points raised in the
+introduction are the following:
+– the marginally detected shift between the astrometric track
+of Gravity Collaboration et al. (2018) and the center of mass
+might be due to the impact of the quiescent radiation of the
+background accretion flow;
+– a dynamical coupling between the plasmoid and the inner
+accretion flow through closed magnetic field lines might
+naturally account for the super-Keplerian speed obtained
+by Gravity Collaboration et al. (2018);
+– in general, a large plasmoid due to magnetic reconnection in
+a thin current sheet in the black hole magnetosphere is a rea-
+sonable model to account for the main features of the Gravity
+Collaboration et al. (2018) observables.
+Sect. 3.3 shows that the temperature, density, and κ param-
+eters of the plasmoid are degenerate. This degeneracy might be
+removed by simultaneous observations of NIR and X-ray flares.
+Moreover, synchrotron cooling leads to a translation of the elec-
+trons from the NIR-emitting band to the millimeter-emitting
+band, which could explain the sub-mm flare and its time lag with
+respect to NIR. We thus intend to consider the multi-wavelength
+properties of our plasmoid model in future work, in order to bet-
+ter assess its ability to account for the complete flare data set of
+Sgr A*.
+A crucial recent observable of Sgr A* flares are the polariza-
+tion QU loops (Gravity Collaboration et al. 2018, 2020d; Wiel-
+gus et al. 2022b). We also intend to study the polarized properties
+of our plasmoid model and compare it to these recent constraints.
+Acknowledgements. NA and FHV are very grateful to B. Cerutti, B. Crinquand,
+S. von Fellenberg, S. Gillessen, S. Masson, B. Ripperda, N. Scepi, and M. Wiel-
+gus for fruitful discussions. This work was granted access to the HPC resources
+of MesoPSL financed by the Region Ile de France and the project Equip@Meso
+(reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir
+supervised by the Agence Nationale pour la Recherche.
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+
+A&A proofs: manuscript no. main
+Appendix A: Torus - Jet model for the quiescent
+state
+We used the Torus-jet model of Vincent et al. (2019). Their jet
+model is restricted to an emitting sheath with an empty funnel in
+agreement with GRMHD simulations (e.g. Mo´scibrodzka & Fal-
+cke 2013; Ressler et al. 2017; Davelaar et al. 2018; Porth et al.
+2019). In their model, Vincent et al. (2019) define an opening
+and closing angle θ1 and θ2 respectively and a base height zb to
+define the geometry of the jet. The number density and the tem-
+perature are defined by their values at the base height of the jet
+(nJ
+e and T J
+e respectively) and their profiles along the jet. The pro-
+file of the number density is fixed (∝ r−2
+cyl with rcyl the projected
+radius in the equatorial plane) and the one of the temperature is
+set by the temperatures slope sT (∝ z−sT with z the height along
+the vertical/spin axis). The jet emits synchrotron radiation from
+a κ electron distribution.
+The torus is defined by its central density and temperature
+(nT
+e and T T
+e respectively). The profiles of these two quantities in
+the torus are governed by the polytropic index k and its geome-
+try. The latter is defined by the inner radius rin and the angular
+momentum l but also on the metric (see Vincent et al. (2019) for
+more details). Contrary to the jet, we consider that the electron
+distribution of the torus is purely thermal.
+We use the same algorithm as in Vincent et al. (2019) after
+the correction of a small technical issue leading to an overestima-
+tion of the number density and temperature. However, we change
+the choice of the magnetization parameter in the jet sheath. As
+illustrated e.g. by Porth et al. (2019), the jet sheath, which cor-
+responds to the dominating emission region of the jet, coincides
+with the transition between the highly-magnetized (σ ≫ 1) fun-
+nel and the less-magnetized (σ ≪ 1) main disk body. Conse-
+quently, we fix the magnetization to σ = 1 in the emitting jet
+sheath, while Vincent et al. (2019) used a low magnetization both
+in the jet and in the torus. Our choice leads to a smaller density
+in the jet sheath compared to Vincent et al. (2019). We found a
+best-fit with a χ2
+red = 0.91 using the same data points as Vincent
+et al. (2019). The values are reported in Table 1 and Fig. 2 shows
+the associated spectrum and the image at 2.2 µm. We obtain a
+magnetic field strength of 257 G for the jet and 212 G at the
+center of the torus.These values are higher than in Bower et al.
+(2019) who considers a full thermal electron population with a
+higher temperature but are of the same order as in Scepi et al.
+(2022).
+Appendix B: Ray-tracing setup
+We consider a Kerr black hole with dimensionless spin param-
+eter a = 0, described in Boyer-Lindquist (t, r, θ, ϕ) coordinates.
+We work in units where the gravitational constant and the speed
+of light are equal to 1, G = c = 1. Radii are thus expressed in
+units of the black hole mass M.
+We use the backward ray-tracing code GYOTO3 (Vincent et al.
+2011; Paumard et al. 2019) to compute images of our models
+at different epochs. Each pixel of our image corresponds to a
+direction on sky. For each pixel of the image (i.e. each direction),
+a null geodesic is integrated backwards in time from the observer
+towards the black hole, integrating along this path the radiative
+transfer equation
+dIν
+ds = −ανIν + jν
+(B.1)
+3 https://gyoto.obspm.fr
+using the synchrotron emissivity jν and absorptivity αν coef-
+ficients, considering various electron distribution functions. This
+allows us to determine the flux centroid for each epoch and trace
+its motion. In addition to astrometry we also determine the total
+flux emitted as the sum of the intensity weighted by the element
+of solid angle subtended by each pixel, again, for each epoch
+which allows us to plot the light curve.
+The images produced are 1000x1000 pixels with a field of
+view of 300 µas vertically and horizontally which makes a reso-
+lution < 0.1 µas2/pixel. This high resolution is needed to resolve
+properly the secondary image which has a very important role in
+both astrometry and light curve (see Sect. 2).
+We model the quiescent state of Sgr A* at 2.2 µm with a jet
+(see Sect. 2.1). However, computing an image of the jet is ∼ 200
+times longer than an image of the flare source (i.e. the hot spot or
+the plasmoid, Sect. 2 and 3 respectively) because the jet is much
+more extended, and integrating the radiative transfer equation is
+thus much longer. The absorption in the jet is negligible thus the
+flux emitted by the flare which crosses the jet is fully transmit-
+ted. We can compute a single image of the jet that we add to
+each images of the hot spot a posteriori. We then calculate the
+total flux by summing the jet flux with the one of the flare at a
+given time. The final centroid position is calculated by a simple
+barycenter of the two centroids (jet and flare).
+Appendix C: Intrinsic emission of the Plasmoid
+Appendix C.1: Tests on the kinetic simulations
+In our model, we follow the evolution of the electron distribu-
+tion taking into account the injection of accelerated electrons by
+the merging of small plasmoids into our large plasmoid and their
+cooling through synchrotron radiation. The emissivity jν and ab-
+sorptivity αν coefficients, needed to integrate the radiative trans-
+fer Eq. B.1, are computed through the formula of Chiaberge &
+Ghisellini (1999); Rybicki & Lightman (1986) (with our nota-
+tion)
+jν(t) = 1
+4π
+� γmax
+γmin
+dγNe(γ, t)Ps(ν, γ),
+(C.1)
+αν(t) = −
+1
+8πmeν2
+� γmax
+γmin
+Ne(γ, t)
+γl
+d
+dγ[γlPs(ν, γ)]
+(C.2)
+with
+Ps(ν, γ) = 3
+√
+3
+π
+σTcUB
+νB
+x2
+�
+K4/3(x)K1/3(x) − 3
+5 x[K2
+4/3(x) − K2
+1/3(x)]
+�
+(C.3)
+where l = (γ2 − 1)1/2 is the electron momentum in units of
+mec, x = ν/(3γ2νB), νB = eB/(2πmec) and Ka(t) is the modified
+Bessel function of order a. We note that the Eq. C.3, is already
+averaged over pitch angle. For standard distributions as thermal,
+power-law and κ-distributions, these formulae are equivalent to
+the fits of Pandya et al. (2016) (see Appendix D) that we used
+for computing the quiescent synchrotron flux.
+As electrons start to cool as soon as they are injected in the
+plasmoid, the full distribution is no more a κ-distribution. How-
+ever, turning off the cooling during the growth phase allows us to
+compare the results of EMBLEM to the fitting formulae of Pandya
+et al. (2016). As we inject electrons following their definition of
+Article number, page 14 of 20
+
+N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+2
+4
+6
+8
+10
+12
+14
+16
+18
+0
+1
+2
+3
+4
+5
+6
+7
+I , max, erg cm
+2 s
+1 sr
+1 Hz
+1
+1e
+6
+Pandya+16
+2
+EMBLEM (without cooling)
+Fig. C.1. Specific intensity at the end of the growth phase (t = tgrowth =
+75 rg/c) of a κ-distribution with ne = 5 × 106 cm−3, B = 10 G, κ = 4 for
+a range of Θe computed from the full fitting formulae of Pandya et al.
+(2016) (black curve), by the EMBLEM code (red dots) and with the high-
+frequency limit analytical expression (dashed blue curve). We overplot
+in light blue the range of Θe where Xκ > 2000 i.e. where the relative
+error between the high frequency limit and the full formula is lower
+than 20%.
+the κ-distribution with a linear increase of the number density,
+the two approach show similar results (see Appendix D). In our
+cases, the absorption is very low, thus we neglect the absorption
+in these tests. We can derive an analytical formula for the spe-
+cific intensity from the high frequency limit emissivity Eq. D.3
+depending on the number density ne, the electron temperature
+Θe and the magnetic field B in case the cooling is switched off
+during growth. We find in the case without cooling that, keeping
+κ constant
+Iν,max ∝ ne,max Θ κ−2 B κ/2, if Xκ > 2000.
+(C.4)
+We show the relative error of the maximum specific intensity
+between the EMBLEM code (red dots) and the formulae of Pandya
+et al. (2016) (black curve) depending on the electron tempera-
+ture and the magnetic field in Fig. C.1 and C.2. We fix the others
+parameters to ne = 5 × 106 cm−3, κ = 4, tgrowth = 75 rg/c. The
+values of EMBLEM are in good agreement with the previous ana-
+lytical expression (Eq. C.4) for low values of Θe and B. For high
+values, we are beyond the validity of our approximations (in the
+intermediate frequency regime of the fitting formula, see Ap-
+pendix D). Comparing the results of EMBLEM (without cooling)
+with the full fitting formula of Pandya et al. (2016) (black curves)
+results in an error lower than 5% showing the good agreement
+between the two approaches.
+Appendix C.2: Analytical estimate of the intrinsic light curve
+Next, we compute the light curve emitted by the plasmoid which
+will be affected by the relativistic effects. To reproduce a given
+light curve, we can estimate the values of the parameters through
+characteristic scales. The growth time, which is a "free" param-
+eter of the model, can be estimated from the light curve taking
+into account the beaming effect and thus, depends on the orbital
+parameters. The synchrotron cooling time of an electron with
+Lorentz factor γ in a magnetic field B reads
+tcool = 3
+4
+mec
+σTUBγ
+(C.5)
+0
+5
+10
+15
+20
+25
+30
+B (in Gauss)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+I , max, erg cm
+2 s
+1 sr
+1 Hz
+1
+1e
+5
+Pandya+16
+B2
+EMBLEM (without cooling)
+Fig. C.2. Same as in Fig.C.1 for a range of B and with Θe = 10.
+with σT the electron Thomson cross section and UB the magnetic
+energy density. In a Dirac spectrum approximation, the Lorentz
+factor of an electron emitting an IR photon at 2.2 µm is (Rybicki
+& Lightman 1979)
+¯γ =
+�νmec
+ηeB
+�1/2
+(C.6)
+with η = (0.29 × 3)/(2π) a dimensionless numerical factor (see
+Appendix E.2). One can thus constrain the magnetic field from
+the synchrotron cooling time as
+tcool = 19 ×
+� B
+20G
+�−1.5 �
+λ
+2.2µm
+�0.5
+min.
+(C.7)
+Taking into account the cooling of the electrons during the
+growth phase leads to a lower flux than what we estimate from
+Eq. C.4. Indeed, as electrons start to cool directly after being in-
+jected, the integral of the distribution in Eq. C.1 and so the emis-
+sivity will always be lower than without cooling. The key param-
+eter of synchrotron cooling is the cooling time scale (Eq. C.5),
+which depends on the magnetic field strength and the initial en-
+ergy of the electrons. It has to be compared to the growth time.
+Indeed, with a low growth time, only high-energy electrons have
+the time to cool. Increasing the growth time will allow lower en-
+ergy electrons to cool and so decrease even more the maximum
+flux of the light curve.
+With some approximations (see Appendix E for the details),
+one can estimate the flux with cooling at t = tgrowth
+νFsyn
+ν (ν, t) =
+neR3
+b ¯γmec2
+12D2tgrowth κθ2
+��Ψ(¯γ) − Ψ(ξ(¯γ, t))� ,
+for ν < ˜ν(t)
+Ψ(¯γ),
+for ν ≥ ˜ν(t)
+(C.8)
+where ˜ν(t) = (ηeB)/(mecb2
+ct2) is the frequency corresponding
+to the condition ¯γ = 1/(bct) and
+Ψ(x) =
+�
+1 + x − 1
+κθ
+�−κ �
+x2(κ − 1) + 2x(κθ − 1) + 2θ(κθ − 2)
+�
+.
+(C.9)
+We plot the maximum light curve evolution relative to the
+magnetic field with EMBLEM with (blue crosses) and without (red
+Article number, page 15 of 20
+
+A&A proofs: manuscript no. main
+0
+10
+20
+30
+40
+50
+60
+B (in Gauss)
+0
+1
+2
+3
+4
+5
+F , max, erg cm
+2 s
+1
+1e
+12
+Analytic
+EMBLEM (without cooling)
+EMBLEM (with cooling)
+Fig. C.3. Evolution of the maximum flux νFν(tgrowth) (at the end of the
+growth phase tgrowth = 75 rg/c) as a function of the magnetic field.
+We show the results of EMBLEM without cooling (red crosses) as in
+Fig. C.2. Allowing the cooling during the growth phase results in a
+lower maximum flux (blue crosses). One can estimates the maximum
+flux with cooling through Eq. C.8 (see Appendix E for details). This
+equation is divided in two regimes, the equilibrium regime where the
+magnetic field is strong enough to compensate the injection and creates
+a stationary state (B ≥ 16.2 G) and non stationary regime where not all
+electrons has cooled at tgrowth (B < 16.2 G). The relative error between
+the analytical formula and the results of EMBLEM (with cooling) is
+below 30% in the whole domain and below 7% in the non stationary
+regime.
+crosses) cooling during the growth phase and the previous an-
+alytical expression in Fig. C.3 (black line). As expected, the
+cooling becomes more significant with a strong magnetic field
+until the maximum flux starts to decrease for very high values
+(B > 100 G). The two regimes of Eq. C.8 are clearly visible in
+Fig. C.3 with a turning point at B = 16.2 G. This approximation
+has a maximum relative error lower than 30% compared to the
+results of EMBLEM in the stationary regime and below 7% for the
+non stationary regime making it a good approximation estimate
+the peak light curve flux.
+Appendix D: Computation of the synchrotron
+coefficients for the Plasmoid
+Appendix D.1: Fitting formulae of Pandya et al. (2016)
+In the hot spot model and for the test of EMBLEM, we used the
+fitting formula of Pandya et al. (2016) to compute the emissivity
+jν and absorptivity αν considering a well defined κ-distribution.
+This distribution has two regimes, the low and high frequency
+regimes.
+In the low frequency limit, the emissivity is
+jν,low = nee2νB
+c
+X1/3
+κ
+sin(θ) 4πΓ(κ − 4/3)
+37/3Γ(κ − 2)
+(D.1)
+and the absorption coefficient is
+αν,low = nee2
+νmec X−2/3
+κ
+31/6 10
+41
+2π
+(Θe κ)10/3−κ
+(κ − 2)(κ − 1)κ
+3κ − 1
+× Γ
+�5
+3
+�
+2
+F1
+�
+κ − 1
+3, κ + 1, κ + 2
+3, −Θe κ
+�
+(D.2)
+10
+7
+10
+4
+10
+1
+102
+105
+108
+Xk
+0
+1
+2
+3
+4
+5
+6
+7
+Js
+Js, lo
+Js
+0
+1
+2
+3
+4
+5
+6
+7
+Js
+Js, hi
+Js
+Fig. D.1. Relative error between the low frequency regime (in red) -
+resp. the high frequency regime (in blue) - fit formulae Js,lo (resp. Js,hi)
+of Pandya et al. (2016) and the full fit formula of the emission coefficient
+Js in function of Xκ =
+ν
+νκ with νκ = νB (Θeκ)2.
+where 2F1 is the hypergeometric function.
+In the high-frequency limit, the emissivity is
+jν,high = nee2νB
+c
+X−(κ−2)/2
+κ
+sin(θ) 3(κ−1)/2
+× (κ − 2)(κ − 1)
+4
+Γ
+�κ
+4 − 1
+3
+�
+Γ
+�κ
+4 + 4
+3
+�
+(D.3)
+and the absorption coefficient is
+αν,high = nee2
+νmec X−(1+κ)/2
+κ
+π3/2
+3
+(κ − 2)(κ − 1)κ
+(Θe κ)3
+×
+�2Γ(2 + κ/2)
+2 + κ
+− 1
+� �������
+�3
+κ
+�19/4
++ 3
+5
+������� .
+(D.4)
+The final approximations for the emissivity and absorption
+coefficient are
+jν =
+�
+j
+−xj
+ν,low + j
+−xj
+ν,high
+�−1/x j
+(D.5)
+αν =
+�
+α−xα
+ν,low + α−xα
+ν,high
+�−1/xα
+(D.6)
+with x j = 3κ−3/2 and xα =
+�
+− 7
+4 + 8
+5κ
+�−43/50.
+The two frequency limits do not have the same dependence
+on the parameters. The frequency regime is defined by the di-
+mensionless parameter Xκ = ν/νκ, with νκ = νB(Θeκ)2. Fig. D.1
+shows the relative error of the two regimes (the low frequency in
+red and the high frequency in blue) compared to the final emis-
+sion coefficient. While at very high (respectively very low) Xκ,
+the high frequency (resp. low frequency) fitting formulae work
+very well, there is a large frequency regime (10−2 ≲ Xκ ≲ 103),
+hereafter intermediate regime, where both limits are needed. At
+2.2 µm, Xκ > 1, while Θe κ ≲ 103 which correspond to our typ-
+ical set of parameters. This is why we used the high frequency
+regime for our test our EMBLEM.
+Article number, page 16 of 20
+
+N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+Appendix D.2: Chiaberge & Ghisellini (1999) approximation
+Modeling the synchrotron cooling of the electrons with a ther-
+mal, power-law or κ distribution is not trivial. Indeed, the evolu-
+tion of the energy of an electron which emits synchrotron radia-
+tion is (e.g., Rybicki & Lightman (1986))
+γ(t) = γ0(1 + Aγ0t)−1
+(D.7)
+with A = 4
+3
+σT B2
+8πmec,
+(D.8)
+γ the Lorentz factor of the electron at time t, and γ0 the ini-
+tial Lorentz factor. The energy evolution strongly depends on
+the initial energy. The higher the initial energy of the electron,
+the faster it will cool. Thus, the initial distribution we could im-
+pose will quickly be deformed (see top-left panel of Fig. 6) and
+could not be modeled by one (or more) of the three distribution
+of Pandya et al. (2016) (thermal, power-law and/or κ).
+In order to properly model the cooling of electrons, we sim-
+ulate the evolution of the electron distribution with injection
+and synchrotron cooling (Sect. 3). These simulations give us the
+electron distribution Ne(γ, t) at different times. We compute the
+emissivity jν and the absorptivity αν associated for a range of
+frequencies from 106 to 1021 Hz following the formula of Chi-
+aberge & Ghisellini (1999) (with our notation)
+jν(t) = 1
+4π
+� γmax
+γmin
+dγNe(γ, t)Ps(ν, γ)
+(D.9)
+and the absorption coefficient follows
+αν(t) = −
+1
+8πmeν2
+� γmax
+γmin
+Ne(γ, t)
+γp
+d
+dγ[γpPs(ν, γ)],
+(D.10)
+where p = (γ2 − 1)1/2 is the electron momentum in units of mec
+and Ps is the emissivity of a single electron (see C.3).
+In order to obtain the emissivity and absorption coefficient at
+any time and any frequency (to account the relativistic Doppler
+effect for example), we made a bilinear interpolation.
+Appendix E: Analytical approximation for Sgr A*
+flare peak flux
+Here we derive an analytical expression to compute the time-
+dependent flux from Sgr A* flares during the growth phase, and
+obtain an analytical formula for the peak flare flux. For that, we
+first obtain the approximate analytical form of the varying elec-
+tron spectrum during the growth phase by solving the kinetic
+equation, and then compute the approximate synchrotron SED
+associated to the time-dependent electron spectrum.
+Appendix E.1: Deriving time-dependent electron spectrum
+during the growth phase
+The kinetic equation describing the evolution of the electron
+spectrum during the growth phase is given by Eq. 5:
+∂Ne(γ, t)
+∂t
+= ∂
+∂γ
+�
+bcγ2Ne(γ, t)
+�
++ Qinj(γ, ne, θ, κ)
+(E.1)
+with the injection term Qinj(γ) given by Eq. 7 and Eq. 8, and
+synchrotron cooling term ˙γsyn = −bc(γ2 − 1) (see Eq. 6), where
+bc = (4σTUB)/(3mec). We use here an approximation ˙γsyn ≈
+−bcγ2, as the bulk of the electrons producing the flare emission
+are highly relativistic.
+We use the method of characteristics to solve the kinetic
+equation. We search for characteristic curves in the γ-t space,
+along which our differential equation in partial derivatives be-
+comes an ordinary differential equation. Let us rewrite the ki-
+netic equation in the following form, expanding the derivative
+on the Lorentz factor:
+∂Ne(γ, t)
+∂t
++ (−1)bcγ2 ∂Ne(γ, t)
+∂γ
+= Qinj(γ) + 2γbcNe(γ, t)
+(E.2)
+When restricting our equation to the characteristic curve
+(γ(t),t), the full derivative of the electron spectrum over time,
+by the chain rule, is:
+dNe(γ, t)
+dt
+= ∂Ne(γ, t)
+∂t
++ dγ
+dt
+∂Ne(γ, t)
+∂γ
+(E.3)
+Comparing this to Eq. E.2, we identify (−1)bcγ2 = dγ
+dt , and
+therefore along the chosen characteristic curve, our equation is
+split into a system of two ordinary differential equations:
+�dγ/dt = −bcγ2
+dNe(γ, t)/dt = Qinj(γ) + 2γbcNe(γ, t)
+(E.4)
+The solution of the first equation is (applying the initial con-
+dition that γ(t = 0) = ξ):
+γ(t) =
+1
+bct + 1/ξ
+(E.5)
+This equation defines a characteristic curve in the γ-t space.
+We have chosen the initial point of the characteristic curve as
+(ξ, 0). The physical meaning of ξ is the initial value of the
+Lorentz factor of an electron before it starts undergoing the cool-
+ing process. Eq. E.5 is equivalent to Eq. D.7, and describes how
+the Lorentz factor of an individual electron evolves in time due
+to synchrotron cooling. From this equation, the initial Lorentz
+factor ξ is:
+ξ = ξ(γ, t) =
+1
+1/γ − bct
+(E.6)
+This formula defines the initial Lorentz factor of the char-
+acteristic curve that passes through a point (γ,t). We denote the
+function Ne(γξ(t), t) = u(t) (electron spectrum along the charac-
+teristic curve), and solve the second equation in the system:
+du/dt − 2bcγ(t)u = Qinj(γ(t))
+(E.7)
+The generic solution of this linear differential equation is:
+u(t) =
+1
+µ(t)
+� t
+0
+µ(t′) Qinj(γ(t′)) dt′ +
+C
+µ(t)
+(E.8)
+Article number, page 17 of 20
+
+A&A proofs: manuscript no. main
+with C being the integration constant, and the function µ(t)
+being the integration factor, which is equal to:
+µ(t) = exp
+��
+−2bcγ(t)dt
+�
+=
+1
+(bct + 1/ξ)2
+(E.9)
+As the electron spectrum at t = 0 is zero, we set the initial
+condition u(t = 0) = 0, which results in C = 0. Therefore, the
+solution for u(t) is:
+u(t) = (bct + 1/ξ)2
+� t
+0
+(bct′ + 1/ξ)−2 Qinj(γ(t′)) dt′
+(E.10)
+Now we have to return back from u(t) to Ne(γ, t), which is
+achieved by substitution of the equation for ξ = ξ(γ, t) (Eq. E.6)
+to the expression for u(t). After doing that, we obtain an expres-
+sion for the electron spectrum at a moment of time t:
+Ne(γ, t) = 1
+γ2
+� t
+0
+Γ2 Qinj(Γ) dt′
+(E.11)
+with Γ = Γ(γ, t, t′) = �1/γ + bc(t′ − t)�−1. We use here an
+approximation for Qinj(Γ), and more specifically, for the kappa
+distribution, to enable analytical integration. As we are in the
+relativistic regime, and the peak of the injection spectrum in our
+case typically occurs at Lorentz factors γ ≫ 1, we can substitute
+γ(γ2 − 1)1/2 with γ2 in the Eq. 8. This leads to some inaccu-
+racies only at very low Lorentz factors, which virtually do not
+contribute to the integral value, and do not contribute to the light
+curve flux. We therefore use for the injected spectrum:
+Qinj(γ, ne, θ, κ) ≈
+N
+tgrowth
+γ2
+�
+1 + γ − 1
+κθ
+�−(κ+1)
+(E.12)
+Now we can perform the analytical integration. We use the
+variable substitution from t′ to Γ(γ, t, t′). In this case, the differ-
+ential dt′ = −b−1
+c Γ−2dΓ. Our integral (Eq. E.11) then becomes:
+Ne(γ, t) =
+N
+γ2tgrowth
+� t
+0
+Γ4
+�
+1 + Γ − 1
+κθ
+�−(κ+1)
+dt′ =
+= −
+N
+bcγ2tgrowth
+� t
+0
+Γ2
+�
+1 + Γ − 1
+κθ
+�−(κ+1)
+dΓ
+(E.13)
+To solve the integral, we perform integration by parts, and
+we obtain:
+� t
+0
+Γ2
+�
+1 + Γ − 1
+κθ
+�−(κ+1)
+dΓ = −
+θκ
+(κ − 2)(κ − 1)Ψ(Γ)
+�����
+t
+0
+(E.14)
+with
+Ψ(x) =
+�
+1 + x − 1
+κθ
+�−κ �
+x2(κ − 1) + 2x(κθ − 1) + 2θ(κθ − 2)
+�
+(E.15)
+We substitute the variable back from Γ to t′, with Γ(t′ =
+0) = (1/γ − bct)−1 = ξ(γ, t) and Γ(t′ = t) = γ, as well as sub-
+stitute the expression for the injection spectrum normalization,
+N = (1/2)ne(κ − 2)(κ − 1)κ−2θ−3 (see Eq. 8), and obtain:
+Ne(γ, t) =
+ne
+2κθ2bcγ2tgrowth
+�Ψ(γ) − Ψ(ξ(γ, t))�
+(E.16)
+One has to consider separately a special case when the bct ≥
+1/γ, as this leads to either ξ → ∞ or ξ < 0. Obviously, the latter
+situation is non-physical, as the Lorentz factor cannot be less
+than unity. Qualitatively, bct ≥ 1/γ → t ≥ 1/(bcγ) means that
+the evolution time of an electron is longer than its cooling time-
+scale, and in this regime the equilibrium between the injection
+and cooling is already reached. Therefore, one can easily see
+that the time-dependent electron spectrum in the Lorentz factor
+domain γ ≥ 1/(bct) will be “frozen” at the steady-state one. A
+steady-state solution corresponds to ξ → ∞, which results in
+Ψ(ξ) → 0 (in case κ > 2). Therefore, the final solution for the
+time-dependent electron spectrum during the growth phase, is:
+Ne(γ, t) =
+ne
+2κθ2bcγ2tgrowth
+��Ψ(γ) − Ψ(ξ(γ, t))� ,
+for γ < (bct)−1
+Ψ(γ),
+for γ ≥ (bct)−1
+(E.17)
+It is worth to note, that one can obtain the same steady-state
+solution (the case γ ≥ 1/(bct)) by directly solving the kinetic
+equation (Eq. E.1) with ∂Ne
+∂t = 0. To find the electron spectrum
+at the peak of the flare, i.e. at the moment when the injection is
+stopped, one simply calculates Ne(γ, t = tgrowth).
+Appendix E.2: Deriving time-dependent synchrotron SED
+during the growth phase
+Now let us proceed to the SED and light curve computation. We
+use the so-called δ-approximation for the electron synchrotron
+emissivity coefficient. This approximation assumes that a sin-
+gle electron with a Lorentz factor γ emits at a single frequency,
+rather than a broad spectrum (Rybicki & Lightman 1979):
+ωpeak ≃ 0.29ωc
+(E.18)
+with ωc = 3γ2eB/(mec) (averaged over pitch angles), e being
+the electron charge, and B being the magnetic field (CGS units).
+From this expression, one obtains:
+νpeak = ηeγ2B
+mec
+(E.19)
+where η = (0.29 × 3)/(2π) ≈ 0.14 is a dimensionless numer-
+ical factor. For a distribution of electrons, the synchrotron SED
+in δ-approximation, is given by (Dermer & Schlickeiser 2002):
+νFsyn
+ν (λ) = 4
+3πR3
+b
+cσTUB
+6πD2 ¯γ3Ne(¯γ)
+(E.20)
+where Rb is the radius of the emitting region, D is the dis-
+tance between the observer and the source, and ¯γ is the Lorentz
+Article number, page 18 of 20
+
+N. Aimar et al.: Magnetic reconnection plasmoid model for Sagittarius A* flares
+factor of electrons emitting synchrotron photons with the fre-
+quency ν. We obtain this Lorentz factor by expressing it from
+Eq. E.19:
+¯γ =
+�mecν
+ηeB
+�1/2
+(E.21)
+Substituting the expression for Ne(γ, t) (Eq. E.17), and the
+expression bc = (4σTUB)/(3mec), into the Eq. E.20, we finally
+obtain the time-dependent SED during the growth phase in δ-
+approximation:
+νFsyn
+ν (ν, t) =
+neR3
+b ¯γmec2
+12D2tgrowth κθ2
+��Ψ(¯γ) − Ψ(ξ(¯γ, t))� ,
+for ν < ˜ν(t)
+Ψ(¯γ),
+for ν ≥ ˜ν(t)
+(E.22)
+where ˜ν(t) = (ηeB)/(mecb2
+ct2) is the frequency corresponding
+to the condition ¯γ = 1/(bct).
+Appendix E.3: Evaluating the peak light curve flux
+To obtain a light curve during the growth phase at a specific fre-
+quency of interest ν∗, one has to compute νFsyn
+ν (ν = ν∗, t). To
+compute the peak light curve flux, one evaluates the quantity
+νFsyn
+ν (ν = ν∗, t = tgrowth).
+Appendix F: Additional Setup for July 22 flare
+We also find another setup which reproduce well the July 22
+flare data. In such scenario, the magnetic reconnection and so
+the plasmoid growth phase occurs way before the observing time
+and the flare is due to the beaming effect combined to the slow
+decrease of the cooling phase. The peak due to the growth phase
+occurs during the negative beaming part of the orbit resulting in
+a low flux comparable to the quiescent state.
+Parameter
+Symbol
+July 22 bis
+Plasmoid
+time in EMBLEM at zero observing time [min]
+temblem
+obs,0
+−53
+initial orbital radius [GM/c2]
+r0
+15
+polar angle [◦]
+θ
+135
+initial azimuthal angle [◦]
+ϕ0
+240
+initial radial velocity [c]
+vr,0
+0.01
+initial azimuthal velocity [c]
+vϕ,0
+0.5
+X position of Sgr A* [µas]
+x0
+0
+Y position of Sgr A* [µas]
+y0
+0
+PALN [◦]
+Ω
+160
+Table F.1. Second orbital parameters of the plasmoid model following
+a conical motion used for the comparison of the July 22 flares observed
+by Gravity Collaboration et al. (2018).
+Article number, page 19 of 20
+
+A&A proofs: manuscript no. main
+Fig. E.1. Time evolution of the electron distribution with EMBLEM (full lines) from t = 0 to t = tgrowth = 75 rg/c injecting a κ-distribution with
+Θe = 10 and κ = 4 and ˙ne = 5.106/tgrowth. The magnetic field strength is set to 30 Gauss resulting in a stationary regime for γ > 104 from the very
+beginning. This regime extends to lower γ values as time growths. For the estimation of the peak flux, we approximate the whole distribution (at
+t = tgrowth) by a simple Dirac at ¯γ represented by the dashed grey line.
+150
+100
+50
+0
+50
+100
+150
+X ( as)
+150
+100
+50
+0
+50
+100
+150
+Y ( as)
+Astrometry
+0
+5
+10
+15
+20
+25
+30
+t (min)
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+F/F(S2)
+Light Curve
+observed
+intrinsic
+data
+Fig. F.1. Data and plasmoid models of the flares from July 22, 2018. The left panels shows the astrometry of the flare while the right panel shows
+the light curves. Note that this is not the result of a fit. Contrary to the setup for Fig. 8, the growth time is shorter tgrowth = 50rg/c resulting into a two
+peak light curve with the first one occurring at t = −22 min but being mitigate by the negative beaming effect. The secondary peak which match
+the observed flare data shown here is due to the positive beaming during the cooling phase (as shown by the intrinsic light curve). The parameter
+set for this model is similar to the set of July 22 and is listed in Table F.1 with the same physical parameter as in Table 3 but with tgrowth = 50rg/c,
+Θe = 72 and B = 10 G.
+Article number, page 20 of 20
+
+105
+t = 3.0 GM/c3
+t = 15.0 GM/c3
+t = 27.0 GM/c3
+103
+t = 39.0 GM/c3
+t = 51.0 GM/c3
+t = 63.0 GM/c3
+101
+t = 75.0 GM/c3
+3
+, cm
+ 10-1
+10-3
+10-5
+10-7
+10-9
+100
+102
+105
+106
+103
+104
+107
+101
+Y
\ No newline at end of file
diff --git a/dNFKT4oBgHgl3EQfqS7x/content/tmp_files/load_file.txt b/dNFKT4oBgHgl3EQfqS7x/content/tmp_files/load_file.txt
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@@ -0,0 +1,1712 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf,len=1711
+page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  ©ESO 2023  January 30, 2023  Magnetic reconnection plasmoid model for Sagittarius A* flares  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar1, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Dmytriiev2, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Vincent1, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' El Mellah3, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Paumard1, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Perrin1, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Zech4  1 LESIA, Observatoire de Paris, Université PSL, CNRS, Sorbonne Université, Université Paris Cité, 5 place Jules Janssen, 92195  Meudon, France  e-mail: nicolas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='aimar@obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='fr  2 Centre for Space Research, North-West University, Potchefstroom, 2531, South Africa  3 Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  4 LUTH, Observatoire de Paris, CNRS, Université Paris Diderot, 5 place Jules Janssen, 92190 Meudon, France  Received September 09, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' accepted January 27, 2023  ABSTRACT  Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Sagittarius A*, the supermassive black hole at the center of our galaxy, exhibits episodic near-infrared flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The recent  monitoring of three such events by the GRAVITY instrument has shown that some flares are associated with orbital motions in the  close environment of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The GRAVITY data analysis points at super-Keplerian azimuthal velocity, while (sub-)Keplerian  velocity is expected for the hot flow surrounding the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We develop a semi-analytic model of Sagittarius A* flares based on an ejected large plasmoid, inspired by recent particle-in-  cell global simulations of black hole magnetospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We model the infrared astrometric and photometric signatures associated to this  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We consider a spherical macroscopic hot plasma region, that we call a large plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This structure is ejected along a  conical orbit in the vicinity of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This plasmoid is assumed to be formed by successive mergers of smaller plasmoids  produced through magnetic reconnection that we do not model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Non-thermal electrons are injected in the plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We compute the  evolution of the electron-distribution function under the influence of synchrotron cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We solve the radiative transfer problem  associated to this scenario and transport the radiation along null geodesics of the Schwarzschild spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We also take into account  the quiescent radiation of the accretion flow, on top of which the flare evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For the first time, we successfully account for the astrometric and flux variations of the GRAVITY data with a flare model  that incorporates an explicit modeling of the emission mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We find good agreement between the prediction of our model and  the recent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In particular, the azimuthal velocity of the plasmoid is set by the magnetic field line it belongs to, which is anchored in  the inner parts of the accretion flow, hence the super-Keplerian motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The astrometric track is also shifted with respect to the center  of mass due to the quiescent radiation, in agreement with the difference measured with the GRAVITY data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' These results support the picture of magnetic reconnection in a black hole magnetosphere as a viable model for Sagit-  tarius A* infrared flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Accretion, accretion disk - Magnetic reconnection - Black hole physics - Relativistic processes - Radiative transfer -  Radiation mechanisms: non-thermal  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Introduction  The Galactic Center hosts the compact radio source Sagittar-  ius A* (Sgr A*) with an estimated mass of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='297 million solar  masses at a distance of only 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='277 kpc (GRAVITY Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This makes the compact object associated to Sgr A*  the closest supermassive black hole (SMBH) candidate to Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Sgr A* is a low-luminosity accretion flow with an accretion rate  of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5) × 10−9M⊙ yr−1 and a bolometric luminosity of  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8 − 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2) × 1035 erg s−1 (Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Event Horizon  Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022b) and thus is accreting at a  very sub-Eddington rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' It has been the subject of numerous ob-  serving campaigns over the past two decades in order to test the  massive black hole (MBH) paradigm (see Gravity Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2020b)) and study the physics of radiatively inefficient ac-  cretion flows (RIAF) around SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Sgr A* shows a slow and low amplitude variability in ra-  dio (Lo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 1975;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Backer 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Krichbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Falcke  1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Michail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021b), in millimetre  and submillimetre (Mauerhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Macquart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Yusef-Zadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Marrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Brinkerink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Wielgus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022a), but also large amplitude and rapid  variability in near infrared (NIR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ghez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Hornstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Hora et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2014) and in X-rays  (Baganoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Nowak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Neilsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Barrière et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ponti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The flux distribution in  the NIR of Sgr A* has been the subject of numerous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Some claim a single state modeled by rednoise (Witzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2019) for the variability of Sgr A* while others  claim that there are two states for Sgr A* (Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Dodds-Eden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Witzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021): a continuously low amplitude variable state  called "quiescent state" and the "flare state" described by short  and bright flux with a typical timescale of 30 minutes to 1 hour  with a rate of ∼ 4 a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Multi-wavelength studies show that  when an X-ray flare is observed, there is a counterpart in NIR  suggesting a common origin but the reverse is not true (Fazio  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Moreover, the flare can also be observed in sub-  mm but with a time lag of several minutes (Eckart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2008,  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Dodds-Eden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Michail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Witzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 1 of 20  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='11874v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='HE]  27 Jan 2023  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  2021) following a dimming (Wielgus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Recently, the GRAVITY instrument (Gravity Collaboration  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Eisenhauer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2008, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Paumard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2008)  was able to resolve the motion of the NIR centroid during three  bright flare events, showing a clockwise, continuous rotation at  low inclination close to face-on (i ∼ 20◦) consistent with a re-  gion of emission located at a few gravitational radii rg = GM/c2  from the central black hole (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  These flares are thus powered very close to the event horizon  of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The exploration of a relativistic accretion re-  gion as close to the event horizon with high-precision astrometry  and imaging techniques like GRAVITY and the Event Horizon  Telescope (EHT) (Event Horizon Telescope Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2022a) promises important information for physics and astron-  omy, including new tests of the MBH paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Significant efforts have been made to explain the flares of  Sgr A*: rednoise (Do et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2009), hot spot (Hamaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Broderick & Loeb 2006), ejected blob (Vin-  cent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2014), star-disk interaction (Nayakshin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2004)  and disk instability (Tagger & Melia 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The GRAVITY  observations in 2018 (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018) sup-  port the hot spot model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' However, the physical origin of such  hot spots remains an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Instabilities in black hole  accretion disks are a candidate, for instance the triggering of  Rossby Waves Instabilities (RWI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Tagger & Melia 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Vin-  cent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Alternatively, it could originate from the dis-  sipation of electromagnetic energy through magnetic reconnec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This modification of the magnetic field topology results  from the inversion of the magnetic field orientation across a cur-  rent sheet which eventually breaks into magnetic islands called  plasmoids (Komissarov 2004, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Komissarov & McKinney  2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Loureiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Sironi & Spitkovsky 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Parfrey  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In the past  years, numerical simulations have repeatedly highlighted the  ubiquity of magnetic reconnection in BH magnetospheres, what-  ever the physical point of view adopted: global particle-in-cell  (PIC) simulations in Kerr metrics (El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' ), re-  sistive general-relativistic magneto-hydrodynamics (GRMHD)  simulations (Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Dexter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020a,b) or resis-  tive force-free simulations (Parfrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' PIC simulations  show that magnetic reconnection in the collisionless corona of  spinning BHs can accelerate leptons up to relativistic Lorentz  factors of γ ∼ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7 (El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022), sufficiently high  to generate the variable IR (and X-ray) emission (Rowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Scepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The GRMHD and PIC frameworks each have different lim-  itations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' GRMHD simulations describe the evolution of the ac-  cretion flow over long time scales, typically of the order of sev-  eral 100,000 rg/c, but they rely on a fluid representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Conse-  quently, they cannot self-consistently capture the kinetic effects  which are important to constrain dissipation, particle accelera-  tion and subsequent non-thermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' On the other hand,  PIC simulations provide an accurate description of the micro-  physics but at the cost of simulations which can only span a few  100rg/c in time and with limited scale separation between global  scales and plasma scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We develop a semi-analytical model, fed by the knowledge  accumulated by recent GRMHD and GRPIC simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The  aim is to condense into a reasonably small set of simple param-  eters the complex physics of GRMHD and GRPIC models, and  thus allow to probe a large parameter space within a reasonable  computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We also want to remain as agnostic as possi-  ble regarding the initial conditions of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In this context,  we discuss the interpretation of the Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2018) flare data paying particular attention to the following di-  agnostics:  – the marginally detected shift between the astrometric data  and the center-of-mass location;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – the tension between the data and the hot spot model used  by Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018), which assumes a Ke-  plerian orbit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – the physical origin of the rising and decaying phases of the  flare light curve in the context of magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The first point can be discussed in the context of a very sim-  ple hot-spot model and is the main topic of Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3 is  the core of our study and focuses on the second and third points  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' It presents a semi-analytical large plasmoid model due to  magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' It highlights in particular the impact of  considering a self-consistent evolution of the electron distribu-  tion function through kinetic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This section shows that  our plasmoid model is able to reasonably account for the Grav-  ity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) flare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The limitations of our  plasmoid model are discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The conclusions and  perspectives are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Quiescent flow impact on astrometry: shifting  and rotating the orbit  Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) used a hot spot model in an  equatorial circular orbit to fit the astrometry of three bright flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  They considered a constant radiation flux from the emitting re-  gion orbiting the black hole to fit the orbital motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The effect  of out-of-plane motion and orbital shear have also been stud-  ied by Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2020c) to model the flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  However, the impact of the quiescent radiation surrounding the  hot spot was not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The aim of this section is to  show that taking into account the quiescent radiation can lead to  shifting and rotating the orbit on sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note a 1-σ difference  between the center of the orbit of the hot spot and the center of  mass derived from the orbit of S2 in Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2018) which makes this shift marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  In this section, we will use a simplified hot spot model that  is sufficient to highlight the main effects of the quiescent radia-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This simple model will also allow us to introduce the most  important relativistic effects at play, that were already studied in  many previous works (Broderick & Loeb 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Hamaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' These reminders will be helpful when we turn to a more  complex hot spot model in the Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3, which is the main aim of  this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Simple hot spot + quiescent model for the flaring Sgr A*  The quiescent radiation of Sgr A* is modeled by means of the  torus-jet model as derived in Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019), to which we  refer for all details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Figure 1 resumes the main features of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The torus emits thermal synchrotron radiation, while the  flux emitted by the jet follows a κ distribution (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' a thermal  core with a power-law tail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The multi-wavelength spectrum of  the quiescent Sgr A* is well fit with this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The κ distribu-  tion emission from the jets dominates at most wavelength except  at the sub-mm bump where the flux comes mostly from the ther-  mal disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We summarize the best-fit parameters in Table 1, and  the resulting best-fit quiescent spectrum is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' More  details on the fitting procedure are given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' With  Article number, page 2 of 20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Scheme of the torus-jet model for the quiescent state in blue and flares in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Two trajectories are considered for the flare, which can either  rotate in the torus (hot spot model) or be ejected along the jet sheath (plasmoid model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The jet is parametrized by the angles θ1 and θ2 that describe  the angular opening of the radiation-emitting sheath, by the base height zb, the constant Lorentz factor Γ j, and the temperature power-law index  sT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The jet is symmetrical with respect to the equatorial plane, and axisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  these parameters, the flux of the torus-jet model at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 µm is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1  mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' It is in perfect agreement with the median quiescent dered-  dened flux provided by Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2020a) of  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3 mJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' At this wavelength, the torus is optically thin and  its emission is negligible compared to the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In the remainder  of this paper, where we focus only on the infrared band, we will  thus neglect the torus and consider a pure jet quiescent model,  unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The only relevant features of our quiescent model for the  rest of this paper are the location of its infrared centroid and  its NIR flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' As depicted in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2, the centroid  of our jet-dominated model lies very close to the mass center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We have checked that considering a disk-dominated model only  very marginally changes the position of the quiescent centroid  at low inclination (see the blue and green dots in the left panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Our conclusions are thus not biased by our particular  choice of a jet-dominated quiescent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The hot spot model is composed of a plasma sphere of radius  1 rg (fixed) with a uniform but time-dependent κ-distribution for  the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The emissivity jν and absorptivity αν coefficients  depend on the density, temperature, and magnetic field which we  considered uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We use the fitting formula of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2016) to compute these coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The typical light curve of  a flare is characterized by a phase with increasing flux and one  with decreasing flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We model this behavior by a Gaussian time  modulation on the density and temperature as follows  ne(t) = nhs  e exp  �������−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5 ×  �t − tre f  tσ  �2������� ,  (1)  Te(t) = T hs  e exp  �������−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5 ×  �t − tre f  tσ  �2�������  (2)  where tσ is the typical duration of the flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' As ne varies over  time (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 1), the magnetic field strength also varies since we set  a constant magnetization σ = B2/4πmpc2ne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Contrary to Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2020c), we keep the  circular equatorial orbit of Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) as  we assume that the hot spot is formed in the equatorial plane and  we do not take into account any shearing effect and assume a  constant spherical geometry of the hot spot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We summarize all  the input parameters of the hot spot in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Shifting the orbit on sky  Figure 3 shows the impact of taking into account the quiescent  radiation on the astrometry of the flare, considering the trivial  Article number, page 3 of 20  20  Jet  sheath  [j  Te(z) α (Z/z)  01  S-  10  EjectedPlasmoid  (aun w)  Torus  N  Orbiting Hot spot  10  20  0  5  10  15  20  X (M unit)A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  1010  1012  1014  1016  1018  [Hz]  1032  1033  1034  1035  L  [erg/s]  150  100  50  0  50  100  150  X [ as]  150  100  50  0  50  100  150  Y [ as]  10  27  10  26  10  25  10  24  10  23  I  [erg cm  2 s  1 sr  1 Hz  1]  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Left: Spectrum associated to the best-fit of the torus-jet model (see Table 1) for the quiescent state of Sgr A* (χ2  red = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='91 with ndof=27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The data are taken from Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2015) for ν < 50 GHz, Brinkerink et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2015) for the 2 points around 100 GHz, Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016) for the  492 GHz point, Marrone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2006) for the 690 GHz point, von Fellenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) for the far infrared upper limits, Witzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) for  the mid infrared data, and Baganoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2001) for the X-ray bow-tie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note that as in Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019), the X-ray data are not fitted as we  do not take into account bremsstrahlung nor Comptonized emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Right: Best-fit image at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 µm of the torus-jet model with a field of view  of 150 µas seen with an inclination of 20◦ and a Position Angle of the Line of Nodes (PALN) of π rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The color bar gives the values of specific  intensity in cgs units in log-scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The outer region emission comes from the backward jet’s part while the emission close to the center comes from  the forward part of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The centroid of the jet is represented by the blue dot at ∼(0, −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Parameter  Symbol  Value  Black Hole  mass [M⊙]  M  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='297 × 106  distance [kpc]  d  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='277  spin  a  0  inclination [◦]  i  20  Torus  angular momentum [rg/c]  l  4  inner radius [rg]  rin  8  polytropic index  k  5/3  central density [cm−3]  nT  e  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 × 109  central temperature [K]  T T  e  7 × 109  magnetization parameter  σT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='002  Jet  inner opening angle [◦]  θ1  20  outer opening angle [◦]  θ2  θ1 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5  jet base height [rg]  zb  2  bulk Lorentz factor  Γ j  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='15  base number density [cm−3]  nJ  e  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5 × 106  base temperature [K]  T J  e  3 × 1010  temperature slope  sT  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='21  κ index  κJ  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5  magnetization parameter  σJ  (fixed) 1  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Best fit parameters of the torus+jet quiescent model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We keep  the same geometrical parameters, bulk Lorentz factor and κ-index as  Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019) and we fit the base number density, base temper-  ature and temperature slope of the jet considering the correction (see  bellow) and the new value of the jet magnetization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The pa-  rameters of the torus are unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  case of a constant-emission hot spot, as well as the varying-  emission hot spot introduced in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Obviously, whether or not the hot spot intrinsic emission  varies, the first effect of adding a quiescent radiation is to shrink  the orbit’s size, because the overall centroid is moved towards  the quiescent radiation’s centroid, which always lies close to the  mass center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  A slightly less obvious effect is that, when the hot spot emis-  sion varies in time, the orbit can shift in the plane of sky, and no  longer be centered at the center-of-mass location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This is clearly  apparent on the solid-red orbit of the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This is  simply due to the time variation of the intensity ratio between the  quiescent and the hot spot radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' At early and late times, the  hot spot has a weaker emission than the quiescent component,  and the overall centroid coincides with the quiescent centroid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  As the hot spot emission increases and dominates, the overall  centroid will be driven towards it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Such a shift between the as-  trometric data and the center-of-mass position is visible at 1-σ  significance in the the Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We note another non-trivial effect appearing in the varying-  emission hot-spot orbit without any quiescent radiation (red-  dotted orbit in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The orbit is not closing, due to the time  delay between the primary and secondary images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Indeed, at the  end of the simulation, the flux from the secondary image is in-  trinsically higher than the primary (the emission times of the pri-  mary and secondary are different), and is amplified by the beam-  ing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' When the centroid is computed, the secondary image  has a larger impact at this time than before, resulting in a closer  centroid position relative to the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This astrometric im-  pact of the secondary image was already discussed by Hamaus  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Rotating the orbit on sky  It is not an easy task to disentangle the intrinsic time variability  of the hot spot from the variability due to the relativistic beam-  ing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Figure 4 illustrates the impact on astrometry and light  curve of playing with the relative influence between the intrin-  sic and beaming-related variability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Here, we simply change the  initial azimuthal coordinate ϕ0 of the hot spot along its orbit,  in order to change the dephasing between the time of the max-  imum intrinsic emission (t = tref) which is fixed, and the time  Article number, page 4 of 20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  100  75  50  25  0  25  50  75  100  X ( as)  100  75  50  25  0  25  50  75  100  Y ( as)  Astrometry  constant emission  Gaussian emission  no quiescent  with quiescent  0  10  20  30  40  50  60  t (min)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6  F/F(S2)  Light Curve  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Astrometry (left) and light curves (right) of the hot spot - jet model with two values for the quiescent state corresponding to no quiescent  (dashed lines) and the with quiescent state (full lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In shade of blue, the hot spot has a nearly constant emission (tσ >> torbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The effect of  beaming is reflected in the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In shade of red, the hot spot has a Gaussian time emission with tσ = 30 min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The parameters of the hot  spot are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We synchronise the maximum of beaming and the intrinsic maximum of the Gaussian modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The black, blue and  green dots in the left panels represent the position of Sgr A*, the jet’s centroid and the disk’s centroid respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Parameter  Symbol  Value  Hot spot  number density max [cm−3]  nhs  e  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='05 × 107  temperature max [K]  T hs  e  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='03 × 1010  time Gaussian sigma [min]  tσ  30  magnetization parameter  σhs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='01  κ-distribution index  κhs  5  orbital radius [rg]  Rhs  9  initial azimuth angle [◦]  ϕhs  0  90  Position Angle of the Line of Nodes [◦]  Ω  160  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Summary of parameters of the hot spot model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note that we  used the maximum number density and temperature of the jet best-fit in  Table 1 as reference and scale them for the hot spot by a factor 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  of the maximum constructive beaming effect (when the hot spot  moves towards the observer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The orbit rotates around the quies-  cent centroid following the variation of ϕ0 (left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The light curve is also strongly affected, reaching much brighter  levels when the intrinsic emission maximum is in phase with the  constructive beaming effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Here we show that the quiescent state of Sgr A* can have  significant impact on the observed astrometry by shrinking the  apparent orbit, creating a shift between the center of the latter  and the position of the mass center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' One should have these effects  in mind for the comparison to the flare data at the end of the  following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Plasmoid model from magnetic reconnection  In this section, we develop a semi-analytical hot-spot-like model  in order to interpret the rise and decay of Sgr A* flares, thus  going one step further with respect to the model we used in sec-  tion 2, where a Gaussian modulation of the emission is enforced  without physical motivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Black hole magnetospheres naturally lead to the develop-  ment of equatorial current sheets corresponding to a strong spa-  tial gradient of the magnetic field which changes sign at the  equator (Komissarov 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Komissarov & McKinney 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Par-  frey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Such a configuration  results in magnetic reconnection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' a change of the topology  of the field lines forming X points (Komissarov 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Loureiro  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Sironi & Spitkovsky 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This process is intrin-  sically non-ideal and thus can only be captured either by re-  sistive MHD or kinetic simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For suitable values of the  magnetic diffusivity, the reconnecting current sheet can break  into chains of plasmoids, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' magnetic islands separated by X  points (Loureiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Parfrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The reconnection rate (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' the typical rate at which magnetic  energy is dissipated into particle kinetic energy) is equal to the  ratio vrec/vout with vrec the velocity of matter injected into the  reconnection region, and vout the bulk outflow velocity of parti-  cles accelerated by the reconnection event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The outflow velocity  is of the order of the Alfven speed, vout ≈ vA, which is itself  of the order of the speed of light, vA ≈ c, for strongly magne-  tized environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The reconnection rate has been shown to be  rather independent of the details of the chosen parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For  PIC simulations, it lies around 10%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' vrec,PIC ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1vA ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1c,  for magnetized collisionless plasmas (Sironi & Spitkovsky 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2015), which are the typical  conditions in the inner flow surrounding Sgr A*1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' GRRMHD  simulations point towards a slower rate of around 1%, so that  vrec,MHD ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='01c (see the discussion in Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022),  1 It is likely that the accretion flow surrounding Sgr A* is in a Magnet-  ically Arrested Disk (MAD, see Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2003) regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' with  strong poloidal magnetic fields in the inner regions (Gravity Collabora-  tion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Dexter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 5 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  100  75  50  25  0  25  50  75  100  X ( as)  100  75  50  25  0  25  50  75  100  Y ( as)  Astrometry  orbit  0  5  10  15  20  25  30  t (min)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6  F/F(S2)  Light Curve  0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  0=90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  0=180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  0=270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  intrinsic  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Astrometry (left) and light curves (right) of the hot spot - jet model for four initial azimuthal angle ϕ0 of 0◦ in blue, 90◦ in orange, 180◦  in green and 270◦ in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The dashed black line shows the primary image centroid track with no quiescent jet (clock-wise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The jet dominates the  beginning and the end of the flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The observed centroids thus start and end close to the one of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The apparent orbits rotate around the latter  with ϕ0 as the maximum of emission occurs at different ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The Gaussian modulation which has a typical duration of tσ = 15 min (grey dashed line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  which is the same for the four the curves) is affected by relativistic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For negative X (right part of the astrometry), the beaming, combined  with relativistic Doppler effect, amplify the flux from the hot spot while in the positive X (left part of the figure), they lower it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The black dot in  the left panels represents the position of Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  but this applies to collisional environments, thus less similar to  Sgr A* vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Fresh plasma flows into the current sheet at the reconnec-  tion rate vrec, is accelerated by the electric field generated in  the current sheet, usually giving rise to power-law energy dis-  tributions of electrons (Sironi & Spitkovsky 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Inside the current sheet, the particles get trapped in the  plasmoids which act as particle reservoirs (Sironi & Spitkovsky  2014) which can merge in a macroscopic magnetic island, that  is, a large plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022), magnetic flux dis-  sipation through reconnection last for ∼ 100rg/c ∼ 30 min and  the resulting hot spot orbits for ∼ 500rg/c ∼ 150 min before it  disappears by losing its coherence through interaction with the  surrounding flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  In the global PIC simulation of El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022), the  authors study magnetic reconnection in the sheath of relativis-  tic jet working with magnetic field loops coupling the BH to the  accretion disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The resulting plasmoids evolve off-plane, propa-  gate away from the BH and are prone to merge with each other to  form macroscopic plasmoids susceptible to radiate high amounts  of energy under the form of non-thermal radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The under-  lying mechanism, first described by Uzdensky (2005) and de  Gouveia dal Pino & Lazarian (2005), relies on the accretion of  poloidal magnetic field loops onto a spinning BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Once the in-  ner footpoint of the loop reaches the BH ergosphere, the mag-  netic field line experiences strong torques due to the frame drag-  ging effect while its other footpoint on the disk rotates at the  local Keplerian speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Thus, the toroidal component of the mag-  netic field quickly grows in the innermost regions, propagates  upstream along the field line and leads to the opening of the  magnetic loop above a certain magnetic loop size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' On the out-  ermost closed magnetic field line (called the separatrix), a Y-  point appears where plasmoids form and flow away along an  inclined current sheet above the disk (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In the PIC sim-  ulations of El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022), a cone-shaped reconnecting  current sheet forms where vivid particle acceleration takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Electrons and positrons pile up into outflowing plasmoids where  they cool through synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This topological con-  figuration where some magnetic field lines anchored in the disk  close within the event horizon is coherent with what is seen in  resistive GRMHD simulations during the short episodes of flux  repulsion which separates different accretion regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' During  approximately 100 rg/c, these simulations show an essentially  force-free funnel surrounded by merging plasmoids formed in  the jet sheath and in the equatorial plane Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Chashkina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Plasmoid model from magnetic reconnection  The aim of this section is to develop a semi-analytic large plas-  moid model (which will be named plasmoid model), inspired  from the reconnection literature reviewed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The interest of  such a model, compared to state-of-the-art GRMHD or GRPIC  modeling, is twofold:  – it allows to remain as agnostic as possible regarding the  physical conditions close to Sgr A* and encapsulate into a  single model a large parameter space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – it allows to perform simulations within a limited computing  time, allowing to explore the large parameter space and com-  pare to astrometric and photometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Our hope is that such a model can not only be fed with the re-  sults of more elaborated simulations, but also bring constraints to  these simulations by determining what features of the modeling  are important in order to explain the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 6 of 20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  The main features of our model are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' in-  spired by the recent GRPIC results of El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Here, we consider a single plasmoid, which is modeled as a  sphere of hot plasma with a constant radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This macroscopic  plasmoid is understood as the end product of a sequence of mi-  croscopic plasmoid mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The spherical geometry is chosen  only for simplicity, given that current data are certainly unable  to make a difference between various geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Plasmoid motion  We consider that the magnetic reconnection event occurs close  to the black hole, and the resulting plasmoid is ejected along the  jet sheath (Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Thus, we  define a conical motion (as in Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2021)) defined by a con-  stant polar angle θ = θ0 and the initial conditions r0, θ0, ϕ0, vr0,  and vϕ0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The subscript 0 reflects the initial value of a given pa-  rameter in Boyer-Lindquist coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' As in Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2021),  we set a constant radial velocity vr = vr0 and the azimuthal veloc-  ity is defined through the conservation of the Newtonian angular  momentum  vϕ(t) = vϕ0  r2  0  r(t)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (3)  The azimuthal angle is obtained by integrating the previous  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3  ϕ(t) = ϕ0 + r2  0  vϕ0  vr  ( 1  r0  − 1  r(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (4)  In the GRPIC simulations performed by El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2022), the plasmoids are formed in the vicinity of the black hole  at the Y-point and are ejected into the black hole magnetosphere,  we thus restrict our study to vr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  An important feature of our model is the fact that the initial  azimuthal velocity of the plasmoid is naturally super-Keplerian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Indeed, the Y-point, from which the plasmoid is generated, is an-  chored to the equatorial plane of the accretion flow through the  separatrix field line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' So it will typically rotate at the Keplerian  speed corresponding to the foot point of the line, thus at a veloc-  ity higher than the Keplerian velocity corresponding to the initial  cylindrical radius of the plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Growth and cooling phases  We consider two phases in the lifetime of the plasmoid that aim  at modeling the ascending and descending phases of the ob-  served flare light curves:  – during the growth phase, which lasts a total time tgrowth, the  plasmoid continuously receives fresh accelerated particles at  a constant rate resulting from the merging of microscopic  plasmoids from reconnection into our large plasmoid which  mix with "old" electrons cooled by synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The growth time tgrowth corresponds to the lifetime of the  reconnection engine, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' the duration of magnetic flux dis-  sipation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – after t = tgrowth, the plasmoid enters the cooling phase: we as-  sume that magnetic reconnection is quenched and plasmoids  no longer merge so injection of fresh plasma stops and the  plasmoid cools by emitting synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We ne-  glect particle escape and adiabatic losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The duration of the growth phase is set both by the recon-  nection rate and the speed at which magnetic flux is advected by  the accretion flow into the current sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In Parfrey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2015),  the accretion of successive magnetic loops of opposite polarity  activates this process, with typical duration of transition of the  order of 100rg/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This duration is representative of the dissipa-  tion of the magnetic flux of one magnetic loop which is set by  both the size of the loop and the accretion speed, that the authors  fix to 2rg and c/200 respectively, and the reconnection rate, fixed  by the prescribed resistivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Resistive GRMHD simulations of  magnetically arrested disks (Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2003) bring support  to these values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020) but fail at reaching re-  connection rates realistically high (Bransgrove et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In  the more ab initio PIC simulations of El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022), the  reconnection rate is more realistic (∼ 10%, Sironi & Spitkovsky  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018) but due to the high computational cost  of the simulations, they did not work over duration long enough  to model the inward drift of the magnetic footpoints on the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  As a consequence, the reconnection rate is accurate but the fuel-  ing magnetic flux is artificially steady and act as an infinite reser-  voir over the ∼200rg/c covered by the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' A coupling  between GRMHD, force-free and PIC simulations to jointly de-  scribe the disk, the corona and the current sheet respectively is  still missing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In this context, we considered duration tgrowth of the  growth phase of the order of 100rg/c, corresponding to a typical  episode of magnetic flux dissipation set by the two rates at which  magnetic flux is advected into the current sheet and dissipated by  magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Evolution of the electron distribution  Next, we prescribe the emission/absorption mechanism in the  plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Most studies use chosen electron distributions, with  analytical prescriptions for their evolution at best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2021) use a fixed thermal distribution with a linear increase of  the number density for the rising part of the light curve and a  decrease of the temperature following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7 for the cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Scepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022) use a kappa distribution with an exponential  cutoff and a synchrotron cooling break for the plasma emission  generating X-ray flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' While their evolution of the plasma pa-  rameters (number density, temperature, magnetic field) is more  elaborate than in our model, their approximation is valid only  while injection and cooling are balanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' When the injection  stops, the shape of the electron distribution changes rapidly (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Here, we choose a different approach by evolving the  electron distribution in the plasmoid by solving the kinetic equa-  tion  ∂Ne(γ, t)  ∂t  = ∂  ∂γ  �  −˙γsyn Ne(γ, t)  �  + Qinj(γ),  (5)  where γ is the Lorentz factor of the electrons, Qinj is the injection  rate and Ne = dne/dγ is the electron number density distribution,  using the EMBLEM code (Dmytriiev et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The term  −˙γsynNe = 4σTUB  3mec (γ2 − 1)Ne  (6)  of the right hand side describes the synchrotron cooling of the  plasmoid particles, with UB = B2/(8π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In our approach, we do  not model the details of the magnetic reconnection process but  instead describe the supply of freshly accelerated particles to the  plasmoid by magnetic reconnection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Therefore, for the injection  rate Qinj(γ) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5, we use the following expression, assuming  Article number, page 7 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Sketch of magnetic reconnection in the black hole magneto-  sphere as shown by El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022) on which our plasmoid  model is built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' There are three types of magnetic field lines: the ones  threading the event horizon which goes to infinity, the ones anchored  in the disk which also go to infinity, and the separatrix which link the  disk and the black hole event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The latter form a Y-point and a  current sheet where chain of plasmoids are formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Here we model a  single plasmoid as the result of multiple mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  a constant injection rate:  Qinj(γ) =  ���������  4πNκ  e(γ)  tgrowth  in the growth phase,  0  in the cooling phase,  (7)  where Nκ  e(γ) is the distribution of the injected particles which  follows the kappa distribution i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' a thermal core with a power-  law tail following:  Nκ  e(γ) = N  4πγ(γ2 − 1)1/2  �  1 + γ − 1  κΘe  �−(κ+1)  (8)  with a normalization factor N = 1/2ne(κ−2)(κ−1)κ−2Θ−3  e , where  ne and Θe are the density and dimensionless temperature of the  injected plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The index κ is defined as  κ = p + 1 = Ap + Bp tanh (Cp βb) + 1  (9)  where  Ap = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7/ √σb, Bp = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7 σ−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='19  b  ,Cp = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4 σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='26  b  ,  (10)  following Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018), where p is  the powerlaw index of the non-thermal part of the distribution,  σb  ≫  1 is the plasma magnetization of the accelerating site  and βb ≪ 1 is the ratio of proton thermal pressure to magnetic  pressure of the accelerating site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' If the magnetization at the ac-  celerating site satisfies σb ≥ 100 (Crinquand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021), then κ  is in the range of [2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4] depending on βb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This implies that the  spectral index α (νFν ∝ να) is between −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5 which is in  perfect agreement with the measured indices for flares (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 32  in Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note that realistic values for the mag-  netization in the funnel region of Sgr A* can be orders of mag-  nitude higher than 100 (see Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022) which result in  a smaller parameter space for κ, closer to the low boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The bounds of the electron Lorentz factor are chosen to sat-  isfy γmin = 1 and γmax = 106 (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3 of Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  When solving the kinetic Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5, we assume that the density of  the plasmoid particles follows  ne(t) =  �  nmax  e  × t/tgrowth  in the growth phase,  nmax  e  in the cooling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (11)  Such high maximum Lorentz factor is needed to also power X-  ray flares with only synchrotron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' However, Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022)  suggest a lower maximum Lorentz factor γmax ∼ 104 in the plas-  moid as electrons cool during their travel time between the accel-  eration site and the later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Such lower value results in a marginally  lower flux in NIR, as most of the emission at this wavelength  comes from lower energy electrons, which can be compensated  with a slightly higher maximum number density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The temperature of the injected particles remains fixed in the  growth phase, and we define a uniform and time-independent  tangled magnetic field in the plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This is also a simplifying  assumption, and we intend to consider in future work the impact  of the magnetic field geometry on the polarized observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The EMBLEM code does not only solve for the evolu-  tion of the electron distribution, it also provides the associated  synchrotron emission and absorption coefficients of the plas-  moid particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We can thus compute an image of our plas-  moid scenario by backwards integrating null geodesics in the  Schwarzschild spacetime from a distant observer screen, and in-  tegrate the radiative transfer equation through the plasmoid by  reading the tabulated emission and absorption provided by EM-  BLEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This step is performed by means of the GYOTO2 ray-  tracing code (see Appendix B for details;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Paumard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The input parameters that we used for the  code are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' With these values of density and mag-  netic field strength, we obtain a magnetization inside the plas-  moid of σp ∼ 10−2 from the end of the growth phase since nei-  ther the density nor the magnetic field evolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Importance of evolving the electron distribution  One of the most important aspects of our model is the self-  consistent evolution of the electron distribution function, and  corresponding radiative transfer, in the plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Here, we il-  lustrate the importance of taking into account the evolution of  the electron distribution by comparing our model with another  reconnection plasmoid model inspired by Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We show in the top-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6 the evolution of the  electron distribution in our plasmoid model, for the parameters  listed in Tables 3 and 4 (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3 for details) and the associ-  ated spectral energy density (SED) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' During the growth  phase (tobs < 10 min), for γ > 103 the distribution is stationary  as the injection is balanced by the cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' After the end of the  growth phase, the shape of the distribution changes rapidly as  there is only cooling left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We show in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6  the light curve obtained with our model (in red) and with a model  inspired by Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2021) who do not take into account the  2 https://gyoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='fr  Article number, page 8 of 20  Current  sheet  Synchrotron  cooling  Plasmoid  v(rcyl)  yr  Injection of  O  accelerated  electrons  Merging  Y point  magnetic  islands  B lines  L  rcyl  rcyl  footpoint  plasmoid  (depends  on spin)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  non-thermal electrons, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' using a thermal distribution, with a  linear increase of the number density with a fixed temperature  during the growth phase and an analytical prescription for the  temperature decrease during the cooling phase using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7 and  assuming Θe = γ/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' While this model gives a similar intrinsic  light curve as our model, the dimensionless temperature required  is twice as high as ours (Θe = 109) with a magnetic field of  B = 20 G to cool faster lower energy electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The evolution of  the distribution with this model is shown in the bottom left panel  of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We do not need such high a temperature as most of  the emission comes from high-energy electrons which are non-  thermal in our model as suggested by PIC simulations (Rowan  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Our temperature could be even lower with a harder (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  lower) κ-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The cooling of the electron distribution through  synchrotron radiation is difficult to model properly and needs a  kinetic approach as we do in our plasmoid model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Comparing GRAVITY 2018 flare data with our plasmoid  model  This paper aims at checking whether we can reproduce with our  plasmoid model the general features of the observed light curve  and astrometry of the July 22, 2018 flare reported by Gravity  Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8 a comparison be-  tween the July 22, 2018 flare data observed by GRAVITY (in  black) and our plasmoid model (red line) with the parameters  listed in Tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For comparison, we show the intrinsic  light curve (dashed line) obtained by removing all the relativistic  effects (Doppler effect, beaming, secondary image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  This comparison is not the result of a fit and was obtained  by estimating the relevant parameters using simple physical ar-  guments:  – The rise time and slope of the light curve are mainly moni-  tored by (i) the growth time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (ii) our choice of linear evolu-  tion of the electron density (which enters the injection func-  tion),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (iii) the relativistic beaming effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' and thus (iv) the  initial azimuthal position of the plasmoid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' ϕ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' which has a  strong impact on beaming as illustrated in the right panel of  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – The decaying part of the light curve is monitored by the syn-  chrotron cooling time, thus by the magnetic field strength,  and by the beaming effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – The maximum of the light curve can be estimated by means  of an analytical formula that we derive in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  This maximum depends mainly on the maximum number  density ne,max, as well as on the temperature and κ-index of  the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' These parameters are degenerate and thus  not constrained with only the NIR flare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Nevertheless,  GRMHD (Dexter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Scepi  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022) and GRPIC simulations (El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022)  of magnetic reconnection suggest that the density in the plas-  moid is higher than its close environment in the current sheet,  of the order of the density at the base of the jet, close to the  event horizon, but lower than in the disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Still, the two re-  maining parameters (Θe, κ) which describe the shape of the  distribution are fully degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – The initial position and velocity of the plasmoid have a  strong impact on the astrometric trace on sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We guess the  initial azimuthal velocity based on the following reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The Keplerian velocity of the plasmoid at its initial cylindri-  cal radius is vKep ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='31c (for our choice of initial cylin-  drical radius given in Table 4, rcyl = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6 rg).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' However, as  Parameter  Symbol  Value  Plasmoid  magnetic field [G]  Bp  15  plasmoid radius [rg]  Rp  1  minimal Lorentz factor  γmin  1  maximum Lorentz factor  γmax  106  kappa distribution index  κ  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  kappa distribution temperature  Θe  50  maximum electron number density [cm−3]  ne,max  5 × 106  growth timescale [rg/c]  tgrowth  120  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Input parameters of the EMBLEM code for the simulation of the  electron distribution evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' These parameters are used for the July  22 flare of Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Parameter  Symbol  July 22  Plasmoid  time in EMBLEM at zero observing time [min]  temblem  obs,0  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6  initial cylindrical radius [rg]  rcyl,0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6  polar angle [◦]  θ  135  initial azimuthal angle [◦]  ϕ0  280  initial radial velocity [c]  vr,0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='01  initial azimuthal velocity [c]  vϕ,0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='45  X position of Sgr A* [µas]  x0  0  Y position of Sgr A* [µas]  y0  0  PALN [◦]  Ω  160  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Orbital parameters of the plasmoid model following a conical  motion used for the comparison of the July 22, 2018 flares observed by  Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  discussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1, our model naturally leads to a super-  Keplerian initial velocity to the plasmoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Indeed, the plas-  moid initial azimuthal velocity is that of the footpoint of the  separatrix (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Based on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8 of El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2022), we can determine the radius of the footpoint, rf p, of  a separatrix giving rise to a Y point located at a cylindrical  radius of ≈ 10 rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We find r f p = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5) rg, which trans-  lates in an orbital velocity vϕ,0 between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='41c and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='45c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The  upper bound of this interval compares well with the July 22  flare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note that this estimate of the initial azimuthal  velocity is anchored in the model of El Mellah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2022)  and thus depends on their choice of initial condition, in par-  ticular on the initial profile of their magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We note that the fiducial values proposed in Tables 3 and  4 represent a set of parameters with values that are inspired by  numerical simulations of reconnection which reproduce the key  observational features of the July 22 flare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This setup is not  unique and is not the result of a fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We let the exploration of the  full parameters space (freeing some fixed/constrained parame-  ters like maximum number density, growth time, inclination) to  a future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Nevertheless, our model disfavor low growth time  (tgrowth < 50rg/c) for this particular flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Overall, our plasmoid model jointly describes the astrom-  etry and the flux variation of the 22 July 2018 flare measured  by (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018) for the first time, consid-  ering a model with a specific emission prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Magnetic  reconnection is thus a viable scenario to explain Sgr A* flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 9 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  100  101  102  103  104  105  106  10  9  10  6  10  3  100  103  ne [cm  3]  kappa  Distribution  Thermal  PL (p=3)  100  101  102  103  104  105  106  10  9  10  6  10  3  100  103  ne [cm  3]  tobs=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='71 min  tobs=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='53 min  tobs=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='70 min  tobs=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='17 min  tobs=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='46 min  tobs=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='93 min  tobs=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='46 min  tobs=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='93 min  tobs=26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='81 min  tobs=29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='28 min  30  20  10  0  10  20  30  40  50  tobs [min]  0  10  20  30  40  50  60  70  80  tintrinsic [min]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0  F  [erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='cm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='s  1]  1e  11  Observing time  tgrowth  growth phase  cooling phase  EMBLEM intrinsic LC  Thermal model intrinsic LC  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (Top-left) Evolution of the electron distribution in our model with EMBLEM at each observing time of the July 22, 2018 flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The black  dotted line correspond to the injected κ electron distribution composed of a thermal core with a power law tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The parameters used are listed in  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (Bottom-left) Evolution of the electron distribution in the Thermal model inspired by Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2021) at each observing time of the July  22, 2018 flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The parameters used for this distribution are the same as in our model (listed in Table 3) but with the dimensionless temperature of  Θe = 109 and the magnetic field of B = 20 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (Right) Full intrinsic light curves of the two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Note that in this panel we plot the light curve  from the beginning of the growth phase while in the left panels we plot the distribution at the observed time of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8 (tobs = tintrinsic − 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6 min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  1010  1012  1014  1016  1018  1020  1022  [Hz]  10  19  10  17  10  15  10  13  10  11  F  [erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='cm  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='s  1]  tobs=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='71 min  tobs=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='53 min  tobs=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='70 min  tobs=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='17 min  tobs=14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='46 min  tobs=16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='93 min  tobs=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='46 min  tobs=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='93 min  tobs=26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='81 min  tobs=29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='28 min  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Intrinsic Spectral Energy Density (SED) evolution from radio to X-rays of our plasmoid model with the parameters listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Color  code the time as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Grey lines are the SED out of the observing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The SED peak occurs in NIR with this set of parameters, but with a  lower κ-index, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' a harder power law tail, the peak could reach X-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Here, X-rays flux drops quickly compare to the typical timescale of X-ray  flares as we consider only synchrotron cooling and not Synchrotron-Self Compton (SSC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note that the flux emitted by our plasmoid at 230  GHz is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 Jy which is the good order of magnitude of the variability associated to flares in sub-mm (Wielgus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 10 of 20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  150  100  50  0  50  100  150  X ( as)  150  100  50  0  50  100  150  Y ( as)  Astrometry  0  5  10  15  20  25  30  t (min)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='65  F/F(S2)  Light Curve  observed  intrinsic  data  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Data and plasmoid models of the flares from July 22, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The left panels shows the astrometry of the flare while the right panel shows  the observed (full line) and intrinsic (dashed line) light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The parameters of the model are listed in Tables 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Note that this is not the  result of a fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The black dot in the left panels represents the position of Sgr A* in GYOTO and the orange cross represent the position of Sgr A*  measured through the orbit of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Limitations of our plasmoid model  Our plasmoid model is vastly simplified with respect to the com-  plexity of realistic magnetic reconnection events in the environ-  ment of black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We review here its main limitations:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We consider a single plasmoid while the instability of thin  current sheets gives rise to a dynamic flow of merging  magnetic islands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Our argument for this simplification is  that the merging process is certainly very dependent on the  unknown initial conditions, and that the final, bigger and  brighter product of the cascade is likely to dominate the  observed signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The initial condition on the plasmoid’s velocity is simply  imposed for the radial motion, and based on a particular  GRPIC model as regards the azimuthal motion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The evolution of the plasma parameters (density, temper-  ature, magnetic field) are chosen to be either constant or  linear, so very simplified compared to a realistic scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  However, we consider that these evolutions are very likely  to be strongly dependent on the initial conditions of the flow,  so that they are weakly constrained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The values of almost all the parameters except mass and  distance of Sgr A* are poorly constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We choose a set  of values which are reasonable according to simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Future work is needed to investigate the details of the  parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We model the plasmoid by a homogeneous sphere for  simplicity from the circle plasmoid seen in 2D GRMHD  (Nathanail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2021) and PIC simulations (Rowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Werner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The 3D aspect of such plasmoid  is cylindrical (flux ropes) both in GRMHD (Bransgrove  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Nathanail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022)  and PIC (Nättilä & Beloborodov 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021)  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Thus, a realistic geometry of the flare source  is likely more complex than in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note that the  exact geometry of the flare is not relevant as we only track  the centroid position, as much the flare source is not too  extended, and we consider tangled magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' However,  the coherence time of the structure might be shorter in 3D  and might have an impact on the rise time of the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Further 3D simulations studies are needed to better model  the shape of the flux ropes and their evolution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We neglect any shearing of the plasmoid and consider  that it remains identical to itself throughout the simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Differential rotation is however likely to stretch the plasmoid  over its orbit and destroy its coherence (Hamaus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2020c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  vii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We consider a tangled magnetic field in the plasmoid  and thus do not consider the impact of the magnetic field  geometry on the observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The magnetic field geometry  of the quiescent flow is likely to be ordered and vertical if  Sgr A* is strongly magnetized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The magnetic field in the  plasmoid, which is our interest here, could be either helical  (plasmoids) or vertical for large flux tubes (Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  viii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' During a flare, the quiescent state can change in a non  axisymmetrical way (Ripperda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This will push  the centroid position of the quiescent further away from  the center-of-mass location which will affect the offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We  choose to use a static and axisymmetric quiescent model  during flare to avoid adding more degrees of freedom  which would lead to higher degeneracies, rather than clearer  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 11 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  ix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We choose a high maximum Lorentz factor γmax = 106 to  be able to power X-ray flares (but without any constrain  for this study).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' However, high energy photons lead to pair-  production and thus increase the number density in the plas-  moid which we do not take into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Despite these many limitations, we consider that our model  is very interesting for fitting flare data, because it allows to cover  a much broader set of physical scenarios than more elaborate  simulations that strongly depend on their assumptions regarding  the relevant physics and the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Conclusion and perspectives  This paper is mainly focused at developing a new plasmoid  model for Sgr A* flares, inspired by magnetic reconnection in  black hole environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Our semi-analytic model allows to  study a broad parameter space within a reasonable computing  time, thus being well suited for data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Our model considers non-thermal electrons accelerated by  magnetic reconnection and injected into a spherical large plas-  moid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We evolve the electron distribution through a kinetic equa-  tion taking into account synchrotron cooling and particle injec-  tion at a constant rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We show in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3) the  importance of taking into account the cooling of the electrons al-  ready in plasmoid during the growth phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Our model also natu-  rally accounts for a super-Keplerian velocity of the flare source,  through the dynamical coupling between the plasmoid and the  inner regions of the accretion flow through magnetic field lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  One of the main results of this paper is that for the first time  we model the astrometry and lightcurve of the flares measured  by (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018) by explicit modeling of  the emission zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Our conclusions regarding the three main points raised in the  introduction are the following:  – the marginally detected shift between the astrometric track  of Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) and the center of mass  might be due to the impact of the quiescent radiation of the  background accretion flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – a dynamical coupling between the plasmoid and the inner  accretion flow through closed magnetic field lines might  naturally account for the super-Keplerian speed obtained  by Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  – in general, a large plasmoid due to magnetic reconnection in  a thin current sheet in the black hole magnetosphere is a rea-  sonable model to account for the main features of the Gravity  Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018) observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3 shows that the temperature, density, and κ param-  eters of the plasmoid are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This degeneracy might be  removed by simultaneous observations of NIR and X-ray flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Moreover, synchrotron cooling leads to a translation of the elec-  trons from the NIR-emitting band to the millimeter-emitting  band, which could explain the sub-mm flare and its time lag with  respect to NIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We thus intend to consider the multi-wavelength  properties of our plasmoid model in future work, in order to bet-  ter assess its ability to account for the complete flare data set of  Sgr A*.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  A crucial recent observable of Sgr A* flares are the polariza-  tion QU loops (Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018, 2020d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Wiel-  gus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2022b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We also intend to study the polarized properties  of our plasmoid model and compare it to these recent constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' NA and FHV are very grateful to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Cerutti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Crinquand,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' von Fellenberg, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Gillessen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Masson, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ripperda, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Scepi, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Wiel-  gus for fruitful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This work was granted access to the HPC resources  of MesoPSL financed by the Region Ile de France and the project Equip@Meso  (reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir  supervised by the Agence Nationale pour la Recherche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' 2019, ApJS, 243, 26  Porth, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=', Younsi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=', Quataert, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' 2017, MNRAS,  467, 3604  Ripperda, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Bacchini, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' 2017, ApJ, 850, 29  Rybicki, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' & Spitkovsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2014, ApJ, 783, L21  Tagger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' 2006, ApJ, 636, L33  Uzdensky, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' 2018, ApJ, 862, 129  Werner, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content='  2018, MNRAS, 473, 4840  Wielgus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Marchili, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Martí-Vidal, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=' 2022a, ApJ, 930, L19  Wielgus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Moscibrodzka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018, ApJ, 863, 15  Witzel, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Martinez, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Willner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021, ApJ, 917, 73  Yusef-Zadeh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Roberts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Wardle, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Heinke, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', & Bower, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2006,  ApJ, 650, 189  Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Sironi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', & Giannios, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2021, ApJ, 922, 261  Article number, page 13 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  Appendix A: Torus - Jet model for the quiescent  state  We used the Torus-jet model of Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Their jet  model is restricted to an emitting sheath with an empty funnel in  agreement with GRMHD simulations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Mo´scibrodzka & Fal-  cke 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Ressler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Davelaar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In their model, Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019) define an opening  and closing angle θ1 and θ2 respectively and a base height zb to  define the geometry of the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The number density and the tem-  perature are defined by their values at the base height of the jet  (nJ  e and T J  e respectively) and their profiles along the jet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The pro-  file of the number density is fixed (∝ r−2  cyl with rcyl the projected  radius in the equatorial plane) and the one of the temperature is  set by the temperatures slope sT (∝ z−sT with z the height along  the vertical/spin axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The jet emits synchrotron radiation from  a κ electron distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The torus is defined by its central density and temperature  (nT  e and T T  e respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The profiles of these two quantities in  the torus are governed by the polytropic index k and its geome-  try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The latter is defined by the inner radius rin and the angular  momentum l but also on the metric (see Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019) for  more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Contrary to the jet, we consider that the electron  distribution of the torus is purely thermal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We use the same algorithm as in Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019) after  the correction of a small technical issue leading to an overestima-  tion of the number density and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' However, we change  the choice of the magnetization parameter in the jet sheath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' As  illustrated e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' by Porth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019), the jet sheath, which cor-  responds to the dominating emission region of the jet, coincides  with the transition between the highly-magnetized (σ ≫ 1) fun-  nel and the less-magnetized (σ ≪ 1) main disk body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Conse-  quently, we fix the magnetization to σ = 1 in the emitting jet  sheath, while Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019) used a low magnetization both  in the jet and in the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Our choice leads to a smaller density  in the jet sheath compared to Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We found a  best-fit with a χ2  red = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='91 using the same data points as Vincent  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The values are reported in Table 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2 shows  the associated spectrum and the image at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We obtain a  magnetic field strength of 257 G for the jet and 212 G at the  center of the torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='These values are higher than in Bower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2019) who considers a full thermal electron population with a  higher temperature but are of the same order as in Scepi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix B: Ray-tracing setup  We consider a Kerr black hole with dimensionless spin param-  eter a = 0, described in Boyer-Lindquist (t, r, θ, ϕ) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We work in units where the gravitational constant and the speed  of light are equal to 1, G = c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Radii are thus expressed in  units of the black hole mass M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We use the backward ray-tracing code GYOTO3 (Vincent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Paumard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2019) to compute images of our models  at different epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Each pixel of our image corresponds to a  direction on sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For each pixel of the image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' each direction),  a null geodesic is integrated backwards in time from the observer  towards the black hole, integrating along this path the radiative  transfer equation  dIν  ds = −ανIν + jν  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1)  3 https://gyoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='obspm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='fr  using the synchrotron emissivity jν and absorptivity αν coef-  ficients, considering various electron distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This  allows us to determine the flux centroid for each epoch and trace  its motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In addition to astrometry we also determine the total  flux emitted as the sum of the intensity weighted by the element  of solid angle subtended by each pixel, again, for each epoch  which allows us to plot the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The images produced are 1000x1000 pixels with a field of  view of 300 µas vertically and horizontally which makes a reso-  lution < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 µas2/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This high resolution is needed to resolve  properly the secondary image which has a very important role in  both astrometry and light curve (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We model the quiescent state of Sgr A* at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 µm with a jet  (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' However, computing an image of the jet is ∼ 200  times longer than an image of the flare source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' the hot spot or  the plasmoid, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 2 and 3 respectively) because the jet is much  more extended, and integrating the radiative transfer equation is  thus much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The absorption in the jet is negligible thus the  flux emitted by the flare which crosses the jet is fully transmit-  ted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We can compute a single image of the jet that we add to  each images of the hot spot a posteriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We then calculate the  total flux by summing the jet flux with the one of the flare at a  given time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The final centroid position is calculated by a simple  barycenter of the two centroids (jet and flare).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix C: Intrinsic emission of the Plasmoid  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1: Tests on the kinetic simulations  In our model, we follow the evolution of the electron distribu-  tion taking into account the injection of accelerated electrons by  the merging of small plasmoids into our large plasmoid and their  cooling through synchrotron radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The emissivity jν and ab-  sorptivity αν coefficients, needed to integrate the radiative trans-  fer Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1, are computed through the formula of Chiaberge &  Ghisellini (1999);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Rybicki & Lightman (1986) (with our nota-  tion)  jν(t) = 1  4π  � γmax  γmin  dγNe(γ, t)Ps(ν, γ),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1)  αν(t) = −  1  8πmeν2  � γmax  γmin  Ne(γ, t)  γl  d  dγ[γlPs(ν, γ)]  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2)  with  Ps(ν, γ) = 3  √  3  π  σTcUB  νB  x2  �  K4/3(x)K1/3(x) − 3  5 x[K2  4/3(x) − K2  1/3(x)]  �  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3)  where l = (γ2 − 1)1/2 is the electron momentum in units of  mec, x = ν/(3γ2νB), νB = eB/(2πmec) and Ka(t) is the modified  Bessel function of order a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We note that the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3, is already  averaged over pitch angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For standard distributions as thermal,  power-law and κ-distributions, these formulae are equivalent to  the fits of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016) (see Appendix D) that we used  for computing the quiescent synchrotron flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  As electrons start to cool as soon as they are injected in the  plasmoid, the full distribution is no more a κ-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' How-  ever, turning off the cooling during the growth phase allows us to  compare the results of EMBLEM to the fitting formulae of Pandya  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' As we inject electrons following their definition of  Article number, page 14 of 20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  2  4  6  8  10  12  14  16  18  0  1  2  3  4  5  6  7  I , max, erg cm  2 s  1 sr  1 Hz  1  1e  6  Pandya+16  2  EMBLEM (without cooling)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Specific intensity at the end of the growth phase (t = tgrowth =  75 rg/c) of a κ-distribution with ne = 5 × 106 cm−3, B = 10 G, κ = 4 for  a range of Θe computed from the full fitting formulae of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (2016) (black curve), by the EMBLEM code (red dots) and with the high-  frequency limit analytical expression (dashed blue curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We overplot  in light blue the range of Θe where Xκ > 2000 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' where the relative  error between the high frequency limit and the full formula is lower  than 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  the κ-distribution with a linear increase of the number density,  the two approach show similar results (see Appendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In our  cases, the absorption is very low, thus we neglect the absorption  in these tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We can derive an analytical formula for the spe-  cific intensity from the high frequency limit emissivity Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3  depending on the number density ne, the electron temperature  Θe and the magnetic field B in case the cooling is switched off  during growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We find in the case without cooling that, keeping  κ constant  Iν,max ∝ ne,max Θ κ−2 B κ/2, if Xκ > 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4)  We show the relative error of the maximum specific intensity  between the EMBLEM code (red dots) and the formulae of Pandya  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016) (black curve) depending on the electron tempera-  ture and the magnetic field in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We fix the others  parameters to ne = 5 × 106 cm−3, κ = 4, tgrowth = 75 rg/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The  values of EMBLEM are in good agreement with the previous ana-  lytical expression (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4) for low values of Θe and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For high  values, we are beyond the validity of our approximations (in the  intermediate frequency regime of the fitting formula, see Ap-  pendix D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Comparing the results of EMBLEM (without cooling)  with the full fitting formula of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016) (black curves)  results in an error lower than 5% showing the good agreement  between the two approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2: Analytical estimate of the intrinsic light curve  Next, we compute the light curve emitted by the plasmoid which  will be affected by the relativistic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' To reproduce a given  light curve, we can estimate the values of the parameters through  characteristic scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The growth time, which is a "free" param-  eter of the model, can be estimated from the light curve taking  into account the beaming effect and thus, depends on the orbital  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The synchrotron cooling time of an electron with  Lorentz factor γ in a magnetic field B reads  tcool = 3  4  mec  σTUBγ  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5)  0  5  10  15  20  25  30  B (in Gauss)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='00  I , max, erg cm  2 s  1 sr  1 Hz  1  1e  5  Pandya+16  B2  EMBLEM (without cooling)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Same as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 for a range of B and with Θe = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  with σT the electron Thomson cross section and UB the magnetic  energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In a Dirac spectrum approximation, the Lorentz  factor of an electron emitting an IR photon at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 µm is (Rybicki  & Lightman 1979)  ¯γ =  �νmec  ηeB  �1/2  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6)  with η = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='29 × 3)/(2π) a dimensionless numerical factor (see  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' One can thus constrain the magnetic field from  the synchrotron cooling time as  tcool = 19 ×  � B  20G  �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5 �  λ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2µm  �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5  min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7)  Taking into account the cooling of the electrons during the  growth phase leads to a lower flux than what we estimate from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Indeed, as electrons start to cool directly after being in-  jected, the integral of the distribution in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 and so the emis-  sivity will always be lower than without cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The key param-  eter of synchrotron cooling is the cooling time scale (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5),  which depends on the magnetic field strength and the initial en-  ergy of the electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' It has to be compared to the growth time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Indeed, with a low growth time, only high-energy electrons have  the time to cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Increasing the growth time will allow lower en-  ergy electrons to cool and so decrease even more the maximum  flux of the light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  With some approximations (see Appendix E for the details),  one can estimate the flux with cooling at t = tgrowth  νFsyn  ν (ν, t) =  neR3  b ¯γmec2  12D2tgrowth κθ2  ��Ψ(¯γ) − Ψ(ξ(¯γ, t))� ,  for ν < ˜ν(t)  Ψ(¯γ),  for ν ≥ ˜ν(t)  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8)  where ˜ν(t) = (ηeB)/(mecb2  ct2) is the frequency corresponding  to the condition ¯γ = 1/(bct) and  Ψ(x) =  �  1 + x − 1  κθ  �−κ �  x2(κ − 1) + 2x(κθ − 1) + 2θ(κθ − 2)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='9)  We plot the maximum light curve evolution relative to the  magnetic field with EMBLEM with (blue crosses) and without (red  Article number, page 15 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  0  10  20  30  40  50  60  B (in Gauss)  0  1  2  3  4  5  F , max, erg cm  2 s  1  1e  12  Analytic  EMBLEM (without cooling)  EMBLEM (with cooling)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Evolution of the maximum flux νFν(tgrowth) (at the end of the  growth phase tgrowth = 75 rg/c) as a function of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We show the results of EMBLEM without cooling (red crosses) as in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Allowing the cooling during the growth phase results in a  lower maximum flux (blue crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' One can estimates the maximum  flux with cooling through Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8 (see Appendix E for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This  equation is divided in two regimes, the equilibrium regime where the  magnetic field is strong enough to compensate the injection and creates  a stationary state (B ≥ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 G) and non stationary regime where not all  electrons has cooled at tgrowth (B < 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The relative error between  the analytical formula and the results of EMBLEM (with cooling) is  below 30% in the whole domain and below 7% in the non stationary  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  crosses) cooling during the growth phase and the previous an-  alytical expression in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3 (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' As expected, the  cooling becomes more significant with a strong magnetic field  until the maximum flux starts to decrease for very high values  (B > 100 G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The two regimes of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8 are clearly visible in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3 with a turning point at B = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This approximation  has a maximum relative error lower than 30% compared to the  results of EMBLEM in the stationary regime and below 7% for the  non stationary regime making it a good approximation estimate  the peak light curve flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix D: Computation of the synchrotron  coefficients for the Plasmoid  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1: Fitting formulae of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016)  In the hot spot model and for the test of EMBLEM, we used the  fitting formula of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016) to compute the emissivity  jν and absorptivity αν considering a well defined κ-distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  This distribution has two regimes, the low and high frequency  regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  In the low frequency limit, the emissivity is  jν,low = nee2νB  c  X1/3  κ  sin(θ) 4πΓ(κ − 4/3)  37/3Γ(κ − 2)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1)  and the absorption coefficient is  αν,low = nee2  νmec X−2/3  κ  31/6 10  41  2π  (Θe κ)10/3−κ  (κ − 2)(κ − 1)κ  3κ − 1  × Γ  �5  3  �  2  F1  �  κ − 1  3, κ + 1, κ + 2  3, −Θe κ  �  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2)  10  7  10  4  10  1  102  105  108  Xk  0  1  2  3  4  5  6  7  Js  Js, lo  Js  0  1  2  3  4  5  6  7  Js  Js, hi  Js  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Relative error between the low frequency regime (in red) -  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' the high frequency regime (in blue) - fit formulae Js,lo (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Js,hi)  of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016) and the full fit formula of the emission coefficient  Js in function of Xκ =  ν  νκ with νκ = νB (Θeκ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  where 2F1 is the hypergeometric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  In the high-frequency limit, the emissivity is  jν,high = nee2νB  c  X−(κ−2)/2  κ  sin(θ) 3(κ−1)/2  × (κ − 2)(κ − 1)  4  Γ  �κ  4 − 1  3  �  Γ  �κ  4 + 4  3  �  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3)  and the absorption coefficient is  αν,high = nee2  νmec X−(1+κ)/2  κ  π3/2  3  (κ − 2)(κ − 1)κ  (Θe κ)3  ×  �2Γ(2 + κ/2)  2 + κ  − 1  � �������  �3  κ  �19/4  + 3  5  ������� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4)  The final approximations for the emissivity and absorption  coefficient are  jν =  �  j  −xj  ν,low + j  −xj  ν,high  �−1/x j  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5)  αν =  �  α−xα  ν,low + α−xα  ν,high  �−1/xα  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6)  with x j = 3κ−3/2 and xα =  �  − 7  4 + 8  5κ  �−43/50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  The two frequency limits do not have the same dependence  on the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The frequency regime is defined by the di-  mensionless parameter Xκ = ν/νκ, with νκ = νB(Θeκ)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1  shows the relative error of the two regimes (the low frequency in  red and the high frequency in blue) compared to the final emis-  sion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' While at very high (respectively very low) Xκ,  the high frequency (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' low frequency) fitting formulae work  very well, there is a large frequency regime (10−2 ≲ Xκ ≲ 103),  hereafter intermediate regime, where both limits are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' At  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2 µm, Xκ > 1, while Θe κ ≲ 103 which correspond to our typ-  ical set of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This is why we used the high frequency  regime for our test our EMBLEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 16 of 20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2: Chiaberge & Ghisellini (1999) approximation  Modeling the synchrotron cooling of the electrons with a ther-  mal, power-law or κ distribution is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Indeed, the evolu-  tion of the energy of an electron which emits synchrotron radia-  tion is (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=', Rybicki & Lightman (1986))  γ(t) = γ0(1 + Aγ0t)−1  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7)  with A = 4  3  σT B2  8πmec,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8)  γ the Lorentz factor of the electron at time t, and γ0 the ini-  tial Lorentz factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The energy evolution strongly depends on  the initial energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The higher the initial energy of the electron,  the faster it will cool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Thus, the initial distribution we could im-  pose will quickly be deformed (see top-left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6) and  could not be modeled by one (or more) of the three distribution  of Pandya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2016) (thermal, power-law and/or κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  In order to properly model the cooling of electrons, we sim-  ulate the evolution of the electron distribution with injection  and synchrotron cooling (Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' These simulations give us the  electron distribution Ne(γ, t) at different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We compute the  emissivity jν and the absorptivity αν associated for a range of  frequencies from 106 to 1021 Hz following the formula of Chi-  aberge & Ghisellini (1999) (with our notation)  jν(t) = 1  4π  � γmax  γmin  dγNe(γ, t)Ps(ν, γ)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='9)  and the absorption coefficient follows  αν(t) = −  1  8πmeν2  � γmax  γmin  Ne(γ, t)  γp  d  dγ[γpPs(ν, γ)],  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='10)  where p = (γ2 − 1)1/2 is the electron momentum in units of mec  and Ps is the emissivity of a single electron (see C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  In order to obtain the emissivity and absorption coefficient at  any time and any frequency (to account the relativistic Doppler  effect for example), we made a bilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix E: Analytical approximation for Sgr A*  flare peak flux  Here we derive an analytical expression to compute the time-  dependent flux from Sgr A* flares during the growth phase, and  obtain an analytical formula for the peak flare flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For that, we  first obtain the approximate analytical form of the varying elec-  tron spectrum during the growth phase by solving the kinetic  equation, and then compute the approximate synchrotron SED  associated to the time-dependent electron spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1: Deriving time-dependent electron spectrum  during the growth phase  The kinetic equation describing the evolution of the electron  spectrum during the growth phase is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 5:  ∂Ne(γ, t)  ∂t  = ∂  ∂γ  �  bcγ2Ne(γ, t)  �  + Qinj(γ, ne, θ, κ)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1)  with the injection term Qinj(γ) given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 7 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8, and  synchrotron cooling term ˙γsyn = −bc(γ2 − 1) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 6), where  bc = (4σTUB)/(3mec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We use here an approximation ˙γsyn ≈  −bcγ2, as the bulk of the electrons producing the flare emission  are highly relativistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We use the method of characteristics to solve the kinetic  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We search for characteristic curves in the γ-t space,  along which our differential equation in partial derivatives be-  comes an ordinary differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Let us rewrite the ki-  netic equation in the following form, expanding the derivative  on the Lorentz factor:  ∂Ne(γ, t)  ∂t  + (−1)bcγ2 ∂Ne(γ, t)  ∂γ  = Qinj(γ) + 2γbcNe(γ, t)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2)  When restricting our equation to the characteristic curve  (γ(t),t), the full derivative of the electron spectrum over time,  by the chain rule, is:  dNe(γ, t)  dt  = ∂Ne(γ, t)  ∂t  + dγ  dt  ∂Ne(γ, t)  ∂γ  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3)  Comparing this to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2, we identify (−1)bcγ2 = dγ  dt , and  therefore along the chosen characteristic curve, our equation is  split into a system of two ordinary differential equations:  �dγ/dt = −bcγ2  dNe(γ, t)/dt = Qinj(γ) + 2γbcNe(γ, t)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4)  The solution of the first equation is (applying the initial con-  dition that γ(t = 0) = ξ):  γ(t) =  1  bct + 1/ξ  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5)  This equation defines a characteristic curve in the γ-t space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  We have chosen the initial point of the characteristic curve as  (ξ, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The physical meaning of ξ is the initial value of the  Lorentz factor of an electron before it starts undergoing the cool-  ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5 is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7, and describes how  the Lorentz factor of an individual electron evolves in time due  to synchrotron cooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' From this equation, the initial Lorentz  factor ξ is:  ξ = ξ(γ, t) =  1  1/γ − bct  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6)  This formula defines the initial Lorentz factor of the char-  acteristic curve that passes through a point (γ,t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We denote the  function Ne(γξ(t), t) = u(t) (electron spectrum along the charac-  teristic curve), and solve the second equation in the system:  du/dt − 2bcγ(t)u = Qinj(γ(t))  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7)  The generic solution of this linear differential equation is:  u(t) =  1  µ(t)  � t  0  µ(t′) Qinj(γ(t′)) dt′ +  C  µ(t)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='8)  Article number, page 17 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  with C being the integration constant, and the function µ(t)  being the integration factor, which is equal to:  µ(t) = exp  ��  −2bcγ(t)dt  �  =  1  (bct + 1/ξ)2  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='9)  As the electron spectrum at t = 0 is zero, we set the initial  condition u(t = 0) = 0, which results in C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Therefore, the  solution for u(t) is:  u(t) = (bct + 1/ξ)2  � t  0  (bct′ + 1/ξ)−2 Qinj(γ(t′)) dt′  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='10)  Now we have to return back from u(t) to Ne(γ, t), which is  achieved by substitution of the equation for ξ = ξ(γ, t) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6)  to the expression for u(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' After doing that, we obtain an expres-  sion for the electron spectrum at a moment of time t:  Ne(γ, t) = 1  γ2  � t  0  Γ2 Qinj(Γ) dt′  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='11)  with Γ = Γ(γ, t, t′) = �1/γ + bc(t′ − t)�−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We use here an  approximation for Qinj(Γ), and more specifically, for the kappa  distribution, to enable analytical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' As we are in the  relativistic regime, and the peak of the injection spectrum in our  case typically occurs at Lorentz factors γ ≫ 1, we can substitute  γ(γ2 − 1)1/2 with γ2 in the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This leads to some inaccu-  racies only at very low Lorentz factors, which virtually do not  contribute to the integral value, and do not contribute to the light  curve flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We therefore use for the injected spectrum:  Qinj(γ, ne, θ, κ) ≈  N  tgrowth  γ2  �  1 + γ − 1  κθ  �−(κ+1)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='12)  Now we can perform the analytical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We use the  variable substitution from t′ to Γ(γ, t, t′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In this case, the differ-  ential dt′ = −b−1  c Γ−2dΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Our integral (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='11) then becomes:  Ne(γ, t) =  N  γ2tgrowth  � t  0  Γ4  �  1 + Γ − 1  κθ  �−(κ+1)  dt′ =  = −  N  bcγ2tgrowth  � t  0  Γ2  �  1 + Γ − 1  κθ  �−(κ+1)  dΓ  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='13)  To solve the integral, we perform integration by parts, and  we obtain:  � t  0  Γ2  �  1 + Γ − 1  κθ  �−(κ+1)  dΓ = −  θκ  (κ − 2)(κ − 1)Ψ(Γ)  �����  t  0  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='14)  with  Ψ(x) =  �  1 + x − 1  κθ  �−κ �  x2(κ − 1) + 2x(κθ − 1) + 2θ(κθ − 2)  �  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='15)  We substitute the variable back from Γ to t′, with Γ(t′ =  0) = (1/γ − bct)−1 = ξ(γ, t) and Γ(t′ = t) = γ, as well as sub-  stitute the expression for the injection spectrum normalization,  N = (1/2)ne(κ − 2)(κ − 1)κ−2θ−3 (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8), and obtain:  Ne(γ, t) =  ne  2κθ2bcγ2tgrowth  �Ψ(γ) − Ψ(ξ(γ, t))�  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='16)  One has to consider separately a special case when the bct ≥  1/γ, as this leads to either ξ → ∞ or ξ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Obviously, the latter  situation is non-physical, as the Lorentz factor cannot be less  than unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Qualitatively, bct ≥ 1/γ → t ≥ 1/(bcγ) means that  the evolution time of an electron is longer than its cooling time-  scale, and in this regime the equilibrium between the injection  and cooling is already reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Therefore, one can easily see  that the time-dependent electron spectrum in the Lorentz factor  domain γ ≥ 1/(bct) will be “frozen” at the steady-state one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' A  steady-state solution corresponds to ξ → ∞, which results in  Ψ(ξ) → 0 (in case κ > 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Therefore, the final solution for the  time-dependent electron spectrum during the growth phase, is:  Ne(γ, t) =  ne  2κθ2bcγ2tgrowth  ��Ψ(γ) − Ψ(ξ(γ, t))� ,  for γ < (bct)−1  Ψ(γ),  for γ ≥ (bct)−1  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='17)  It is worth to note, that one can obtain the same steady-state  solution (the case γ ≥ 1/(bct)) by directly solving the kinetic  equation (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1) with ∂Ne  ∂t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' To find the electron spectrum  at the peak of the flare, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' at the moment when the injection is  stopped, one simply calculates Ne(γ, t = tgrowth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2: Deriving time-dependent synchrotron SED  during the growth phase  Now let us proceed to the SED and light curve computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We  use the so-called δ-approximation for the electron synchrotron  emissivity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This approximation assumes that a sin-  gle electron with a Lorentz factor γ emits at a single frequency,  rather than a broad spectrum (Rybicki & Lightman 1979):  ωpeak ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='29ωc  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='18)  with ωc = 3γ2eB/(mec) (averaged over pitch angles), e being  the electron charge, and B being the magnetic field (CGS units).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  From this expression, one obtains:  νpeak = ηeγ2B  mec  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='19)  where η = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='29 × 3)/(2π) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='14 is a dimensionless numer-  ical factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For a distribution of electrons, the synchrotron SED  in δ-approximation, is given by (Dermer & Schlickeiser 2002):  νFsyn  ν (λ) = 4  3πR3  b  cσTUB  6πD2 ¯γ3Ne(¯γ)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='20)  where Rb is the radius of the emitting region, D is the dis-  tance between the observer and the source, and ¯γ is the Lorentz  Article number, page 18 of 20  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Aimar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' : Magnetic reconnection plasmoid model for Sagittarius A* flares  factor of electrons emitting synchrotron photons with the fre-  quency ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' We obtain this Lorentz factor by expressing it from  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='19:  ¯γ =  �mecν  ηeB  �1/2  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='21)  Substituting the expression for Ne(γ, t) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='17), and the  expression bc = (4σTUB)/(3mec), into the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='20, we finally  obtain the time-dependent SED during the growth phase in δ-  approximation:  νFsyn  ν (ν, t) =  neR3  b ¯γmec2  12D2tgrowth κθ2  ��Ψ(¯γ) − Ψ(ξ(¯γ, t))� ,  for ν < ˜ν(t)  Ψ(¯γ),  for ν ≥ ˜ν(t)  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='22)  where ˜ν(t) = (ηeB)/(mecb2  ct2) is the frequency corresponding  to the condition ¯γ = 1/(bct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3: Evaluating the peak light curve flux  To obtain a light curve during the growth phase at a specific fre-  quency of interest ν∗, one has to compute νFsyn  ν (ν = ν∗, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' To  compute the peak light curve flux, one evaluates the quantity  νFsyn  ν (ν = ν∗, t = tgrowth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Appendix F: Additional Setup for July 22 flare  We also find another setup which reproduce well the July 22  flare data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' In such scenario, the magnetic reconnection and so  the plasmoid growth phase occurs way before the observing time  and the flare is due to the beaming effect combined to the slow  decrease of the cooling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The peak due to the growth phase  occurs during the negative beaming part of the orbit resulting in  a low flux comparable to the quiescent state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Parameter  Symbol  July 22 bis  Plasmoid  time in EMBLEM at zero observing time [min]  temblem  obs,0  −53  initial orbital radius [GM/c2]  r0  15  polar angle [◦]  θ  135  initial azimuthal angle [◦]  ϕ0  240  initial radial velocity [c]  vr,0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='01  initial azimuthal velocity [c]  vϕ,0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5  X position of Sgr A* [µas]  x0  0  Y position of Sgr A* [µas]  y0  0  PALN [◦]  Ω  160  Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Second orbital parameters of the plasmoid model following  a conical motion used for the comparison of the July 22 flares observed  by Gravity Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 19 of 20  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' main  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Time evolution of the electron distribution with EMBLEM (full lines) from t = 0 to t = tgrowth = 75 rg/c injecting a κ-distribution with  Θe = 10 and κ = 4 and ˙ne = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='106/tgrowth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The magnetic field strength is set to 30 Gauss resulting in a stationary regime for γ > 104 from the very  beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' This regime extends to lower γ values as time growths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' For the estimation of the peak flux, we approximate the whole distribution (at  t = tgrowth) by a simple Dirac at ¯γ represented by the dashed grey line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  150  100  50  0  50  100  150  X ( as)  150  100  50  0  50  100  150  Y ( as)  Astrometry  0  5  10  15  20  25  30  t (min)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='7  F/F(S2)  Light Curve  observed  intrinsic  data  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Data and plasmoid models of the flares from July 22, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The left panels shows the astrometry of the flare while the right panel shows  the light curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Note that this is not the result of a fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' Contrary to the setup for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' 8, the growth time is shorter tgrowth = 50rg/c resulting into a two  peak light curve with the first one occurring at t = −22 min but being mitigate by the negative beaming effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The secondary peak which match  the observed flare data shown here is due to the positive beaming during the cooling phase (as shown by the intrinsic light curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content=' The parameter  set for this model is similar to the set of July 22 and is listed in Table F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='1 with the same physical parameter as in Table 3 but with tgrowth = 50rg/c,  Θe = 72 and B = 10 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='  Article number, page 20 of 20  105  t = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0 GM/c3  t = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0 GM/c3  t = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0 GM/c3  103  t = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0 GM/c3  t = 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0 GM/c3  t = 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0 GM/c3  101  t = 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
+page_content='0 GM/c3  3  , cm  10-1  10-3  10-5  10-7  10-9  100  102  105  106  103  104  107  101  Y' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFKT4oBgHgl3EQfqS7x/content/2301.11874v1.pdf'}
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+arXiv:2301.01591v1  [math.CA]  4 Jan 2023
+Extremal polynomials on the n-grid
+Arno B.J. Kuijlaars
+January 5, 2023
+Abstract
+The n-grid En consists of n equally spaced points in [−1, 1] includ-
+ing the endpoints ±1. The extremal polynomial p∗
+n is the polynomial
+that maximizes the uniform norm ∥p∥[−1,1] among polynomials p of
+degree ≤ αn that are bounded by one on En. For every α ∈ (0, 1),
+we determine the limit of 1
+n log ∥p∗
+n∥[−1,1] as n → ∞. The interest in
+this limit comes from a connection with an impossibility theorem on
+stable approximation on the n-grid.
+1
+Statement of result
+The n-grid from the title refers to n equally spaced points in [−1, 1]
+En = {ξk,n = 2k−n−1
+n−1
+| k = 1, . . . , n},
+(1.1)
+ranging from −1 to 1. Let 0 < α < 1. This paper is about the determination
+of the limit of the expression
+1
+n log
+sup
+deg p≤αn
+∥p∥[−1,1]
+∥p∥En
+(1.2)
+as n → ∞, where the norms are uniform norms over the indicated sets, and
+the supremum is over univariate polynomials p of degrees at most αn that do
+not vanish identically. The result was already announced in the paper [12]
+from 2011. Renewed interest in it is due to [9].
+1
+
+Theorem 1.1. For every α ∈ (0, 1) the limit
+lim
+n→∞
+1
+n log
+sup
+deg p≤αn
+∥p∥[−1,1]
+∥p∥En
+= C(α)
+(1.3)
+exists and is equal to
+C(α) = (1 + α) log(1 + α) + (1 − α) log(1 − α)
+2
+.
+(1.4)
+The limit C(α) is positive and strictly increasing with α. There is a nice
+Taylor expansion
+C(α) =
+∞
+�
+k=1
+α2k
+2k(2k − 1).
+The odd Taylor coefficients vanish since the power series defines an odd
+function. The even Taylor coefficients are positive and therefore C(α) ≥ 1
+2α2.
+It is also worth noticing that C(α) → log 2 as α → 1−. For α = 1 however,
+we have that the limit in (1.3) is +∞, since for each n, there is a non-zero
+polynomial of degree n that vanishes on the n-grid.
+Discussion
+The limit (1.3) shows that a polynomial of degree ≤ αn that is
+bounded on En can be exponentially large somewhere in the interval [−1, 1].
+Namely, if |p(ξk,n)| ≤ 1 for each k = 1, . . . , n, then |p(x)| at some x ∈ [−1, 1]
+can be as large as en(C(α)+o(1)) as n → ∞, and the constant C(α) is sharp. The
+result is related to earlier work of Coppersmith and Rivlin [5] who showed
+that there exist universal constants C2 > C1 > 1 such that for n large enough,
+and for every d ≤ n − 1,
+Cd2/n
+1
+≤ sup
+deg p≤d
+∥p∥[−1,1]
+∥p∥En
+≤ Cd2/n
+2
+.
+(1.5)
+The inequalities (1.5) show that polynomials of degree d ≤ c√n that are
+bounded by one on En are uniformly bounded on [−1, 1] with a constant
+that only depends on c. However, if d grows proportionally with n then (1.5)
+shows that polynomials that are bounded by one on En may be exponentially
+large on [−1, 1], and this behavior is made more precise in the limit (1.3).
+The comparisons of the two uniform norms ∥ · ∥En and ∥ · ∥[−1,1] arises
+naturally when studying approximation or interpolation methods for analytic
+functions based on function values on the n-grid. There is a trade-off between
+2
+
+convergence and stability properties that was made precise in the impossi-
+bility theorem of [12, Theorem 3.1]. For example, exponential convergence
+as n → ∞ comes together with exponential instability. The proof of the
+impossibility theorem in [12] relies on the Coppersmith-Rivlin inequalities
+(1.5). For recent work in this direction we refer to [1, 9].
+It was observed in [12, section 4] that the phenomenon that polynomials of
+degree d ≈ αn can be much larger on [−1, 1] than on En may be understood
+in terms of potential theory. Here one thinks of a polynomial in terms of
+its zeros. A polynomial may be small at certain gridpoints in En by simply
+having a zero very close to these gridpoints. However, since there are more
+gridpoints than zeros this cannot happen for every gridpoint. The extremal
+polynomial p∗
+n for (1.2) will place a certain fraction of its zeros extremely
+close to gridpoints lying in a subset S of [−1, 1] (with S depending on α).
+Then p∗
+n is small at the gridpoints in S but not necessarily in between, and
+in fact it has high oscillations in S. Following [2, 7] we call S the saturated
+region. The non-saturated region [−1, 1]\S has considerably fewer zeros than
+gridpoints. The extremal polynomial p∗
+n is not only small at the gridpoints
+in [−1, 1] \ S but it is of comparable size over the full set [−1, 1] \ S, see [14]
+for very precise estimates.
+This phenomenon was first described by Rakhmanov [13] for orthogonal
+polynomials on the n-grid, or more generally, for polynomials that minimize
+a discrete Lp norm on En. These polynomials have their zeros in [−1, 1] and
+they are separated by the gridpoints, in the sense that in between any two
+distinct zeros there is at least one gridpoint. In the limit n → ∞ the zeros of
+the extremal polynomials of degree ⌊αn⌋ considered in [13] have a limiting
+distribution µα (depending only on α) that is characterized by a constrained
+equilibrium problem from potential theory. The measure µα has a density
+with respect to Lebesgue measure on [−1, 1] with dµα
+dx ≤ 1
+2 where 1
+2 is the
+limiting density of the gridpoints as n → ∞.
+The saturated region S is
+where the equality dµα
+dx = 1
+2. holds. We give details in section 2.2 below. The
+extremal polynomials p∗
+n for the extremal problem in (1.3) have the same
+limiting zero distribution µα as n → ∞, as we will show in this paper. Also
+in other aspects they behave similarly to the Lp-extremal polynomials on the
+n-grid, and this will be the clue to the proof of Theorem 1.1.
+3
+
+Outline of the proof
+The extremal polynomial for (1.2) is a polynomial
+p∗
+n of degree ≤ αn such that
+∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+=
+sup
+deg p≤αn
+∥p∥[−1,1]
+∥p∥En
+.
+(1.6)
+The proof of (1.3) then naturally comes in two steps. In the first step we
+prove the lower bound
+lim inf
+n→∞
+1
+n log ∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+≥ C(α)
+(1.7)
+and in the second step the corresponding upper bound
+lim sup
+n→∞
+1
+n log ∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+≤ C(α).
+(1.8)
+The lower bound comes from considering the L∞-extremal polynomials P ∗
+n
+on En, where P ∗
+n is the monic polynomial of degree ⌊αn⌋ that minimizes the
+uniform norm ∥ · ∥En. Using the results from [7, 13] and some additional
+calculations we prove in section 2 that
+lim
+n→∞
+1
+n log ∥P ∗
+n∥[−1,1]
+∥P ∗n∥En
+= C(α),
+(1.9)
+and this implies the lower bound (1.7).
+The upper bound (1.8) is proved in section 3. It comes from a study
+of the zeros of the extremal polynomials p∗
+n satisfying (1.6). We show in
+Lemma 3.1 that the zeros are real and simple and at least ⌊αn⌋ − 1 zeros
+are in [−1, 1] where they are separated by the gridpoints. Note that one zero
+could be in R \ [−1, 1]. We then use potential theoretic arguments to show
+that the limiting distribution of the zeros of p∗
+n is equal to the constrained
+equilibrium measure µα. Along the way we prove that µα is the maximizer
+of a functional J that we define in (3.5) with J(µα) = C(α), which leads to
+the upper bound (1.8).
+We finally note that discrete orthogonal polynomials and the constrained
+equilibrium problem also play a role in the analysis of iterative methods from
+numerical linear algebra [3, 10, 11], and the asymptotic analysis of integrable
+systems [6], random matrices and random tiling models [2, 4].
+4
+
+2
+Proof of the lower bound
+2.1
+Extremal polynomials on the n-grid
+As explained above, we are going to consider the monic polynomial P ∗
+n of
+degree ⌊αn⌋ such that
+∥P ∗
+n∥En =
+min
+deg P = ⌊αn⌋
+P is monic
+∥P∥En.
+(2.1)
+We are going to show that the limit (1.9) holds.
+Rakhmanov [13] considered polynomials P of degree n that are monic
+(leading coefficient equal to 1) and that minimize either the uniform norm
+∥P∥EN, or a discrete p-norm on EN among all such polynomials. The interest
+is in their asymptotic behavior as both n, N → ∞ with n/N → c < 1. The
+equispaced n-grid (1.1) is actually only a special case of far more general
+discrete sets that are considered in [13]. Compared to [13] we change N �→ n,
+n �→ ⌊αn⌋, c �→ α.
+2.2
+Limiting behavior of zeros
+The zeros of P ∗
+n are real and simple. They belong to the interval (−1, 1),
+where they are separated by the nodes in En, see [7, 13]. To P ∗
+n we associate
+the normalized zero counting measure
+νn = 1
+n
+⌊αn⌋
+�
+k=1
+δxk,n,
+where xk,n for k = 1, . . . , ⌊αn⌋, denote the zeros of P ∗
+n. Note that we nor-
+malize with the factor 1/n while the degree of P ∗
+n is ⌊αn⌋. Thus νn is not a
+probability measure but rather has a total mass ⌊αn⌋
+n
+Rakhmanov [13, Theorem 2], see also [7, Theorem 3.3], proved that the
+weak∗ limit
+νn
+∗→ µα
+(2.2)
+exists, where µα is the measure on [−1, 1] with density [13, Theorem 1]
+dµα
+dx =
+
+
+
+
+
+
+
+1
+2,
+for x ∈ [−1, −r] ∪ [r, 1],
+1
+π arctan
+�
+α
+√
+r2 − x2
+�
+,
+for x ∈ [−r, r],
+(2.3)
+5
+
+where
+r = r(α) =
+√
+1 − α2.
+(2.4)
+The density on [−r, r] can alternatively be written as
+dµα
+dx = 1
+2 − 1
+π arccos
+�
+α
+√
+1 − x2
+�
+,
+for x ∈ [−r, r].
+(2.5)
+The measure µα belongs to the class
+Mα,σ := {µ | ∫ dµ = α, 0 ≤ µ ≤ σ}
+(2.6)
+where
+dσ = 1
+2χ[−1,1](x)dx
+(2.7)
+denotes the Lebesgue measure restricted to [−1, 1] with normalization such
+that
+´
+dσ = 1. The upper constraint µα ≤ σ comes from the fact that the
+zeros of P ∗
+n are separated by the nodes ξk,n in the equispaced grid En.
+Rakhmanov also characterized µα in terms of notions from logarithmic
+potential theory [15]. Let
+I(µ) =
+¨
+log
+1
+|x − y|dµ(x)dµ(y)
+(2.8)
+and
+Uµ(x) =
+ˆ
+log
+1
+|x − y|dµ(y)
+(2.9)
+denote the logarithmic energy and the logarithmic potential of a measure µ,
+respectively. Then
+I(µα) =
+min
+µ∈Mα,σ I(µ)
+(2.10)
+and µα is the unique minimizer within the class (2.6). Furthermore, there is
+a constant ℓα such that
+Uµα(x)
+�
+= ℓα,
+for x ∈ supp(σ − µα),
+≤ ℓα,
+for x ∈ [−1, 1],
+(2.11)
+and µα is the only measure µ in Mα,σ such that Uµ(x) = ℓ is constant on
+supp(σ − µ) and Uµ(x) ≤ ℓ on [−1, 1] for a certain constant ℓ.
+Because of the upper constraint µ ≤ σ, the measure µα is called a con-
+strained equilibrium measure, see [2, 7]. The saturated region S is where
+dµα
+dx = dσ
+dx = 1
+2 and in view of (2.3) we have S = [−1, −r] ∪ [r, 1]. The con-
+straint is not active in the region where dµα
+dx < 1
+2. This is the non-saturated
+region and its closure is supp(σ − µα) = [−r, r].
+6
+
+2.3
+Two lemmas
+The connection between potential theory and the asymptotics theory of poly-
+nomials is well-known. If P is a monic polynomial and
+ν = 1
+n
+�
+x:P (x)=0
+δx
+is its normalized zero counting measure (each zero is included in the sum
+according to its multiplicity), then
+1
+n log |P(x)| = −Uν(x).
+If (Pn)n is a sequence of monic polynomials, and (νn)n is the corresponding
+sequence of normalized zero counting measures then the convergence of (νn)n
+contains information on the nth root asymptotic behavior of the polynomials.
+We need two such results.
+Lemma 2.1.
+(a) Let (Pn)n be a sequence of monic polynomials, deg Pn ≤
+αn, having real and simple zeros, such that the zeros of Pn are separated
+by the points of En for every n. Suppose that the sequence of normalized
+zero counting measures (νn)n where νn = 1
+n
+�
+x:Pn(x)=0 δx, has a weak∗
+limit µ as n → ∞. Then µ ≤ σ, and
+lim inf
+n→∞
+1
+n log ∥Pn∥En ≥ −
+min
+x∈supp(σ−µ) Uµ(x).
+(2.12)
+(b) If (P ∗
+n)n is the sequence of extremal polynomials satisfying (2.1) then
+νn
+∗→ µα as n → ∞, and equality holds
+lim
+n→∞
+1
+n log ∥P ∗
+n∥En = −
+min
+x∈supp(σ−µ) Uµα(x).
+(2.13)
+Proof. Part (a) is Lemma 4.2 of [13], where it is stated under the assumption
+that the zeros are in [−1, 1]. See Lemma 5.5 in [7] for the statement without
+this extra assumption. Part (b) is in [13, Theorem 2] or [7, Theorem 3.3].
+Part (b) of Lemma 2.1 will be used in the proof of the lower bound,
+while part (a) will be used in the proof of the upper bound, see the proof of
+Proposition 3.2.
+7
+
+Remark 2.2. Note that the logarithmic potential Uµ of a positive measure
+µ is a lower semi-continuous function [15] and therefore its minimum over a
+compact (as in (2.12) and (2.13), as well as in (2.14) below) exists.
+In the situation of Lemma 2.1, however, the logarithmic potential Uµ
+is actually continuous.
+This follows from µ ≤ σ and the fact that Uσ is
+continuous.
+Indeed, Uσ−µ is lower semi-continuous, and therefore Uµ =
+Uσ − Uσ−µ is upper semi-continuous as well, hence continuous.
+The second lemma is probably well-known, but I could not find an ap-
+propriate reference for it.
+Lemma 2.3. Suppose (Pn)n is a sequence of monic polynomials, deg Pn ≤
+αn, such that the zeros of all Pn are in a compact set. Suppose that the
+sequence of normalized zero counting measures (νn)n has a weak∗ limit µ as
+n → ∞. Then
+lim
+n→∞
+1
+n log ∥Pn∥[−1,1] = − min
+x∈[−1,1] Uµ(x).
+(2.14)
+Proof. By the principle of descent [15] we have
+Uµ(x∗) ≤ lim inf
+n→∞ Uνn(xn)
+whenever xn → x∗. Since |Pn(x)| = e−nUνn(x), this means that
+lim sup
+n→∞
+1
+n log |Pn(xn)| ≤ −Uµ(x∗) ≤ − min
+x∈[−1,1] Uµ(x),
+whenever (xn)n is a convergent sequence with a limit x∗ ∈ [−1, 1]. Taking
+xn ∈ [−1, 1] with |Pn(xn)| = ∥Pn∥[−1,1] and passing to convergent subse-
+quences if necessary, we then find
+lim sup
+n→∞
+1
+n log ∥Pn∥[−1,1] ≤ − min
+x∈[−1,1] Uµ(x).
+(2.15)
+By the lower envelope theorem [15] we have
+Uµ(x) = lim
+n→∞ Uνn(x)
+q.e.
+(2.16)
+where q.e. means quasi everywhere, i.e., the exceptional set is a polar set (a
+small set for potential theory). The limit (2.16) means
+lim
+n→∞
+1
+n log |Pn(x)| = −Uµ(x)
+q.e.
+(2.17)
+8
+
+Let x0 ∈ [−1, 1] be such that
+Uµ(x0) =
+min
+x∈[−1,1] Uµ(x).
+(2.18)
+Note that the minimum exists since Uµ is lower semicontinuous. Let ε > 0.
+Then we claim that
+{x ∈ [−1, 1] | Uµ(x) < Uµ(x0) + ε}
+(2.19)
+is not a polar set. This is easy to see if Uµ is a continuous function, since
+then (2.19) contains a non-empty interval and this is not a polar set, see e.g.
+[8, Example 5.2.7]. If Uµ is not continuous, then we can come to the same
+conclusion, if we use certain more advanced results from potential theory, in
+particular around thinness and the fine topology, which we will not explain
+here. The set {x ∈ C | Uµ(x) < Uµ(x0) + ε} is an open neighborhood of x0
+in the fine topology, and [−1, 1] is not thin at x0 ∈ [−1, 1], see [8, Corollary
+6.7.8]. Therefore (2.19) is not a polar set.
+Knowing that (2.19) is not polar, we conclude from (2.17) that there
+exists x1 ∈ [−1, 1] with Uµ(x1) < Uµ(x0) + ε and the limit (2.17) holds for
+x = x1. Then by the above and (2.18)
+lim inf
+n→∞
+1
+n log ∥Pn∥[−1,1] ≥ lim
+n→∞
+1
+n log |Pn(x1)| = −Uµ(x1)
+> −Uµ(x0) − ε = − min
+x∈[−1,1] Uµ(x) − ε.
+(2.20)
+Then the lemma follows from (2.15) and (2.20), since ε > 0 is arbitrary.
+2.4
+Conclusion of the proof of the lower bound
+We apply the two lemmas to the extremal polynomials P ∗
+n satisfying (2.1).
+By Lemma 2.1 (b) we have
+lim
+n→∞
+1
+n log ∥P ∗
+n∥En = − min
+x∈[−r,r] Uµα(x),
+(2.21)
+since supp(σ − µα) = [−r, r] by (2.3). From Lemma 2.3 and (2.2) we get
+lim
+n→∞
+1
+n log ∥P ∗
+n∥[−1,1] = − min
+x∈[−1,1] Uµα(x).
+(2.22)
+9
+
+Combining (2.21) and (2.22) we obtain
+lim
+n→∞
+1
+n log ∥P ∗
+n∥[−1,1]
+∥P ∗
+n∥En
+= min
+x∈[−r,r] Uµα(x) − min
+x∈[−1,1] Uµα(x).
+(2.23)
+The limit (1.9) and thereby the lower bound (1.7) follows from (2.23) and
+the following proposition.
+Proposition 2.4. We have
+C(α) = min
+x∈[−r,r] Uµα(x) − min
+x∈[−1,1] Uµα(x)
+(2.24)
+with C(α) as in (1.4) above.
+Proof. The derivative of Uµα is a principal value integral that can be calcu-
+lated explicitly. The result is
+d
+dxUµα(x) = −
+ dµα(y)
+x − y
+= 1
+2 log
+�
+1 − x2�
+− log
+�
+α +
+√
+x2 − r2
+�
+,
+for r < x < 1.
+(2.25)
+We give the details of the calculations for (2.25) later, after finishing the
+main line of the argument.
+The derivative (2.25) is negative for r < x < 1.
+Therefore (and by
+symmetry) the minimum of Uµα(x) over [−1, −r] ∪ [r, 1] is at x = ±1. Also
+Uµα is constant on [−r, r] by (2.11). Hence
+min
+x∈[−r,r]Uµα(x) − min
+x∈[−1,1] Uµα(x) = Uµα(r) − Uµα(1).
+In view of (2.25) and the fundamental theorem of calculus, we arrive at (2.24)
+provided that
+C(α) =
+ˆ 1
+r
+�
+log
+�
+α +
+√
+x2 − r2
+�
+− 1
+2 log
+�
+1 − x2��
+dx,
+(2.26)
+with r = r(α) =
+√
+1 − α2. Thus the proof of the proposition is complete up
+to the verification of the two identities (2.25) and (2.26) to which we turn
+next.
+10
+
+Proof of the identity (2.25).
+By (2.3) and (2.5) the principal value integral
+in (2.25) (with x ∈ (r, 1)) splits into two parts
+ dµα(y)
+x − y = 1
+2
+ 1
+−1
+1
+x − ydy − 1
+π
+ˆ r
+−r
+1
+x − y arccos
+�
+α
+�
+1 − y2
+�
+dy
+= 1
+2 log(1 + x) − 1
+2 log(1 − x) − Iα(x)
+(2.27)
+where
+Iα(x) = 1
+π
+ˆ r
+−r
+1
+x − y arccos
+�
+α
+�
+1 − y2
+�
+dy
+is a usual integral (not a principal value integral) that converges for every
+x > r. We integrate by parts
+Iα(x) = −α
+π
+ˆ r
+−r
+log(x − y)
+y
+1 − y2
+1
+�
+r2 − y2dy,
+x > r,
+and then compute the derivative
+d
+dxIα(x) = −α
+π
+ˆ r
+−r
+1
+x − y
+y
+1 − y2
+1
+�
+r2 − y2dy
+= −
+αx
+1 − x2
+1
+√
+x2 − r2 − 1
+2
+1
+x − 1 + 1
+2
+1
+x + 1
+by first turning the integral into an integral on a contour around the interval
+[−r, r] in the complex plane, and then evaluating it by the residue theorem
+for the exterior domain. The result can be integrated again to give
+Iα(x) = − log
+�
+α +
+√
+x2 − r2
+�
++ log(1 + x),
+x > r,
+(2.28)
+where we note that the constant of integration vanishes since Iα(x) → 0 as
+x → +∞. Using this in (2.27) we obtain (2.25).
+Proof of the identity (2.26).
+Observe that (2.26) holds for α = 0 since then
+both sides are equal to 0. Thus it is enough to show that the α-derivatives
+of the two sides agree.
+For the left-hand side of (2.26) we have by (1.4)
+d
+dαC(α) = log(1 + α) − log(1 − α)
+2
+.
+(2.29)
+11
+
+For the right-hand side we first compute the α-derivative of the integrand.
+Using r = r(α) =
+√
+1 − α2 we find by direct calculation
+d
+dα
+�
+log
+�
+α +
+√
+x2 − r2
+�
+− 1
+2 log
+�
+1 − x2��
+=
+1
+√
+x2 − r2.
+The integrand of (2.26) vanishes at x = r, and thus we obtain the following
+derivative of the right hand side of (2.26)
+ˆ 1
+r
+1
+√
+x2 − r2dx = log
+�
+x +
+√
+x2 − r2
+����
+x=1
+x=r
+= log
+�
+1 +
+√
+1 − r2
+�
+− log r
+= log
+�
+1 + α2�
+− 1
+2 log(1 − α)
+which after simplification agrees with (2.29).
+3
+Proof of the upper bound
+To prove the upper bound (1.8) we start by showing that the extremal poly-
+nomial p∗
+n has only real zeros that are separated by the n-grid En.
+3.1
+Zeros of the extremal polynomial
+We fix 0 < α < 1. For each n, we take a polynomial p∗
+n of degree ≤ αn as in
+(1.6) that we normalize such that
+∥p∗
+n∥[−1,1] = 1 = p∗
+n(x∗
+n)
+(3.1)
+for some x∗
+n ∈ [−1, 1]. It is clear that x∗
+n ̸∈ En since otherwise ∥pn∥En = 1,
+and the polynomial would not maximize the ratio.
+Lemma 3.1.
+(a) The polynomial p∗
+n minimizes ∥p∥En among all polyno-
+mials p of degree ≤ αn with p(x∗
+n) = 1.
+(b) The polynomial q∗
+n defined by
+q∗
+n(x) = x⌊αn⌋p∗
+n
+�
+x∗
+n + 1
+x
+�
+(3.2)
+12
+
+is a monic polynomial of degree ⌊αn⌋ that minimizes the weighted uni-
+form norm
+max
+x∈Σn
+��x−⌊αn⌋q(x)
+�� ,
+(3.3)
+among all monic polynomials of degree ⌊αn⌋, where Σn is the trans-
+formed grid,
+Σn = {(x − x∗
+n)−1 | x ∈ En}.
+(3.4)
+(c) p∗
+n has only simple real zeros with at least ⌊αn⌋ − 1 zeros in [−1, 1],
+(d) The zeros of p∗
+n are separated by the points in the n-grid En.
+Proof. (a) Suppose p is a polynomial of degree αn with p(x∗
+n) = 1 and
+∥p∥En < ∥p∗
+n∥En. Since x∗
+n ∈ [−1, 1] and p(x∗
+n) = 1, we then have ∥p∥[−1,1] ≥
+1 = ∥p∗
+n∥[−1,1], and therefore
+∥p∥[−1,1]
+∥p∥En
+> ∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+which contradicts the extremal property (1.6) of p∗
+n.
+(b) It is easy to see that (3.2) is indeed a polynomial of degree ⌊αn⌋ and
+it is monic because p∗
+n(x∗
+n) = 1.
+Likewise, we associate to any polynomial p of degree ≤ αn with p(x∗
+n) = 1
+the monic polynomial q(x) = x⌊αn⌋p(x∗
+n + 1
+x) of degree ⌊αn⌋. Then
+p(x) = (x − x∗
+n)⌊αn⌋q
+�
+1
+x−x∗n
+�
+and, with Σn as in (3.3)
+∥p∥En = max
+x∈En
+���(x − x∗
+n)⌊αn⌋q
+�
+1
+x−x∗n
+���� = max
+x∈Σn |w(x)q(x)|
+with
+w(x) = |x|−⌊αn⌋.
+Because of part (a) we see that q∗
+n minimizes ∥wq∥Σn among monic polyno-
+mials of degree ⌊αn⌋.
+(c) q∗
+n has only real zeros. Indeed if z0 were a non-real zero of q∗
+n then
+q(x) = x − Re z0
+x − z0
+q∗
+n(x)
+13
+
+would be a monic polynomial of the same degree satisfying |q(x)| < |q∗
+n(x)|
+for every real x that is not a zero of q∗
+n. This would lead to a contradiction
+with part (b).
+Also the zeros of q∗
+n are simple, since if x0 is a higher order real zero then
+for small enough ε > 0 the monic polynomial
+q(x) = (x − x0 − ε)(x − x0 + ε)
+(x − x0)2
+q∗
+n(x)
+would have a smaller weighted norm ∥wq∥Σn than q∗
+n. Because of (3.2) it
+then follows that p∗
+n has only simple real zeros as well, since any zero x0 ̸= 0
+of q∗
+n corresponds to the zero x∗
+n + 1
+x0 of p∗
+n.
+Since q∗
+n has ⌊αn⌋ simple real zeros, at least ⌊αn⌋−1 of them are different
+from 0, and thus p∗
+n has at least that number of simple real zeros. In particular
+p∗
+n has degree ≥ ⌊αn⌋ − 1.
+(d) The zeros of q∗
+n are separated by the points of Σn. Indeed if x1 < x2
+are two zeros of q∗
+n and the interval [x1, x2] would not contain any points of
+Σn then
+x �→ (x − x1 + ε)(x − x2 − ε)
+(x − x1)(x − x2)
+q∗
+n(x)
+would be a monic polynomial of the same degree with a strictly smaller
+weighted norm ∥wqn∥Σn < ∥wq∗
+n∥Σn, provided ε > 0 is small enough, which
+would contradict part (b). This in turn implies that in between any two zeros
+of p∗
+n there is a gridpoint of En, which gives part (d).
+3.2
+The functional J
+Recall from (2.10) that µα minimizes I(µ) among measures µ ∈ Mα,σ. We
+consider another functional
+J(µ) =
+min
+x∈supp(σ−µ) Uµ(x) − min
+x∈[−1,1] Uµ(x)
+(3.5)
+on measures µ ∈ Mα,σ. Note that by Proposition 2.4 we have
+J(µα) = C(α) > 0,
+(3.6)
+since supp(σ − µα) = [−r, r].
+14
+
+Proposition 3.2. We have
+lim sup
+n→∞
+1
+n log ∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+≤
+sup
+µ∈Mα,σ
+J(µ).
+Proof. We start by taking a subsequence N ⊂ N such that
+lim
+N ∋n→∞
+1
+n log ∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+= lim sup
+n→∞
+1
+n log ∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+.
+(3.7)
+The zeros of p∗
+n may not be uniformly bounded. However, by Lemma
+3.1(c), there is at most one zero outside [−2, 2]. If there is such a zero, say
+x0, then we set
+�pn(x) = κ−1
+n
+p∗
+n(x)
+x − x0
+,
+(3.8)
+where κn is the leading coefficient of p∗
+n. Otherwise we set
+�pn(x) = κ−1
+n p∗
+n(x).
+(3.9)
+Then �pn is a monic polynomial of degree ⌊αn⌋ or ⌊αn⌋ − 1. In case (3.9) we
+clearly have
+∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+= ∥�pn∥[−1,1]
+∥�pn∥En
+,
+(3.10)
+while in case (3.8) we can claim that
+1
+3
+∥p∗
+n∥[−1,1]
+∥p∗n∥En
+≤ ∥�pn∥[−1,1]
+∥�pn∥En
+≤ 3∥p∗
+n∥[−1,1]
+∥p∗n∥En
+.
+(3.11)
+To obtain (3.11) we note that since |x0| > 2 we have |x0| − 1 ≤ |x − x0| ≤
+|x0| + 1 for x ∈ [−1, 1], so that
+|κ−1
+n ||pn ∗ (x)|
+|x0| + 1 ≤ |�pn(x)| ≤ |κ−1
+n | |p∗
+n(x)|
+|x0| − 1,
+for x ∈ [−1, 1].
+Taking the supremum over x ∈ [−1, 1] and over x ∈ En, we obtain
+|κ−1
+n |
+|x0| + 1∥p∗
+n∥[−1,1] ≤ ∥�pn∥[−1,1] ≤
+|κ−1
+n |
+|x0| − 1∥p∗
+n∥[−1,1],
+|κ−1
+n |
+|x0| + 1|κ−1
+n |∥p∗
+n∥En ≤ ∥�pn∥En ≤
+|κ−1
+n |
+|x0| − 1∥p∗
+n∥En.
+15
+
+Taking ratios of these inequalities leads to (3.11) since |x0|+1
+|x0|−1 < 3 for |x0| > 2.
+From (3.7) (3.10), (3.11) it then follows that
+lim sup
+n→∞
+1
+n
+∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+=
+lim
+N ∋n→∞
+1
+n log ∥�pn∥[−1,1]
+∥�pn∥En
+.
+(3.12)
+Next, by taking a further subsequence if necessary, we may also assume
+that the sequence (νn)n of normalized zero counting measures, i.e.,
+νn = 1
+n
+�
+x:�pn(x)=0
+δx
+converges in the weak∗ sense as n → ∞ with n ∈ N . Here we use Helly’s
+selection theorem, and Lemma 3.1 (c). By Lemma 3.1 (d) the weak∗ limit,
+say µ, belongs to Mα,σ.
+Now we apply Lemmas 2.1 and 2.3 to the polynomials �pn. From part (a)
+of Lemma 2.1 we get
+lim inf
+N ∋n→∞
+1
+n log ∥�pn∥En ≥ −
+min
+x∈supp(σ−µ) Uµ(x)
+and from Lemma 2.3
+lim
+N ∋n→∞
+1
+n log ∥�pn∥[−1,1] = − min
+x∈[−1,1] Uµ(x).
+These limits and (3.12) then imply that
+lim sup
+n→∞
+1
+n log ∥p∗
+n∥[−1,1]
+∥p∗n∥En
+≤ J(µ)
+and the proposition since µ ∈ Mα,σ.
+3.3
+Conclusion of the proof of the upper bound
+In view of Proposition 3.2 it remains to show that
+sup
+µ∈Mα,σ
+J(µ) = C(α) in
+order to obtain (1.8). This is the final result of the paper.
+Proposition 3.3. For every µ ∈ Mα,σ with µ ̸= µα we hae
+J(µ) < J(µα) = C(α).
+(3.13)
+16
+
+Proof. We already noted in (3.6) that J(µα) = C(α) > 0.
+Let µ ∈ Mα,σ with µ ̸= µα. Take x0 ∈ [−1, 1] with
+Uµ(x0) =
+min
+x∈[−1,1] Uµ(x).
+(3.14)
+If x0 ∈ supp(σ − µ), then the minimimu of Uµ over supp(σ − µ) is also
+attained at x0, and it would follow from (3.5) that J(µ) = 0. Then the strict
+inequality (3.13) holds. Hence we may assume that x0 ∈ [−1, 1]\supp(σ−µ).
+Since µ ≤ σ and µα ≤ σ we see that both Uµ and Uµα are continuous
+functions on C, see also Remark 2.2, and they are both harmonic in \[−1, 1].
+Also µ ≥ µα on [−1, 1] \ supp(σ − µ), and therefore Uµ−µα is superharmonic
+on C \ supp(σ − µ). It has a finite limit at infinity since µ and µα have the
+same total mass. The minimum principle for superharmonic functions [8, 15]
+then tells us that the minimum of Uµ−µα is taken on supp(σ − µ) only. In
+particular, since x0 ̸∈ supp(σ − µ)
+Uµ−µα(x0) >
+min
+x∈supp(σ−µ) Uµ−µα(x).
+(3.15)
+Combining (3.15) with the obvious inequality (since supp(σ − µ) ⊂ [−1, 1])
+min
+x∈supp(σ−µ) Uµ−µα(x) ≥
+min
+x∈supp(σ−µ) Uµ(x) − max
+x∈[−1,1] Uµα(x),
+we obtain
+Uµ(x0) − Uµα(x0) >
+min
+x∈supp(σ−µ) Uµ(x) − max
+x∈[−1,1] Uµα(x),
+which leads to
+min
+x∈supp(σ−µ) Uµ(x) − Uµ(x0) < max
+x∈[−1,1] Uµα(x) − Uµα(x0)
+≤ max
+x∈[−1,1] Uµα(x) − min
+x∈[−1,1] Uµα(x).
+(3.16)
+The left-hand side of (3.16) is equal to J(µ) because of (3.5) and (3.14). For
+the right-hand side, we note that by the special property (2.11) of Uµα we
+have
+max
+x∈[−1,1] Uµα(x) = ℓα =
+min
+x∈supp(σ−µα) Uµα(x),
+and therefore the right-hand side of (3.16) is equal to J(µα). Thus J(µ) <
+J(µα) and the proposition is proved.
+17
+
+Remark 3.4. According to Proposition 3.3 the constrained equilibrium mea-
+sure µα is the unique maximizer of J(µ) among measures µ ∈ Mα,σ. We may
+conclude from this that the sequence of normalized zero counting measures
+of the extremal polynomials p∗
+n tends to the constrained equilibrium measure
+µα as n → ∞. This follows from the proof of Proposition 3.2, combined with
+the proven fact that
+lim
+n→∞
+1
+n log ∥p∗
+n∥[−1,1]
+∥p∗
+n∥En
+= C(α),
+as this gives that the weak∗ limit of any convergent subsequence is a measure
+µ ∈ Mα,σ with J(µ) = C(α). Because of (3.13) this limit has to be µα, and
+thus by a compactness argument the full sequence tends to µα indeed.
+Acknowledgement
+I want to thank Daan Huybrechs and Nick Trefethen for their interest in this
+work, for useful discussions, and for stimulating me to write the details of
+the proof of Theorem 1.1.
+The author was supported by the long term structural funding ”Methusalem
+grant of the Flemish Government” and by FWO Flanders projects EOS
+30889451 and G.0910.20.
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+19
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf,len=528
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='01591v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='CA]  4 Jan 2023  Extremal polynomials on the n-grid  Arno B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Kuijlaars  January 5, 2023  Abstract  The n-grid En consists of n equally spaced points in [−1, 1] includ-  ing the endpoints ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The extremal polynomial p∗  n is the polynomial  that maximizes the uniform norm ∥p∥[−1,1] among polynomials p of  degree ≤ αn that are bounded by one on En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For every α ∈ (0, 1),  we determine the limit of 1  n log ∥p∗  n∥[−1,1] as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The interest in  this limit comes from a connection with an impossibility theorem on  stable approximation on the n-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  1  Statement of result  The n-grid from the title refers to n equally spaced points in [−1, 1]  En = {ξk,n = 2k−n−1  n−1  | k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' , n},  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1)  ranging from −1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Let 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' This paper is about the determination  of the limit of the expression  1  n log  sup  deg p≤αn  ∥p∥[−1,1]  ∥p∥En  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2)  as n → ∞, where the norms are uniform norms over the indicated sets, and  the supremum is over univariate polynomials p of degrees at most αn that do  not vanish identically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The result was already announced in the paper [12]  from 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Renewed interest in it is due to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  1  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For every α ∈ (0, 1) the limit  lim  n→∞  1  n log  sup  deg p≤αn  ∥p∥[−1,1]  ∥p∥En  = C(α)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3)  exists and is equal to  C(α) = (1 + α) log(1 + α) + (1 − α) log(1 − α)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4)  The limit C(α) is positive and strictly increasing with α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' There is a nice  Taylor expansion  C(α) =  ∞  �  k=1  α2k  2k(2k − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The odd Taylor coefficients vanish since the power series defines an odd  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The even Taylor coefficients are positive and therefore C(α) ≥ 1  2α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  It is also worth noticing that C(α) → log 2 as α → 1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For α = 1 however,  we have that the limit in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3) is +∞, since for each n, there is a non-zero  polynomial of degree n that vanishes on the n-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Discussion  The limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3) shows that a polynomial of degree ≤ αn that is  bounded on En can be exponentially large somewhere in the interval [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Namely, if |p(ξk,n)| ≤ 1 for each k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' , n, then |p(x)| at some x ∈ [−1, 1]  can be as large as en(C(α)+o(1)) as n → ∞, and the constant C(α) is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The  result is related to earlier work of Coppersmith and Rivlin [5] who showed  that there exist universal constants C2 > C1 > 1 such that for n large enough,  and for every d ≤ n − 1,  Cd2/n  1  ≤ sup  deg p≤d  ∥p∥[−1,1]  ∥p∥En  ≤ Cd2/n  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5)  The inequalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5) show that polynomials of degree d ≤ c√n that are  bounded by one on En are uniformly bounded on [−1, 1] with a constant  that only depends on c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' However, if d grows proportionally with n then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5)  shows that polynomials that are bounded by one on En may be exponentially  large on [−1, 1], and this behavior is made more precise in the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The comparisons of the two uniform norms ∥ · ∥En and ∥ · ∥[−1,1] arises  naturally when studying approximation or interpolation methods for analytic  functions based on function values on the n-grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' There is a trade-off between  2  convergence and stability properties that was made precise in the impossi-  bility theorem of [12, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For example, exponential convergence  as n → ∞ comes together with exponential instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The proof of the  impossibility theorem in [12] relies on the Coppersmith-Rivlin inequalities  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For recent work in this direction we refer to [1, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  It was observed in [12, section 4] that the phenomenon that polynomials of  degree d ≈ αn can be much larger on [−1, 1] than on En may be understood  in terms of potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Here one thinks of a polynomial in terms of  its zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' A polynomial may be small at certain gridpoints in En by simply  having a zero very close to these gridpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' However, since there are more  gridpoints than zeros this cannot happen for every gridpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The extremal  polynomial p∗  n for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2) will place a certain fraction of its zeros extremely  close to gridpoints lying in a subset S of [−1, 1] (with S depending on α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Then p∗  n is small at the gridpoints in S but not necessarily in between, and  in fact it has high oscillations in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Following [2, 7] we call S the saturated  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The non-saturated region [−1, 1]\\S has considerably fewer zeros than  gridpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The extremal polynomial p∗  n is not only small at the gridpoints  in [−1, 1] \\ S but it is of comparable size over the full set [−1, 1] \\ S, see [14]  for very precise estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  This phenomenon was first described by Rakhmanov [13] for orthogonal  polynomials on the n-grid, or more generally, for polynomials that minimize  a discrete Lp norm on En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' These polynomials have their zeros in [−1, 1] and  they are separated by the gridpoints, in the sense that in between any two  distinct zeros there is at least one gridpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' In the limit n → ∞ the zeros of  the extremal polynomials of degree ⌊αn⌋ considered in [13] have a limiting  distribution µα (depending only on α) that is characterized by a constrained  equilibrium problem from potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The measure µα has a density  with respect to Lebesgue measure on [−1, 1] with dµα  dx ≤ 1  2 where 1  2 is the  limiting density of the gridpoints as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The saturated region S is  where the equality dµα  dx = 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We give details in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The  extremal polynomials p∗  n for the extremal problem in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3) have the same  limiting zero distribution µα as n → ∞, as we will show in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Also  in other aspects they behave similarly to the Lp-extremal polynomials on the  n-grid, and this will be the clue to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  3  Outline of the proof  The extremal polynomial for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2) is a polynomial  p∗  n of degree ≤ αn such that  ∥p∗  n∥[−1,1]  ∥p∗  n∥En  =  sup  deg p≤αn  ∥p∥[−1,1]  ∥p∥En  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6)  The proof of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3) then naturally comes in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' In the first step we  prove the lower bound  lim inf  n→∞  1  n log ∥p∗  n∥[−1,1]  ∥p∗  n∥En  ≥ C(α)  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7)  and in the second step the corresponding upper bound  lim sup  n→∞  1  n log ∥p∗  n∥[−1,1]  ∥p∗  n∥En  ≤ C(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8)  The lower bound comes from considering the L∞-extremal polynomials P ∗  n  on En, where P ∗  n is the monic polynomial of degree ⌊αn⌋ that minimizes the  uniform norm ∥ · ∥En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Using the results from [7, 13] and some additional  calculations we prove in section 2 that  lim  n→∞  1  n log ∥P ∗  n∥[−1,1]  ∥P ∗n∥En  = C(α),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='9)  and this implies the lower bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8) is proved in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' It comes from a study  of the zeros of the extremal polynomials p∗  n satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We show in  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1 that the zeros are real and simple and at least ⌊αn⌋ − 1 zeros  are in [−1, 1] where they are separated by the gridpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Note that one zero  could be in R \\ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We then use potential theoretic arguments to show  that the limiting distribution of the zeros of p∗  n is equal to the constrained  equilibrium measure µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Along the way we prove that µα is the maximizer  of a functional J that we define in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5) with J(µα) = C(α), which leads to  the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  We finally note that discrete orthogonal polynomials and the constrained  equilibrium problem also play a role in the analysis of iterative methods from  numerical linear algebra [3, 10, 11], and the asymptotic analysis of integrable  systems [6], random matrices and random tiling models [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  4  2  Proof of the lower bound  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1  Extremal polynomials on the n-grid  As explained above, we are going to consider the monic polynomial P ∗  n of  degree ⌊αn⌋ such that  ∥P ∗  n∥En =  min  deg P = ⌊αn⌋  P is monic  ∥P∥En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1)  We are going to show that the limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='9) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Rakhmanov [13] considered polynomials P of degree n that are monic  (leading coefficient equal to 1) and that minimize either the uniform norm  ∥P∥EN, or a discrete p-norm on EN among all such polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The interest  is in their asymptotic behavior as both n, N → ∞ with n/N → c < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The  equispaced n-grid (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1) is actually only a special case of far more general  discrete sets that are considered in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Compared to [13] we change N �→ n,  n �→ ⌊αn⌋, c �→ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2  Limiting behavior of zeros  The zeros of P ∗  n are real and simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' They belong to the interval (−1, 1),  where they are separated by the nodes in En, see [7, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' To P ∗  n we associate  the normalized zero counting measure  νn = 1  n  ⌊αn⌋  �  k=1  δxk,n,  where xk,n for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' , ⌊αn⌋, denote the zeros of P ∗  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Note that we nor-  malize with the factor 1/n while the degree of P ∗  n is ⌊αn⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Thus νn is not a  probability measure but rather has a total mass ⌊αn⌋  n  Rakhmanov [13, Theorem 2], see also [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3], proved that the  weak∗ limit  νn  ∗→ µα  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2)  exists, where µα is the measure on [−1, 1] with density [13, Theorem 1]  dµα  dx =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  1  2,  for x ∈ [−1, −r] ∪ [r, 1],  1  π arctan  �  α  √  r2 − x2  �  ,  for x ∈ [−r, r],  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3)  5  where  r = r(α) =  √  1 − α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4)  The density on [−r, r] can alternatively be written as  dµα  dx = 1  2 − 1  π arccos  �  α  √  1 − x2  �  ,  for x ∈ [−r, r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5)  The measure µα belongs to the class  Mα,σ := {µ | ∫ dµ = α, 0 ≤ µ ≤ σ}  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6)  where  dσ = 1  2χ[−1,1](x)dx  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7)  denotes the Lebesgue measure restricted to [−1, 1] with normalization such  that  ´  dσ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The upper constraint µα ≤ σ comes from the fact that the  zeros of P ∗  n are separated by the nodes ξk,n in the equispaced grid En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Rakhmanov also characterized µα in terms of notions from logarithmic  potential theory [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Let  I(µ) =  ¨  log  1  |x − y|dµ(x)dµ(y)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8)  and  Uµ(x) =  ˆ  log  1  |x − y|dµ(y)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='9)  denote the logarithmic energy and the logarithmic potential of a measure µ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Then  I(µα) =  min  µ∈Mα,σ I(µ)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='10)  and µα is the unique minimizer within the class (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Furthermore, there is  a constant ℓα such that  Uµα(x)  �  = ℓα,  for x ∈ supp(σ − µα),  ≤ ℓα,  for x ∈ [−1, 1],  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='11)  and µα is the only measure µ in Mα,σ such that Uµ(x) = ℓ is constant on  supp(σ − µ) and Uµ(x) ≤ ℓ on [−1, 1] for a certain constant ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Because of the upper constraint µ ≤ σ, the measure µα is called a con-  strained equilibrium measure, see [2, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The saturated region S is where  dµα  dx = dσ  dx = 1  2 and in view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3) we have S = [−1, −r] ∪ [r, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The con-  straint is not active in the region where dµα  dx < 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' This is the non-saturated  region and its closure is supp(σ − µα) = [−r, r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3  Two lemmas  The connection between potential theory and the asymptotics theory of poly-  nomials is well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' If P is a monic polynomial and  ν = 1  n  �  x:P (x)=0  δx  is its normalized zero counting measure (each zero is included in the sum  according to its multiplicity), then  1  n log |P(x)| = −Uν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  If (Pn)n is a sequence of monic polynomials, and (νn)n is the corresponding  sequence of normalized zero counting measures then the convergence of (νn)n  contains information on the nth root asymptotic behavior of the polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  We need two such results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (a) Let (Pn)n be a sequence of monic polynomials, deg Pn ≤  αn, having real and simple zeros, such that the zeros of Pn are separated  by the points of En for every n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Suppose that the sequence of normalized  zero counting measures (νn)n where νn = 1  n  �  x:Pn(x)=0 δx, has a weak∗  limit µ as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Then µ ≤ σ, and  lim inf  n→∞  1  n log ∥Pn∥En ≥ −  min  x∈supp(σ−µ) Uµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='12)  (b) If (P ∗  n)n is the sequence of extremal polynomials satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1) then  νn  ∗→ µα as n → ∞, and equality holds  lim  n→∞  1  n log ∥P ∗  n∥En = −  min  x∈supp(σ−µ) Uµα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='13)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Part (a) is Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2 of [13], where it is stated under the assumption  that the zeros are in [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' See Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5 in [7] for the statement without  this extra assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Part (b) is in [13, Theorem 2] or [7, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Part (b) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1 will be used in the proof of the lower bound,  while part (a) will be used in the proof of the upper bound, see the proof of  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  7  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Note that the logarithmic potential Uµ of a positive measure  µ is a lower semi-continuous function [15] and therefore its minimum over a  compact (as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='13), as well as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='14) below) exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  In the situation of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1, however, the logarithmic potential Uµ  is actually continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  This follows from µ ≤ σ and the fact that Uσ is  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Indeed, Uσ−µ is lower semi-continuous, and therefore Uµ =  Uσ − Uσ−µ is upper semi-continuous as well, hence continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The second lemma is probably well-known, but I could not find an ap-  propriate reference for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Suppose (Pn)n is a sequence of monic polynomials, deg Pn ≤  αn, such that the zeros of all Pn are in a compact set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Suppose that the  sequence of normalized zero counting measures (νn)n has a weak∗ limit µ as  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Then  lim  n→∞  1  n log ∥Pn∥[−1,1] = − min  x∈[−1,1] Uµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='14)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' By the principle of descent [15] we have  Uµ(x∗) ≤ lim inf  n→∞ Uνn(xn)  whenever xn → x∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Since |Pn(x)| = e−nUνn(x), this means that  lim sup  n→∞  1  n log |Pn(xn)| ≤ −Uµ(x∗) ≤ − min  x∈[−1,1] Uµ(x),  whenever (xn)n is a convergent sequence with a limit x∗ ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Taking  xn ∈ [−1, 1] with |Pn(xn)| = ∥Pn∥[−1,1] and passing to convergent subse-  quences if necessary, we then find  lim sup  n→∞  1  n log ∥Pn∥[−1,1] ≤ − min  x∈[−1,1] Uµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='15)  By the lower envelope theorem [15] we have  Uµ(x) = lim  n→∞ Uνn(x)  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='16)  where q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' means quasi everywhere, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=', the exceptional set is a polar set (a  small set for potential theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='16) means  lim  n→∞  1  n log |Pn(x)| = −Uµ(x)  q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='17)  8  Let x0 ∈ [−1, 1] be such that  Uµ(x0) =  min  x∈[−1,1] Uµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='18)  Note that the minimum exists since Uµ is lower semicontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Let ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Then we claim that  {x ∈ [−1, 1] | Uµ(x) < Uµ(x0) + ε}  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='19)  is not a polar set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' This is easy to see if Uµ is a continuous function, since  then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='19) contains a non-empty interval and this is not a polar set, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  [8, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' If Uµ is not continuous, then we can come to the same  conclusion, if we use certain more advanced results from potential theory, in  particular around thinness and the fine topology, which we will not explain  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The set {x ∈ C | Uµ(x) < Uµ(x0) + ε} is an open neighborhood of x0  in the fine topology, and [−1, 1] is not thin at x0 ∈ [−1, 1], see [8, Corollary  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='19) is not a polar set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Knowing that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='19) is not polar, we conclude from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='17) that there  exists x1 ∈ [−1, 1] with Uµ(x1) < Uµ(x0) + ε and the limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='17) holds for  x = x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Then by the above and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='18)  lim inf  n→∞  1  n log ∥Pn∥[−1,1] ≥ lim  n→∞  1  n log |Pn(x1)| = −Uµ(x1)  > −Uµ(x0) − ε = − min  x∈[−1,1] Uµ(x) − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='20)  Then the lemma follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='20), since ε > 0 is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4  Conclusion of the proof of the lower bound  We apply the two lemmas to the extremal polynomials P ∗  n satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1 (b) we have  lim  n→∞  1  n log ∥P ∗  n∥En = − min  x∈[−r,r] Uµα(x),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='21)  since supp(σ − µα) = [−r, r] by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2) we get  lim  n→∞  1  n log ∥P ∗  n∥[−1,1] = − min  x∈[−1,1] Uµα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='22)  9  Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='21) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='22) we obtain  lim  n→∞  1  n log ∥P ∗  n∥[−1,1]  ∥P ∗  n∥En  = min  x∈[−r,r] Uµα(x) − min  x∈[−1,1] Uµα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='23)  The limit (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='9) and thereby the lower bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7) follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='23) and  the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We have  C(α) = min  x∈[−r,r] Uµα(x) − min  x∈[−1,1] Uµα(x)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='24)  with C(α) as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The derivative of Uµα is a principal value integral that can be calcu-  lated explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The result is  d  dxUµα(x) = −  dµα(y)  x − y  = 1  2 log  �  1 − x2�  − log  �  α +  √  x2 − r2  �  ,  for r < x < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25)  We give the details of the calculations for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25) later, after finishing the  main line of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The derivative (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25) is negative for r < x < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Therefore (and by  symmetry) the minimum of Uµα(x) over [−1, −r] ∪ [r, 1] is at x = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Also  Uµα is constant on [−r, r] by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Hence  min  x∈[−r,r]Uµα(x) − min  x∈[−1,1] Uµα(x) = Uµα(r) − Uµα(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  In view of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25) and the fundamental theorem of calculus, we arrive at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='24)  provided that  C(α) =  ˆ 1  r  �  log  �  α +  √  x2 − r2  �  − 1  2 log  �  1 − x2��  dx,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='26)  with r = r(α) =  √  1 − α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Thus the proof of the proposition is complete up  to the verification of the two identities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='26) to which we turn  next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  10  Proof of the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5) the principal value integral  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25) (with x ∈ (r, 1)) splits into two parts  dµα(y)  x − y = 1  2  1  −1  1  x − ydy − 1  π  ˆ r  −r  1  x − y arccos  �  α  �  1 − y2  �  dy  = 1  2 log(1 + x) − 1  2 log(1 − x) − Iα(x)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='27)  where  Iα(x) = 1  π  ˆ r  −r  1  x − y arccos  �  α  �  1 − y2  �  dy  is a usual integral (not a principal value integral) that converges for every  x > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We integrate by parts  Iα(x) = −α  π  ˆ r  −r  log(x − y)  y  1 − y2  1  �  r2 − y2dy,  x > r,  and then compute the derivative  d  dxIα(x) = −α  π  ˆ r  −r  1  x − y  y  1 − y2  1  �  r2 − y2dy  = −  αx  1 − x2  1  √  x2 − r2 − 1  2  1  x − 1 + 1  2  1  x + 1  by first turning the integral into an integral on a contour around the interval  [−r, r] in the complex plane, and then evaluating it by the residue theorem  for the exterior domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The result can be integrated again to give  Iα(x) = − log  �  α +  √  x2 − r2  �  + log(1 + x),  x > r,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='28)  where we note that the constant of integration vanishes since Iα(x) → 0 as  x → +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Using this in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='27) we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Proof of the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Observe that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='26) holds for α = 0 since then  both sides are equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Thus it is enough to show that the α-derivatives  of the two sides agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  For the left-hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='26) we have by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4)  d  dαC(α) = log(1 + α) − log(1 − α)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='29)  11  For the right-hand side we first compute the α-derivative of the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Using r = r(α) =  √  1 − α2 we find by direct calculation  d  dα  �  log  �  α +  √  x2 − r2  �  − 1  2 log  �  1 − x2��  =  1  √  x2 − r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The integrand of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='26) vanishes at x = r, and thus we obtain the following  derivative of the right hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='26)  ˆ 1  r  1  √  x2 − r2dx = log  �  x +  √  x2 − r2  ����  x=1  x=r  = log  �  1 +  √  1 − r2  �  − log r  = log  �  1 + α2�  − 1  2 log(1 − α)  which after simplification agrees with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  3  Proof of the upper bound  To prove the upper bound (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8) we start by showing that the extremal poly-  nomial p∗  n has only real zeros that are separated by the n-grid En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1  Zeros of the extremal polynomial  We fix 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For each n, we take a polynomial p∗  n of degree ≤ αn as in  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6) that we normalize such that  ∥p∗  n∥[−1,1] = 1 = p∗  n(x∗  n)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1)  for some x∗  n ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' It is clear that x∗  n ̸∈ En since otherwise ∥pn∥En = 1,  and the polynomial would not maximize the ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (a) The polynomial p∗  n minimizes ∥p∥En among all polyno-  mials p of degree ≤ αn with p(x∗  n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (b) The polynomial q∗  n defined by  q∗  n(x) = x⌊αn⌋p∗  n  �  x∗  n + 1  x  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2)  12  is a monic polynomial of degree ⌊αn⌋ that minimizes the weighted uni-  form norm  max  x∈Σn  ��x−⌊αn⌋q(x)  �� ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3)  among all monic polynomials of degree ⌊αn⌋, where Σn is the trans-  formed grid,  Σn = {(x − x∗  n)−1 | x ∈ En}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4)  (c) p∗  n has only simple real zeros with at least ⌊αn⌋ − 1 zeros in [−1, 1],  (d) The zeros of p∗  n are separated by the points in the n-grid En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' (a) Suppose p is a polynomial of degree αn with p(x∗  n) = 1 and  ∥p∥En < ∥p∗  n∥En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Since x∗  n ∈ [−1, 1] and p(x∗  n) = 1, we then have ∥p∥[−1,1] ≥  1 = ∥p∗  n∥[−1,1], and therefore  ∥p∥[−1,1]  ∥p∥En  > ∥p∗  n∥[−1,1]  ∥p∗  n∥En  which contradicts the extremal property (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6) of p∗  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (b) It is easy to see that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2) is indeed a polynomial of degree ⌊αn⌋ and  it is monic because p∗  n(x∗  n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Likewise, we associate to any polynomial p of degree ≤ αn with p(x∗  n) = 1  the monic polynomial q(x) = x⌊αn⌋p(x∗  n + 1  x) of degree ⌊αn⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Then  p(x) = (x − x∗  n)⌊αn⌋q  �  1  x−x∗n  �  and, with Σn as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3)  ∥p∥En = max  x∈En  ���(x − x∗  n)⌊αn⌋q  �  1  x−x∗n  ���� = max  x∈Σn |w(x)q(x)|  with  w(x) = |x|−⌊αn⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Because of part (a) we see that q∗  n minimizes ∥wq∥Σn among monic polyno-  mials of degree ⌊αn⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (c) q∗  n has only real zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Indeed if z0 were a non-real zero of q∗  n then  q(x) = x − Re z0  x − z0  q∗  n(x)  13  would be a monic polynomial of the same degree satisfying |q(x)| < |q∗  n(x)|  for every real x that is not a zero of q∗  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' This would lead to a contradiction  with part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Also the zeros of q∗  n are simple, since if x0 is a higher order real zero then  for small enough ε > 0 the monic polynomial  q(x) = (x − x0 − ε)(x − x0 + ε)  (x − x0)2  q∗  n(x)  would have a smaller weighted norm ∥wq∥Σn than q∗  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Because of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2) it  then follows that p∗  n has only simple real zeros as well, since any zero x0 ̸= 0  of q∗  n corresponds to the zero x∗  n + 1  x0 of p∗  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Since q∗  n has ⌊αn⌋ simple real zeros, at least ⌊αn⌋−1 of them are different  from 0, and thus p∗  n has at least that number of simple real zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' In particular  p∗  n has degree ≥ ⌊αn⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (d) The zeros of q∗  n are separated by the points of Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Indeed if x1 < x2  are two zeros of q∗  n and the interval [x1, x2] would not contain any points of  Σn then  x �→ (x − x1 + ε)(x − x2 − ε)  (x − x1)(x − x2)  q∗  n(x)  would be a monic polynomial of the same degree with a strictly smaller  weighted norm ∥wqn∥Σn < ∥wq∗  n∥Σn, provided ε > 0 is small enough, which  would contradict part (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' This in turn implies that in between any two zeros  of p∗  n there is a gridpoint of En, which gives part (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2  The functional J  Recall from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='10) that µα minimizes I(µ) among measures µ ∈ Mα,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We  consider another functional  J(µ) =  min  x∈supp(σ−µ) Uµ(x) − min  x∈[−1,1] Uµ(x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5)  on measures µ ∈ Mα,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Note that by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4 we have  J(µα) = C(α) > 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6)  since supp(σ − µα) = [−r, r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  14  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We have  lim sup  n→∞  1  n log ∥p∗  n∥[−1,1]  ∥p∗  n∥En  ≤  sup  µ∈Mα,σ  J(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We start by taking a subsequence N ⊂ N such that  lim  N ∋n→∞  1  n log ∥p∗  n∥[−1,1]  ∥p∗  n∥En  = lim sup  n→∞  1  n log ∥p∗  n∥[−1,1]  ∥p∗  n∥En  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7)  The zeros of p∗  n may not be uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' However, by Lemma  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1(c), there is at most one zero outside [−2, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' If there is such a zero, say  x0, then we set  �pn(x) = κ−1  n  p∗  n(x)  x − x0  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8)  where κn is the leading coefficient of p∗  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Otherwise we set  �pn(x) = κ−1  n p∗  n(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='9)  Then �pn is a monic polynomial of degree ⌊αn⌋ or ⌊αn⌋ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' In case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='9) we  clearly have  ∥p∗  n∥[−1,1]  ∥p∗  n∥En  = ∥�pn∥[−1,1]  ∥�pn∥En  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='10)  while in case (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8) we can claim that  1  3  ∥p∗  n∥[−1,1]  ∥p∗n∥En  ≤ ∥�pn∥[−1,1]  ∥�pn∥En  ≤ 3∥p∗  n∥[−1,1]  ∥p∗n∥En  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='11)  To obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='11) we note that since |x0| > 2 we have |x0| − 1 ≤ |x − x0| ≤  |x0| + 1 for x ∈ [−1, 1], so that  |κ−1  n ||pn ∗ (x)|  |x0| + 1 ≤ |�pn(x)| ≤ |κ−1  n | |p∗  n(x)|  |x0| − 1,  for x ∈ [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Taking the supremum over x ∈ [−1, 1] and over x ∈ En, we obtain  |κ−1  n |  |x0| + 1∥p∗  n∥[−1,1] ≤ ∥�pn∥[−1,1] ≤  |κ−1  n |  |x0| − 1∥p∗  n∥[−1,1],  |κ−1  n |  |x0| + 1|κ−1  n |∥p∗  n∥En ≤ ∥�pn∥En ≤  |κ−1  n |  |x0| − 1∥p∗  n∥En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  15  Taking ratios of these inequalities leads to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='11) since |x0|+1  |x0|−1 < 3 for |x0| > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='7) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='11) it then follows that  lim sup  n→∞  1  n  ∥p∗  n∥[−1,1]  ∥p∗  n∥En  =  lim  N ∋n→∞  1  n log ∥�pn∥[−1,1]  ∥�pn∥En  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='12)  Next, by taking a further subsequence if necessary, we may also assume  that the sequence (νn)n of normalized zero counting measures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=',  νn = 1  n  �  x:�pn(x)=0  δx  converges in the weak∗ sense as n → ∞ with n ∈ N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Here we use Helly’s  selection theorem, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1 (d) the weak∗ limit,  say µ, belongs to Mα,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Now we apply Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3 to the polynomials �pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' From part (a)  of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1 we get  lim inf  N ∋n→∞  1  n log ∥�pn∥En ≥ −  min  x∈supp(σ−µ) Uµ(x)  and from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3  lim  N ∋n→∞  1  n log ∥�pn∥[−1,1] = − min  x∈[−1,1] Uµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  These limits and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='12) then imply that  lim sup  n→∞  1  n log ∥p∗  n∥[−1,1]  ∥p∗n∥En  ≤ J(µ)  and the proposition since µ ∈ Mα,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3  Conclusion of the proof of the upper bound  In view of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2 it remains to show that  sup  µ∈Mα,σ  J(µ) = C(α) in  order to obtain (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' This is the final result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For every µ ∈ Mα,σ with µ ̸= µα we hae  J(µ) < J(µα) = C(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='13)  16  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We already noted in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='6) that J(µα) = C(α) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Let µ ∈ Mα,σ with µ ̸= µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Take x0 ∈ [−1, 1] with  Uµ(x0) =  min  x∈[−1,1] Uµ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='14)  If x0 ∈ supp(σ − µ), then the minimimu of Uµ over supp(σ − µ) is also  attained at x0, and it would follow from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5) that J(µ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Then the strict  inequality (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='13) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Hence we may assume that x0 ∈ [−1, 1]\\supp(σ−µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Since µ ≤ σ and µα ≤ σ we see that both Uµ and Uµα are continuous  functions on C, see also Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2, and they are both harmonic in \\[−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Also µ ≥ µα on [−1, 1] \\ supp(σ − µ), and therefore Uµ−µα is superharmonic  on C \\ supp(σ − µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' It has a finite limit at infinity since µ and µα have the  same total mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' The minimum principle for superharmonic functions [8, 15]  then tells us that the minimum of Uµ−µα is taken on supp(σ − µ) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' In  particular, since x0 ̸∈ supp(σ − µ)  Uµ−µα(x0) >  min  x∈supp(σ−µ) Uµ−µα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='15)  Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='15) with the obvious inequality (since supp(σ − µ) ⊂ [−1, 1])  min  x∈supp(σ−µ) Uµ−µα(x) ≥  min  x∈supp(σ−µ) Uµ(x) − max  x∈[−1,1] Uµα(x),  we obtain  Uµ(x0) − Uµα(x0) >  min  x∈supp(σ−µ) Uµ(x) − max  x∈[−1,1] Uµα(x),  which leads to  min  x∈supp(σ−µ) Uµ(x) − Uµ(x0) < max  x∈[−1,1] Uµα(x) − Uµα(x0)  ≤ max  x∈[−1,1] Uµα(x) − min  x∈[−1,1] Uµα(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='16)  The left-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='16) is equal to J(µ) because of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' For  the right-hand side, we note that by the special property (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='11) of Uµα we  have  max  x∈[−1,1] Uµα(x) = ℓα =  min  x∈supp(σ−µα) Uµα(x),  and therefore the right-hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='16) is equal to J(µα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Thus J(µ) <  J(µα) and the proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  17  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' According to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='3 the constrained equilibrium mea-  sure µα is the unique maximizer of J(µ) among measures µ ∈ Mα,σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' We may  conclude from this that the sequence of normalized zero counting measures  of the extremal polynomials p∗  n tends to the constrained equilibrium measure  µα as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' This follows from the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='2, combined with  the proven fact that  lim  n→∞  1  n log ∥p∗  n∥[−1,1]  ∥p∗  n∥En  = C(α),  as this gives that the weak∗ limit of any convergent subsequence is a measure  µ ∈ Mα,σ with J(µ) = C(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content=' Because of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='13) this limit has to be µα, and  thus by a compactness argument the full sequence tends to µα indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  Acknowledgement  I want to thank Daan Huybrechs and Nick Trefethen for their interest in this  work, for useful discussions, and for stimulating me to write the details of  the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='  The author was supported by the long term structural funding ”Methusalem  grant of the Flemish Government” and by FWO Flanders projects EOS  30889451 and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='0910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNAzT4oBgHgl3EQfn_3n/content/2301.01591v1.pdf'}
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+Large Low Background kTon-Scale Liquid Argon Time
+Projection Chambers
+T. Bezerra1, A. Borkum1, E. Church2, C. Cuesta3, Z. Djurcic4, J. Genovesi5,
+J. Haiston5, C. M. Jackson2, I. Lazanu6, B. Monreal7, S. Munson2, C. Ortiz8,
+M. Parvu6, S. J. M. Peeters1, D. Pershey8, S. S. Poudel2, J. Reichenbacher5,
+R. Saldanha2, K. Scholberg8, G. Sinev5, S. Westerdale9, J. Zennamo10
+1University of Sussex, Brighton, BN1 9RH, United Kingdom
+2Pacific Northwest National Laboratory, Richland, WA 99352, USA
+3CIEMAT, E-28040 Madrid, Spain
+4Argonne National Laboratory, Argonne, IL 60439, USA
+5South Dakota School of Mines and Technology, Rapid City, SD 57701, USA
+6University of Bucharest, Bucharest, Romania
+7Case Western Reserve University, Cleveland, Ohio 44106, USA
+8Duke University, Durham, NC 27708, USA
+9Princeton University, Princeton, NJ 08544, USA
+10Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
+E-mail: eric.church@pnnl.gov, christopher.jackson@pnnl.gov
+Abstract: We find that it is possible to increase sensitivity to low energy physics in a third
+or fourth DUNE-like module with careful controls over radiopurity and targeted modifications
+to a detector similar to the DUNE Far Detector design. In particular, sensitivity to supernova
+and solar neutrinos can be enhanced with improved MeV-scale reach. A neutrinoless double beta
+decay search with 136Xe loading appears feasible. Furthermore, sensitivity to Weakly-Interacting
+Massive Particle (WIMP) Dark Matter (DM) becomes competitive with the planned world program
+in such a detector, offering a unique seasonal variation detection that is characteristic for the nature
+of WIMPs.
+arXiv:2301.11878v1  [hep-ex]  27 Jan 2023
+
+Large Low Background kTon-Scale LArTPCs
+2
+1. Introduction
+In this study we introduce and discuss a dedicated low background module that would enhance the
+physics program of next-generation experiments such as the planned Deep Underground Neutrino
+Experiment (DUNE). Such a low background DUNE-like module could be installed as either
+module 3 or module 4, the so-called “Module of Opportunity” in DUNE. Such a module would
+increase the physics reach of supernova and solar neutrino physics, and could potentially host
+a next-generation neutrinoless double beta decay (0νββ) or Weakly Interacting Massive Particle
+(WIMP) dark matter search. We refer to this design as the Sanford Underground Low background
+Module (SLoMo).
+The physics reach would be enhanced by lowering the nominal energy threshold of a DUNE-
+like experiment from the anticipated 5-10 MeV to levels necessary to address three potential
+physics targets, listed in order of increasing difficulty:
+• ∼ 3.5 MeV energy threshold. This threshold is set by the Q-value of the 42K (daughter of
+42Ar) in the detector target. By reducing neutron captures, alpha-emitting radon daughters
+and pileup events above this threshold, supernova burst neutrino sensitivity could be increased
+in distance, energy and time. Sensitivity to solar neutrinos would also be enhanced, allowing
+explorations of interesting solar-reactor oscillation tensions and Non-Standard Interactions.
+• ∼ 0.5 MeV energy threshold. This threshold is set by the decay Q-value of the 39Ar in the
+detector target. With reduction of electron and photon backgrounds in the target, particularly
+if the 42Ar content is reduced through use of underground argon (UAr), sensitivity to low
+energy solar neutrinos from the CNO process would allow a precision measurement to be
+made. Such a detector will be sensitive to 0νββ search with loading of 136Xe,.
+• < 100 keV energy threshold.
+This threshold could be achieved by enhancing the light
+collection within the detector and by lowering the 39Ar background by deploying UAr. With
+rejection of electron recoil backgrounds (using timing based pulse shape discrimination), a
+sensitive WIMP dark matter search could take place, and interesting phenomena such as
+a supernova coherent elastic neutrino-nucleus scattering signal (a CEνNS glow) could be
+studied.
+These low background targets are achievable due to the unprecedented size and also the increased
+radiopurity of the module, allowing significant fiducialization and hence less stringent radioactive
+background requirements than current world-leading dark matter searches. The light enhancements
+are achievable with current production techniques. As described in Section 2.2.3, production
+of UAr at the scale required for a DUNE module is potentially achievable, though it requires a
+dedicated effort to identify the potential argon source and work with commercial gas suppliers
+In this paper in Section 2 we outline the design of the module and discuss potential paths to
+achieve the detector requirements. In Section 3 we present our initial studies of physics reach of
+this detector.
+
+Large Low Background kTon-Scale LArTPCs
+3
+Figure 1. Summary of physics targets of this low background module and the primary radiological
+backgrounds.
+2. Detector Design
+This section outlines the proposed design for the low background module and describes in detail
+the radioactive background control and photon detection system enhancements required to enable
+physics measurements outlined in the introduction. We note this module design is not necessarily
+endorsed by the DUNE collaboration, as the so-called “Phase II” process that includes building the
+final two far detector modules is in its early stages.
+2.1. Module Layout
+A low background module and its attendant physics goals are enabled most simply by minimizing
+the detector components in the bulk of the argon. The resulting module must still allow for very
+good light detection efficiency and for charge detection efficiency similar to existing designs of
+large LArTPCs [1, 2, 3]. For the benefit of conforming to the longstanding plans for the Long
+Baseline Neutrino Facility (LBNF) far detector complex and cavern layouts, we also want to
+use the same commercial cryostat concept and existing module designs to the greatest extent
+possible and perturb them only where necessary. Starting with the existing DUNE modules, simple
+modifications assure minimal disruption to the main long baseline neutrino oscillation program to
+measure remaining parameters in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.
+2.1.1. Single phase
+We therefore start with DUNE’s Far Detector Vertical Drift Module (VD) [4],
+sometimes referred to as Module 2, and consider design modifications to suit our low background
+purposes. We show a working design in Figure 2.
+Water in “bricks” which are imagined to be nestled among the I-beam support structure
+
+WIMPs
+Ovβ
+Solar Neutrinos
+Supernova Neutrinos
+39Ar
+42 Ar
+Internal Alphas/Betas/Gammas
+NeutronsLarge Low Background kTon-Scale LArTPCs
+4
+Figure 2. Shown is the base design for the proposed low background detector. Blue shows external
+water “bricks”. The top and bottom yellow planes are the Charge Readout Panels unchanged from
+the Vertical Detector design. The central cathode is in green. The white box of acrylic (full interior
+volume) is of thickness 1 inch and has x,y,z extent of 6,12,40 m. The black points are SiPM modules
+shown here at a coverage of 10%, while some studies in this paper use up to 80% coverage. A
+proposed fiducial volume totaling 2 kTon is shown in the two beige boxes. This paper also considers
+a 3 kTon volume.
+achieve large external neutron reduction, as discussed in Section 2.2.1.
+We keep the Charge
+Readout Planes of the VD unaltered. Similarly the central cathode of the VD is preserved with
+the exception that the sparse and mostly distant photon detection modules, known as X-Arapucas,
+are swapped out for the SiPM modules mounted within the cathode plane – at least on that part
+of the cathode in the inner region of this detector. An acrylic box of one inch thickness with
+x,y,z extent of 6,12,40 m serves to mount SiPM modules and reflective WLS foils. Here x is the
+horizontal dimension, y is the vertical dimension, z is in the beam direction. SiPM modules are
+mounted on the inside of the acrylic box at anywhere from 10-80% area coverage. We envision two
+1 or 1.5 kTon long skinny fiducial volumes, depending on the study, that have a stand-off of 1.5m
+from the central cathode. We also discuss, alternatively, a 3 kTon fiducial volume in this paper in
+studies where backgrounds from the central cathode are thought to be small or events from it can
+be reconstructed and cut away.
+The bulk volume of this module consists mostly of argon, with only small material
+contributions such as the slender support structures for the cathode plane panels. As in the VD,
+there are two 6 m vertical drifts.
+
+Large Low Background kTon-Scale LArTPCs
+5
+(a)
+Figure 3.
+Interactions are shown above 100 keV for neutrons emanating from the cold cryostat
+stainless steel at 2 · 10−10 neutrons/cm3/sec for a 1.4 yr exposure. We show a possible ∼3kTon
+fiducial volume pair, avoiding the cathode, looking down the beam line. (The vertical bands are
+interactions in the acrylic.)
+2.2. Radioactive Backgrounds
+To enable the physics targets for this module, improvements in control of internal and external
+radioactive background levels are required. Making a module of this size low background will
+require significant quality and materials controls beyond what has been been attempted by previous
+experiments, and certainly beyond what is required for the Phase I DUNE program. However, we
+note that due to the increased size of this module self-shielding in the argon allows the background
+requirements to be less stringent than those expected to be reached by the current Generation 2
+(G2) dark matter experiments. Thus future research and development will need to focus on how to
+scale the techniques successfully deployed to low background dark matter or neutrinoless double
+beta decay searches to the kTon scale.
+A particular concern for this module will be neutron-induced background events, which will
+be the main background to the neutrino searches above 3.5 MeV thresholds. A neutron capture in
+40Ar produces a 6.1 MeV gamma cascade which can Compton scatter or pair produce electrons
+which can mimic the charged current neutrino interactions in the argon. Captures on 36Ar can
+produce 8.8 MeV gamma cascades. Neutrons are also the primary backgrounds for the lowest
+energy searches for WIMP dark matter, where nuclear recoils can mimic the signal. Below 3.5
+MeV the primary backgrounds will come from alpha, beta and gamma emitting isotopes within
+the argon or detector materials.
+
+6000
+400
+4000
+350
+300
+2000
+250
+y [mm]
+200
+150
+-2000
+100
+4000
+50
+-6000
+6000
+4000
+-2000
+2000
+4000
+6000
+x[mm]Large Low Background kTon-Scale LArTPCs
+6
+2.2.1. Cavern Neutron Backgrounds
+The most significant source of neutrons will likely be those
+induced by spontaneous fission or (α, n) interactions from the uranium-238 or thorium-232 decay
+chains within the surrounding rock and shotcrete of the detector cavern. As reported in [5], the
+external neutron rate at SURF (a likely hosting laboratory for this low background module) is
+assumed to be 1.0 × 10−5 n/cm2/s. As proposed in References [6, 7], it is possible to add water
+shielding to a DUNE-like cryostat, taking advantage of the space between the structural supports
+even when the detector is located within a cavern at SURF with limited space around the detector.
+Those references shows that a 40 cm water shield, located within the support structure, is enough
+to lower the external neutron rate by three orders of magnitude. We assume this is achievable for
+this module.
+The strategy to control the radioactive backgrounds from the detector components such as the
+cryostat will have three parts:
+• improvements to material selection,
+• additional internal neutron shielding,
+• and advanced event selections and analysis tools,
+with the aim of lowering the internal neutron rate within the detector by at least three orders of
+magnitude to match the levels of the external rates.
+Aside from external neutrons from the cavern, the main source of neutrons in a DUNE-like
+detector cryostat is likely to be spontaneous fission or (α, n) interactions from the uranium-238 or
+thorium-232 decay chains in the order 1 kTon of stainless steel that makes up the I-beam support
+structure. Research and development will be required to lower the internal background rates by
+the three orders of magnitude required, for example by careful selection of the raw ingredients
+and/or control of the manufacturing process. It should be noted that the “Generation 2” dark
+matter experiments expect to reach neutron rates from their steel a further two orders of magnitude
+beyond this, so the goal is achievable (even if the scale is much larger than previously attempted).
+Another area of R&D required to support this goal is improvements in knowledge of (α, n) cross
+sections, as highlighted in a recent IAEA workshop [8].
+Another approach to reduce neutron background from the internal shielding within the
+detector would be by adding higher density rigid polyurethane foam (R-PUF) insulation and/or
+boron, lithium or gadolinium loaded material layers within the membrane cryostat structure.
+Our studies show that this could easily reduce the neutron capture rate in LAr by one order of
+magnitude. One approach, as planned by the DarkSide collaboration, would be to use the additional
+planes of Gd-doped acrylic to act as a neutron absorber. DarkSide-20k [9] intends to use multiple
+layers within a ProtoDUNE-style cryostat for their dark matter search. Another design choice
+might be to take advantage of the existing cryostat but replace some materials such as the insulating
+foam with a borated version, e.g., to reduce backgrounds from the support structure.
+Analysis based cuts can also be used to remove events. For example, with the low threshold
+of this planned detector, neutron induced multiple scatters could be tagged and rejected. This
+
+Large Low Background kTon-Scale LArTPCs
+7
+takes advantage of the excellent (∼ 10 mm) position resolution of a TPC. Studies [10] to identify
+dark matter nuclear recoil backgrounds show ∼ 30% reduction at 100 keV threshold and ∼ 90%
+reduction at 50 keV, due to the increased probability of detecting an additional scatter at lower
+thresholds.
+2.2.2. Radon and other internal argon backgrounds
+Radon is an important background that must
+be controlled as it can diffuse throughout the detector, entering the fiducial volume. The radon
+decays to a number of daughter isotopes that can be direct backgrounds (for example 214Bi for a
+neutrinoless double beta decay search) or that can produce neutrons through (α, n) interactions.
+For this low background module we set a radon target level in the liquid argon of 2 µBq/kg. This is
+about three orders of magnitude below the expected DUNE radon level of 1 mBq/kg [11, 12, 13].
+2 µBq/kg has been achieved in liquid argon by the DarkSide-50 experiment [14], and exceeded
+by DEAP-3600 which achieved a level of 0.2 µBq/kg [15]. It should be noted that the higher
+volume-to-(radon-emanating)-surface ratio in a large detector such as DUNE compared to dark
+matter detectors will help achieve this target.
+To reach this level in a kTon-scale detector will require research and development to
+implement a combination of the following techniques:
+• Radon removal during purification via an inline radon trap.
+No radon removal is
+in the current design of the purification system for the baseline DUNE. Dark matter and
+neutrinoless double beta decay search experiments typically use cooled, activated charcoal
+radon traps to remove radon directly from the recirculating target. The most sensitive dark
+matter experiments typically purify the argon in the gaseous phase [16, 17], however such
+an approach would be impractical for a kTon-scale experiment. Borexino used a charcoal
+radon trap to purify liquid nitrogen [18], and such an approach could be adopted and scaled
+appropriately for a low background module. New materials such as Metal-organic frameworks
+could improve the capture-potential beyond charcoal, allowing a potential shrinkage of
+footprint of a radon-capture facility to fit the existing cavern designs. Recent evidence from
+MicroBoone [19] indicates that a copper filter purification system similar to that planned for
+DUNE may remove greater than 97% or 99.999% of the radon (depending on whether slowed
+or trapped) in the system without the need for additional removal techniques.
+• Emanation measurement materials campaign. All materials used in detector construction
+are known to emanate radon at some level.
+A large-scale emanation assay campaign to
+identify materials suitable for construction, similar to the QA/QC campaign described above
+will be required to ensure the detector can meet the target. A topic for R&D will be how
+to increase throughput of samples, as emanation measurements typically take two weeks per
+sample.
+• Surface treatments. For large components such as the cryostat where it may be impractical
+and costly to make significant improvements to the radiopurity, surface treatments can be
+used to lower emanation rates. It is known that acid leaching and electropolishing lowers
+
+Large Low Background kTon-Scale LArTPCs
+8
+emanation rates. Coating the inner surface of the cryostat with a radon barrier could lower
+emanation rates from this significant source.
+• Dust control. Dust is a significant radon source and cleanliness standards will be higher
+in this low background module than the baseline DUNE design. Cleanliness protocols and
+requirements R&D will be necessary to develop automated techniques applicable to the
+thousands of m2’s of surface area of a DUNE-like cryostat, for example.
+• Radon reduction system during installation and operation: The mine air underground
+is radon laden up to 1,000 Bq/m3 and radon daughter plate-out during installation, filling
+and operation must be controlled. An upscaled vacuum swing system with large charcoal
+columns in parallel to remove radon from the ambient air and to provide radon-free air to the
+cleanroom and cryostat would be suitable. Vacuum swing systems providing radon-free air
+have been successfully employed by e.g. Borexino [20], LZ [21] and SuperCDMS [22].
+• Drifting of charged daughters to cathode. Several daughters in the radon chain are charged
+and will drift towards the cathode and out of the fiducial volume. This effect may be countered
+by mixing effects of the purification system however.
+• Alpha tagging through pulse shape discrimination. Alpha events in the radon chain that
+produce neutrons directly in the argon may be taggable, by identifying the alpha track before
+the (α,n) event.
+Though the amount of light may be relatively small, the timing profile
+is distinct and may allow pulse shape discrimination on this module with enhanced optical
+systems.
+2.2.3. Underground Argon
+Atmospheric argon (AAr) consists mostly of 40Ar which is stable.
+There are some long-lived radioactive isotopes-39Ar (T1/2=269y, Qβ=565 keV), 37Ar (T1/2=35d,
+Q=813 keV), 42Ar (T1/2=32.9y, Qβ=599 keV) [23] which are, in atmosphere, produced primarily
+by cosmic ray-induced reactions in 40Ar. The use of atmospheric argon in a low-threshold multi-
+ton scale argon detector has limitations due to high 39Ar activity (1 Bq per kg of argon [24]) in
+atmospheric argon (AAr). Radiogenic and cosmic-ray muon-induced interactions, especially on K
+and Ca isotopes, can produce 39Ar underground. The dominant production channels are negative
+muon capture on 39K and (α, n)-induced (n,p) reactions on 39K. 39Ar production underground
+decreases significantly with depth[25] as muon flux decreases. DarkSide-50, the only experiment
+to use underground argon (UAr), measured the 39Ar activity of 0.73 mBq/kg [26], a factor of
+1400 smaller than in AAr. With the ARIA project[27], which is planning for a throughput for
+39Ar processing of ∼ 10 kg/day, the DarkSide collaboration is planning to further reduce the 39Ar
+present in UAr through large-scale isotopic separation by cryogenic distillation.
+42Ar decays in the bulk argon volume will produce 42K isotopes.
+While the 42Ar beta-
+spectrum has an endpoint of 599 keV, Betas from 42K-decays span a much larger energy range
+up to 3.5 MeV, and can be problematic backgrounds. Since UAr should be heavily depleted of
+42Ar, significant suppression of 42K decay backgrounds is achievable with UAr. In the atmosphere,
+the 42Ar concentration is ∼ 10−20 42Ar per 40Ar atom [28],[29], which is four orders of magnitude
+
+Large Low Background kTon-Scale LArTPCs
+9
+smaller than the concentration of 39Ar. The daughter isotope of 42Ar, 42K (T1/2= 12 h) has two
+major decay modes : 1) direct beta-decay to the ground state of 42Ca (Qβ= 3525 keV, BR=81
+%), and 2) Beta-decay (Qβ= 2001 keV) to an excited state of 42Ca followed by a prompt 1524
+keV gamma emission. In the atmosphere 42Ar is primarily produced by 40Ar(α, 2p)42Ar occurring
+in the upper atmosphere [29], where energetic alphas are readily available from cosmic-ray muon
+interactions. Production through two-step neutron capture is also possible but greatly sub-dominant
+due to the short-lived intermediate isotope 41Ar [30]. The 42Ar production rate underground is
+not known, but it is expected to be several orders of magnitude smaller than in AAr. Particle
+interactions on isotopes of K, Ca, and Ti can produce some 42Ar in the earth’s crust, given the
+relatively high abundance of the elements (by mass-fraction[31]:K-2.09%,Ca-4.15%,Ti-0.565%).
+However, the reaction thresholds are high, making 42Ar production energetically not possible by
+fission, (α,n)-neutrons or alphas from the natural radioactivity chains of 238U, 235U, and 232Th.
+Energetic particles from cosmic ray muon-induced interactions can produce 42Ar.
+But at the
+depths at which underground argon is usually extracted, the cosmic ray muon-flux should be hugely
+suppressed, so 42Ar production should be negligible. Based on GERDA’s findings[32], following
+42Ar decays, 42K nuclei could retain the positive charge long enough to drift in the influence of
+electric field. So, we expect 42K ions to drift and move towards the cathode plane, which suggests
+an additional suppression of 42K backgrounds is achievable through fiducialisation.
+The 85Kr isotope, predominantly a β-emitter, has a half-life of 10.7 years and Q-value of 687
+keV[23]. Primary modes of 85Kr production are spontaneous fission of uranium and plutonium
+isotopes, neutron capture on 84Kr, and human-induced nuclear fissions in nuclear reactors[33]. We
+would expect 85Kr to be present at some level in AAr. However, its concentration can vary across
+argon extraction sites. Using the UAr data, DarkSide-50 measured 85Kr activity of 2 mBq/kg[26],
+a few orders of magnitude smaller than in AAr.
+85Kr concentration in UAr should also vary
+depending on the location of the gas reservoir and gas origin (mantle-like or crustal-like). DEAP
+sees no evidence of 85Kr in its AAr after filtering in a charcoal trap with 3.3 tonnes of LAr [34].
+We expect argon gas extracted from an underground source to be highly depleted of 39Ar,
+42Ar and 85Kr.
+There is evidence that air infiltration during the UAr extraction could have
+contributed to the DarkSide-50’s 39Ar - actual 39Ar content in the UAr could be significantly
+smaller (on the order of few tens of µBq/kg)[35].
+85Kr and 42Ar content could also be much
+smaller. Unlike stable gas isotopes such as 40Ar, which can collect at gas wells over time, isotopes
+such as 39Ar(T1/2=269y), 42Ar (T1/2=32.9y) and 85Kr(T1/2=10.7y) diffusing through rocks and
+collecting in a significant number at the underground gas wells is less likely.
+However, air-
+infiltration and cosmogenic activation in the argon bulk could introduce these isotopes in the
+extracted UAr [36, 37]. Greater care, perhaps, is necessary to ensure avoiding contamination of
+the UAr during extraction, processing, transport and storage.
+While UAr is desirable, it requires a dedicated effort to identify the potential argon source
+and procure argon on a large enough scale necessary for this project. The Urania plant [9] in
+southwestern Colorado, USA is expected to produce underground argon from CO2 gas wells at
+
+Large Low Background kTon-Scale LArTPCs
+10
+a rate of ∼ 300 kg/day (at full rate) for DarkSide. The argon source and the production rate is
+not large enough for a kiloton scale experiment. The authors are in the process of identifying
+alternative gas wells with enriched argon streams. Discussions with three potential commercial
+suppliers are ongoing, however the underground source samples are not yet tested and low levels
+of radioactive isotopes must be proven. Initial gas analysis indicates the mantle origin of this
+sample, with suppressed cosmogenic production compared to a crust source. Based on estimates
+by the gas suppliers, the production cost could be as low as three times the cost of atmospheric
+argon and 5 kTon of argon production per year could be achievable.
+2.2.4. Surface storage and spallation issues
+During above-ground storage of our UAr, before
+placement into the underground module, 39Ar is produced by cosmogenic neutrons in the reactions:
+40Ar(n,2n)39Ar and 38Ar(n,γ)39Ar. Production cross sections for 39Ar have been measured in [38].
+42Ar is produced primarily in 40Ar(α, 2p)42Ar reactions [29]. In a previous work [39], to which
+we refer the interested reader, the cross sections for these reactions were obtained from nuclear
+reaction codes and confronted with data where they exist. An evaluation of cosmogenic 39Ar and
+42Ar production in UAr stored on the surface can be found in [40]. A full study of expected
+spallation and pileup backgrounds during the detector operation is in reference [6].
+2.3. Light Collection Enhancement
+The light collection for this low background module will be enhanced to enable two main goals:
+• lower the energy threshold and improve the resolution at these lower neutrino energies;
+• improve pulse shape discrimination for radioactive background rejection.
+In this section we describe the improvements to the light detection system required to enable this.
+2.3.1. Light Collection Efficiency Impacts on Energy Resolution
+Charged particles that traverse
+the liquid argon deposit energy and excite and ionize the argon atoms. This process results in
+the electrons recombining with the ions generating unstable argon dimers, which decay and emit
+scintillation light. If an electric field is applied, a fraction of the electrons will drift away before
+recombining. These ionization electrons are collected by the anode plane and create the charge
+signal by removing electrons that would have recombined yields an anticorrelation between the
+light and charge signals observed in LArTPCs. The small number of ionized argon atoms at low
+energies means that fluctuations in this recombination process can smear the amount of charge
+observed to energy deposits. The Noble Element Simulation Technique (NEST) collaboration has
+explored the precision of LArTPC for 1 MeV electrons as a function of charge readout signal-
+to-noise ratio and the efficiency of collecting light [41]. They predict that using only the charge
+signal in a LArTPC, the most precise one can reconstruct the energy for a 1 MeV electron is 5%
+
+Large Low Background kTon-Scale LArTPCs
+11
+and MicroBooNE has validated this prediction [42]. To improve the energy reconstruction, further
+one needs to include measurements of scintillation light.
+The NEST collaboration explored the expected energy resolution enhancements that a
+LArTPC can achieve by including light signals along with the charge measurement. Ref. [41]
+shows that the energy resolution for a 1 MeV electron for LArTPCs with different signal-to-noise
+ratios (SNR) and varying light collection efficiency. In particular, we see that a LArTPC with SNR
+near 40 needs 50% of the scintillation light to measure energy deposits with 1% precision.
+2.3.2. Photosensors
+As a baseline design our plan is to use 24 cm2, Darkside-20k style [9],
+SiPM tiles. 50% quantum efficiency at visible wavelengths. We assume here another 50% wave
+length shifter (WLS) efficiency from, e.g., TPB on the tile surface, for a total efficiency of 25%.
+For maximum light detection we envision covering inside the cathode (between the two planes of
+cathode wires, as will be done in module 2) and the interior acrylic walls at up to 80% coverage. We
+assume the Module 2 power-over-fiber concerns [4] to be solved to allow our SiPM coverage of the
+cathode. The resulting number of SiPM modules for 10% coverage – a value which current studies
+naively show is sufficient for a dark matter search using pulse shape discrimination – is ∼ 50000,
+to be compared to Darkside20k’s planned ∼ 10000. However, as discussed in Section 2.3.1, to
+achieve the energy resolution required for a neutrinoless double beta decay search will require
+significantly more coverage. It is likely possible to optimize the placement of the tiles around the
+fiducial volume, to reduce the total number required, and work is ongoing to study this.
+2.3.3. Reflectors
+To maximize the light capture in this module the surface of the acrylic box will
+be coated with a reflector to create a light-tight inner volume. If a PTFE reflector is used, as in
+DarkSide-50, reflectivities of 97% should be possible.
+2.3.4. Argon Purity
+The baseline requirement for DUNE is < 25 ppm of nitrogen to ensure
+that photon propagation in the argon is not attenuated. For our simulation studies we assume
+that attenuation (absorption) lengths of order 50 m are achievable, which corresponds to nitrogen
+contamination levels of 1-2 ppm [43].
+We note that dedicated dark matter experiments have
+achieved ppb levels of purity.
+3. Physics Studies
+This section outlines key physics goals of this module and describes the studies that have been
+performed.
+
+Large Low Background kTon-Scale LArTPCs
+12
+3.1. Argon-39 Studies
+Reduction of the rate of 39Ar is desirable for a number of reasons. This 600 keV endpoint beta
+emitter decays at a 1 Bq/l rate in ordinary atmospheric argon. It may mask low energy physics by
+creating optical signals that confuse the reconstruction process. It also represents a serious hurdle
+in the detector triggering considerations.
+Trigger primitives, which constitute a 6 PByte/year data source, not necessarily to be stored
+into perpetuity, are still an onerous data flow to deal with during steady data-taking. That number
+drops to a far more manageable tens of TBytes if the 39Ar is reduced by a factor of 1400. In reality
+of course, much of the data budget will perhaps be consumed by low-threshold activity in this
+detector.
+Perhaps the more pernicious practical problem is that 39Ar beta decays may reconstruct
+as optical ”hits” which may, in turn, comprise ”flashes” which then confuse the charge-light
+association for reconstructed physics objects – especially at low energy and far from the light
+detectors.
+(Hits are simply individual light signals over threshold and the parameters that
+characterize them, and flashes are collections of hits in narrow ranges of time, hypothesized to be of
+the same physical origin.) We have performed a study in a Module 1-like environment, not shown
+here, that gives the not very surprising conclusion that supernova flashes become unambiguously
+matched with a x1400 39Ar reduction.
+Third, the highly desirable property of pulse shape discrimination (PSD) in Argon which
+would allow to subtract out the nuisance 39Ar contribution for our, e.g., dark matter search
+ambitions, is not practicable if pile-up is too intense. We show in [10] and discuss later in this
+paper that a x1400 reduction of 39Ar just allows for a search in our fiducial volume down to 100
+keV thresholds and perhaps lower.
+Here we mention that even a x1400 reduction of 39Ar does not allow this detector to get
+to arbitrarily low thresholds for searches of physics with electronic signatures – as apposed to
+neutron-like interactions. We expect that reductions by this amount will still leave 39Ar as an
+overwhelming background to low-rate, low-energy solar processes, for example.
+3.2. Simulation
+Almost all studies in this paper are carried out in a standalone Geant4[44] simulation. Source
+code and build instructions are found at Reference [45]. In that simulation is proper isotope decay
+and neutron physics, along with optical physics. The volume is basically a 10 kTon box of liquid
+argon, but with a reasonable model of the cryostat walls on all six sides with charge readout planes
+(CRPs) made of G10 on the floor and ceiling, as in the VD. Further, there is an acrylic box inside,
+open on top and bottom and tiled with 24 cm2 SiPM modules at an 80% coverage. SiPM modules
+with that same coverage viewing both upper and lower volumes also tile the central cathode plane.
+See Figure 2 for a representation of our simulated geometry. We use an after-the-fact 25% total
+
+Large Low Background kTon-Scale LArTPCs
+13
+quantum efficiency by merely scaling by that fraction of the detected hits. There is no SiPM
+electronics response applied. We include a 96% reflectivity of the acrylic surface and a 44%
+reflectivity for the CRPs (as the holes in each CRP are about 56% of the surface area), and impose
+an argon attenuation (absorption) length of 50m, and a Rayleigh scattering length of 90cm. All
+simulations generate their 128 nm photons ab initio from the charge particles which create them
+and are propagated on the fly, with no lookup libraries, until their end point on a SiPM where they
+are counted or they disappear through absorption.
+3.3. Optical Studies
+The simulation described in Section 3.2 was used to study the minimal requirements for the
+optical system to allow pulse shape discrimination in a dark matter search, including the required
+reflectivity of the acrylic box and anode readout surfaces as a function of SiPM tile coverage.
+The minimal photon counting requirement for the optical system was set at 400 photons reaching
+the SiPM surface, which results in a total of 100 photons being detected due to the assumed 25%
+efficiency described in Section 2.3.2. The 400 (100) photon requirement was chosed as the minimal
+amount of photons required to perform a pulse shape discrimination analysis such as described in
+Section 3.7.
+The results of the simulation are shown in Figure 4. The studies were performed varying
+acrylic and anode reflectivity between 0 % and 100 % (a realistic 97 % is highlighted) and then
+varying the SiPM coverage on the walls and cathode plane to count the accepted photons. Both the
+standard acrylic box described above and also a maximally sized box where the walls are moved
+out to the edges of the detector were simulated. This large box would be the worst-case optics
+scenario, maximising the path length of the photons. The studies show that a relatively modest
+amount of SiPM coverage of 10-20 % is required even in the worst-case scenarios to reach the
+target.
+The effect of attenuation with the liquid argon was also studied. Figure 5 shows the number of
+photons detected at the SiPMs as a function of SiPM coverage for a variety of different attenuation
+lengths. Assuming the relation between nitrogen contamination of the argon versus attenuation
+found in [43], this study shows that the 10-20% SiPM coverage is sufficient to tolerate 0.5-5 ppm
+levels of nitrogen within the argon.
+3.4. Supernova Neutrino Physics
+In this section we present several supernova neutrino burst studies that could be enhanced by this
+module. This includes increased sensitivity to lower energies, later times and sources from greater
+distances. We also discuss the CEνNS glow sensitivity.
+
+Large Low Background kTon-Scale LArTPCs
+14
+(a)
+(b)
+(c)
+(d)
+Figure 4.
+Number of photons detected at the SiPMs as a function of coverage, varying both
+reflectivity of the surrounding acrylic and anode plane walls, and the size of the acrylic box
+containing the inner volume, from Original Size (6x12x20 m3) to Max Size (at fiducial boundaries
+at 12x12x60 m3).
+
+Acrylic Reflectivity (Original Size Acrylic Box)
+Mean SiPM Hits on Y-axis
+006
+800
+700
+600
+500
+400
+Target
+300
+200
++100%
+■97%
+≥0%
+100
+0
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+% of SiPM CoverageAcrylic Reflectivity (Max Size Acrylic Box)
+900
+Mean SiPM Hits on Y-axis
+800
+700
+600
+500
+400
+Target
+300
+200
++100%
+■97%
+ 0%
+100
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+% of SipM CoverageAnode Reflectivity (Original Size Acrylic Box)
+006
+Mean SiPM Hits on Y-axis
+800
+700
+600
+500
+Target
+300
+200
++100%
+■44%
+≥0%
+100
+0
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+% of siPM CoverageAnode Reflectivity (Max Size Acrylic Box)
+Mean SiPM Hits on Y-axis
+900
+800
+700
+600
+009
+Target
+000
+200
++100%
+■44%
+≥ 0%
+100
+0
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+% of SiPM CoverageLarge Low Background kTon-Scale LArTPCs
+15
+Figure 5. Number of photons detected as a function of SiPM coverage when varying the attenuation
+length within the argon.
+3.4.1. Supernova Energy Spectrum
+We wish to explore the potential improvement provided by
+a low background large LArTPC to that achievable in current DUNE Far Detector designs for
+sensitivity to supernova explosion detection. To this end we simulate a ten-second exposure of our
+detector to a flux of electron neutrinos from the Livermore [46] supernova model and run them
+through the MARLEY [47] event generator to produce the final state particles in liquid argon.
+MARLEY considers all candidate νe processes, including coherent, elastic, and charged current
+processes. The detector response is provided by the simulation described in previous section 3.2.
+Using that simulation we count SiPM hits for an 80% coverage and convert to energy using a
+simple conversion. That conversion comes from running 3 MeV electrons uniformly through the
+detector and counting SiPM hits in a coarse x,y,z binning of the production point. We emphasize
+that this study is simplified and that refinements including TPC charge response as well as better
+SIPM energy resolution will result in improvements. Events must originate in the inner 3 kTon
+fiducial volume. The result is shown in Figure 6. The most evident feature is that the elastic
+scattering component of the νe flux becomes dominant at low energies inaccessible to the baseline
+DUNE far detectors – and it does so because neutron rates are required to be low from the cold
+cryoskin stainless steel and the 42Ar is at very low levels in UAr. The threshold here can go all the
+way down to 600 keV, which is a factor of roughly 18 lower than the baseline DUNE far detector
+design.
+The low detector threshold allows access to a significant number of elastic scatter events
+within the liquid argon.
+This opens up the possibility of reconstructing the position of the
+supernova with a pointing analysis. Liquid argon TPCs, with the excellent track resolution, are
+well suited to making this measurement. With the expected reduction in backgrounds in this
+sample, a clean elastic scatter sample will dominate at thresholds below 5 MeV (see Figure 6),
+though pointing with these reconstructed tracks is challenging.
+3.4.2. Supernova Trigger
+The trigger system in a DUNE-like LAr detector relies on the so-called
+Trigger Primitives (i.e. hits, TPs) generated from the electronics connected to the wires or photo-
+
+Ar Attenuation Length (Original Size Acrylic Box)
+900
+Mean SiPM Hits on Y-axis
+800
+700
+600
+500
+400
+Target
+300
+200
+→100m =50m ^20m 10m
+100
+0
+4
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+15
+16
+17
+18
+19
+20
+21
+22
+% of SiPM CoverageLarge Low Background kTon-Scale LArTPCs
+16
+Figure 6.
+Energy deposited from neutrinos from supernova located at 10 kpc, assuming the
+Livermore model, during a ten second detector exposure window. Here we presume no Rn222
+contribution and a likely too conservative x500 Ar42 suppression in UAr with respect to that
+achievable in atmospheric argon. Reaching 5 MeV SN detection is straightforward in this module.
+sensors. They are simple objects constructed from the signal waveform. A stream of TPs arrives
+in the trigger module of the DAQ, which will use them to form a Trigger Activity (i.e. a cluster of
+hits, TAs), which is an association in time and space done by a trigger algorithm. A TA is related
+to each sub-module/component of the detector (e.g. an APA module), and multiple TAs form a
+Trigger Candidate (TC). Having a positive TC, the readout system stores the requested data. For
+low energy physics, which does not have an external trigger like beam events, it is essential to
+understand the detector backgrounds (e.g. radiological and electronics noise), to avoid triggers
+issued by undesired data. Thus, a Low Background LAr detector can perform better than the
+current designs.
+The bottleneck for designing efficient supernova neutrino burst (SNB) trigger algorithms is
+the data transfer and storage resources available for the detector. To get the most of an SNB event,
+around 100 seconds of full detector readout is desirable, which means about 150 TB of raw data for
+a 10 kTon LAr detector. It will take about one hour to transfer the data from this trigger event from
+
+livCC
+livES
+103
+.0kTonne-10sec/1.0MeV
+Ar39 Bg/L/1500
+Ar42Bq/L/1500/5E8
+neutrons 1e-11/cm3/s
+101
+Events
+10-5
+0
+5
+10
+15
+20
+25
+Energy [MeV]Large Low Background kTon-Scale LArTPCs
+17
+the detector caverns to the storage on the surface and several additional hours to transmit those
+data to the primary storage centre. Therefore, while the effective threshold must be set low enough
+to satisfy the requirements on SNB detection efficiency, it is crucial to not fire too frequently on
+background fluctuations. A requirement on the fake trigger rate of once per month is determined by
+these limits on data-handling, for a DUNE-like LAr detector. Additionally, the triggering decision
+needs to be made within 10 seconds since this is the typical amount of data buffered in the DAQ.
+SNB triggers are likely to be both TPC- and Photon Detection System (PDS)-based in far detector
+modules one and two, though at the lower energy thresholds we concern ourselves with in this
+paper a PDS-only trigger will be required.
+In current studies both TPC and PDS information gives a tagging efficiency of about 20-30%
+for a single neutrino interaction [48]. Reducing the estimated neutron capture rate in the LAr
+volume by a factor of ten (which is the principal background for SNB triggering, given the higher
+de-excitation gamma energy of about 6 MeV when compared to other radiological components),
+this efficiency improves to 70%, which translates to a 100% (20%) SNB triggering efficiency
+for a Milky Way (Magellanic Clouds) SNB. The trigger strategy described above is a “counting”
+method. If we utilise the integrated charge of each TP, it is possible to construct a distribution
+of the TAs raw energy (SNB signal with backgrounds) and compare it to the background-only
+one. The “shape” triggering method improves the efficiency to tag Magellan Clouds’ SNB. The
+efficiency figure for such a SNB producing ten neutrino interactions (see Ref. [48]) shows the
+efficiency as 70% in a ten kTons DUNE-like LAr detector using the standard DUNE background
+model described in Reference [1] and a shape triggering algorithm that keeps the fake trigger rate
+to one per month.
+With the Low Background LAr detector, a less stringent selection can be used, increasing
+the signal efficiency while keeping the fake trigger rate at the requirement level. Thus, higher
+efficiencies will be reached with a lower number of SN events, it then being possible to achieve
+100% efficiency to trigger a Magellan Cloud SNB. Identifying Andromeda’s SNBs is more
+challenging even with this design since they do not produce enough interactions in the whole
+detector volume in a 10 seconds window (see the supernova sensitivity plot in Ref.
+[48]), and
+lowering the TA requirements would encounter the 39Ar activities.
+3.4.3.
+Late Time Supernova Neutrinos The neutrino flux from a core-collapse supernova is
+expected to cool over a few tens of seconds, with the late time events getting lower and lower
+in energy. The tail end of the burst, where a black-hole-formation cutoff may be present, will be
+challenging to observe. A lower energy threshold extends the time range with which a large liquid
+argon detector can follow the evolution of the supernova burst [49].
+3.4.4.
+Pre-supernova Neutrino Signal Another benefit of lowering the threshold is potential
+sensitivity to presupernova neutrinos [50, 51, 52, 53, 54, 55, 56]. The final stages (hours to days)
+of stellar burning before the core collapse are expected to be associated with an uptick in neutrino
+
+Large Low Background kTon-Scale LArTPCs
+18
+production and energy, and observation of these could provide a true early warning of a core-
+collapse supernova. The presupernova flux is expected to be small, and energies are typically
+less than 10 MeV; nevertheless they may be observable in a large liquid argon detector with low
+threshold [53] for progenitors within a few kpc nearing the ends of their lives.
+3.4.5. CEvNS Glow
+Coherent elastic neutrino-nucleus scattering (CEvNS) [57, 58] is a process
+that occurs when a neutrino interacts coherently with the total weak nuclear charge, causing the
+ground state nucleus to recoil elastically. The cross section is large compared to the inelastic
+charged- and neutral-current interactions, but resulting nuclear recoil energies are in the few tens
+of keV range. CEvNS has now been observed in argon by the COHERENT collaboration using the
+stopped-pion neutrino flux from the Spallation Neutron Source [59] with a cross section on order
+22 · 10−40cm2 for an incoming average flux of < Eν >≈ 30 MeV from pion decay at rest.
+In a large LArTPC there will be a high rate of CEvNS in a core-collapse supernova burst—
+approximately 30-100 times more events with respect to νeCC (the dominant inelastic channel),
+depending on expected supernova spectrum. However, each event is individually not likely to
+produce more than a few detected photons, and sub-50-keV thresholds need to be achieved to find
+these events, if they are to be found one by one. Even with depleted argon, the 39Ar rate down in
+this range in our proposed detector is by far overwhelming. However, an alternative is to identify
+a “CEvNS glow,” [60] in which the excess rate of detected photons can be tracked statistically
+above the 39Ar rate as a function of time in a characteristic explosion. Fig. 7 shows that simulated
+supernova-induced activity from a burst at 10 kpc over 10 seconds in an inner fiducial 3 kTon gives
+three distributions: broadly-distributed-in-time, higher-energy charged-current (CC) component,
+a burst of low-hit-multiplicity CEvNS events at about 0.01 sec, and then the absolutely flat 39Ar
+activity. In ongoing work, we propose to subtract the reconstructed CC events and fit to the CEvNS
+bump above the background. We note that in principle, an excess of collected ionization from
+CEvNS events is observable as a “CEvNS buzz” in coincidence with the CEvNS photon glow.
+3.5. Solar Neutrino Physics
+In this section we present our enhanced sensitivity to solar neutrino searches including lower
+energies and enhanced oscillation sensitivity to ∆m2
+21. This allows exploration of solar-reactor
+oscillation tensions and Non-Standard Interactions. It also allows a precision CNO solar neutrino
+measurement and a measurement of the 3He+p solar flux. A first study of DUNE as the next-
+generation solar neutrino experiment is presented in reference [7]. That study is dominantly one
+of higher energy 8B CC processes, but in this section we widen the discussion and point out what
+a lower threshold would offer to solar neutrino studies.
+3.5.1. Low Threshold Gains and Elastic scattering
+Figure 8 shows the number of expected ES
+interactions over threshold for a 3-kTon·year exposure. One sees that reducing the threshold to 1
+
+Large Low Background kTon-Scale LArTPCs
+19
+(a)
+(b)
+Figure 7. The CEvNS glow is the statistically significant increase observed from the low SiPM
+hit-count CEvNS Supernova events, shown in bottom population of figure (b) that sits on top of the
+rate of 39Ar seen in figure (a).
+MeV makes pep and CNO neutrinos observable. A threshold of 0.5 MeV could add 7Be neutrinos,
+and 0.1 MeV could allow for detection of pp neutrinos, though this threshold encroaches into
+the large 39Ar background, even in UAr. The total number of available neutrinos is important for
+possible studies discussed in Section 3.5.3. This amounts to 9,200 neutrinos for a 1-MeV threshold,
+130,000 neutrinos for a 0.5-MeV threshold, and 820,000 for a 0.1-MeV threshold.
+In addition to the previously discussed methods for reducing backgrounds, we are
+investigating directionality with ES for all solar neutrinos and Cherenkov radiation for more
+energetic 8B neutrinos as ways to enhance neutrino signal over backgrounds.
+
+SiPMHitsvs.Time-Ar39
+2500
+1.4
+2000
+1.2
+1.0
+1500
+events/hits/s
+SiPM Hits
+0.8
+1000
+0.6
+0.4
+500
+0.2
+0.0
+m
+4
+8
+time (s)SiPMHits vs.Time -CEvNS + CC
+105
+103
+102
+104
+Count
+101
+/hits/
+Hit
+103
+100
+/ents/
+SiPM I
+10-1
+102
+10~2
+101
+10~3
+10~4
+10~3
+10-
+10~1
+100
+time (s)Large Low Background kTon-Scale LArTPCs
+20
+Figure 8. Number of events over threshold for solar-neutrino ES interactions in argon for a 3-
+kTon·year exposure with contributions from different solar fluxes.
+3.5.2.
+Solar Neutrino Oscillations Current measurements of the neutrino mixing parameter,
+∆m2
+21, using solar neutrinos from SNO and Super-Kamiokande are currently discrepant at 1.4 σ
+with measurements from KamLAND using neutrinos from nuclear reactors [61]. Differing results
+from these two methods would suggest new physics possibly involved with exotic matter effects
+as the neutrino passes through the Sun and Earth. Further data from DUNE will further investigate
+this discrepancy with high statistical significance. The sensitivity comes from the “day-night”
+effect, a partial regeneration of the νe solar flux due to matter effects in Earth which depends on
+∆m2
+21, neutrino energy, and nadir angle. This is an advantageous strategy allowing for constraints
+of uncertainties using daytime data.
+Solar neutrinos are much lower energy than typically observed in DUNE making
+reconstruction of these events challenging. Also, at these low energies, radiological backgrounds,
+principally neutron capture with a contribution from 40Ar(α, γ), dominate analysis backgrounds.
+A low-background DUNE-like module with enhanced light collection will help with both of these
+challenges.
+Improved energy resolution from increased photodetector coverage would significantly
+improve DUNE-like sensitivity to ∆m2
+21. This would both improve reconstruction of the dominant
+neutron background around the 6.1 MeV total visible energy of the neutron capture on argon-
+40, and better measure the dependence of the νe flux regeneration on neutrino energy. A single
+10-kTon, low-background module could discern between SNO/SK and KamLAND best fits at
+over 6 σ. Lower neutron background levels would also improve the signal-to-background ratio
+for the measurement, which could also lower the energy threshold for detecting and analyzing
+solar neutrino events. An estimate of sensitivity to ∆m2
+21 with 100 kTon-yrs of exposure with
+a low-background module is compared to DUNE’s sensitivity with 400 kTon-yrs of data from
+
+107
+106
+105
+104
+103
+102
+101
+hep
+17F
+150
+7Be
+ES
+100
+eo +eF
+8B
+pep
+total
+eN
+13N
+7Be
+10-1
+10-2
+10-1
+100
+101
+threshold Ethr (MeV)Large Low Background kTon-Scale LArTPCs
+21
+nominal, horizontal drift modules in Fig. 9. We use a larger 10-kTon fiducial volume for the
+reduced background/threshold curves in that figure, increased from other studies in this paper.
+Figure 9. Sensitivity of a low-background module to the neutrino mixing parameter ∆m2
+21 assuming
+a true value of the solar best fit, 5.13·10−5 eV2 [61].
+Colored contours show sensitivity after
+100 kTon-yrs for various detector configurations compared to 400 kTon-yrs of DUNE data with
+horizontal drift design, shown in black and dominated by CC events.
+3.5.3.
+Non-Standard Neutrino Interactions Non-standard neutrino interactions (NSI) could
+modify neutrino oscillations in the Sun and result in a different number of neutrinos observed
+compared to the one predicted by the Standard Model [62] (see also studies with different
+parameter definitions in [63] and [64]).
+The NSI Hamiltonian (for neutral currents only) relevant for solar-neutrino oscillations can
+be written in the following form:
+HNSI
+ν
+=
+√
+2GF(nu + nd)
+� −ϵD
+ϵN
+ϵ∗
+N
+ϵD
+�
+,
+(1)
+where GF is the Fermi constant, nu and nd are the up- and down-quark densities, respectively,
+and ϵD and ϵN are the diagonal and off-diagonal NSI couplings. These couplings affect νe solar
+survival probability (see Figure 10). The diagonal coupling can mimic different vacuum ∆m2
+values, resulting in its incorrect measurement when NSI are not taken into account.
+Detection of ES in the proposed module (see Section 3.5.1) allows for a great opportunity
+to have an almost NSI-independent anchor point in the oscillation probability near 0.1 MeV in
+addition to investigating NSI in solar-neutrino oscillations at several MeV energies where changes
+in oscillation probability could be large but not much experimental data exists, yet.
+
+120
+10 kt 2% resolution + lowered
+100
+bkg + lowered threshold
+80
+10 kt 1% resolution
+10 kt 2% resolution
++ lowered bkg
+60
+40
+10 kt 2% resolution
+20
+40 kt HD design
+10-3
+8.04
+0.05
+0.06
+0.07
+0.08
+0.09
+Am2 (eV2)Large Low Background kTon-Scale LArTPCs
+22
+Figure 10. 2-flavor electron-neutrino survival probability for solar neutrinos for the Standard Model
+and several different NSI couplings.
+An example of possible constraints on the NSI couplings is shown in Figure 11. This plot
+was obtained with the assumption of no backgrounds or systematic errors, an energy threshold of
+1 MeV, and an exposure of 3 kTon·years. For comparison, see the larger constraints using current
+neutrino data available in [61].
+3.5.4. Precision Measurement of CNO flux
+The CNO flux has been measured [65] recently by
+the Borexino collaboration to 3.5σ above 0, though it yields indistinct information about the high
+or low metallicity solution. Per [65], the CNO neutrino flux scales with the metal abundance in
+the solar core, which probes the initial chemical composition of the Sun at its formation. The metal
+abundance in the core is decoupled from the surface by a radiative zone, and CNO neutrinos are
+the only probe of the initial condition.
+Here we want to investigate if an energy window exists where a precise measurement of
+the CNO flux can be performed in our low background module. We use our by-now standard
+simulation tools to count true deposited energy for a variety of backgrounds and solar neutrino
+
+0.60
+0.55
+0.50
+0.45
+solar
+0.40
+Standard Model
+2-flavor
+ED = 0.1, EN = 0
+ED = - 0.1,EN = 0
+0.35
+ED=O,EN=O.1
+ED=0,EN=-0.1
+0.30
+10-1
+100
+101
+true neutrino energy Etrue (MeV)Large Low Background kTon-Scale LArTPCs
+23
+Figure 11. Possible NSI constraints for 3 kTon·years obtained with the proposed module.
+sources in a 3 kTon fiducial volume. We impose a 2% energy smearing, as is reasonable by
+arguments in section 2.3.1. Fluxes come from [66] with a 0.3 survival probability applied, as
+appropriate to this low energy range. We use the assumption that the 42Ar content will be at a rate
+that is reduced by a further factor 100,000 compared to atmospheric argon. We emphasize again
+per section 2.2.3 this is likely very conservative. The result is we see in a window about 0.2 MeV
+wide just above the pep cutoff that the CNO signal sticks up above other solar neutrino sources.
+This is merely an illustration; a real analysis will likely do a templated fit above the 7Be peak.
+Neutrons from the cold cryoskin stainless steel are forced to be low, as usual; radon is taken as
+controlled. The 210Bi background which Borexino took exquisite care to control and measure [65]
+is yet to be thoroughly investigated here. Nevertheless, there would appear to be a window where
+the high and low metallicity solutions are statistically separable. By comparison, CNO sensitivity
+in the two-phase LArTPC program, which studies are further along than the current work, can be
+seen in [67].
+Triggering for CNO neutrinos
+Initial studies into measuring the CNO flux with TPC triggering
+are underway, taking into account the full complement of radiological backgrounds predicted in
+the DUNE detector. In these early studies, TPC triggering requires sufficiently large, coincident
+clusters on the collection plane and at least one induction plane. We expect future work will in fact
+lead to a much higher-efficiency, light-based trigger being implemented for most work discussed
+in this paper. The dominant background in the 1.4 - 2.0 MeV energy window is radon in this
+
+2
+1
+0
+1g
+90% CL
+-1
+2g
+99% CL
+3g
+-2
+minimum x2
+-2
+-1
+0
+1
+2
+EDLarge Low Background kTon-Scale LArTPCs
+24
+early TPC triggering study, not in fact solar neutrinos – despite the rate reduction by a factor of
+103 predicted in the low background module. In order to perform the true CNO detection there
+is clearly much work to be done in offline processing to accurately characterize and implement
+rejection by alpha detection, Bi-Po coincidence, disproportionate activity near the cathode from
+drifting cation decay products, and emanation properties of the various detector materials.
+Figure 12. CNO solar neutrinos with backgrounds taking radon to be solved and negligible and
+neutrons constrained. This is 2% smeared true deposited energy in our inner 3 kTon fiducial volume
+in a year. Note that a likely, low 42Ar level is shown that reveals that a CNO fit is possible above
+the 7Be peak and explicitly by a simple counting experiment in roughly the region shown in yellow.
+This is a lower 42Ar level than shown in Fig 1, but still realistic. The inset zooms in on the 1.25-
+1.45 above the pep region to show that the signal statistics are large enough to favor either high- or
+low-metallicity after a year.
+3.5.5. Precision measurement of hep flux
+The hep solar neutrino process (3He+p →4 He+e+ +
+νe) produces the highest energy neutrinos, though they have the lowest flux and have not yet been
+observed. This low background module will be able to measure tens of these neutrinos per year
+via CC events, with a significant reduction in background due to radon and neutron interactions
+within the liquid argon.
+
+1010
+CNOhiM
+1.25-1.45 MeV region
+CNOloM
+80
+7Be
+60
+MeV
+108
+8B
+40 
+pep
+20 
+Ar39 Bg/L/1500
+106
+Ar42Bg/L/1500*9.2E-5/1E5
+1.25
+1.30
+1.35
+1.40
+1.45
+Deposited Energy [MeV]
+104
+102
+100
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+1.75
+2.00
+Deposited Energy[Mev]Large Low Background kTon-Scale LArTPCs
+25
+3.6. Neutrinoless Double Beta Decay
+A discovery of neutrinoless double beta decay (0ν2β) is the most straightforward way to prove the
+Majorana nature of neutrinos. It would be parsimonious to be able to search for 0ν2β in the same
+detector we are suggesting to use for the other measurements mentioned in this paper. 136Xe is one
+isotope in which much work is being performed worldwide to try to make this discovery [68, 69].
+Loading large LArTPCs with few-percent level Xe and measuring neutrinoless double beta
+decay has been suggested in [70] and [71] among other places. In this subsection we show that
+naively a discovery is likely possible with a signal 136Xe half-life of 5 · 1028 years, and is quite
+apparent at 1·1028 years, provided energy resolution requirements can be met. We imagine, say, a
+five-year search campaign for 0ν2β at the end of the prosecution of the baseline physics program
+of this module, since any dark matter search requiring PSD would necessarily be compromised by
+xenon loading.
+We start with a 0ν2β signal calculated using a Gaussian 1.5 (3) % energy resolution at 2.435
+MeV with 1/
+√
+E dependence for top (bottom) plots in Figure 13. Energy resolution ambitions
+are consistent [72] with what we may expect in a LArTPC from charge energy collection in
+combination with the photon readout system, as well as discussion in section 2.3.1. We similarly
+smear the 2ν double-beta true spectrum by these resolutions.
+And for the 208Tl background
+emanating from the G10 of the charge readout planes we use only the expected rate from the
+simulation, creating the actual spectrum by likewise smearing the 2.6 MeV gamma by 1.5 (3)% in
+top (bottom) plots. For the solar and 42Ar backgrounds previously discussed, on the other hand, we
+use the resolution from the simulation that uses only the poorer light-only collection from the SiPM
+hits. These are flat backgrounds which extend to low energy where signal will likely be obscured
+in the noise of the charge readout, hence the need to rely on only the SiPMs. For both plots we use
+a 3% concentration of 136Xe in our inner volume, using a 2.0 kTon fiducial volume, and propose
+to run for five years. We plot a signal corresponding to a (5)1 · 1028 year 0νββ half-life.
+Among the assumptions made for our figures is that underground argon can give a 42Ar
+suppression of a factor of 10 beyond that in atmospheric Ar. See section 2.2.3 which indicates this
+is likely easily achievable, up to processing concerns. We also take in Fig 13 that the irreducible
+solar 8B elastic scattering may be suppressed by a factor of two (conservatively down from three
+assumed in Reference [73, 74] ) from inspecting Chernkov/Scintillation light ratios in single versus
+double electron events. We impose no efficiency hit due to this cut, whereas Reference [73]
+uses an efficiency of 0.75.
+More careful event reconstruction studies are needed to bear out
+the reasonableness of this cut. We assign the 208Tl in the G10 charge collection planes a Th
+concentration of 50 mBq/kg. Charged current solar neutrino events, are in principle reducible to
+zero, due to the excited state gammas that are emitted. And similarly, neutrons shall be almost
+entirely removed using pulse-shape discrimination. We again take the radon issue to be solved for
+the sake of this study. A 2σ band to either side of the 0νββ energy of 2.435 MeV is shown.
+We take Eres ∼ 1.5 % in the top plot at the Q value, even though, as we have said, it is not
+
+Large Low Background kTon-Scale LArTPCs
+26
+immediately obvious we can achieve that (nor in fact are we currently confident we can reach the
+2.5% of the bottom plot) with our charge readout plus 80% SiPM coverage, open as it is at the top
+and bottom. That study is beyond the scope of this paper. However, we see that there is sensitivity
+in this detector to 0νββ discovery at lifetimes that stretch the reach of coming experiments, despite
+the large, irreducible solar 8B neutrinos which elastically scatter off the mostly argon target.
+3.7. Dark Matter
+It is known that a large amount of dark matter exists within the Universe, that has so far only
+been observed by gravitation interactions. One popular candidate for the dark matter is the Weakly
+Interacting Massive Particle, or WIMP, that is the focus of several current and future experiments.
+The potential of using this low background module to search for WIMP dark matter was studied
+in Reference [10]. This study assumed a dual phase TPC design, with a single fiducial volume at
+the center of the detector (rather than the split fiducial volumes described above). The criteria for
+achieving a sensitive WIMP dark matter search was set as requiring:
+• a 50-100 keV nuclear recoil threshold;
+• O(10) background events;
+• O(100) photons detected per event to allow pulse shape discrimination (PSD).
+Assuming that 1250 photons per 100 keV of prompt scintillation light is emitted (as measured
+by SCENE for 500 V/cm fields [75]), the studies in Section 3.3 show that reaching 100 photons
+per event is realistic. A pseudo-Monte Carlo simulation of the Poisson distributed light output
+was performed to determine the width of a typical PSD variable f90, defined as the ratio of light
+detector detected in the first 90 ns of an event to the light detected in the second 90 ns of an event,
+for the 39Ar decays. The results taken from measurements from the SCENE experiment [75].
+Reference [10] shows PSD is expected to reach the 1010 rejection level for electron/gamma
+backgrounds. We also direct the interested reader to the PSD study [76] to suppress the 39Ar
+decay background in DEAP.
+All other electron/gamma backgrounds are expected to be subdominant to the 39Ar and will
+thus be removed by PSD. Neutron backgrounds were managed as described above. The main
+background will be from irreducible atmospheric neutrinos at the so-called neutrino floor. A full
+background table from the study is shown in Ref. [10].
+The background rates are used to set a 90% C.L. sensitivity to WIMP dark matter Ref. [10]
+shows that a three-year search with this detector will have comparable sensitivity to planned
+next-generation detectors, which have expected run times of 10 years. This timescale allows a
+rapid cross-check of any signals discovered in these detectors, in particular for the liquid argon
+experiments such as DarkSide-20k [9] or ARGO [77].
+
+Large Low Background kTon-Scale LArTPCs
+27
+Figure 13. An optimistic background/resolution scenario for a 136Xe 0νββ half-life of 5·28 years.
+Detector energy resolution is 1.5% on top. Backgrounds are as discussed in the text. On the bottom
+is the same with a reduced resolution of 3% and for a half-life of 1E28 years.
+
+140
+Onubb
+MeV
+2nubb
+TI208 50 mBq/kg
+Events/0.189kTonneXe136-5yr /0.02[
+120
+8BES
+Ar42Bq/L/1500/5E8/10
+100
+pseudo-data
+80
+60
+40
+20
+2.30
+2.35
+2.40
+2.45
+2.50
+2.55
+2.60
+Energy [MeV]140
+Onubb
+Events /0.189 kTonneXe136-5yr/0.02 MeV
+2nubb
+Tl20850mBq/kg
+120
+8BES
+Ar42Bq/L/1500/5E8/10
+100
+pseudo-data
+80
+60
+40
+20
+2.30
+2.35
+2.40
+2.45
+2.50
+2.55
+2.60
+Energy [MeV]Large Low Background kTon-Scale LArTPCs
+28
+3.7.1. Seasonal Variation of Rate for WIMP Dark Matter
+The prominent model for dark matter
+(DM) is the so-called Standard Halo Model (SHM) [78] featuring WIMP DM. The SHM describes
+a basic isometric spherical distribution of WIMP DM around our galaxy and has been used due to
+it being a good trade-off between realism and simplicity. The relative velocity of our solar system
+of 233 km/s [79] at which the Sun moves through the gas-like halo of WIMP DM induces as a
+”WIMP wind”. Figure 14 illustrates the Sun’s rotation in our galaxy together with Earth’s solar
+orbit into and out of the WIMP wind. We modeled the Sun’s rotational and peculiar velocities
+into one combined effective velocity of 233 km/s in the galactic plane and accounted for the 60◦
+inclined plane of Earth’s orbit. We were then able to accurately describe the annual modulation of
+detectable WIMP rate R on Earth by one simple periodic sinusoidal function with one amplitude
+parameter A and a maximal rate on June 1 of each year[79] [80]:
+R( [d−1] ) = A [d−1] × cos
+� 2π
+T[d] × ( t[d] − tJune1[d] )
+�
++ Ravg [d−1]
+(2)
+The constant term Ravg is the average annual rate. Earth’s period T is 365.2422 days and the phase
+corresponding to the maximal rate Rmax observable on June 1 of each year is 2π · tJune1/T =
+2π · 0.415.
+Galactic WIMPs in the halo are assumed to have a Maxwell-Boltzmann-like velocity
+distribution.
+The effective WIMP velocity distribution is shifted up when Earth is flowing
+maximally into the WIMP wind and shifted down when Earth is flowing maximally with the
+WIMP wind. Due to this aspect, an analysis on the annual modulation of the detection rate R
+could provide a powerful tool for identifying WIMP DM. Due to the unrivaled large fiducial mass
+of 3 kTons and a potentially very long DUNE operation of one decade (or even several), this
+concept can offer a unique detection of the seasonal variation of the detectable WIMP rate R at
+a sufficient statistical significance for providing a smoking gun signature for the WIMP nature of
+DM. This would be particularly of interest in case upcoming generation-2 DM experiments like
+LZ [81] and/or XENONnT [82] have evidence for WIMPs near their sensitivity. It would be nearly
+impossible for the planned generation-3 DM experiments [83] [84] to make such a smoking gun
+detection proving the WIMP nature of DM.
+The differential rate of interactions in an arbitrary detector for the SHM is described in
+Equation 3:
+dR
+dER
+= σSI
+N
+A2mANTρχ
+2mχµ2
+N
+F 2(ER)
+∞
+�
+νmin(ER)
+d3−→ν
+ν
+f⊕(−→ν , −→
+νobs)
+(3)
+where σSI
+N is the spin-independent-nucleon cross-section for WIMPs, A is the atomic number of
+argon, mA is the mass of argon, NT is the number of target nuclei, ρχ is the local dark matter
+density (0.3 GeV
+cm3 ), mχ is the mass of a WIMP, µN is the reduced WIMP-nucleus mass, F(ER) is
+the the nuclear form-factor, νmin is minimum WIMP velocity to produce a recoil of energy ER,
+ν is WIMP velocity, and νobs is the observer velocity with respect to the galaxy. When Earth is
+
+Large Low Background kTon-Scale LArTPCs
+29
+Figure 14. Galactic WIMP wind as it relates to Earth’s orbital plane employing an illustrative
+rendition [79] of our Milky Way galaxy. Our solar system’s velocity in reference to our galaxy
+has contributions from both a rotational aspect with a tangential velocity of 220 km/s and from a
+minor solar peculiar aspect with velocity components (U, V, W) = (10, 13, 7) km/s [80]. In our
+model the combined relative velocity of our solar system is then 233 km/s at which the Sun moves
+through a gas-like halo of WIMP dark matter assumed to have a Maxwell-Boltzmann-like velocity
+distribution. This induces what we experience as a “WIMP wind”. This WIMP wind is at an angle
+compared to Earth rotation around the Sun as pictured in the zoomed in diagram, with the effective
+WIMP velocity distribution shifted up when flowing maximally into the wind and shifted down
+when flowing maximally with the wind. Due to this aspect, an analysis of the annual modulation of
+detectable WIMP rate on Earth could provide a powerful tool for identifying the WIMP nature of
+dark matter.
+moving into or with the WIMP wind, it affects the differential WIMP rate, which in turn would
+affect our −→
+νobs. For ease, we can define:
+∞
+�
+νmin(ER)
+d3−→ν
+ν
+f⊕(−→ν , −→
+νobs) = ζ(ER),
+(4)
+where
+f(−→ν ) = 1
+N
+�
+e
+−ν2
+ν2
+0 − e
+−ν2esc
+ν2
+0
+�
+,
+(5)
+and
+f⊕(−→ν , −→
+νobs) = f(−→ν + −→
+νobs)
+(6)
+
+OurMilky Way Galaxy
+WIMP
+max,detectable
+wind
+WIMP rate
+Galactic Center
+on Jun. 1
+Earth
+UO+U
+Galactic
+0094
+Plane
+1AU
+<1 AU
+'Sun
+TAU
+WIMP
+Solar Rotational Velocity
+wind
+ = (0, 220, 0) km/s
+ = (10, 13, 7) km/s
+J. Genovesi, J. Reichenbacher 2022 - SD Mines - v02/f 1/22Large Low Background kTon-Scale LArTPCs
+30
+ht
+Figure 15. Example NEST[85] simulation of the annual modulation for 40 GeV/c2 WIMP dark
+matter with a cross section of 4 × 10−48 cm2 for a 10 year measurement time with 3 kTon LAr at
+50 keV threshold. On top a Likelihood-fit result for a 10 year period and on bottom the same data
+combined into a single annual period starting on January 1 of each year fitted with a χ2-method.
+The ideal case without background is assumed in this study.
+
+[counts / (20 days)]
+R
+0
+500
+1000
+1500
+2000
+2500
+3000
+3500
+Elapsed Timet (in days since startof experimentAnnualModHist
+30
+Integral
+277
+x? / ndf
+22.37 / 17
+25
+po
+3.827 ± 1.144
+p1
+13.53 ± 0.85
+20
+15
+10
+0
+50
+100
+150
+200
+250
+300
+350
+Elapsed Time t (in days since 1st of January of each year)Large Low Background kTon-Scale LArTPCs
+31
+with N as a normalization factor, ν0 is the expected WIMP velocity, and νesc is the escape
+velocity of the galaxy. Using Equation 5, we can solve for individual cases of WIMPs in certain
+speed brackets.
+As mentioned earlier, Earth’s orbit can play a crucial role for differential WIMP rate
+predictions in the SHM. As the solar system travels through our galaxy, observers on Earth would
+observe WIMPs dominantly from a certain direction in a windshield-to-rain like effect. This is
+due to the distribution of WIMPs being treated as a gas with a Maxwell-Boltzmann-like velocity
+distribution with the stars in our galaxy moving through the dark matter due to their orbits around
+the galactic nucleus. In addition to the standard rotational contribution from the solar system, a
+minor peculiar velocity exists of our solar system traveling through the galaxy. This WIMP wind
+would be at an angle of 60◦ towards Earth’s orbital plane. This means that during June 1, the Earth
+will experience a maximum effective flux of WIMPs while on December 1 the Earth will have
+experienced the lowest effective WIMP rate, as one can see again in Figure 14.
+An example fit using Equation 2 of an annual modulation signal simulated in NEST [85]
+without background is shown in Figure 15 on top for a 10 year period, and on bottom the same
+data combined into a single annual period starting on January 1 of each year. This is for a 40 GeV/c2
+WIMP with a cross section of 4 × 10−48 cm2, which is very close to the limit of sensitivity of the
+upcoming generation-2 xenon dark matter experiments LZ [81] and XENONnT [82]. We further
+assumed a 3 kTon × 10 year exposure of our proposed low background LAr module with a 50 keV
+threshold resulting in 277 events. The bottom plot of Figure 15 shows a good χ2-fit result for
+the amplitude A = 3.827 ± 1.144 from Equation 2. It confirms the possible measurement of the
+seasonal variation of the WIMP rate at a sufficient statistical significance for providing a smoking
+gun signature for the WIMP nature of DM. This will be uniquely possible with this detector, due
+to the unrivaled large mass of 3 kTons vs. only 300 tons of argon for ARGO [84] and 100 tons for
+a Gen-3 xenon experiment [83]. Moreover, the annual modulation effect in xenon is significantly
+smaller due to the relatively lower energies of nuclear recoils in xenon compared to argon. Last
+but not least, the logistics of a decade long operation with this detector can utilize strong synergies
+with the DUNE long-baseline physics, including the cavern availability and occupancy.
+3.8. Additional Topics
+This module, with its unprecedented combination of low background and size, also can explore
+several other topics. We describe several examples in this section.
+3.8.1. Atmospheric neutrinos
+The detector will measure approximately 10 CEνNS events due to
+atmospheric neutrinos. These events have not yet been observed and this would allow a cross-check
+of background rates from the upcoming generation of dark matter experiments.
+
+Large Low Background kTon-Scale LArTPCs
+32
+3.8.2.
+Strangelets Recently, the paper “Can strangelets be detected in a large LAr neutrino
+detector?” [86], predicted that a LArTPC detector is able to detect and discriminate light strangelets
+with (Z, A) between (2,14) and (7,70) for energies up to 10 GeV in the presence of radioactive
+background found at the surface. When operated underground the detection limits are expected
+to be extended due to lower background levels and, combined with the increased dimensions of
+the detector module, will improve the event rates with a factor of 40. In the case of strangelets
+the main uncertainties are due to the estimations of their survival probability deep underground.
+The presence of 39Ar masks both ionization and scintillation signals from strangelets and induces
+false signals in the collected charge from ionization. The use of underground argon in this module
+allows a cleaner detection signal.
+3.8.3. Charged micro-black holes and Superheavy dark matter Hawking [87] suggested that
+unidentified tracks in the photographs taken in old bubble chamber detectors could be explained
+as signals of gravitationally “collapsed objects” (µBH). The small black holes are expected to be
+unstable due to Hawking radiation, but the evaporation is not well-understood at masses of the
+order of the Planck scale. Certain inflationary models naturally assume the formation of a large
+number of small black holes [88] and the generalized uncertainty principle may indeed prevent
+total evaporation of small black holes by dynamics and not by symmetry, just like the hydrogen
+atom is prevented from collapse by the standard uncertainty principle [89]. Given the profound
+nature of the issues addressed, some disagreement and controversy exist.
+In principle the direct detection of charged micro black holes with masses around and upward
+of the Planck scale (10−5 g), ensuring a classical gravitational treatment of these objects, is possible
+in huge LAr detectors. It has been shown that the signals (ionization and scintillation) produced
+in LAr enable the discrimination between micro black holes (with masses between 10−5 - 10−4
+grams, and velocities in the range 250 - 1000 km/s) and other particles [90]. It is expected that
+the trajectories of these micro black holes will appear as crossing the whole active medium, in any
+direction, producing uniform ionization and scintillation on the whole path.
+Along these lines, an analysis looking for multiple co-linear nuclear recoils can also probe
+ultra heavy dark matter beyond the Plank scale, as described in Ref.[91, 92]. Sensitivity to the
+heaviest dark matter candidates is limited by the number density of the dark matter, which is
+inversely proportional to the mass, as the ability to detect heavy dark matter with a high cross
+section is set by the probability that a dark matter particle enters the detector. As such, sensitivity
+to the highest masses scales with the detector’s surface area, and would leverage the large size of
+this module compared to DEAP-3600, which has a 1.7 m diameter.
+Similarly, in the direct detection of the charged micro-black holes, unlike in traditional WIMP
+detection, there will exist both ionization and scintillation signals from direct interactions and from
+recoiling nuclei. The capability to perform pulse shape discrimination in this detector will allow
+these tracks to be identified. Natural radioactivity is the main source of background in this case
+and the reduced number of free electrons (and photons) from beta decays of 39Ar will allow a
+
+Large Low Background kTon-Scale LArTPCs
+33
+significant improvement of the capability of the detector to correctly identify the micro-black hole
+signals.
+3.8.4. Other topics
+This detector would have sensitivity to a small number of CEνNS events
+within the neutrino beam. It may have applications to geologic tomography. The improved energy
+resolution for low energy could have applications in searches for other exotics and beyond the
+standard model physics phenomena. Though the optimal search region for a diffuse supernova
+neutrino background is above the energy of the solar neutrinos [93], and thus the radioactive
+backgrounds, the improved energy resolution of this detector will again likely improve the search
+sensitivity.
+4. Conclusion
+We have presented a design in this paper for a low background kTon-scale LArTPC to potentially
+expand the current physics program for such detectors. The design is based on the vertical drift
+detector planned for DUNE’s second far detector module. The module discussed is a candidate for
+a third or fourth DUNE “Module of Opportunity.” It is realized by providing additional shielding,
+stringent radioactive background control and enhanced light detection to the nominal vertical
+drift module.
+Energy resolution will benefit in all energy ranges due to event reconstruction
+and topology classification improvements from the superior light detection system and the quiet
+detector, which will allow to capture more cascade gammas[13] and thereby improve the hadronic
+component of neutrino-nucleus interactions.
+The physics goals achieved by the SLoMo design extend the capability of large LArTPCs to
+search for solar and supernova neutrinos, neutrino-less double beta-decay, and WIMP dark matter.
+At the same time the design proposed here, by the nature of its small perturbations to the vertical
+drift module, assures continuing strong support to the long-baseline neutrino oscillation program
+to measure remaining parameters in the PMNS matrix.
+Acknowledgments
+Pacific Northwest National Laboratory (PNNL) is operated by Battelle for the United States
+Department of Energy (DOE) under Contract no. DE-AC05-76RL01830. Parts of this study at
+PNNL were supported by the DOE, USA Office of High Energy Physics Advanced Technology
+R&D subprogram and other parts by the Open Call Initiative, under the Laboratory Directed
+Research and Development Program. In the United Kingdom this work was supported by STFC.
+For IL and MP this work was performed with the financial support of the Romanian Program
+PNCDI III, Programme 5, Module CERN-RO, under contract no. 04/2022. KS is supported by
+the Department of Energy and the National Science Foundation. ZD acknowledges the support
+
+Large Low Background kTon-Scale LArTPCs
+34
+of U.S. Department of Energy Office of Science under contract number DE- AC02-06CH11357.
+South Dakota School of Mines and Technology acknowledges the support of Department of Energy
+through award number DE-SC0014223, as well as DE-AC02-07CH11359 through subaward from
+Fermi National Accelerator Laboratory subcontract no. 664706. JZ gratefully acknowledges using
+the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.S. Department of
+Energy, Office of Science, HEP User Facility. Fermilab is managed by Fermi Research Alliance,
+LLC (FRA), acting under Contract No. DEAC02-07CH11359.
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+page_content=' Cuesta3, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Djurcic4, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Genovesi5,  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Haiston5, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Jackson2, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Lazanu6, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Monreal7, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Munson2, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Ortiz8,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Parvu6, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Peeters1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Pershey8, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Poudel2, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Reichenbacher5,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Saldanha2, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Scholberg8, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Sinev5, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Westerdale9, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Zennamo10  1University of Sussex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Brighton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' United Kingdom  2Pacific Northwest National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Richland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' E-28040 Madrid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Spain  4Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Argonne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' USA  5South Dakota School of Mines and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Rapid City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' USA  6University of Bucharest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Bucharest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Romania  7Case Western Reserve University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Cleveland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' USA  8Duke University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Durham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' USA  9Princeton University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Princeton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' NJ 08544,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' USA  10Fermi National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Batavia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' IL 60510,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' USA  E-mail: eric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='church@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='gov, christopher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='jackson@pnnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='gov  Abstract: We find that it is possible to increase sensitivity to low energy physics in a third  or fourth DUNE-like module with careful controls over radiopurity and targeted modifications  to a detector similar to the DUNE Far Detector design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In particular, sensitivity to supernova  and solar neutrinos can be enhanced with improved MeV-scale reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A neutrinoless double beta  decay search with 136Xe loading appears feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Furthermore, sensitivity to Weakly-Interacting  Massive Particle (WIMP) Dark Matter (DM) becomes competitive with the planned world program  in such a detector, offering a unique seasonal variation detection that is characteristic for the nature  of WIMPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='11878v1  [hep-ex]  27 Jan 2023  Large Low Background kTon-Scale LArTPCs  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Introduction  In this study we introduce and discuss a dedicated low background module that would enhance the  physics program of next-generation experiments such as the planned Deep Underground Neutrino  Experiment (DUNE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Such a low background DUNE-like module could be installed as either  module 3 or module 4, the so-called “Module of Opportunity” in DUNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Such a module would  increase the physics reach of supernova and solar neutrino physics, and could potentially host  a next-generation neutrinoless double beta decay (0νββ) or Weakly Interacting Massive Particle  (WIMP) dark matter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We refer to this design as the Sanford Underground Low background  Module (SLoMo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The physics reach would be enhanced by lowering the nominal energy threshold of a DUNE-  like experiment from the anticipated 5-10 MeV to levels necessary to address three potential  physics targets, listed in order of increasing difficulty:  ∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 MeV energy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This threshold is set by the Q-value of the 42K (daughter of  42Ar) in the detector target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' By reducing neutron captures, alpha-emitting radon daughters  and pileup events above this threshold, supernova burst neutrino sensitivity could be increased  in distance, energy and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Sensitivity to solar neutrinos would also be enhanced, allowing  explorations of interesting solar-reactor oscillation tensions and Non-Standard Interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 MeV energy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This threshold is set by the decay Q-value of the 39Ar in the  detector target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' With reduction of electron and photon backgrounds in the target, particularly  if the 42Ar content is reduced through use of underground argon (UAr), sensitivity to low  energy solar neutrinos from the CNO process would allow a precision measurement to be  made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Such a detector will be sensitive to 0νββ search with loading of 136Xe,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  < 100 keV energy threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  This threshold could be achieved by enhancing the light  collection within the detector and by lowering the 39Ar background by deploying UAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' With  rejection of electron recoil backgrounds (using timing based pulse shape discrimination), a  sensitive WIMP dark matter search could take place, and interesting phenomena such as  a supernova coherent elastic neutrino-nucleus scattering signal (a CEνNS glow) could be  studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  These low background targets are achievable due to the unprecedented size and also the increased  radiopurity of the module, allowing significant fiducialization and hence less stringent radioactive  background requirements than current world-leading dark matter searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The light enhancements  are achievable with current production techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' As described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3, production  of UAr at the scale required for a DUNE module is potentially achievable, though it requires a  dedicated effort to identify the potential argon source and work with commercial gas suppliers  In this paper in Section 2 we outline the design of the module and discuss potential paths to  achieve the detector requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In Section 3 we present our initial studies of physics reach of  this detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Large Low Background kTon-Scale LArTPCs  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Summary of physics targets of this low background module and the primary radiological  backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Detector Design  This section outlines the proposed design for the low background module and describes in detail  the radioactive background control and photon detection system enhancements required to enable  physics measurements outlined in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We note this module design is not necessarily  endorsed by the DUNE collaboration, as the so-called “Phase II” process that includes building the  final two far detector modules is in its early stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Module Layout  A low background module and its attendant physics goals are enabled most simply by minimizing  the detector components in the bulk of the argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The resulting module must still allow for very  good light detection efficiency and for charge detection efficiency similar to existing designs of  large LArTPCs [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For the benefit of conforming to the longstanding plans for the Long  Baseline Neutrino Facility (LBNF) far detector complex and cavern layouts, we also want to  use the same commercial cryostat concept and existing module designs to the greatest extent  possible and perturb them only where necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Starting with the existing DUNE modules, simple  modifications assure minimal disruption to the main long baseline neutrino oscillation program to  measure remaining parameters in the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Single phase  We therefore start with DUNE’s Far Detector Vertical Drift Module (VD) [4],  sometimes referred to as Module 2, and consider design modifications to suit our low background  purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We show a working design in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Water in “bricks” which are imagined to be nestled among the I-beam support structure  WIMPs  Ovβ  Solar Neutrinos  Supernova Neutrinos  39Ar  42 Ar  Internal Alphas/Betas/Gammas  NeutronsLarge Low Background kTon-Scale LArTPCs  4  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Shown is the base design for the proposed low background detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Blue shows external  water “bricks”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The top and bottom yellow planes are the Charge Readout Panels unchanged from  the Vertical Detector design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The central cathode is in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The white box of acrylic (full interior  volume) is of thickness 1 inch and has x,y,z extent of 6,12,40 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The black points are SiPM modules  shown here at a coverage of 10%, while some studies in this paper use up to 80% coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A  proposed fiducial volume totaling 2 kTon is shown in the two beige boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This paper also considers  a 3 kTon volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  achieve large external neutron reduction, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  We keep the Charge  Readout Planes of the VD unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Similarly the central cathode of the VD is preserved with  the exception that the sparse and mostly distant photon detection modules, known as X-Arapucas,  are swapped out for the SiPM modules mounted within the cathode plane – at least on that part  of the cathode in the inner region of this detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' An acrylic box of one inch thickness with  x,y,z extent of 6,12,40 m serves to mount SiPM modules and reflective WLS foils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Here x is the  horizontal dimension, y is the vertical dimension, z is in the beam direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' SiPM modules are  mounted on the inside of the acrylic box at anywhere from 10-80% area coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We envision two  1 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 kTon long skinny fiducial volumes, depending on the study, that have a stand-off of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5m  from the central cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We also discuss, alternatively, a 3 kTon fiducial volume in this paper in  studies where backgrounds from the central cathode are thought to be small or events from it can  be reconstructed and cut away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The bulk volume of this module consists mostly of argon, with only small material  contributions such as the slender support structures for the cathode plane panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' As in the VD,  there are two 6 m vertical drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Large Low Background kTon-Scale LArTPCs  5  (a)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Interactions are shown above 100 keV for neutrons emanating from the cold cryostat  stainless steel at 2 · 10−10 neutrons/cm3/sec for a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4 yr exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We show a possible ∼3kTon  fiducial volume pair, avoiding the cathode, looking down the beam line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' (The vertical bands are  interactions in the acrylic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=')  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Radioactive Backgrounds  To enable the physics targets for this module, improvements in control of internal and external  radioactive background levels are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Making a module of this size low background will  require significant quality and materials controls beyond what has been been attempted by previous  experiments, and certainly beyond what is required for the Phase I DUNE program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' However, we  note that due to the increased size of this module self-shielding in the argon allows the background  requirements to be less stringent than those expected to be reached by the current Generation 2  (G2) dark matter experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Thus future research and development will need to focus on how to  scale the techniques successfully deployed to low background dark matter or neutrinoless double  beta decay searches to the kTon scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  A particular concern for this module will be neutron-induced background events, which will  be the main background to the neutrino searches above 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 MeV thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A neutron capture in  40Ar produces a 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1 MeV gamma cascade which can Compton scatter or pair produce electrons  which can mimic the charged current neutrino interactions in the argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Captures on 36Ar can  produce 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='8 MeV gamma cascades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Neutrons are also the primary backgrounds for the lowest  energy searches for WIMP dark matter, where nuclear recoils can mimic the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5  MeV the primary backgrounds will come from alpha, beta and gamma emitting isotopes within  the argon or detector materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  6000  400  4000  350  300  2000  250  y [mm]  200  150  2000  100  4000  50  6000  6000  4000  2000  2000  4000  6000  x[mm]Large Low Background kTon-Scale LArTPCs  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Cavern Neutron Backgrounds  The most significant source of neutrons will likely be those  induced by spontaneous fission or (α, n) interactions from the uranium-238 or thorium-232 decay  chains within the surrounding rock and shotcrete of the detector cavern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' As reported in [5], the  external neutron rate at SURF (a likely hosting laboratory for this low background module) is  assumed to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0 × 10−5 n/cm2/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' As proposed in References [6, 7], it is possible to add water  shielding to a DUNE-like cryostat, taking advantage of the space between the structural supports  even when the detector is located within a cavern at SURF with limited space around the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Those references shows that a 40 cm water shield, located within the support structure, is enough  to lower the external neutron rate by three orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We assume this is achievable for  this module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The strategy to control the radioactive backgrounds from the detector components such as the  cryostat will have three parts:  improvements to material selection,  additional internal neutron shielding,  and advanced event selections and analysis tools,  with the aim of lowering the internal neutron rate within the detector by at least three orders of  magnitude to match the levels of the external rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Aside from external neutrons from the cavern, the main source of neutrons in a DUNE-like  detector cryostat is likely to be spontaneous fission or (α, n) interactions from the uranium-238 or  thorium-232 decay chains in the order 1 kTon of stainless steel that makes up the I-beam support  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Research and development will be required to lower the internal background rates by  the three orders of magnitude required, for example by careful selection of the raw ingredients  and/or control of the manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It should be noted that the “Generation 2” dark  matter experiments expect to reach neutron rates from their steel a further two orders of magnitude  beyond this, so the goal is achievable (even if the scale is much larger than previously attempted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Another area of R&D required to support this goal is improvements in knowledge of (α, n) cross  sections, as highlighted in a recent IAEA workshop [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Another approach to reduce neutron background from the internal shielding within the  detector would be by adding higher density rigid polyurethane foam (R-PUF) insulation and/or  boron, lithium or gadolinium loaded material layers within the membrane cryostat structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Our studies show that this could easily reduce the neutron capture rate in LAr by one order of  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' One approach, as planned by the DarkSide collaboration, would be to use the additional  planes of Gd-doped acrylic to act as a neutron absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' DarkSide-20k [9] intends to use multiple  layers within a ProtoDUNE-style cryostat for their dark matter search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Another design choice  might be to take advantage of the existing cryostat but replace some materials such as the insulating  foam with a borated version, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=', to reduce backgrounds from the support structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Analysis based cuts can also be used to remove events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For example, with the low threshold  of this planned detector, neutron induced multiple scatters could be tagged and rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This  Large Low Background kTon-Scale LArTPCs  7  takes advantage of the excellent (∼ 10 mm) position resolution of a TPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Studies [10] to identify  dark matter nuclear recoil backgrounds show ∼ 30% reduction at 100 keV threshold and ∼ 90%  reduction at 50 keV, due to the increased probability of detecting an additional scatter at lower  thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Radon and other internal argon backgrounds  Radon is an important background that must  be controlled as it can diffuse throughout the detector, entering the fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The radon  decays to a number of daughter isotopes that can be direct backgrounds (for example 214Bi for a  neutrinoless double beta decay search) or that can produce neutrons through (α, n) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  For this low background module we set a radon target level in the liquid argon of 2 µBq/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This is  about three orders of magnitude below the expected DUNE radon level of 1 mBq/kg [11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2 µBq/kg has been achieved in liquid argon by the DarkSide-50 experiment [14], and exceeded  by DEAP-3600 which achieved a level of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2 µBq/kg [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It should be noted that the higher  volume-to-(radon-emanating)-surface ratio in a large detector such as DUNE compared to dark  matter detectors will help achieve this target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  To reach this level in a kTon-scale detector will require research and development to  implement a combination of the following techniques:  Radon removal during purification via an inline radon trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  No radon removal is  in the current design of the purification system for the baseline DUNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Dark matter and  neutrinoless double beta decay search experiments typically use cooled, activated charcoal  radon traps to remove radon directly from the recirculating target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The most sensitive dark  matter experiments typically purify the argon in the gaseous phase [16, 17], however such  an approach would be impractical for a kTon-scale experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Borexino used a charcoal  radon trap to purify liquid nitrogen [18], and such an approach could be adopted and scaled  appropriately for a low background module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' New materials such as Metal-organic frameworks  could improve the capture-potential beyond charcoal, allowing a potential shrinkage of  footprint of a radon-capture facility to fit the existing cavern designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Recent evidence from  MicroBoone [19] indicates that a copper filter purification system similar to that planned for  DUNE may remove greater than 97% or 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='999% of the radon (depending on whether slowed  or trapped) in the system without the need for additional removal techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Emanation measurement materials campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' All materials used in detector construction  are known to emanate radon at some level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  A large-scale emanation assay campaign to  identify materials suitable for construction, similar to the QA/QC campaign described above  will be required to ensure the detector can meet the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A topic for R&D will be how  to increase throughput of samples, as emanation measurements typically take two weeks per  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Surface treatments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For large components such as the cryostat where it may be impractical  and costly to make significant improvements to the radiopurity, surface treatments can be  used to lower emanation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It is known that acid leaching and electropolishing lowers  Large Low Background kTon-Scale LArTPCs  8  emanation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Coating the inner surface of the cryostat with a radon barrier could lower  emanation rates from this significant source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Dust control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Dust is a significant radon source and cleanliness standards will be higher  in this low background module than the baseline DUNE design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Cleanliness protocols and  requirements R&D will be necessary to develop automated techniques applicable to the  thousands of m2’s of surface area of a DUNE-like cryostat, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Radon reduction system during installation and operation: The mine air underground  is radon laden up to 1,000 Bq/m3 and radon daughter plate-out during installation, filling  and operation must be controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' An upscaled vacuum swing system with large charcoal  columns in parallel to remove radon from the ambient air and to provide radon-free air to the  cleanroom and cryostat would be suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Vacuum swing systems providing radon-free air  have been successfully employed by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Borexino [20], LZ [21] and SuperCDMS [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Drifting of charged daughters to cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Several daughters in the radon chain are charged  and will drift towards the cathode and out of the fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This effect may be countered  by mixing effects of the purification system however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Alpha tagging through pulse shape discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Alpha events in the radon chain that  produce neutrons directly in the argon may be taggable, by identifying the alpha track before  the (α,n) event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Though the amount of light may be relatively small, the timing profile  is distinct and may allow pulse shape discrimination on this module with enhanced optical  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Underground Argon  Atmospheric argon (AAr) consists mostly of 40Ar which is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  There are some long-lived radioactive isotopes-39Ar (T1/2=269y, Qβ=565 keV), 37Ar (T1/2=35d,  Q=813 keV), 42Ar (T1/2=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='9y, Qβ=599 keV) [23] which are, in atmosphere, produced primarily  by cosmic ray-induced reactions in 40Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The use of atmospheric argon in a low-threshold multi-  ton scale argon detector has limitations due to high 39Ar activity (1 Bq per kg of argon [24]) in  atmospheric argon (AAr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Radiogenic and cosmic-ray muon-induced interactions, especially on K  and Ca isotopes, can produce 39Ar underground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The dominant production channels are negative  muon capture on 39K and (α, n)-induced (n,p) reactions on 39K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 39Ar production underground  decreases significantly with depth[25] as muon flux decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' DarkSide-50, the only experiment  to use underground argon (UAr), measured the 39Ar activity of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='73 mBq/kg [26], a factor of  1400 smaller than in AAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' With the ARIA project[27], which is planning for a throughput for  39Ar processing of ∼ 10 kg/day, the DarkSide collaboration is planning to further reduce the 39Ar  present in UAr through large-scale isotopic separation by cryogenic distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  42Ar decays in the bulk argon volume will produce 42K isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  While the 42Ar beta-  spectrum has an endpoint of 599 keV, Betas from 42K-decays span a much larger energy range  up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 MeV, and can be problematic backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Since UAr should be heavily depleted of  42Ar, significant suppression of 42K decay backgrounds is achievable with UAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In the atmosphere,  the 42Ar concentration is ∼ 10−20 42Ar per 40Ar atom [28],[29], which is four orders of magnitude  Large Low Background kTon-Scale LArTPCs  9  smaller than the concentration of 39Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The daughter isotope of 42Ar, 42K (T1/2= 12 h) has two  major decay modes : 1) direct beta-decay to the ground state of 42Ca (Qβ= 3525 keV, BR=81  %), and 2) Beta-decay (Qβ= 2001 keV) to an excited state of 42Ca followed by a prompt 1524  keV gamma emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In the atmosphere 42Ar is primarily produced by 40Ar(α, 2p)42Ar occurring  in the upper atmosphere [29], where energetic alphas are readily available from cosmic-ray muon  interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Production through two-step neutron capture is also possible but greatly sub-dominant  due to the short-lived intermediate isotope 41Ar [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The 42Ar production rate underground is  not known, but it is expected to be several orders of magnitude smaller than in AAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Particle  interactions on isotopes of K, Ca, and Ti can produce some 42Ar in the earth’s crust, given the  relatively high abundance of the elements (by mass-fraction[31]:K-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='09%,Ca-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='15%,Ti-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='565%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  However, the reaction thresholds are high, making 42Ar production energetically not possible by  fission, (α,n)-neutrons or alphas from the natural radioactivity chains of 238U, 235U, and 232Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Energetic particles from cosmic ray muon-induced interactions can produce 42Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  But at the  depths at which underground argon is usually extracted, the cosmic ray muon-flux should be hugely  suppressed, so 42Ar production should be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Based on GERDA’s findings[32], following  42Ar decays, 42K nuclei could retain the positive charge long enough to drift in the influence of  electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' So, we expect 42K ions to drift and move towards the cathode plane, which suggests  an additional suppression of 42K backgrounds is achievable through fiducialisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The 85Kr isotope, predominantly a β-emitter, has a half-life of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='7 years and Q-value of 687  keV[23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Primary modes of 85Kr production are spontaneous fission of uranium and plutonium  isotopes, neutron capture on 84Kr, and human-induced nuclear fissions in nuclear reactors[33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We  would expect 85Kr to be present at some level in AAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' However, its concentration can vary across  argon extraction sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Using the UAr data, DarkSide-50 measured 85Kr activity of 2 mBq/kg[26],  a few orders of magnitude smaller than in AAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  85Kr concentration in UAr should also vary  depending on the location of the gas reservoir and gas origin (mantle-like or crustal-like).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' DEAP  sees no evidence of 85Kr in its AAr after filtering in a charcoal trap with 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3 tonnes of LAr [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  We expect argon gas extracted from an underground source to be highly depleted of 39Ar,  42Ar and 85Kr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  There is evidence that air infiltration during the UAr extraction could have  contributed to the DarkSide-50’s 39Ar - actual 39Ar content in the UAr could be significantly  smaller (on the order of few tens of µBq/kg)[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  85Kr and 42Ar content could also be much  smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Unlike stable gas isotopes such as 40Ar, which can collect at gas wells over time, isotopes  such as 39Ar(T1/2=269y), 42Ar (T1/2=32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='9y) and 85Kr(T1/2=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='7y) diffusing through rocks and  collecting in a significant number at the underground gas wells is less likely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  However, air-  infiltration and cosmogenic activation in the argon bulk could introduce these isotopes in the  extracted UAr [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Greater care, perhaps, is necessary to ensure avoiding contamination of  the UAr during extraction, processing, transport and storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  While UAr is desirable, it requires a dedicated effort to identify the potential argon source  and procure argon on a large enough scale necessary for this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The Urania plant [9] in  southwestern Colorado, USA is expected to produce underground argon from CO2 gas wells at  Large Low Background kTon-Scale LArTPCs  10  a rate of ∼ 300 kg/day (at full rate) for DarkSide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The argon source and the production rate is  not large enough for a kiloton scale experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The authors are in the process of identifying  alternative gas wells with enriched argon streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Discussions with three potential commercial  suppliers are ongoing, however the underground source samples are not yet tested and low levels  of radioactive isotopes must be proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Initial gas analysis indicates the mantle origin of this  sample, with suppressed cosmogenic production compared to a crust source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Based on estimates  by the gas suppliers, the production cost could be as low as three times the cost of atmospheric  argon and 5 kTon of argon production per year could be achievable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Surface storage and spallation issues  During above-ground storage of our UAr, before  placement into the underground module, 39Ar is produced by cosmogenic neutrons in the reactions:  40Ar(n,2n)39Ar and 38Ar(n,γ)39Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Production cross sections for 39Ar have been measured in [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  42Ar is produced primarily in 40Ar(α, 2p)42Ar reactions [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In a previous work [39], to which  we refer the interested reader, the cross sections for these reactions were obtained from nuclear  reaction codes and confronted with data where they exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' An evaluation of cosmogenic 39Ar and  42Ar production in UAr stored on the surface can be found in [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A full study of expected  spallation and pileup backgrounds during the detector operation is in reference [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Light Collection Enhancement  The light collection for this low background module will be enhanced to enable two main goals:  lower the energy threshold and improve the resolution at these lower neutrino energies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  improve pulse shape discrimination for radioactive background rejection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  In this section we describe the improvements to the light detection system required to enable this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Light Collection Efficiency Impacts on Energy Resolution  Charged particles that traverse  the liquid argon deposit energy and excite and ionize the argon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This process results in  the electrons recombining with the ions generating unstable argon dimers, which decay and emit  scintillation light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' If an electric field is applied, a fraction of the electrons will drift away before  recombining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' These ionization electrons are collected by the anode plane and create the charge  signal by removing electrons that would have recombined yields an anticorrelation between the  light and charge signals observed in LArTPCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The small number of ionized argon atoms at low  energies means that fluctuations in this recombination process can smear the amount of charge  observed to energy deposits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The Noble Element Simulation Technique (NEST) collaboration has  explored the precision of LArTPC for 1 MeV electrons as a function of charge readout signal-  to-noise ratio and the efficiency of collecting light [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' They predict that using only the charge  signal in a LArTPC, the most precise one can reconstruct the energy for a 1 MeV electron is 5%  Large Low Background kTon-Scale LArTPCs  11  and MicroBooNE has validated this prediction [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' To improve the energy reconstruction, further  one needs to include measurements of scintillation light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The NEST collaboration explored the expected energy resolution enhancements that a  LArTPC can achieve by including light signals along with the charge measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' [41]  shows that the energy resolution for a 1 MeV electron for LArTPCs with different signal-to-noise  ratios (SNR) and varying light collection efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In particular, we see that a LArTPC with SNR  near 40 needs 50% of the scintillation light to measure energy deposits with 1% precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Photosensors  As a baseline design our plan is to use 24 cm2, Darkside-20k style [9],  SiPM tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 50% quantum efficiency at visible wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We assume here another 50% wave  length shifter (WLS) efficiency from, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=', TPB on the tile surface, for a total efficiency of 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  For maximum light detection we envision covering inside the cathode (between the two planes of  cathode wires, as will be done in module 2) and the interior acrylic walls at up to 80% coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We  assume the Module 2 power-over-fiber concerns [4] to be solved to allow our SiPM coverage of the  cathode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The resulting number of SiPM modules for 10% coverage – a value which current studies  naively show is sufficient for a dark matter search using pulse shape discrimination – is ∼ 50000,  to be compared to Darkside20k’s planned ∼ 10000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' However, as discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1, to  achieve the energy resolution required for a neutrinoless double beta decay search will require  significantly more coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It is likely possible to optimize the placement of the tiles around the  fiducial volume, to reduce the total number required, and work is ongoing to study this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Reflectors  To maximize the light capture in this module the surface of the acrylic box will  be coated with a reflector to create a light-tight inner volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' If a PTFE reflector is used, as in  DarkSide-50, reflectivities of 97% should be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Argon Purity  The baseline requirement for DUNE is < 25 ppm of nitrogen to ensure  that photon propagation in the argon is not attenuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For our simulation studies we assume  that attenuation (absorption) lengths of order 50 m are achievable, which corresponds to nitrogen  contamination levels of 1-2 ppm [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  We note that dedicated dark matter experiments have  achieved ppb levels of purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Physics Studies  This section outlines key physics goals of this module and describes the studies that have been  performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Large Low Background kTon-Scale LArTPCs  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Argon-39 Studies  Reduction of the rate of 39Ar is desirable for a number of reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This 600 keV endpoint beta  emitter decays at a 1 Bq/l rate in ordinary atmospheric argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It may mask low energy physics by  creating optical signals that confuse the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It also represents a serious hurdle  in the detector triggering considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Trigger primitives, which constitute a 6 PByte/year data source, not necessarily to be stored  into perpetuity, are still an onerous data flow to deal with during steady data-taking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' That number  drops to a far more manageable tens of TBytes if the 39Ar is reduced by a factor of 1400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In reality  of course, much of the data budget will perhaps be consumed by low-threshold activity in this  detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Perhaps the more pernicious practical problem is that 39Ar beta decays may reconstruct  as optical ”hits” which may, in turn, comprise ”flashes” which then confuse the charge-light  association for reconstructed physics objects – especially at low energy and far from the light  detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  (Hits are simply individual light signals over threshold and the parameters that  characterize them, and flashes are collections of hits in narrow ranges of time, hypothesized to be of  the same physical origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=') We have performed a study in a Module 1-like environment, not shown  here, that gives the not very surprising conclusion that supernova flashes become unambiguously  matched with a x1400 39Ar reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Third, the highly desirable property of pulse shape discrimination (PSD) in Argon which  would allow to subtract out the nuisance 39Ar contribution for our, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=', dark matter search  ambitions, is not practicable if pile-up is too intense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We show in [10] and discuss later in this  paper that a x1400 reduction of 39Ar just allows for a search in our fiducial volume down to 100  keV thresholds and perhaps lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Here we mention that even a x1400 reduction of 39Ar does not allow this detector to get  to arbitrarily low thresholds for searches of physics with electronic signatures – as apposed to  neutron-like interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We expect that reductions by this amount will still leave 39Ar as an  overwhelming background to low-rate, low-energy solar processes, for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Simulation  Almost all studies in this paper are carried out in a standalone Geant4[44] simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Source  code and build instructions are found at Reference [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In that simulation is proper isotope decay  and neutron physics, along with optical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The volume is basically a 10 kTon box of liquid  argon, but with a reasonable model of the cryostat walls on all six sides with charge readout planes  (CRPs) made of G10 on the floor and ceiling, as in the VD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Further, there is an acrylic box inside,  open on top and bottom and tiled with 24 cm2 SiPM modules at an 80% coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' SiPM modules  with that same coverage viewing both upper and lower volumes also tile the central cathode plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  See Figure 2 for a representation of our simulated geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We use an after-the-fact 25% total  Large Low Background kTon-Scale LArTPCs  13  quantum efficiency by merely scaling by that fraction of the detected hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' There is no SiPM  electronics response applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We include a 96% reflectivity of the acrylic surface and a 44%  reflectivity for the CRPs (as the holes in each CRP are about 56% of the surface area), and impose  an argon attenuation (absorption) length of 50m, and a Rayleigh scattering length of 90cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' All  simulations generate their 128 nm photons ab initio from the charge particles which create them  and are propagated on the fly, with no lookup libraries, until their end point on a SiPM where they  are counted or they disappear through absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Optical Studies  The simulation described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2 was used to study the minimal requirements for the  optical system to allow pulse shape discrimination in a dark matter search, including the required  reflectivity of the acrylic box and anode readout surfaces as a function of SiPM tile coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The minimal photon counting requirement for the optical system was set at 400 photons reaching  the SiPM surface, which results in a total of 100 photons being detected due to the assumed 25%  efficiency described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The 400 (100) photon requirement was chosed as the minimal  amount of photons required to perform a pulse shape discrimination analysis such as described in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The results of the simulation are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The studies were performed varying  acrylic and anode reflectivity between 0 % and 100 % (a realistic 97 % is highlighted) and then  varying the SiPM coverage on the walls and cathode plane to count the accepted photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Both the  standard acrylic box described above and also a maximally sized box where the walls are moved  out to the edges of the detector were simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This large box would be the worst-case optics  scenario, maximising the path length of the photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The studies show that a relatively modest  amount of SiPM coverage of 10-20 % is required even in the worst-case scenarios to reach the  target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The effect of attenuation with the liquid argon was also studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Figure 5 shows the number of  photons detected at the SiPMs as a function of SiPM coverage for a variety of different attenuation  lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Assuming the relation between nitrogen contamination of the argon versus attenuation  found in [43], this study shows that the 10-20% SiPM coverage is sufficient to tolerate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5-5 ppm  levels of nitrogen within the argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Supernova Neutrino Physics  In this section we present several supernova neutrino burst studies that could be enhanced by this  module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This includes increased sensitivity to lower energies, later times and sources from greater  distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We also discuss the CEνNS glow sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Large Low Background kTon-Scale LArTPCs  14  (a)  (b)  (c)  (d)  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Number of photons detected at the SiPMs as a function of coverage, varying both  reflectivity of the surrounding acrylic and anode plane walls, and the size of the acrylic box  containing the inner volume, from Original Size (6x12x20 m3) to Max Size (at fiducial boundaries  at 12x12x60 m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='Acrylic Reflectivity (Original Size Acrylic Box)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='Mean SiPM Hits on Y-axis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='006  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='800  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='700  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='600  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='400  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='Target  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='300  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='200  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='+100%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='■97%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='≥0%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='100  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='Mean SiPM Hits on Y-axis  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='900  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='% of SiPM CoverageLarge Low Background kTon-Scale LArTPCs  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Number of photons detected as a function of SiPM coverage when varying the attenuation  length within the argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Supernova Energy Spectrum  We wish to explore the potential improvement provided by  a low background large LArTPC to that achievable in current DUNE Far Detector designs for  sensitivity to supernova explosion detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' To this end we simulate a ten-second exposure of our  detector to a flux of electron neutrinos from the Livermore [46] supernova model and run them  through the MARLEY [47] event generator to produce the final state particles in liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  MARLEY considers all candidate νe processes, including coherent, elastic, and charged current  processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The detector response is provided by the simulation described in previous section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Using that simulation we count SiPM hits for an 80% coverage and convert to energy using a  simple conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' That conversion comes from running 3 MeV electrons uniformly through the  detector and counting SiPM hits in a coarse x,y,z binning of the production point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We emphasize  that this study is simplified and that refinements including TPC charge response as well as better  SIPM energy resolution will result in improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Events must originate in the inner 3 kTon  fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The result is shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The most evident feature is that the elastic  scattering component of the νe flux becomes dominant at low energies inaccessible to the baseline  DUNE far detectors – and it does so because neutron rates are required to be low from the cold  cryoskin stainless steel and the 42Ar is at very low levels in UAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The threshold here can go all the  way down to 600 keV, which is a factor of roughly 18 lower than the baseline DUNE far detector  design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The low detector threshold allows access to a significant number of elastic scatter events  within the liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  This opens up the possibility of reconstructing the position of the  supernova with a pointing analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Liquid argon TPCs, with the excellent track resolution, are  well suited to making this measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' With the expected reduction in backgrounds in this  sample, a clean elastic scatter sample will dominate at thresholds below 5 MeV (see Figure 6),  though pointing with these reconstructed tracks is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Supernova Trigger  The trigger system in a DUNE-like LAr detector relies on the so-called  Trigger Primitives (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' hits, TPs) generated from the electronics connected to the wires or photo-  Ar Attenuation Length (Original Size Acrylic Box)  900  Mean SiPM Hits on Y-axis  800  700  600  500  400  Target  300  200  →100m =50m ^20m 10m  100  0  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  % of SiPM CoverageLarge Low Background kTon-Scale LArTPCs  16  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Energy deposited from neutrinos from supernova located at 10 kpc, assuming the  Livermore model, during a ten second detector exposure window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Here we presume no Rn222  contribution and a likely too conservative x500 Ar42 suppression in UAr with respect to that  achievable in atmospheric argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Reaching 5 MeV SN detection is straightforward in this module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' They are simple objects constructed from the signal waveform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A stream of TPs arrives  in the trigger module of the DAQ, which will use them to form a Trigger Activity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' a cluster of  hits, TAs), which is an association in time and space done by a trigger algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A TA is related  to each sub-module/component of the detector (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' an APA module), and multiple TAs form a  Trigger Candidate (TC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Having a positive TC, the readout system stores the requested data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For  low energy physics, which does not have an external trigger like beam events, it is essential to  understand the detector backgrounds (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' radiological and electronics noise), to avoid triggers  issued by undesired data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Thus, a Low Background LAr detector can perform better than the  current designs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The bottleneck for designing efficient supernova neutrino burst (SNB) trigger algorithms is  the data transfer and storage resources available for the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' To get the most of an SNB event,  around 100 seconds of full detector readout is desirable, which means about 150 TB of raw data for  a 10 kTon LAr detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It will take about one hour to transfer the data from this trigger event from  livCC  livES  103  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0kTonne-10sec/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0MeV  Ar39 Bg/L/1500  Ar42Bq/L/1500/5E8  neutrons 1e-11/cm3/s  101  Events  10-5  0  5  10  15  20  25  Energy [MeV]Large Low Background kTon-Scale LArTPCs  17  the detector caverns to the storage on the surface and several additional hours to transmit those  data to the primary storage centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Therefore, while the effective threshold must be set low enough  to satisfy the requirements on SNB detection efficiency, it is crucial to not fire too frequently on  background fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A requirement on the fake trigger rate of once per month is determined by  these limits on data-handling, for a DUNE-like LAr detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Additionally, the triggering decision  needs to be made within 10 seconds since this is the typical amount of data buffered in the DAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  SNB triggers are likely to be both TPC- and Photon Detection System (PDS)-based in far detector  modules one and two, though at the lower energy thresholds we concern ourselves with in this  paper a PDS-only trigger will be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  In current studies both TPC and PDS information gives a tagging efficiency of about 20-30%  for a single neutrino interaction [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Reducing the estimated neutron capture rate in the LAr  volume by a factor of ten (which is the principal background for SNB triggering, given the higher  de-excitation gamma energy of about 6 MeV when compared to other radiological components),  this efficiency improves to 70%, which translates to a 100% (20%) SNB triggering efficiency  for a Milky Way (Magellanic Clouds) SNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The trigger strategy described above is a “counting”  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' If we utilise the integrated charge of each TP, it is possible to construct a distribution  of the TAs raw energy (SNB signal with backgrounds) and compare it to the background-only  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The “shape” triggering method improves the efficiency to tag Magellan Clouds’ SNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The  efficiency figure for such a SNB producing ten neutrino interactions (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' [48]) shows the  efficiency as 70% in a ten kTons DUNE-like LAr detector using the standard DUNE background  model described in Reference [1] and a shape triggering algorithm that keeps the fake trigger rate  to one per month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  With the Low Background LAr detector, a less stringent selection can be used, increasing  the signal efficiency while keeping the fake trigger rate at the requirement level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Thus, higher  efficiencies will be reached with a lower number of SN events, it then being possible to achieve  100% efficiency to trigger a Magellan Cloud SNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Identifying Andromeda’s SNBs is more  challenging even with this design since they do not produce enough interactions in the whole  detector volume in a 10 seconds window (see the supernova sensitivity plot in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [48]), and  lowering the TA requirements would encounter the 39Ar activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Late Time Supernova Neutrinos The neutrino flux from a core-collapse supernova is  expected to cool over a few tens of seconds, with the late time events getting lower and lower  in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The tail end of the burst, where a black-hole-formation cutoff may be present, will be  challenging to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A lower energy threshold extends the time range with which a large liquid  argon detector can follow the evolution of the supernova burst [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Pre-supernova Neutrino Signal Another benefit of lowering the threshold is potential  sensitivity to presupernova neutrinos [50, 51, 52, 53, 54, 55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The final stages (hours to days)  of stellar burning before the core collapse are expected to be associated with an uptick in neutrino  Large Low Background kTon-Scale LArTPCs  18  production and energy, and observation of these could provide a true early warning of a core-  collapse supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The presupernova flux is expected to be small, and energies are typically  less than 10 MeV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' nevertheless they may be observable in a large liquid argon detector with low  threshold [53] for progenitors within a few kpc nearing the ends of their lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' CEvNS Glow  Coherent elastic neutrino-nucleus scattering (CEvNS) [57, 58] is a process  that occurs when a neutrino interacts coherently with the total weak nuclear charge, causing the  ground state nucleus to recoil elastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The cross section is large compared to the inelastic  charged- and neutral-current interactions, but resulting nuclear recoil energies are in the few tens  of keV range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' CEvNS has now been observed in argon by the COHERENT collaboration using the  stopped-pion neutrino flux from the Spallation Neutron Source [59] with a cross section on order  22 · 10−40cm2 for an incoming average flux of < Eν >≈ 30 MeV from pion decay at rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  In a large LArTPC there will be a high rate of CEvNS in a core-collapse supernova burst—  approximately 30-100 times more events with respect to νeCC (the dominant inelastic channel),  depending on expected supernova spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' However, each event is individually not likely to  produce more than a few detected photons, and sub-50-keV thresholds need to be achieved to find  these events, if they are to be found one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Even with depleted argon, the 39Ar rate down in  this range in our proposed detector is by far overwhelming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' However, an alternative is to identify  a “CEvNS glow,” [60] in which the excess rate of detected photons can be tracked statistically  above the 39Ar rate as a function of time in a characteristic explosion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 7 shows that simulated  supernova-induced activity from a burst at 10 kpc over 10 seconds in an inner fiducial 3 kTon gives  three distributions: broadly-distributed-in-time, higher-energy charged-current (CC) component,  a burst of low-hit-multiplicity CEvNS events at about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='01 sec, and then the absolutely flat 39Ar  activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In ongoing work, we propose to subtract the reconstructed CC events and fit to the CEvNS  bump above the background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We note that in principle, an excess of collected ionization from  CEvNS events is observable as a “CEvNS buzz” in coincidence with the CEvNS photon glow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Solar Neutrino Physics  In this section we present our enhanced sensitivity to solar neutrino searches including lower  energies and enhanced oscillation sensitivity to ∆m2  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This allows exploration of solar-reactor  oscillation tensions and Non-Standard Interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It also allows a precision CNO solar neutrino  measurement and a measurement of the 3He+p solar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A first study of DUNE as the next-  generation solar neutrino experiment is presented in reference [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' That study is dominantly one  of higher energy 8B CC processes, but in this section we widen the discussion and point out what  a lower threshold would offer to solar neutrino studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Low Threshold Gains and Elastic scattering  Figure 8 shows the number of expected ES  interactions over threshold for a 3-kTon·year exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' One sees that reducing the threshold to 1  Large Low Background kTon-Scale LArTPCs  19  (a)  (b)  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The CEvNS glow is the statistically significant increase observed from the low SiPM  hit-count CEvNS Supernova events, shown in bottom population of figure (b) that sits on top of the  rate of 39Ar seen in figure (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  MeV makes pep and CNO neutrinos observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 MeV could add 7Be neutrinos,  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1 MeV could allow for detection of pp neutrinos, though this threshold encroaches into  the large 39Ar background, even in UAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The total number of available neutrinos is important for  possible studies discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This amounts to 9,200 neutrinos for a 1-MeV threshold,  130,000 neutrinos for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5-MeV threshold, and 820,000 for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1-MeV threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  In addition to the previously discussed methods for reducing backgrounds, we are  investigating directionality with ES for all solar neutrinos and Cherenkov radiation for more  energetic 8B neutrinos as ways to enhance neutrino signal over backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  SiPMHitsvs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='Time-Ar39  2500  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4  2000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0  1500  events/hits/s  SiPM Hits  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='8  1000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4  500  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0  m  4  8  time (s)SiPMHits vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='Time -CEvNS + CC  105  103  102  104  Count  101  /hits/  Hit  103  100  /ents/  SiPM I  10-1  102  10~2  101  10~3  10~4  10~3  10-  10~1  100  time (s)Large Low Background kTon-Scale LArTPCs  20  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Number of events over threshold for solar-neutrino ES interactions in argon for a 3-  kTon·year exposure with contributions from different solar fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Solar Neutrino Oscillations Current measurements of the neutrino mixing parameter,  ∆m2  21, using solar neutrinos from SNO and Super-Kamiokande are currently discrepant at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4 σ  with measurements from KamLAND using neutrinos from nuclear reactors [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Differing results  from these two methods would suggest new physics possibly involved with exotic matter effects  as the neutrino passes through the Sun and Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Further data from DUNE will further investigate  this discrepancy with high statistical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The sensitivity comes from the “day-night”  effect, a partial regeneration of the νe solar flux due to matter effects in Earth which depends on  ∆m2  21, neutrino energy, and nadir angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This is an advantageous strategy allowing for constraints  of uncertainties using daytime data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Solar neutrinos are much lower energy than typically observed in DUNE making  reconstruction of these events challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Also, at these low energies, radiological backgrounds,  principally neutron capture with a contribution from 40Ar(α, γ), dominate analysis backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  A low-background DUNE-like module with enhanced light collection will help with both of these  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Improved energy resolution from increased photodetector coverage would significantly  improve DUNE-like sensitivity to ∆m2  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This would both improve reconstruction of the dominant  neutron background around the 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1 MeV total visible energy of the neutron capture on argon-  40, and better measure the dependence of the νe flux regeneration on neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A single  10-kTon, low-background module could discern between SNO/SK and KamLAND best fits at  over 6 σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Lower neutron background levels would also improve the signal-to-background ratio  for the measurement, which could also lower the energy threshold for detecting and analyzing  solar neutrino events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' An estimate of sensitivity to ∆m2  21 with 100 kTon-yrs of exposure with  a low-background module is compared to DUNE’s sensitivity with 400 kTon-yrs of data from  107  106  105  104  103  102  101  hep  17F  150  7Be  ES  100  eo +eF  8B  pep  total  eN  13N  7Be  10-1  10-2  10-1  100  101  threshold Ethr (MeV)Large Low Background kTon-Scale LArTPCs  21  nominal, horizontal drift modules in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We use a larger 10-kTon fiducial volume for the  reduced background/threshold curves in that figure, increased from other studies in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Sensitivity of a low-background module to the neutrino mixing parameter ∆m2  21 assuming  a true value of the solar best fit, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='13·10−5 eV2 [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Colored contours show sensitivity after  100 kTon-yrs for various detector configurations compared to 400 kTon-yrs of DUNE data with  horizontal drift design, shown in black and dominated by CC events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Non-Standard Neutrino Interactions Non-standard neutrino interactions (NSI) could  modify neutrino oscillations in the Sun and result in a different number of neutrinos observed  compared to the one predicted by the Standard Model [62] (see also studies with different  parameter definitions in [63] and [64]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The NSI Hamiltonian (for neutral currents only) relevant for solar-neutrino oscillations can  be written in the following form:  HNSI  ν  =  √  2GF(nu + nd)  � −ϵD  ϵN  ϵ∗  N  ϵD  �  ,  (1)  where GF is the Fermi constant, nu and nd are the up- and down-quark densities, respectively,  and ϵD and ϵN are the diagonal and off-diagonal NSI couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' These couplings affect νe solar  survival probability (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The diagonal coupling can mimic different vacuum ∆m2  values, resulting in its incorrect measurement when NSI are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Detection of ES in the proposed module (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1) allows for a great opportunity  to have an almost NSI-independent anchor point in the oscillation probability near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1 MeV in  addition to investigating NSI in solar-neutrino oscillations at several MeV energies where changes  in oscillation probability could be large but not much experimental data exists, yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  120  10 kt 2% resolution + lowered  100  bkg + lowered threshold  80  10 kt 1% resolution  10 kt 2% resolution  + lowered bkg  60  40  10 kt 2% resolution  20  40 kt HD design  10-3  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='07  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='09  Am2 (eV2)Large Low Background kTon-Scale LArTPCs  22  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 2-flavor electron-neutrino survival probability for solar neutrinos for the Standard Model  and several different NSI couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  An example of possible constraints on the NSI couplings is shown in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This plot  was obtained with the assumption of no backgrounds or systematic errors, an energy threshold of  1 MeV, and an exposure of 3 kTon·years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For comparison, see the larger constraints using current  neutrino data available in [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Precision Measurement of CNO flux  The CNO flux has been measured [65] recently by  the Borexino collaboration to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5σ above 0, though it yields indistinct information about the high  or low metallicity solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Per [65], the CNO neutrino flux scales with the metal abundance in  the solar core, which probes the initial chemical composition of the Sun at its formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The metal  abundance in the core is decoupled from the surface by a radiative zone, and CNO neutrinos are  the only probe of the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Here we want to investigate if an energy window exists where a precise measurement of  the CNO flux can be performed in our low background module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We use our by-now standard  simulation tools to count true deposited energy for a variety of backgrounds and solar neutrino  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='45  solar  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='40  Standard Model  2-flavor  ED = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1, EN = 0  ED = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1,EN = 0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='35  ED=O,EN=O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1  ED=0,EN=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='30  10-1  100  101  true neutrino energy Etrue (MeV)Large Low Background kTon-Scale LArTPCs  23  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Possible NSI constraints for 3 kTon·years obtained with the proposed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  sources in a 3 kTon fiducial volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We impose a 2% energy smearing, as is reasonable by  arguments in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Fluxes come from [66] with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3 survival probability applied, as  appropriate to this low energy range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We use the assumption that the 42Ar content will be at a rate  that is reduced by a further factor 100,000 compared to atmospheric argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We emphasize again  per section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3 this is likely very conservative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The result is we see in a window about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2 MeV  wide just above the pep cutoff that the CNO signal sticks up above other solar neutrino sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  This is merely an illustration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' a real analysis will likely do a templated fit above the 7Be peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Neutrons from the cold cryoskin stainless steel are forced to be low, as usual;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' radon is taken as  controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The 210Bi background which Borexino took exquisite care to control and measure [65]  is yet to be thoroughly investigated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Nevertheless, there would appear to be a window where  the high and low metallicity solutions are statistically separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' By comparison, CNO sensitivity  in the two-phase LArTPC program, which studies are further along than the current work, can be  seen in [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Triggering for CNO neutrinos  Initial studies into measuring the CNO flux with TPC triggering  are underway, taking into account the full complement of radiological backgrounds predicted in  the DUNE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In these early studies, TPC triggering requires sufficiently large, coincident  clusters on the collection plane and at least one induction plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We expect future work will in fact  lead to a much higher-efficiency, light-based trigger being implemented for most work discussed  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The dominant background in the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4 - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0 MeV energy window is radon in this  2  1  0  1g  90% CL  1  2g  99% CL  3g  2  minimum x2  2  1  0  1  2  EDLarge Low Background kTon-Scale LArTPCs  24  early TPC triggering study, not in fact solar neutrinos – despite the rate reduction by a factor of  103 predicted in the low background module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In order to perform the true CNO detection there  is clearly much work to be done in offline processing to accurately characterize and implement  rejection by alpha detection, Bi-Po coincidence, disproportionate activity near the cathode from  drifting cation decay products, and emanation properties of the various detector materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' CNO solar neutrinos with backgrounds taking radon to be solved and negligible and  neutrons constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This is 2% smeared true deposited energy in our inner 3 kTon fiducial volume  in a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Note that a likely, low 42Ar level is shown that reveals that a CNO fit is possible above  the 7Be peak and explicitly by a simple counting experiment in roughly the region shown in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  This is a lower 42Ar level than shown in Fig 1, but still realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The inset zooms in on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='25-  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='45 above the pep region to show that the signal statistics are large enough to favor either high- or  low-metallicity after a year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Precision measurement of hep flux  The hep solar neutrino process (3He+p →4 He+e+ +  νe) produces the highest energy neutrinos, though they have the lowest flux and have not yet been  observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This low background module will be able to measure tens of these neutrinos per year  via CC events, with a significant reduction in background due to radon and neutron interactions  within the liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  1010  CNOhiM  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='25-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='45 MeV region  CNOloM  80  7Be  60  MeV  108  8B  40  pep  20  Ar39 Bg/L/1500  106  Ar42Bg/L/1500*9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2E-5/1E5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='30  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='45  Deposited Energy [MeV]  104  102  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='00  Deposited Energy[Mev]Large Low Background kTon-Scale LArTPCs  25  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Neutrinoless Double Beta Decay  A discovery of neutrinoless double beta decay (0ν2β) is the most straightforward way to prove the  Majorana nature of neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It would be parsimonious to be able to search for 0ν2β in the same  detector we are suggesting to use for the other measurements mentioned in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 136Xe is one  isotope in which much work is being performed worldwide to try to make this discovery [68, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Loading large LArTPCs with few-percent level Xe and measuring neutrinoless double beta  decay has been suggested in [70] and [71] among other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In this subsection we show that  naively a discovery is likely possible with a signal 136Xe half-life of 5 · 1028 years, and is quite  apparent at 1·1028 years, provided energy resolution requirements can be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We imagine, say, a  five-year search campaign for 0ν2β at the end of the prosecution of the baseline physics program  of this module, since any dark matter search requiring PSD would necessarily be compromised by  xenon loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  We start with a 0ν2β signal calculated using a Gaussian 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 (3) % energy resolution at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='435  MeV with 1/  √  E dependence for top (bottom) plots in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Energy resolution ambitions  are consistent [72] with what we may expect in a LArTPC from charge energy collection in  combination with the photon readout system, as well as discussion in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We similarly  smear the 2ν double-beta true spectrum by these resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  And for the 208Tl background  emanating from the G10 of the charge readout planes we use only the expected rate from the  simulation, creating the actual spectrum by likewise smearing the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='6 MeV gamma by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 (3)% in  top (bottom) plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For the solar and 42Ar backgrounds previously discussed, on the other hand, we  use the resolution from the simulation that uses only the poorer light-only collection from the SiPM  hits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' These are flat backgrounds which extend to low energy where signal will likely be obscured  in the noise of the charge readout, hence the need to rely on only the SiPMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For both plots we use  a 3% concentration of 136Xe in our inner volume, using a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='0 kTon fiducial volume, and propose  to run for five years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We plot a signal corresponding to a (5)1 · 1028 year 0νββ half-life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Among the assumptions made for our figures is that underground argon can give a 42Ar  suppression of a factor of 10 beyond that in atmospheric Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' See section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3 which indicates this  is likely easily achievable, up to processing concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We also take in Fig 13 that the irreducible  solar 8B elastic scattering may be suppressed by a factor of two (conservatively down from three  assumed in Reference [73, 74] ) from inspecting Chernkov/Scintillation light ratios in single versus  double electron events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We impose no efficiency hit due to this cut, whereas Reference [73]  uses an efficiency of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  More careful event reconstruction studies are needed to bear out  the reasonableness of this cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We assign the 208Tl in the G10 charge collection planes a Th  concentration of 50 mBq/kg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Charged current solar neutrino events, are in principle reducible to  zero, due to the excited state gammas that are emitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' And similarly, neutrons shall be almost  entirely removed using pulse-shape discrimination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We again take the radon issue to be solved for  the sake of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A 2σ band to either side of the 0νββ energy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='435 MeV is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  We take Eres ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5 % in the top plot at the Q value, even though, as we have said, it is not  Large Low Background kTon-Scale LArTPCs  26  immediately obvious we can achieve that (nor in fact are we currently confident we can reach the  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5% of the bottom plot) with our charge readout plus 80% SiPM coverage, open as it is at the top  and bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' That study is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' However, we see that there is sensitivity  in this detector to 0νββ discovery at lifetimes that stretch the reach of coming experiments, despite  the large, irreducible solar 8B neutrinos which elastically scatter off the mostly argon target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Dark Matter  It is known that a large amount of dark matter exists within the Universe, that has so far only  been observed by gravitation interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' One popular candidate for the dark matter is the Weakly  Interacting Massive Particle, or WIMP, that is the focus of several current and future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The potential of using this low background module to search for WIMP dark matter was studied  in Reference [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This study assumed a dual phase TPC design, with a single fiducial volume at  the center of the detector (rather than the split fiducial volumes described above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The criteria for  achieving a sensitive WIMP dark matter search was set as requiring:  a 50-100 keV nuclear recoil threshold;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  O(10) background events;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  O(100) photons detected per event to allow pulse shape discrimination (PSD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Assuming that 1250 photons per 100 keV of prompt scintillation light is emitted (as measured  by SCENE for 500 V/cm fields [75]), the studies in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3 show that reaching 100 photons  per event is realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A pseudo-Monte Carlo simulation of the Poisson distributed light output  was performed to determine the width of a typical PSD variable f90, defined as the ratio of light  detector detected in the first 90 ns of an event to the light detected in the second 90 ns of an event,  for the 39Ar decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The results taken from measurements from the SCENE experiment [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Reference [10] shows PSD is expected to reach the 1010 rejection level for electron/gamma  backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We also direct the interested reader to the PSD study [76] to suppress the 39Ar  decay background in DEAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  All other electron/gamma backgrounds are expected to be subdominant to the 39Ar and will  thus be removed by PSD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Neutron backgrounds were managed as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The main  background will be from irreducible atmospheric neutrinos at the so-called neutrino floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' A full  background table from the study is shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The background rates are used to set a 90% C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' sensitivity to WIMP dark matter Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' [10]  shows that a three-year search with this detector will have comparable sensitivity to planned  next-generation detectors, which have expected run times of 10 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This timescale allows a  rapid cross-check of any signals discovered in these detectors, in particular for the liquid argon  experiments such as DarkSide-20k [9] or ARGO [77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Large Low Background kTon-Scale LArTPCs  27  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' An optimistic background/resolution scenario for a 136Xe 0νββ half-life of 5·28 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Detector energy resolution is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='5% on top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Backgrounds are as discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' On the bottom  is the same with a reduced resolution of 3% and for a half-life of 1E28 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  140  Onubb  MeV  2nubb  TI208 50 mBq/kg  Events/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='189kTonneXe136-5yr /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='02[  120  8BES  Ar42Bq/L/1500/5E8/10  100  pseudo-data  80  60  40  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='45  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='60  Energy [MeV]140  Onubb  Events /0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='189 kTonneXe136-5yr/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='02 MeV  2nubb  Tl20850mBq/kg  120  8BES  Ar42Bq/L/1500/5E8/10  100  pseudo-data  80  60  40  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='30  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='35  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='40  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='45  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='60  Energy [MeV]Large Low Background kTon-Scale LArTPCs  28  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Seasonal Variation of Rate for WIMP Dark Matter  The prominent model for dark matter  (DM) is the so-called Standard Halo Model (SHM) [78] featuring WIMP DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The SHM describes  a basic isometric spherical distribution of WIMP DM around our galaxy and has been used due to  it being a good trade-off between realism and simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The relative velocity of our solar system  of 233 km/s [79] at which the Sun moves through the gas-like halo of WIMP DM induces as a  ”WIMP wind”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Figure 14 illustrates the Sun’s rotation in our galaxy together with Earth’s solar  orbit into and out of the WIMP wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We modeled the Sun’s rotational and peculiar velocities  into one combined effective velocity of 233 km/s in the galactic plane and accounted for the 60◦  inclined plane of Earth’s orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We were then able to accurately describe the annual modulation of  detectable WIMP rate R on Earth by one simple periodic sinusoidal function with one amplitude  parameter A and a maximal rate on June 1 of each year[79] [80]:  R( [d−1] ) = A [d−1] × cos  � 2π  T[d] × ( t[d] − tJune1[d] )  �  + Ravg [d−1]  (2)  The constant term Ravg is the average annual rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Earth’s period T is 365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2422 days and the phase  corresponding to the maximal rate Rmax observable on June 1 of each year is 2π · tJune1/T =  2π · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Galactic WIMPs in the halo are assumed to have a Maxwell-Boltzmann-like velocity  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The effective WIMP velocity distribution is shifted up when Earth is flowing  maximally into the WIMP wind and shifted down when Earth is flowing maximally with the  WIMP wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Due to this aspect, an analysis on the annual modulation of the detection rate R  could provide a powerful tool for identifying WIMP DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Due to the unrivaled large fiducial mass  of 3 kTons and a potentially very long DUNE operation of one decade (or even several), this  concept can offer a unique detection of the seasonal variation of the detectable WIMP rate R at  a sufficient statistical significance for providing a smoking gun signature for the WIMP nature of  DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This would be particularly of interest in case upcoming generation-2 DM experiments like  LZ [81] and/or XENONnT [82] have evidence for WIMPs near their sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It would be nearly  impossible for the planned generation-3 DM experiments [83] [84] to make such a smoking gun  detection proving the WIMP nature of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The differential rate of interactions in an arbitrary detector for the SHM is described in  Equation 3:  dR  dER  = σSI  N  A2mANTρχ  2mχµ2  N  F 2(ER)  ∞  �  νmin(ER)  d3−→ν  ν  f⊕(−→ν , −→  νobs)  (3)  where σSI  N is the spin-independent-nucleon cross-section for WIMPs, A is the atomic number of  argon, mA is the mass of argon, NT is the number of target nuclei, ρχ is the local dark matter  density (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3 GeV  cm3 ), mχ is the mass of a WIMP, µN is the reduced WIMP-nucleus mass, F(ER) is  the the nuclear form-factor, νmin is minimum WIMP velocity to produce a recoil of energy ER,  ν is WIMP velocity, and νobs is the observer velocity with respect to the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' When Earth is  Large Low Background kTon-Scale LArTPCs  29  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Galactic WIMP wind as it relates to Earth’s orbital plane employing an illustrative  rendition [79] of our Milky Way galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Our solar system’s velocity in reference to our galaxy  has contributions from both a rotational aspect with a tangential velocity of 220 km/s and from a  minor solar peculiar aspect with velocity components (U, V, W) = (10, 13, 7) km/s [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In our  model the combined relative velocity of our solar system is then 233 km/s at which the Sun moves  through a gas-like halo of WIMP dark matter assumed to have a Maxwell-Boltzmann-like velocity  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This induces what we experience as a “WIMP wind”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This WIMP wind is at an angle  compared to Earth rotation around the Sun as pictured in the zoomed in diagram, with the effective  WIMP velocity distribution shifted up when flowing maximally into the wind and shifted down  when flowing maximally with the wind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Due to this aspect, an analysis of the annual modulation of  detectable WIMP rate on Earth could provide a powerful tool for identifying the WIMP nature of  dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  moving into or with the WIMP wind, it affects the differential WIMP rate, which in turn would  affect our −→  νobs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' For ease, we can define:  ∞  �  νmin(ER)  d3−→ν  ν  f⊕(−→ν , −→  νobs) = ζ(ER),  (4)  where  f(−→ν ) = 1  N  �  e  −ν2  ν2  0 − e  −ν2esc  ν2  0  �  ,  (5)  and  f⊕(−→ν , −→  νobs) = f(−→ν + −→  νobs)  (6)  OurMilky Way Galaxy  WIMP  max,detectable  wind  WIMP rate  Galactic Center  on Jun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=" 1  Earth  UO+U  Galactic  0094  Plane  1AU  <1 AU  'Sun  TAU  WIMP  Solar Rotational Velocity  wind  = (0, 220, 0) km/s  = (10, 13, 7) km/s  J." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Genovesi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Reichenbacher 2022 - SD Mines - v02/f 1/22Large Low Background kTon-Scale LArTPCs  30  ht  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Example NEST[85] simulation of the annual modulation for 40 GeV/c2 WIMP dark  matter with a cross section of 4 × 10−48 cm2 for a 10 year measurement time with 3 kTon LAr at  50 keV threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' On top a Likelihood-fit result for a 10 year period and on bottom the same data  combined into a single annual period starting on January 1 of each year fitted with a χ2-method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The ideal case without background is assumed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [counts / (20 days)]  R  0  500  1000  1500  2000  2500  3000  3500  Elapsed Timet (in days since startof experimentAnnualModHist  30  Integral  277  x?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' / ndf  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='37 / 17  25  po  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='827 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='144  p1  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='85  20  15  10  0  50  100  150  200  250  300  350  Elapsed Time t (in days since 1st of January of each year)Large Low Background kTon-Scale LArTPCs  31  with N as a normalization factor, ν0 is the expected WIMP velocity, and νesc is the escape  velocity of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Using Equation 5, we can solve for individual cases of WIMPs in certain  speed brackets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  As mentioned earlier, Earth’s orbit can play a crucial role for differential WIMP rate  predictions in the SHM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' As the solar system travels through our galaxy, observers on Earth would  observe WIMPs dominantly from a certain direction in a windshield-to-rain like effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This is  due to the distribution of WIMPs being treated as a gas with a Maxwell-Boltzmann-like velocity  distribution with the stars in our galaxy moving through the dark matter due to their orbits around  the galactic nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In addition to the standard rotational contribution from the solar system, a  minor peculiar velocity exists of our solar system traveling through the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This WIMP wind  would be at an angle of 60◦ towards Earth’s orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This means that during June 1, the Earth  will experience a maximum effective flux of WIMPs while on December 1 the Earth will have  experienced the lowest effective WIMP rate, as one can see again in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  An example fit using Equation 2 of an annual modulation signal simulated in NEST [85]  without background is shown in Figure 15 on top for a 10 year period, and on bottom the same  data combined into a single annual period starting on January 1 of each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This is for a 40 GeV/c2  WIMP with a cross section of 4 × 10−48 cm2, which is very close to the limit of sensitivity of the  upcoming generation-2 xenon dark matter experiments LZ [81] and XENONnT [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We further  assumed a 3 kTon × 10 year exposure of our proposed low background LAr module with a 50 keV  threshold resulting in 277 events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The bottom plot of Figure 15 shows a good χ2-fit result for  the amplitude A = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='827 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='144 from Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It confirms the possible measurement of the  seasonal variation of the WIMP rate at a sufficient statistical significance for providing a smoking  gun signature for the WIMP nature of DM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' This will be uniquely possible with this detector, due  to the unrivaled large mass of 3 kTons vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' only 300 tons of argon for ARGO [84] and 100 tons for  a Gen-3 xenon experiment [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Moreover, the annual modulation effect in xenon is significantly  smaller due to the relatively lower energies of nuclear recoils in xenon compared to argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Last  but not least, the logistics of a decade long operation with this detector can utilize strong synergies  with the DUNE long-baseline physics, including the cavern availability and occupancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Additional Topics  This module, with its unprecedented combination of low background and size, also can explore  several other topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' We describe several examples in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Atmospheric neutrinos  The detector will measure approximately 10 CEνNS events due to  atmospheric neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' These events have not yet been observed and this would allow a cross-check  of background rates from the upcoming generation of dark matter experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Large Low Background kTon-Scale LArTPCs  32  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Strangelets Recently, the paper “Can strangelets be detected in a large LAr neutrino  detector?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' [86], predicted that a LArTPC detector is able to detect and discriminate light strangelets  with (Z, A) between (2,14) and (7,70) for energies up to 10 GeV in the presence of radioactive  background found at the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' When operated underground the detection limits are expected  to be extended due to lower background levels and, combined with the increased dimensions of  the detector module, will improve the event rates with a factor of 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In the case of strangelets  the main uncertainties are due to the estimations of their survival probability deep underground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The presence of 39Ar masks both ionization and scintillation signals from strangelets and induces  false signals in the collected charge from ionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The use of underground argon in this module  allows a cleaner detection signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Charged micro-black holes and Superheavy dark matter Hawking [87] suggested that  unidentified tracks in the photographs taken in old bubble chamber detectors could be explained  as signals of gravitationally “collapsed objects” (µBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The small black holes are expected to be  unstable due to Hawking radiation, but the evaporation is not well-understood at masses of the  order of the Planck scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Certain inflationary models naturally assume the formation of a large  number of small black holes [88] and the generalized uncertainty principle may indeed prevent  total evaporation of small black holes by dynamics and not by symmetry, just like the hydrogen  atom is prevented from collapse by the standard uncertainty principle [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Given the profound  nature of the issues addressed, some disagreement and controversy exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  In principle the direct detection of charged micro black holes with masses around and upward  of the Planck scale (10−5 g), ensuring a classical gravitational treatment of these objects, is possible  in huge LAr detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It has been shown that the signals (ionization and scintillation) produced  in LAr enable the discrimination between micro black holes (with masses between 10−5 - 10−4  grams, and velocities in the range 250 - 1000 km/s) and other particles [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It is expected that  the trajectories of these micro black holes will appear as crossing the whole active medium, in any  direction, producing uniform ionization and scintillation on the whole path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Along these lines, an analysis looking for multiple co-linear nuclear recoils can also probe  ultra heavy dark matter beyond the Plank scale, as described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' [91, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Sensitivity to the  heaviest dark matter candidates is limited by the number density of the dark matter, which is  inversely proportional to the mass, as the ability to detect heavy dark matter with a high cross  section is set by the probability that a dark matter particle enters the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' As such, sensitivity  to the highest masses scales with the detector’s surface area, and would leverage the large size of  this module compared to DEAP-3600, which has a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='7 m diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Similarly, in the direct detection of the charged micro-black holes, unlike in traditional WIMP  detection, there will exist both ionization and scintillation signals from direct interactions and from  recoiling nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The capability to perform pulse shape discrimination in this detector will allow  these tracks to be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Natural radioactivity is the main source of background in this case  and the reduced number of free electrons (and photons) from beta decays of 39Ar will allow a  Large Low Background kTon-Scale LArTPCs  33  significant improvement of the capability of the detector to correctly identify the micro-black hole  signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Other topics  This detector would have sensitivity to a small number of CEνNS events  within the neutrino beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' It may have applications to geologic tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The improved energy  resolution for low energy could have applications in searches for other exotics and beyond the  standard model physics phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Though the optimal search region for a diffuse supernova  neutrino background is above the energy of the solar neutrinos [93], and thus the radioactive  backgrounds, the improved energy resolution of this detector will again likely improve the search  sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Conclusion  We have presented a design in this paper for a low background kTon-scale LArTPC to potentially  expand the current physics program for such detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The design is based on the vertical drift  detector planned for DUNE’s second far detector module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' The module discussed is a candidate for  a third or fourth DUNE “Module of Opportunity.” It is realized by providing additional shielding,  stringent radioactive background control and enhanced light detection to the nominal vertical  drift module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Energy resolution will benefit in all energy ranges due to event reconstruction  and topology classification improvements from the superior light detection system and the quiet  detector, which will allow to capture more cascade gammas[13] and thereby improve the hadronic  component of neutrino-nucleus interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  The physics goals achieved by the SLoMo design extend the capability of large LArTPCs to  search for solar and supernova neutrinos, neutrino-less double beta-decay, and WIMP dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  At the same time the design proposed here, by the nature of its small perturbations to the vertical  drift module, assures continuing strong support to the long-baseline neutrino oscillation program  to measure remaining parameters in the PMNS matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Acknowledgments  Pacific Northwest National Laboratory (PNNL) is operated by Battelle for the United States  Department of Energy (DOE) under Contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' DE-AC05-76RL01830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Parts of this study at  PNNL were supported by the DOE, USA Office of High Energy Physics Advanced Technology  R&D subprogram and other parts by the Open Call Initiative, under the Laboratory Directed  Research and Development Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' In the United Kingdom this work was supported by STFC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  For IL and MP this work was performed with the financial support of the Romanian Program  PNCDI III, Programme 5, Module CERN-RO, under contract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 04/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' KS is supported by  the Department of Energy and the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' ZD acknowledges the support  Large Low Background kTon-Scale LArTPCs  34  of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Department of Energy Office of Science under contract number DE- AC02-06CH11357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  South Dakota School of Mines and Technology acknowledges the support of Department of Energy  through award number DE-SC0014223, as well as DE-AC02-07CH11359 through subaward from  Fermi National Accelerator Laboratory subcontract no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 664706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' JZ gratefully acknowledges using  the resources of the Fermi National Accelerator Laboratory (Fermilab), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Department of  Energy, Office of Science, HEP User Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Fermilab is managed by Fermi Research Alliance,  LLC (FRA), acting under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' DEAC02-07CH11359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Abi et al (DUNE Collaboration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Deep Underground Neutrino Experiment (DUNE), Far Detector Technical  Design Report, Volume IV: Far Detector Single-phase Technology, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' arXiv:2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='03010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='. Abud et al (DUNE Collaboration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Design, construction and operation of the ProtoDUNE-SP Liquid  Argon TPC, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1088/1748-0221/17/01/P01005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Amerio et al (ICARUS Collaboration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Design, construction and tests of the ICARUS T600 detector, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='nima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='044.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [4] Xin Qian, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Snowmass 2021 Letter of Interest: “Development of LArTPC Vertical Drift Solutions with  PCB Anode Readouts for DUNE”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='snowmass21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Timmes, and Kai Zuber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='  Large Low Background kTon-Scale LArTPCs  41  Physical Review Letters, 126(1), Jan 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' http://dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='  012002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' Gonzalez-Garcia, Michele Maltoni, Ivan Martinez-Soler, and Jordi Salvado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Updated  constraints on non-standard interactions from global analysis of oscillation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  JHEP, 08:180, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [Addendum: JHEP 12, 152 (2020)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' Shoemaker, and Luca Vecchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='physletb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content='  [65] BOREXINO Collaboration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' Villante, Sarbani Basu, Johannes Bergstr¨om, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+page_content=' Viel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Walding,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Waqar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Ward, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Westerdale, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Willis, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Zu˜niga Reyes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' First Direct Detection Constraints on  Planck-Scale Mass Dark Matter with Multiple-Scatter Signatures Using the DEAP-3600 Detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=', 128:011801, Jan 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  011801.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [92] Daniel Carney, Nirmal Raj, Yang Bai, Joshua Berger, Carlos Blanco, Joseph Bramante, Christopher Cappiello,  Ma´ıra Dutra, Reza Ebadi, Kristi Engel, Edward Kolb, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Patrick Harding, Jason Kumar, Gordan Krnjaic,  Rafael F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Lang, Rebecca K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Leane, Benjamin V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Lehmann, Shengchao Li, Andrew J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Long, Gopolang  Mohlabeng, Ibles Olcina, Elisa Pueschel, Nicholas L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Rodd, Carsten Rott, Dipan Sengupta, Bibhushan Shakya,  Ronald L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Walsworth, and Shawn Westerdale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Snowmass2021 Cosmic Frontier White Paper: Ultraheavy  Particle Dark Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='org/abs/2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='06508, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='  [93] Klaes MØller, Anna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Suliga, Irene Tamborra, and Peter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Denton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Measuring the supernova unknowns  at the next-generation neutrino telescopes through the diffuse neutrino background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' Journal of Cosmology  Large Low Background kTon-Scale LArTPCs  44  and Astroparticle Physics, 2018(05):066–066, may 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
+page_content='1088/1475-7516/  2018/05/066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfrS7d/content/2301.11878v1.pdf'}
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+THE HORNICH-HLAWKA FUNCTIONAL INEQUALITY FOR
+FUNCTIONS WITH POSITIVE DIFFERENCES
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+Version 2
+Abstract. We analyze the role played by n-convexity for the fulfillment of
+a series of linear functional inequalities that extend the Hornich-Hlawka func-
+tional inequality, f (x) + f (y) + f (z) + f (x + y + z) ≥ f (x + y) + f (y + z) +
+f (z + x) + f(0), including extensions to the case of positive operators.
+1. Introduction
+Many noteworthy inequalities are related to the following problem:
+Problem 1. Suppose that S is an abelian additive semigroup with neutral element
+0, f a function defined on S and taking values in an ordered vector space E (or in
+its positive cone E+). For n ≥ 2, find the linear inequalities relating
+�n
+i=1 f(xi),
+�
+1≤i<j≤n f(xi + xj), . . . , f(
+�n
+i=1 xi)
+for all x1, . . . , xn ∈ S.
+Due to its many ramifications, this problem is still the subject of intense activity,
+and the present paper reports some new results in this direction, under the um-
+brella of the Hornich-Hlawka functional inequality. Specifically, we are interested
+in studying conditions under which a continuous function f : S → R+ satisfies
+(1.1)
+f (x) + f (y) + f (z) + f (x + y + z) ≥ f (x + y) + f (y + z) + f (z + x)
+for all x, y, z ∈ S. When f is a real-valued function it is usual to replace (1.1) by
+(1.2) f (x)+f (y)+f (z)+f (x + y + z) ≥ f (x + y)+f (y + z)+f (z + x)+f(0).
+A good start for understanding the Hornich-Hlawka functional inequality is pro-
+vided by the following elementary (but powerful) inequality:
+(1.3) |x| + |y| + |z| + |x + y + z| ≥ |x + y| + |y + z| + |z + x|
+for all x, y, z ∈ R.
+As was noticed by Levi [17], every piecewise linear inequality like (1.3) remains true
+when the real variables x, y, z are replaced by arbitrary vectors x, y, z in RN and
+the absolute value function is replaced by the Euclidean norm,
+(1.4)
+∥x∥ + ∥y∥ + ∥z∥ + ∥x + y + z∥ ≥ ∥x + y∥ + ∥y + z∥ + ∥z + x∥ .
+Date: January 19, 2023.
+2000 Mathematics Subject Classification. Primary 26B25; Secondary 26B35, 26D15, 26A48,
+26A51.
+Key words and phrases. Hornich-Hlawka functional inequality, completely monotone functions,
+function with positive differences, higher order convexity, positive (semi)definite matrix.
+1
+arXiv:2301.08342v1  [math.FA]  19 Jan 2023
+
+2
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+Inequality (1.4) is what is nowadays known as the Hornich-Hlawka inequality. See
+the paper of Hornich [14], which includes the marvelous argument of Hlawka, based
+on the triangle inequality and an identity (due to Fr´echet [10]) which characterizes
+inner product spaces.
+Using a standard technique, one can easily infer from (1.3) that the Hornich-
+Hlawka inequality (1.4) also works for all Lebesgue spaces L1(µ), and so also for all
+spaces that can be embedded linearly and isometrically into an L1(µ). The latter
+comment includes all Lebesgue spaces Lp(µ) with p ∈ [1, 2])—see Lindenstrauss
+and Pe�lczy´nski [19].
+In 1946, Popoviciu [32] proved that every continuous function f : R+ → R that
+vanishes at the origin and admits a nondecreasing derivative of second order on
+(0, ∞), verifies the Hornich-Hlawka functional inequality (1.1). One can easily put
+Popoviciu’s result in full generality by showing that actually all continuous 3-convex
+functions on R+ taking values in an ordered Banach space verify inequality (1.2).
+This fact and its analogue in the case of continuous n-convex functions,
+f
+��n
+i=1 xi
+�
+−
+�
+1≤i1<···<in−1≤n f(xi1 + · · · + xin−1)
++
+�
+1≤i1<···<in−2≤n f(xi1 + · · · + xin−2) − · · · + (−1)n−1 �n
+i=1 f(xi) ≥ f(0),
+will be the subject of Section 3.
+Close to the above inequality is the characterization of the property of n-convexity
+via differences (∆hf) (x) = f(x + h) − f(x), rather than via divided differences as
+is usual.
+See Theorem 5, which expresses the identity of the class of continu-
+ous n-convex functions with the class of continuous functions having positive dif-
+ferences of order n in the sense that ∆x1∆x2 · · · ∆xnf(x) ≥ 0. The connection
+with Popoviciu’s inequality is evident when n is an odd integer because the condi-
+tion ∆x1∆x2 · · · ∆xnf(x) ≥ 0, simply means the introduction of a new variable in
+Popoviciu’s inequality as follows:
+�n
+i=1 f(xi + x) −
+�
+1≤i<j≤n f(xi + xj + x)
++
+�
+1≤i<j<k≤n f(xi + xj + xk + x) − · · ·
++(−1)n−1f(x1 + · · · + xn + x) ≥ f(x).
+It is worth noticing that the functions f : R+ → R+ that have positive differences
+of any order are precisely the absolutely monotonic functions in the terminology
+of Bernstein [4]. Leaving the elegant framework of analysis on intervals one easily
+discovers that n-convexity and the property of having positive differences of order
+n are different concepts. This idea is detailed at the end of Section 3.
+Two important classes of functions that mix a string of properties of n-convexity
+are those of completely monotone functions and of Bernstein functions. See Section
+2 for their definitions and some examples. Sendov and Zitikis [36] prove that these
+functions verify inequalities of the form
+(1.5)
+�n
+i=1 f(xi) −
+�
+1≤i<j≤n f(xi + xj) + · · · + (−1)n−1f
+��n
+i=1 xi
+�
+≥ 0,
+for all x1, . . . , xn in R+ and n ≥ 1. Their proof combines the classical integral repre-
+sentation theorems (respectively the Bernstein theorem and the L´evy–Khintchine
+representation theorem) with some probabilistic considerations. In Section 4 we
+
+3
+extend this result as a double inequality that holds for completely monotone func-
+tions defined on cones.
+Combining this result with [35, Theorem 1.3 (a)], we
+then show that the function f(X) = (det X)−ρ (defined on N × N-real symmet-
+ric positive definite matrices) also verifies the whole string of inequalities (1.5) if
+ρ ∈ {0, 1/2, 1, 3/2, ...} ∪ [(N − 1) /2, ∞).
+Section 5 considers functions defined on cones and having positive differences of a
+certain order n > 0. A surprising result is Theorem 9, which shows that the function
+det has positive differences of any order (though it is not completely monotonic).
+Probably the same happens for other immanants function (like the permanents),
+but we were able to prove only the positivity of differences of order 3. [TODO
+TODO].
+For the reader’s convenience, background on higher order convexity and the
+theory of ordered Banach spaces is summarized in Section 2.
+2. Preliminaries
+The study of higher order convexity was initiated by Hopf [13] and Popoviciu
+[29, 31], who defined it in terms of divided differences of a function. Assuming
+f a real-valued function defined on a real interval I, the divided differences of
+order 0, 1, . . . , n associated to a family x0, x1, . . . , xn of n + 1 distinct points are
+respectively defined by the formulas
+[x0; f] = f(x0)
+[x0, x1; f] = f(x1) − f(x0)
+x1 − x0
+...
+[x0, x1, ..., xn; f] = [x1, x2, ..., xn; f] − [x0, x1, ..., xn−1; f]
+xn − x0
+=
+�n
+j=0
+f(xj)
+�
+k̸=j (xj − xk).
+Notice that all these divided differences are invariant to permutations of the points
+x0, x1, ..., xn. As a consequence, we may always assume that x0 < x1 < · · · < xn.
+A function f is called n-convex (respectively n-concave) if all divided differ-
+ences [x0, x1, . . . , xn; f] are nonnegative (respectively nonpositive). In particular,
+0-convex functions are precisely the nonnegative functions, 1-convex functions the
+nondecreasing ones, while 2-convex functions are simply the usual convex functions.
+If f is n times differentiable, then a repeated application of Lagrange’s mean
+value theorem yields the existence of a point ξ ∈ (mink xk, maxk xk) such that
+[x0, x1, ..., xn; f] = f (n)(ξ)
+n!
+.
+As a consequence, one obtains the following practical criterion of n-convexity.
+Lemma 1. Every continuous function f defined on an interval I which is n times
+differentiable on the interior of I is n-convex provided that f (n) ≥ 0.
+A big source of convex functions of higher order is provided by the Bernstein func-
+tions and the completely monotone functions. Recall that a function f : (0, ∞) →
+R+ is a Bernstein function if it is infinitely differentiable and verifies the condition
+(−1)n+1f (n)(x) ≥ 0
+for all x > 0 and n ≥ 1;
+
+4
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+while, the function f is completely monotone if instead
+(−1)nf (n)(x) ≥ 0
+for all x > 0 and n ≥ 0.
+By definition, a function f : [0, ∞) → R+ is a Bernstein function (respectively
+a completely monotone function) if it is continuous and its restriction of to (0, ∞)
+has the respective property.
+Every Bernstein function is (2n + 1)-convex and every completely monotone
+function is 2n-convex for every n ≥ 0.
+If f : [0, ∞) → R+ is a Bernstein function then so is f −f(0); if f is a completely
+monotone function then f(0) − f is a Bernstein function. Some simple examples of
+Bernstein functions are
+x/(x + 1), 1 − e−αx (for α > 0),
+log(1 + x), (x − 1)/ log x and xα (for 0 < α ≤ 1).
+A nice account of the two aforementioned classes of functions is offered by the
+authoritative monograph of Schilling, Song and Vondraˇcek [34].
+Besides the five examples mentioned above some other examples
+of 3-convex
+functions on R+ are xα (for α ∈ (0, 1] ∪ [2, ∞)), −x2 + √x, −x log x, sinh, cosh,
+− log (Γ(x)) etc.
+The function 1 − (x − 3) + (x−3)3
+6
+is continuous and 3-convex on R+ but not
+n-convex for any n ∈ {0, 1, 2} .
+The polynomials with positive coefficients and the exponential are n-convex for
+every n ≥ 0.
+All polynomials of degree less than or equal to 2 are both 3-convex and 3-concave.
+The following approximation theorem due to Popoviciu [30] (see also [11, Theo-
+rem 1.3.1 (i), pg. 20]) allows us to reduce reasoning with n-convex functions to the
+case where they are also differentiable.
+Theorem 1 (Popoviciu’s approximation theorem). If a continuous function f :
+[0, 1] → R is k-convex, then so are the Bernstein polynomials associated to it,
+Bn(f)(x) =
+n
+�
+i=0
+�n
+i
+�
+xi(1 − x)n−if
+� i
+n
+�
+.
+Moreover, by the well-known property of simultaneous uniform approximation of
+a function and its derivatives by Bernstein polynomials and their derivatives, it
+follows that Bn(f) and any derivative (of any order) of it, converge uniformly to f
+and to its derivatives, correspondingly.
+Using a change of variable, one can easily see that the approximation theorem
+extends to functions defined on compact intervals [a, b] with a < b.
+Lemma 2. (i) The composition of two continuous functions that are increasing,
+concave or 3-convex is a function of the same nature.
+(ii) If f : R+ → R+ is a continuous 3-convex function which is also nondecreas-
+ing and concave, then the same properties hold for f α if α ∈ (0, 1].
+Proof. According to Theorem 1, we may reduce the proof to the case where the
+involved functions are also of class C3. In this case the proof can be completed by
+computing the sign of the derivatives of order 1, 2 and 3.
+□
+
+5
+The n-convex functions taking values in an ordered Banach space can be in-
+troduced in the same manner as real-valued n-convex functions by using divided
+differences. We recall useful definitions below.
+Recall that an ordered Banach space is any Banach space E endowed with the
+ordering ≤ associated to a closed convex cone E+ via the formula
+x ≤ y if and only if y − x ∈ E+,
+such that
+E = E+ − E+,
+(−E+) ∩ E+ = {0} ,
+and
+0 ≤ x ≤ y in E implies ∥x∥ ≤ ∥y∥ .
+The basic facts concerning the theory of ordered Banach spaces are made available
+by the book of Schaefer and Wolff [33]. See [25] for a short overview centered on
+two important particular cases: Rn, the n-dimensional Euclidean space endowed
+with the coordinate-wise ordering, and Sym(n, R) the ordered Banach space of all
+n × n symmetric matrices with real coefficients endowed with the operator norm
+∥A∥ = sup
+∥x∥≤1
+|⟨Ax, x⟩| ,
+and the L¨owner (partial) ordering,
+A ≤ B if and only if ⟨Ax, x⟩ ≤ ⟨Bx, x⟩ for all x ∈ Rn.
+Here the operator norm can be replaced by any Schatten norm, in particular
+with the Frobenius norm,
+∥A∥F =
+��N
+i=1
+�N
+j=1 a2
+ij
+�1/2.
+The Frobenius norm is associated to the trace inner product
+⟨A, B⟩ = trace(AB).
+The positive cone of Rn is the first orthant Rn
++, while the positive cone of
+Sym(n, R) is the set Sym+(n, R) consisting of all positive semi-definite matrices. We
+denote by Rn
+++ and Sym++(n, R) respectively the interior of Rn
++ and Sym+(n, R).
+Remark 1. Much of the study of vector-valued convex functions can be reduced to
+that of real-valued functions. Indeed, in any ordered Banach space E, any inequality
+of the form u ≤ v is equivalent to x∗(u) ≤ x∗(v) for all x∗ ∈ E∗
++.
+As a consequence, a function f : I → E is respectively nondecreasing, convex
+or n-convex if and only if x∗ ◦ f has this property whenever x∗ ∈ E∗ is a positive
+functional. For E = Rn, this inequality reduces to the components of f.
+Combining Remark 1 with Lemma 1 one obtains the following practical test of
+3-convexity for vector-valued differentiable functions:
+Theorem 2. Suppose that f is a continuous function defined on an interval I and
+taking values in an ordered Banach space E. If f is three times differentiable on
+the interior of I and f ′′′ ≥ 0, then f is a 3-convex function.
+An example illustrating Theorem 2 is provided by the function
+f : R+ → Sym(n, R),
+f(t) = −e−tA,
+
+6
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+associated to a positive semi-definite matrix A ∈ Sym(n, R). This function is of
+class C∞ and its first three derivatives are given by the formulas
+f ′(t) = Ae−tA,
+f ′′(t) = −A2e−tA,
+f ′′′(t) = A3e−tA.
+Thus f is nondecreasing, concave and 3-convex (according to the ordering of Sym(n, R)).
+The matrix A3e−tA is positive semidefinite since the product of commuting positive
+semi-definite matrices is also positive semidefinite.
+3. The functional inequality of Popoviciu
+Popoviciu [32] published in 1946 a short note on a functional inequality that we
+restate here in a slightly more general form.
+Theorem 3. Suppose that E is an ordered Banach space and f : [0, A] → E is a
+continuous n-convex function (n ≥ 1). Then
+f(
+�n
+i=1 xi) −
+�
+1≤i1<···<in−1≤n f(xi1 + · · · + xin−1)
++
+�
+1≤i1<···<in−2≤n f(xi1+· · ·+xin−2)−· · ·+(−1)n−1 �n
+i=1 f(xi) ≥ (−1)n−1f(0),
+for all x1, x2, ..., xn ≥ 0 with �n
+i=1 xi ≤ A.
+Notice that this inequality can be reformulated as
+�
+ε1,...,εn∈{0,1}
+(−1)n−(ε1+···+εn) f (ε1x1 + · · · + εnxn) ≥ 0.
+Popoviciu supplied the details only in the case n = 3, for functions f that vanish
+at the origin and admit a nondecreasing second order derivative.
+The proof of Theorem 3 is by induction, starting with the following instance of
+Hardy-Littlewood-P´olya’s majorization inequality (see [26, Theorem 4.1.3, pg. 186]):
+Lemma 3. If g : [a, b] → R is a continuous convex function and c and d are two
+points in [a, b] such that a + b = c + d, then
+g(c) + g(d) ≤ g(a) + g(b).
+Proof of Theorem 3. The case n = 1 is trivial since 1-convexity is equivalent to
+the fact that f is nondecreasing. For n = 2 the inequality under attention reads as
+f(x1) + f(x2) ≤ f(x1 + x2) + f(0),
+which follows from Lemma 3. Suppose thus that the statement of Theorem 3 holds
+for all continuous n-convex functions and all x1, x2, ..., xn ≥ 0 with �n
+i=1 xi ≤ A.
+Let f be a continuous (n + 1)-convex function. According to Remark 1 we may
+assume that f is real-valued, while Popoviciu’s approximation theorem (Theorem 1)
+allows us to restrict ourselves functions of class C1. Then f ′ is continuous and n-
+convex and the same is true for the function ϕ(x) = f ′(x1 + x). According to the
+induction hypothesis, if x1, ..., xn, xn+1 ≥ 0 and x1 + · · · + xn+1 ≤ A, we have
+f ′(
+�n+1
+i=1 xi) −
+�
+2≤i1<···<in−1≤n+1 f ′(x1 + xi1 + · · · + xin−1)
++
+�
+2≤i1<···<in−2≤n+1 f ′(x1 + xi1 + · · · + xin−2)
+− · · · + (−1)n−1 �
+2≤j≤n+1 f ′(x1 + xj) ≥ (−1)n−1f ′(x1).
+
+7
+Similar inequalities occur by permuting the variables.
+Consider x2, x3, . . . , xn+1 fixed in [0, A] and x1 ≥ 0 variable such that x1 + · · · +
+xn+1 ≤ A. The function F defined by the formula
+F (x1) = f(
+�n+1
+i=1 xi) −
+�
+1≤i1<···<in≤n+1
+f(xi1 + · · · + xin) + · · · +
++ (−1)n−1
+�
+1≤i<j≤n+1
+f(xi + xj) + (−1)n �n+1
+i=1 f(xi) + (−1)n+1 f (0)
+is differentiable and, according to the induction hypothesis,
+F ′ (x1) = f ′
+��n+1
+i=1 xi
+�
+−
+�
+2≤i1<···<in+1≤n+1
+f ′(x1 + xi1 + · · · + xin−1)+
++ (−1)n−1
+�
+2≤j≤n+1
+f ′(x1 + xj) + (−1)nf ′(x1) ≥ 0.
+Therefore F is a nondecreasing function, whence F (x1) ≥ F (0) = 0. In conclusion
+F is a nonnegative function and the proof is done.
+□
+It is worth noticing that Popoviciu’s inequality can be turned into a characteri-
+zation of n-convexity using difference operators.
+The difference operators ∆h (of step size h ≥ 0) can be introduced in a large
+category of situations including the case of functions defined on n-dimensional in-
+tervals or on convex cones, ordered abelian semigroups, etc. They associate to each
+such function f, the function ∆hf defined by
+(∆hf) (x) = f(x + h) − f(x),
+for all x and h such that the right-hand side formula makes sense.
+Clearly, difference operators are linear and commute with each other,
+∆h1∆h2 = ∆h2∆h1.
+They also verify the following property of invariance under translation:
+∆h (f ◦ Ta) = (∆hf) ◦ Ta,
+where Ta is the translation defined by the formula Ta(x) = x + a.
+Lemma 4. If n is a positive integer, then the following formula holds:
+∆h1∆h2 · · · ∆hnf(x) =
+�
+ε1,...,εn∈{0,1}
+(−1)n−(ε1+···+εn) f (x + ε1h1 + · · · + εnhn) .
+The proof is immediate, by mathematical induction.
+The property of convexity of a continuous function f defined on an interval I
+can be characterized via the difference operators as follows:
+Theorem 4. A continuous function f : I → R is convex if and only if the following
+inequality holds,
+(3.1)
+∆a∆bf(x) = f(a + b + x) − f(a + x) − f(b + x) + f(x) ≥ 0
+at all interior points x ∈ I and all a, b ≥ 0 for which x + a + b ∈ I.
+In other words, for continuous functions defined on intervals, convexity is equiv-
+alent to the property of having positive differences of second order.
+
+8
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+Proof. The fact that convexity implies the inequality ∆a∆bf ≥ 0 is a consequence
+of the Hardy-Littlewood-P´olya inequality of majorization. See Lemma 3. On the
+other hand, for u < v arbitrarily fixed in I, choosing a = b = (v − u) /2 and x = u,
+we infer from (3.1) that
+f(u) + f(v)
+2
+≥ f
+�u + v
+2
+�
+,
+which is equivalent to convexity since f was assumed to be continuous. See [26,
+Theorem 1.1.8, pg. 5].
+□
+The following result extends Theorem 4 to the case of higher-order convexity
+and originates from an old paper of Boas and Widder [7].
+Theorem 5. Suppose that f : [0, A] → R is a continuous function and n ≥ 1 is an
+integer. Then the following conditions are equivalent:
+(i) f is n-convex;
+(ii) f has positive differences of order n in the sense that
+∆x1∆x2 · · · ∆xnf(t) ≥ 0
+for all points t, x1, x2, ..., xn ≥ 0 such that t + x1 + · · · + xn ≤ A.
+The same works if the interval [0, A] is replaced by R+ and R.
+Proof. The implication (i) =⇒ (ii) follows from Theorem 3, when applied to the
+n-convex function g(x) = f(x + t) − f(t). An alternative proof is made available by
+the paper of Boas and Widder [7].
+The converse implication is immediate and is the objective of [16, Theorem
+15.3.1, pg. 430] in Kuczma’s book. A very short proof of the implication (ii) =⇒ (i)
+when n = 3 can be found in [2, Proposition 1].
+□
+Corollary 1. Suppose that E is an ordered Banach space and f : [0, A] → E is a
+continuous function. Then f is n-convex if and only if it verifies the inequality
+∆x1∆x2 · · · ∆xnf(x) =
+�
+ε1,...,εn∈{0,1}
+(−1)n−(ε1+···+εn) f (x + ε1x1 + · · · + εnxn) ≥ 0
+for all points x, x1, . . . , xn ∈ [0, A] such that x + x1 + · · · + xn ≤ A.
+In particular, a continuous function f : [0, A] → E is 3-convex if and only if it
+verifies the inequality
+f (x + t) + f (y + t) + f (z + t) + f (x + y + z + t)
+≥ f (x + y + t) + f (y + z + t) + f (z + x + t) + f(t)
+for all points x, y, z, t ∈ [0, A] such that x + y + z + t ≤ A.
+Both Theorem 3 and Theorem 5 can be applied successfully to derive a num-
+ber of useful inequalities satisfied by the Bernstein functions and the completely
+monotonic functions on [0, ∞). We will come back to this matter in the next section.
+In higher dimensions, the equivalence between usual convexity and the property
+of having positive differences of second order is no anymore valid. Some simple
+examples are indicated in what follows.
+
+9
+Example 1. Consider the case of the infinitely differentiable function
+f(x, y) = −2 (xy)1/2 ,
+x, y ∈ (0, ∞).
+This function is convex, its Hessian being the positive semidefinite matrix
+H = 1
+2
+�
+x−3/2y1/2
+−x−1/2y−1/2
+−x−1/2y−1/2
+x1/2y−3/2
+�
+.
+However, Φ fails the inequality
+∆A∆Bf(X) ≥ 0
+for A, B, X ∈ (0, ∞) × (0, ∞);
+for example, choose A = (1, 2), B = (2, 1) and X near the origin.
+Example 2. The function
+M : R2
++ → R,
+M (x, y) = min {x, y}
+is continuous and concave. Besides it has positive differences of second order as
+min {x + s + u, y + t + v} − min {x + s, y + t}
+− min {x + u, y + v} + min {x, y} ≥ 0,
+for all s, t, u, v > 0. The function M proves useful in statistics as the Fr´echet-
+Hoeffding upper bound for joint distribution functions of random variables.
+See
+Nelsen [24].
+Popoviciu [29, 31] introduced the concept of higher order convexity for functions
+of several variables using multiple divided differences. To gain some insight, let us
+consider the case of a function f = f(x, y) defined on a product I × J of intervals,
+and let x0, x1, . . . , xm be distinct points in I, and y0, y1, . . . , yn be distinct points
+in J. The divided double differences are defined via the formula
+�
+x0,
+x1,
+. . .
+, xm
+y0,
+y1,
+. . .
+, yn ; f
+�
+= [x0, x2, . . . , xm; [y0, y1, . . . , yn; f((x, ·)]]
+(3.2)
+= [y0, y1, . . . , yn; [x0, x1, . . . , xm; f((·, y)]].
+Notice that this formula is invariant under the permutation of variables xk (and
+also under the permutation of the variables yk).
+Drawing a parallel to the one dimensional case, Popoviciu [29, pg. 78] calls a
+function f : I × J → R convex of order (m, n) if the divided differences
+� x0,
+x1,
+. . .
+, xm
+y0,
+y1,
+. . .
+, yn ; f
+�
+are nonnegative for all distinct points x0, x1, ..., xm ∈ I and y0, y1, ..., yn ∈ J.
+Needless to say, the study of this concept of convexity implies a formidable
+formalism, so little progress was made since the times of Popoviciu. The only one
+recent contribution is [12] that studies the cases m = n = 1 and m = n = 2.
+4. The case of completely monotone functions on cones
+The theory of completely monotone functions can be easily extended to the
+context of several variables using convex analysis. In what follows V denotes a
+finite-dimensional real vector space and C an open convex cone in V with closure
+C. Its dual cone is C∗ = {y ∈ E∗ : ⟨y, x⟩ ≥ 0 for all x ∈ C}. The points in C∗ are
+linear functionals that are nonnegative on C.
+
+10
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+Definition 1. A function f : C → R+ is called completely monotone if f is C∞ on
+C and, for all integers k ≥ 1 and all vectors v1, ..., vk ∈ C, we have
+(4.1)
+(−1)k Dv1 · · · Dvkf(x) ≥ 0
+for all x ∈ C.
+Here Dv denotes the directional derivative along the vector v.
+A function f : C → R+ is called completely monotone if it is the continuous
+extension of a completely monotone function on C.
+When C = (0, ∞)n, the condition (4.1) means that
+(−1)k
+∂kf
+∂xi1∂xi2 · · · ∂xik
+(x) ≥ 0
+for all x ∈ (0, ∞)n and all sets of indices 1 ≤ i1 ≤ i2 ≤ · · · ≤ ik ≤ n of arbitrary
+length k.
+As in the case of completely monotone functions of one real variable, these func-
+tions can be obtained as Laplace transforms of Borel measures on the dual cone.
+Theorem 6. (Bernstein-Hausdorff-Widder-Choquet theorem). Let f be a nonnega-
+tive continuous function on the open convex cone C. Then f is completely monotone
+if and only if it is the Laplace transform of a unique Borel measure µ supported on
+the dual cone C∗, that is,
+f(x) =
+�
+C∗ e−⟨y,x⟩dµ(y)
+for all x ∈ C.
+When f admits a continuous extension to C, the last equality works for all x ∈ C.
+For details, see Choquet [8].
+Remark 2. Finding the positive Borel measure µ that makes the formula of The-
+orem 6 working represents a practical way for checking the complete monotonicity
+of f. So is the case of Riesz kernels: If α1, α2, ..., αN > 0, then
+x−α1
+1
+x−α2
+2
+· · · x−αN
+N
+=
+�
+RN
+++
+e−⟨y,x⟩ x−α1
+1
+x−α2
+2
+· · · x−αN
+N
+Γ(α1)Γ(α2) · · · Γ(αN)dy
+for all x ∈ RN
+++. See [15, Proposition 2.7]. A more subtle case is that of inverse
+powers of the determinant
+f (X) = (det X)−ρ ,
+X ∈ Sym++(N, R),
+for which Scott and Sokal [35] have shown that is completely monotone if and only
+if ρ ∈ {0, 1/2, 1, 3/2, ..., (N − 1) /2} ∪ ((N − 1) /2, ∞). See also [15, Theorem 4.1].
+It is worth noticing that Siegel established in 1929 the formula
+(det A)−ρ =
+�
+Sym++(N,R)
+e− trace AX
+(det X)ρ dX
+πn(n−1)/4Γ(ρ)Γ (ρ − 1/2) · · · Γ (ρ − (n − 1)/2))
+for all A ∈ Sym++(N, R) and ρ ≥ (N + 1) /2. See [38, Hilfssatz 37, pg. 585].
+We next extend (and improve) a result due to Sendov and Zitikis; see [36, The-
+orem 4.1, pg. 76].
+Theorem 7. Every completely monotone function f : C → R+ satisfies
+(4.2)
+�n
+i=1 f(xi) −
+�
+1≤i<j≤n f(xi + xj) + · · · + (−1)n−1f(
+�n
+i=1 xi) ≥ 0,
+
+11
+for every x1, . . . , xn ∈ C and n ≥ 1. When f admits a continuous extension to C,
+then then f satisfies the double inequality
+(4.3) f(0) ≥
+�n
+i=1 f(xi)−
+�
+1≤i<j≤n f(xi +xj)+· · ·+(−1)n−1f(
+�n
+i=1 xi) ≥ 0,
+for every x1, . . . , xn ∈ C and n ≥ 1.
+For n = 3, the first conclusion of Theorem 7 reads as
+�3
+i=1 f(xi) −
+�
+1≤i<j≤3 f(xi + xj) + f(
+�3
+i=1 xi) ≥ 0,
+which is nothing but a Hornich-Hlawka type inequality.
+The proof of Theorem 7 needs the following auxiliary result.
+Lemma 5. We have
+P =
+�n
+i=1 e−αi −
+�
+1≤i<j≤n e−(αi+αj) + · · · + (−1)n+1e− �n
+i=1 αi ≥ 0
+and
+Q =
+�n
+i=1
+�
+1 − e−αi�
+−
+�
+1≤i<j≤n
+�
+1 − e−(αi+αj)�
++ · · · + (−1)n+1 �
+1 − e− �n
+i=1 αi�
+≥ 0,
+whenever α1, . . . , αn ∈ R+ and n ≥ 2.
+Proof. Indeed,
+S = 1 −
+�n
+i=1
+�
+1 − e−αi�
+and Q =
+�n
+i=1(−1)k+1
+�n
+k
+�
+− S = 1 − S.
+□
+Proof of Theorem 7. As per to Theorem 6, f admits the integral representation
+f(x) =
+�
+C∗ e−⟨y,x⟩dµ(y)
+for all x ∈ C,
+where µ is a Borel measure on C∗. Then, taking into account to the first assertion
+of Lemma 5, we have the inequality
+0 ≤
+�
+C∗
+�n
+i=1 e−⟨xi,y⟩ −
+�
+1≤i<j≤n e−⟨xi+xj,y⟩
++ · · · + (−1)n−1e−⟨�n
+i=1 xi,y⟩�
+dµ(y)
+=
+�n
+i=1 f(xi) −
+�
+1≤i<j≤n f(xi + xj) + · · · + (−1)n−1f(
+�n
+i=1 xi).
+The case where f is defined on C can be settled in the same manner, using both
+assertions of Lemma 5.
+□
+Combining Theorem 7 with the aforementioned result of Scott and Sokal (see
+Remark 2) one obtains the following result:
+Corollary 2. If ρ ∈ {0, 1/2, 1, 3/2, ...} ∪ [(N − 1) /2, ∞), then
+�n
+i=1 det−ρ(Ai) −
+�
+1≤i<j≤n det−ρ(Ai + Aj)
++ · · · + (−1)n−1 det−ρ(
+�n
+i=1 Ai) ≥ 0,
+
+12
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+for every A1, . . . , An ∈ Sym++(N, R) and n ≥ 1.
+Given Corollary 2 one may wonder whether Theorem 7 also specializes to el-
+ementary symmetric polynomials. The situation here turns out to be more sub-
+tle, and a qualified answer follows from the discussion below. Recall that for any
+m = 0, 1, ..., N, the m-th elementary symmetric polynomial of x ∈ RN is defined
+by the formula
+Em,N(x) =
+�
+1≤i1<···<im≤N xi1...xim.
+Notice that det and the functions Em,N are hyperbolic polynomials in the sense
+of G˚arding.
+For details concerning this notion see [26, Section 4.6].
+Kozhasov,
+Michalek and Sturmfels provided a constructive proof for the fact that any elemen-
+tary symmetric polynomial Em,n admits a real exponent α′ > 0 such that E−α
+m,n is
+completely monotone on RN
+++ for all α ≥ α′—see [15, Theorem 6.4].
+Remark 3. The restriction of e−x to R+ is a completely monotone function and
+thus it verifies the inequality
+e−x1 + e−x2 + e−x3 + e−(x1+x2+x3) ≥ e−(x1+x2) + e−(x2+x3) + e−(x3+x1).
+However there x1, x2, x3 > 0 such that
+e−x1 + e−x2 + e−x3 + e−(x1+x2+x3) � e−(x1+x2) + e−(x2+x3) + e−(x3+x1) + e0.
+This shows that the inequality
+�3
+i=1 f(xi) −
+�
+1≤i<j≤3 f(xi + xj) + f(
+�3
+i=1 xi) ≥ f(0),
+does not characterize the 3-convexity within the class of continuous functions.
+The fact that the Bernstein functions f : R+ → R+ also verify the inequalities
+(4.2) follows from Lemma 5 and the L´evy-Khintchine theorem (Theorem 3.2, p. 15
+of [34]), which asserts that each such function admits the integral representation
+f(x) = a + bx +
+� ∞
+0
+(1 − e−xt)dµ(t)
+for some constants a, b ≥ 0 and a positive measure µ on [0, ∞) such that
+� ∞
+0
+min {1, t} dµ(t) < ∞.
+As far as we know, no attempt was made to extend the theory of Bernstein
+functions to the framework of functions defined on cones.
+5. Functions with positive differences on cones
+The difference operators ∆hf : x → ∆hf(x) = f(x + h) − f(x) are well defined
+in the case of functions f defined on a convex cone C. Such a function is said to be
+a function with positive differences of order n (n ≥ 1) if
+∆h1∆h2 · · · ∆hnf(x) ≥ 0 for all x, h1, h2, ..., hn ∈ C.
+For convenience, we say that f has positive differences of order 0 if f ≥ 0.
+In the literature, the concept of n-absolute monotonicity is used with the meaning
+that the function under attention has positive differences of order k for all k ∈
+{0, 1, ..., n}.
+
+13
+Theorem 8. If f : C → F+ is a completely monotonic function, then f has positive
+differences of any even order k = 0, 2, 4, ... ; under the same hypothesis, −f has
+positive differences of any odd order k = 1, 3, 5, ... .
+This follows easily from the characterization of monotonicity in terms of Gˆateaux
+differentiability (as was established by Amann [1], Proposition 3.2, p. 184):
+Lemma 6. Suppose that E and F are two ordered Banach spaces, C is a convex
+subset of E with nonempty interior int C and Φ : C → F is a function, continuous
+on C and Gˆateaux differentiable on int C. Then Φ is monotone nondecreasing on C
+if and only if
+DΦ(a)[v] = Φ(a + tv) − Φ(a)
+t
+≥ 0
+for all points a ∈ int C and all vectors v ∈ E+.
+Example 3. Every function of the form
+Φ(x) = f (⟨x, w⟩) ,
+x ∈ RN
++,
+associated to a continuous n-convex function f : R+→ R and a vector w ∈ RN
++ has
+positive differences of order n.
+The proof is straightforward, as evident from the case n = 3. Indeed, in this case,
+for every x, y, z, t ∈ Rn
++ we have
+Φ (x + t) + Φ (y + t) + Φ (z + t) + Φ (x + y + z + t)
+− Φ (x + y + t) − Φ (y + z + t) − Φ (z + x + t) − Φ(t)
+= f (⟨x + t, v⟩) + f (⟨y + t, v⟩) + f (⟨z + t, v⟩) + f (⟨x + y + z + t, v⟩)
+− f (⟨x + y + t, v⟩) − f (⟨y + z + t, v⟩) − f (⟨z + x + t, v⟩) − f(⟨t, v⟩)
+= f (⟨x, v⟩ + ⟨t, v⟩) + f (⟨y, v⟩ + ⟨t, v⟩) + f (⟨z, v⟩ + ⟨t, v⟩)
++ f (⟨x + y + z, v⟩ + ⟨t, v⟩) − f (⟨x + y, v⟩ + ⟨t, v⟩) − f (⟨y + z, v⟩ + t, v⟩)
+− f (⟨z + x, v⟩ + ⟨t, v⟩) − f(⟨t, v⟩) ≥ 0.
+A variant of this example is provided by the map
+Ψ(A) = f (trace(AW)) ,
+A ∈ Sym+(n, R),
+associated to a continuous n-convex function f : R+→ R and to an operator W ∈
+RN
++. The ambient Hilbert space in this case is Sym(n, R), endowed with the Frobenius
+norm and L¨owner ordering.
+Linear algebra offers many examples of functions that have positive differences
+of any order n ≥ 0. So is the case of the determinant function det, restricted to
+Sym+(N, R) (though det it is not completely monotonic).
+Clearly, det(X) ≥ 0 and since det is monotonic on the semidefinite matrices,
+(∆A det) (X) = det(X + A) − det(X) ≥ 0,
+for all A, X ∈ Sym+(N, R). Also simple is the fact that
+(∆A (∆B det)) (X) = det (A + B + X) − det (A + X) − det(B + X) + det(X) ≥ 0
+for all A, B, X ∈ Sym+(N, R). This inequality is mentioned in [42, Problem 36,
+pg. 215]. For X = 0 it reduces to the property of superaditivity of the function det,
+det(A + B) ≥ det(A) + det(B).
+
+14
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+Our next goal is to show the more challenging third-order inequality
+∆A (∆B(∆C det)) ≥ 0.
+We require some preparatory lemmas before stating our proof.
+Lemma 7. Let A, B, C ∈ Sym+(N, R), and let ek(X) denote the k-th elementary
+symmetric function of a matrix X (0 ≤ k ≤ N). Then,
+ek(A) + ek(B) + ek(C) + ek(A + B + C) ≥ ek(A + B) + ek(B + C) + ek(C + A).
+Proof. Recall that for an N × N matrix X, we have ek(X) = tr(∧kX), where ∧
+denotes the anti-symmetric tensor product; moreover, ∧k(X) = P ∗
+k (⊗kX)Pk for
+a suitable projection matrix Pk—see e.g., [5, pg. 18] for these facts. Using Pk in
+Theorem 2.1 in [3], the claimed inequality for ek follows.
+□
+Lemma 8. If A, B, C, X ∈ Sym+(N, R), then
+det (A + X) + det (B + X) + det (C + X) + det(A + B + C + X)
+≥ det (A + B + X) + det (B + C + X) + det (C + A + X) + det X.
+For X = 0 we get the determinantal Hornich-Hlawka inequality
+det A + det B + det C + det(A + B + C)
+≥ det (A + B) + det (B + C) + det (C + A) ,
+first noticed by Lin [18], who provided a proof based on eigenvalue majorization.
+Proof. Without loss of generality we may assume that X is invertible (for example,
+replace X by X + εI if necessary). Then
+det(A + X) = det(X) det(X−1/2AX−1/2 + I),
+where I is the identity matrix; the inequality under attention is then equivalent to
+det(A + I) + det(B + I) + det(C + I) + det(A + B + C + I)
+≥ det(A + B + I) + det(B + C + I) + det(C + A + I) + 1.
+Consider the function f(A) := det(A + I) − 1. Then the above inequality becomes
+(5.1) f(A) + f(B) + f(C) + f(A + B + C) ≥ f(A + B) + f(B + C) + f(C + A).
+Now recall the well-known expansion
+det(A + I) =
+N
+�
+k=0
+ek(A),
+where ek(·) denotes the k-th elementary symmetric polynomial (e0 = 1, e1 =
+tr ..., eN = det)—see [22, Theorem 7.1.2, pg. 197] Thus, f(A) = �N
+k=1 ek(A),
+and inequality (5.1) becomes
+N
+�
+k=1
+�
+ek(A) + ek(B) + ek(C) + ek(A + B + C)
+�
+≥
+N
+�
+k=1
+�
+ek(A + B) + ek(B + C) + ek(C + A)
+�
+.
+The proof ends by applying Lemma 7 for each k ∈ {1, ..., N}.
+□
+
+15
+A similar argument using [3, Corollary 3.4] yields that the function det has
+positive differences of any order.
+Theorem 9. We have
+∆A1 (∆A2(...(∆An det)))(X)
+=
+n
+�
+k=0
+(−1)k+1
+�
+1≤i1<i2<···<ik≤n
+det(Ai1 + · · · + Aik + X) ≥ 0,
+whenever A1, . . . , An, X ∈ Sym+(N, R) and n ≥ 1. In other words, the restriction
+of the det function to Sym+(N, R) has positive differences of any order n ≥ 0.
+Lemma 8 can be extended to a larger class of matrix functions, that of im-
+manants. The immanant function dG
+χ , associated to a subgroup G of the symmetric
+group SN of N letters and to an irreducible character χ of G, is defined via the
+formula
+dG
+χ (A) =
+�
+σ∈G χ(σ)
+�N
+i=1 ai σ(i), A ∈ Sym+(N, R).
+When G = Gm and χ(σ) = sgn σ we have dG
+χ (A) = det A, while for χ(σ) ≡ 1 we
+obtain the permanent of A.
+The following two inequalities
+dG
+χ (A + X) − dG
+χ (X) ≥ 0
+dG
+χ (A + B + X) − dG
+χ (A + X) − dG
+χ (B + X) + dG
+χ (X) ≥ 0
+occur for all A, B, X ∈ Sym+(N, R). See respectively Merris [21], p. 228 and Paksoy,
+Turkmen and Zhang [27]. The fact that dG
+χ has positive differences of third order
+on Sym+(N, R) makes the objective of the following result:
+Theorem 10. If A, B, C, X ∈ Sym+(N, R), then
+dG
+χ (A + X) + dG
+χ (B + X) + dG
+χ (C + X) + dG
+χ (A + B + C + X)
+≥ dG
+χ (A + B + X) + dG
+χ (B + C + X) + dG
+χ (C + A + X) + dG
+χ (X).
+Using arguments from multilinear algebra one can show that there exists a matrix
+ZG,χ such that
+(5.2)
+dG
+χ (X) = Z∗
+G,χ
+�
+⊗NX
+�
+ZG,χ.
+See [20], p. 126. The representation (5.2) turns the assertion of Theorem 10 into
+an immediate consequence of the following general result:
+Theorem 11. (Operator Hornich-Hlawka inequality). If A, B, C, X ∈ Sym+(N, R)
+and p ≥ 1 an integer, then
+⊗p(A + X) + ⊗p(B + X) + ⊗p(C + X) + ⊗p(A + B + C + X)
+≥ ⊗p(A + B + X) + ⊗p(B + C + X) + ⊗p(C + A + X) + ⊗pX,
+(5.3)
+in the L¨owner order. Here ⊗ denotes the usual tensor product.
+The proof of Theorem 11 follows the proof structure of [3, Theorem 2.1], but
+due to the additional X term in (5.3) it turns out to be more intricate and requires
+some preparation. We start by introducing the following convenient notation:
+(5.4)
+Xj ≡ ⊗jX = X⊗j,
+for j ≥ 1,
+and X0 = 1 ∈ N.
+
+16
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+The slight abuse of notation X0 = 1 will be helpful in simplifying the presentation.
+Lemma 9. Let k, l ≥ 0 be integers and X, Y, Z, V ∈ Sym+(N, R). Then,
+Xk ⊗ V ⊗ Xl + (X + Y + Z)k ⊗ V ⊗ (X + Y + Z)l
+≥ (X + Y )k ⊗ V ⊗ (X + Y )l + (X + Z)k ⊗ V ⊗ (X + Z)l,
+in the sense of L¨owner order.
+Proof. The proof is by induction on k and l; we provide the argument for k, which
+holds essentially unchanged for l.
+This approach suffices since for a fixed but
+arbitrary l ≥ 0 we prove that the result holds for all k ≥ 0; similarly, for an
+arbitrarily fixed k ≥ 0 the result holds for all l ≥ 0.
+For the base case of induction, suppose k = 0.
+Then the inequality under
+attention reduces to the following one:
+(5.5)
+V ⊗ Xl + V ⊗ (X + Y + Z)l ≥ V ⊗ (X + Y )l + V ⊗ (X + Z)l.
+We know from [3, Theorem 2.1] that
+(5.6)
+Xl + (X + Y + Z)l ≥ (X + Y )l + (X + Z)l.
+Since tensor product preserves inequalities, taking tensor product with V on both
+sides of the inequality (5.6) one immediately get (5.5).
+Assume therefore that
+inequality in the the statement of Lemma 9 holds for a fixed l and some k > 0.
+Then, consider
+(X + Y + Z)k+1 ⊗ V ⊗ (X + Y + Z)l
+= (X + Y + Z) ⊗
+�
+(X + Y + Z)k ⊗ V ⊗ (X + Y + Z)l�
+≥ (X + Y + Z) ⊗
+�
+(X + Y )k ⊗ V ⊗ (X + Y )l
++ (X + Z)k ⊗ V ⊗ (X + Z)l − Xk ⊗ V ⊗ Xl�
+,
+= (X +Y )k+1 ⊗V ⊗(X +Y )l +(X +Z)k+1 ⊗V ⊗(X +Z)l −Xk+1 ⊗V ⊗Xl +T ,
+where the inequality follows from the induction hypothesis and the elementary
+monotonicity properties of the tensor product. It remains to show that the term
+T = Z⊗(X+Y )k⊗V ⊗(X+Y )l+Y ⊗(X+Z)k⊗V ⊗(X+Z)l−(Y +Z)⊗Xk⊗V ⊗Xl
+is a nonnegative operator.
+Since X, Y, Z ≥ 0, it follows that X + Y ≥ X and
+X + Z ≥ X. Thus, the positive terms in T attached to Y and Z are clearly bigger
+than the respective negative terms, whence T ≥ 0. Inductively, we can conclude
+that for fixed l, the inequality in the statement of Lemma 9 holds for all k ≥ 0.
+Applying a similar argument for l, we conclude that this inequality works in full
+generality.
+□
+We are now in a position to detail the proof of Theorem 11.
+Proof of Theorem 11. Using the auxiliary function
+fp(Z) = (Z + X)p − Xp,
+the inequality of interest (5.3) can be rewritten as
+fp(A) + fp(B) + fp(C) + fp(A + B + C) ≥ fp(A + B) + fp(B + C) + fp(C + A).
+
+17
+Now introduce the function
+gp(Z) = fp(Z)+fp(B)+fp(C)+fp(Z +B+C)−fp(Z +B)−fp(Z +C)−fp(B+C).
+We will show that gp is monotonic (as a map from Sym+(N, R) into itself, under
+the L¨owner order). Once this monotonicity is established we can conclude that
+gp(A) ≥ gp(0) = 0
+for every A ∈ Sym+(N, R),
+a fact equivalent to the assertion of Theorem 11.
+Monotonicity of gp follows from Lemma 6 by considering its derivative. To that
+end, consider the (directional) derivative of the map Φ(Z) = Zp:
+D(Zp)[V ] = V ⊗ Z ⊗ · · · ⊗ Z + Z ⊗ V · · · ⊗ Z + · · · + Z ⊗ · · · ⊗ Z ⊗ V
+=
+p−1
+�
+j=0
+Zj ⊗ V ⊗ Zp−1−j,
+whenever Z ∈ Sym++(N, R) and V ∈ Sym+(N, R). See [6, Eq. (2.13), pg. 44].
+Indeed, applying this formula to fp(Z) = (Z + X)p − Xp we obtain
+Dfp(Z)[V ] =
+p−1
+�
+j=0
+(Z + V )j ⊗ V ⊗ (Z + V )p−1−j,
+which in turn leads to the identity
+Dgp(Z)[V ]
+= Dfp(Z)[V ] + Dfp(Z + B + C)[V ] − Dfp(Z + B)[V ] − Dfp(Z + C)[V ]
+=
+p−1
+�
+j=0
+�
+(Z + V )j ⊗ V ⊗ (Z + V )p−1−j +(Z+B+C+V )j⊗V ⊗(Z+B+C+V )p−1−j
+−(Z+B+V )j⊗V ⊗(Z+B+V )p−1−j −(Z + C + V )j ⊗ V ⊗ (Z + C + V )p−1−j�
+.
+But this sum evaluates to a positive quantity, which follows from Lemma 9 upon
+setting k ← j, l ← p − 1 − j and x ← x + j. Now the proof is complete.
+□
+6. Further comments and extensions
+6.1. Multivariable case for positive operators. Given A1, ..., An, X ∈ Sym+(N, R)
+and p ∈ N, let us consider the matrices
+(6.1)
+Sp
+0 = ⊗pX,
+Sp
+k =
+�
+1≤i1<···<ik≤n
+⊗p(Ai1 + · · · + Aik + X) (1 ≤ k ≤ N).
+Then one can prove the following generalization of the operator Hornich-Hlawka
+inequality (Theorem 11):
+(6.2)
+Sp
+n + Sp
+n−2 + · · · ≥ Sp
+n−1 + Sp
+n−3 + · · · ,
+Theorem 3.3 of [3] proves inequality (6.2) for the special case X = 0. One can prove
+the general case by a suitable monotonicity argument as in Section 5. The details
+are tedious so we omit them, leaving them as an exercise for the interested reader.
+
+18
+CONSTANTIN P. NICULESCU AND SUVRIT SRA
+6.2. Two inequalities for determinants. Serre [37] noticed that det1/2 viewed
+as a function on Sym+(2, R) verifies the opposite of the Hornich-Hlawka inequality,
+(6.3)
+det1/2 A + det1/2 B + det1/2 C + det1/2(A + B + C)
+≤ det1/2 (A + B) + det1/2 (B + C) + det1/2 (C + A) .
+While in [40], the second named author proved the following Minkowski-like in-
+equality for det
+1
+n , by viewing it as a function on Sym+(n, R):
+(6.4)
+det
+1
+n (A + B) det
+1
+n (A + C) ≥ det
+1
+n B det
+1
+n C + det
+1
+n A det
+1
+n (A + B + C).
+Inequality (6.4) is stronger the a usual log-supermodularity inequality for det, given
+the extra det
+1
+n B det
+1
+n C term on its right hand side.
+6.3. Vasic-Adamovic inequalities. We close the paper by briefly mentioning the
+related class of Vasic-Adamovic inequalities. Inspired by the work of D. M. Smiley
+and M. F. Smiley [39] on polygonal inequalities, P. M. Vasi´c and D. D. Adamovi´c [41]
+found an inductive scheme for generating inequalities for any function that verifies
+the functional Hornich-Hlawka inequality. We state here a slightly modified version
+of their result [23, Theorem 2, pg. 528]:
+Theorem 12. Let S be a commutative additive semigroup with 0 and G be an
+ordered abelian group (i.e., an abelian group with an order relation ≤) such that
+x, y, z ∈ G and x ≤ y implies x + z ≤ y + z.
+If ϕ : S → G is a function such that for all x1, x2, x3 ∈ S, we have
+�
+1≤i<j≤3 ϕ
+�
+xi + xj
+�
+≤
+�3
+k=1 ϕ (xk) + ϕ
+��3
+k=1xk
+�
+then for each pair {k, n} of integers with 2 ≤ k < n we also have
+�
+1≤i1<...<ik≤n
+ϕ
+� k
+�
+j=1
+xij
+�
+≤
+�n − 2
+k − 1
+�
+n
+�
+k=1
+ϕ (xk) +
+�n − 2
+k − 2
+�
+ϕ
+� n
+�
+k=1
+xk
+�
+,
+whenever x1, ..., xn ∈ S.
+This claim applies to every function f (defined on a convex cone C and taking
+values in an ordered Banach space E) that has positive differences of the third
+order. Indeed, in their case,
+f (x + t) + f (y + t) + f (z + t) + f (x + y + z + t)
+≥ f (x + y + t) + f (y + z + t) + f (z + x + t) + f(t)
+for all points x, y, z, t ∈ C and performing the change of function ϕ(v) = f(v + t) −
+f(t) we obtain a function that verifies the Hornich-Hlawka inequality
+ϕ (x) + ϕ (y) + ϕ (z) + ϕ (x + y + z) ≥ ϕ (x + y) + ϕ (y + z) + ϕ (z + x) .
+
+19
+References
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+CONSTANTIN P. NICULESCU AND SUVRIT SRA
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+Department of Mathematics, University of Craiova, Craiova 200585, Romania
+Email address: constantin.p.niculescu@gmail.com
+Massachusetts Institute of Technology, Cambridge, MA 02139, USA
+Email address: suvrit@mit.edu
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf,len=853
+page_content='THE HORNICH-HLAWKA FUNCTIONAL INEQUALITY FOR  FUNCTIONS WITH POSITIVE DIFFERENCES  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  Version 2  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We analyze the role played by n-convexity for the fulfillment of  a series of linear functional inequalities that extend the Hornich-Hlawka func-  tional inequality, f (x) + f (y) + f (z) + f (x + y + z) ≥ f (x + y) + f (y + z) +  f (z + x) + f(0), including extensions to the case of positive operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Introduction  Many noteworthy inequalities are related to the following problem:  Problem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Suppose that S is an abelian additive semigroup with neutral element  0, f a function defined on S and taking values in an ordered vector space E (or in  its positive cone E+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' For n ≥ 2, find the linear inequalities relating  �n  i=1 f(xi),  �  1≤i<j≤n f(xi + xj), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , f(  �n  i=1 xi)  for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Due to its many ramifications, this problem is still the subject of intense activity,  and the present paper reports some new results in this direction, under the um-  brella of the Hornich-Hlawka functional inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Specifically, we are interested  in studying conditions under which a continuous function f : S → R+ satisfies  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1)  f (x) + f (y) + f (z) + f (x + y + z) ≥ f (x + y) + f (y + z) + f (z + x)  for all x, y, z ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' When f is a real-valued function it is usual to replace (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1) by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2) f (x)+f (y)+f (z)+f (x + y + z) ≥ f (x + y)+f (y + z)+f (z + x)+f(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  A good start for understanding the Hornich-Hlawka functional inequality is pro-  vided by the following elementary (but powerful) inequality:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3) |x| + |y| + |z| + |x + y + z| ≥ |x + y| + |y + z| + |z + x|  for all x, y, z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  As was noticed by Levi [17], every piecewise linear inequality like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3) remains true  when the real variables x, y, z are replaced by arbitrary vectors x, y, z in RN and  the absolute value function is replaced by the Euclidean norm,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4)  ∥x∥ + ∥y∥ + ∥z∥ + ∥x + y + z∥ ≥ ∥x + y∥ + ∥y + z∥ + ∥z + x∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Date: January 19, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Primary 26B25;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Secondary 26B35, 26D15, 26A48,  26A51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Hornich-Hlawka functional inequality, completely monotone functions,  function with positive differences, higher order convexity, positive (semi)definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='08342v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='FA]  19 Jan 2023  2  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  Inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4) is what is nowadays known as the Hornich-Hlawka inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See  the paper of Hornich [14], which includes the marvelous argument of Hlawka, based  on the triangle inequality and an identity (due to Fr´echet [10]) which characterizes  inner product spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Using a standard technique, one can easily infer from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3) that the Hornich-  Hlawka inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4) also works for all Lebesgue spaces L1(µ), and so also for all  spaces that can be embedded linearly and isometrically into an L1(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The latter  comment includes all Lebesgue spaces Lp(µ) with p ∈ [1, 2])—see Lindenstrauss  and Pe�lczy´nski [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  In 1946, Popoviciu [32] proved that every continuous function f : R+ → R that  vanishes at the origin and admits a nondecreasing derivative of second order on  (0, ∞), verifies the Hornich-Hlawka functional inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' One can easily put  Popoviciu’s result in full generality by showing that actually all continuous 3-convex  functions on R+ taking values in an ordered Banach space verify inequality (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  This fact and its analogue in the case of continuous n-convex functions,  f  ��n  i=1 xi  �  −  �  1≤i1<···<in−1≤n f(xi1 + · · · + xin−1)  +  �  1≤i1<···<in−2≤n f(xi1 + · · · + xin−2) − · · · + (−1)n−1 �n  i=1 f(xi) ≥ f(0),  will be the subject of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Close to the above inequality is the characterization of the property of n-convexity  via differences (∆hf) (x) = f(x + h) − f(x), rather than via divided differences as  is usual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  See Theorem 5, which expresses the identity of the class of continu-  ous n-convex functions with the class of continuous functions having positive dif-  ferences of order n in the sense that ∆x1∆x2 · · · ∆xnf(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The connection  with Popoviciu’s inequality is evident when n is an odd integer because the condi-  tion ∆x1∆x2 · · · ∆xnf(x) ≥ 0, simply means the introduction of a new variable in  Popoviciu’s inequality as follows:  �n  i=1 f(xi + x) −  �  1≤i<j≤n f(xi + xj + x)  +  �  1≤i<j<k≤n f(xi + xj + xk + x) − · · ·  +(−1)n−1f(x1 + · · · + xn + x) ≥ f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  It is worth noticing that the functions f : R+ → R+ that have positive differences  of any order are precisely the absolutely monotonic functions in the terminology  of Bernstein [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Leaving the elegant framework of analysis on intervals one easily  discovers that n-convexity and the property of having positive differences of order  n are different concepts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' This idea is detailed at the end of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Two important classes of functions that mix a string of properties of n-convexity  are those of completely monotone functions and of Bernstein functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See Section  2 for their definitions and some examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Sendov and Zitikis [36] prove that these  functions verify inequalities of the form  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='5)  �n  i=1 f(xi) −  �  1≤i<j≤n f(xi + xj) + · · · + (−1)n−1f  ��n  i=1 xi  �  ≥ 0,  for all x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn in R+ and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Their proof combines the classical integral repre-  sentation theorems (respectively the Bernstein theorem and the L´evy–Khintchine  representation theorem) with some probabilistic considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' In Section 4 we  3  extend this result as a double inequality that holds for completely monotone func-  tions defined on cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Combining this result with [35, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3 (a)], we  then show that the function f(X) = (det X)−ρ (defined on N × N-real symmet-  ric positive definite matrices) also verifies the whole string of inequalities (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='5) if  ρ ∈ {0, 1/2, 1, 3/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='} ∪ [(N − 1) /2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Section 5 considers functions defined on cones and having positive differences of a  certain order n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' A surprising result is Theorem 9, which shows that the function  det has positive differences of any order (though it is not completely monotonic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Probably the same happens for other immanants function (like the permanents),  but we were able to prove only the positivity of differences of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' [TODO  TODO].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  For the reader’s convenience, background on higher order convexity and the  theory of ordered Banach spaces is summarized in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Preliminaries  The study of higher order convexity was initiated by Hopf [13] and Popoviciu  [29, 31], who defined it in terms of divided differences of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Assuming  f a real-valued function defined on a real interval I, the divided differences of  order 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , n associated to a family x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn of n + 1 distinct points are  respectively defined by the formulas  [x0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f] = f(x0)  [x0, x1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f] = f(x1) − f(x0)  x1 − x0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f] = [x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f] − [x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f]  xn − x0  =  �n  j=0  f(xj)  �  k̸=j (xj − xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Notice that all these divided differences are invariant to permutations of the points  x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' As a consequence, we may always assume that x0 < x1 < · · · < xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  A function f is called n-convex (respectively n-concave) if all divided differ-  ences [x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f] are nonnegative (respectively nonpositive).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' In particular,  0-convex functions are precisely the nonnegative functions, 1-convex functions the  nondecreasing ones, while 2-convex functions are simply the usual convex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  If f is n times differentiable, then a repeated application of Lagrange’s mean  value theorem yields the existence of a point ξ ∈ (mink xk, maxk xk) such that  [x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f] = f (n)(ξ)  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  As a consequence, one obtains the following practical criterion of n-convexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Every continuous function f defined on an interval I which is n times  differentiable on the interior of I is n-convex provided that f (n) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  A big source of convex functions of higher order is provided by the Bernstein func-  tions and the completely monotone functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Recall that a function f : (0, ∞) →  R+ is a Bernstein function if it is infinitely differentiable and verifies the condition  (−1)n+1f (n)(x) ≥ 0  for all x > 0 and n ≥ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  4  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  while, the function f is completely monotone if instead  (−1)nf (n)(x) ≥ 0  for all x > 0 and n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  By definition, a function f : [0, ∞) → R+ is a Bernstein function (respectively  a completely monotone function) if it is continuous and its restriction of to (0, ∞)  has the respective property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Every Bernstein function is (2n + 1)-convex and every completely monotone  function is 2n-convex for every n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  If f : [0, ∞) → R+ is a Bernstein function then so is f −f(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' if f is a completely  monotone function then f(0) − f is a Bernstein function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Some simple examples of  Bernstein functions are  x/(x + 1), 1 − e−αx (for α > 0),  log(1 + x), (x − 1)/ log x and xα (for 0 < α ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  A nice account of the two aforementioned classes of functions is offered by the  authoritative monograph of Schilling, Song and Vondraˇcek [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Besides the five examples mentioned above some other examples  of 3-convex  functions on R+ are xα (for α ∈ (0, 1] ∪ [2, ∞)), −x2 + √x, −x log x, sinh, cosh,  − log (Γ(x)) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The function 1 − (x − 3) + (x−3)3  6  is continuous and 3-convex on R+ but not  n-convex for any n ∈ {0, 1, 2} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The polynomials with positive coefficients and the exponential are n-convex for  every n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  All polynomials of degree less than or equal to 2 are both 3-convex and 3-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The following approximation theorem due to Popoviciu [30] (see also [11, Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1 (i), pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 20]) allows us to reduce reasoning with n-convex functions to the  case where they are also differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Theorem 1 (Popoviciu’s approximation theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If a continuous function f :  [0, 1] → R is k-convex, then so are the Bernstein polynomials associated to it,  Bn(f)(x) =  n  �  i=0  �n  i  �  xi(1 − x)n−if  � i  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Moreover, by the well-known property of simultaneous uniform approximation of  a function and its derivatives by Bernstein polynomials and their derivatives, it  follows that Bn(f) and any derivative (of any order) of it, converge uniformly to f  and to its derivatives, correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Using a change of variable, one can easily see that the approximation theorem  extends to functions defined on compact intervals [a, b] with a < b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' (i) The composition of two continuous functions that are increasing,  concave or 3-convex is a function of the same nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  (ii) If f : R+ → R+ is a continuous 3-convex function which is also nondecreas-  ing and concave, then the same properties hold for f α if α ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' According to Theorem 1, we may reduce the proof to the case where the  involved functions are also of class C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' In this case the proof can be completed by  computing the sign of the derivatives of order 1, 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  5  The n-convex functions taking values in an ordered Banach space can be in-  troduced in the same manner as real-valued n-convex functions by using divided  differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We recall useful definitions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Recall that an ordered Banach space is any Banach space E endowed with the  ordering ≤ associated to a closed convex cone E+ via the formula  x ≤ y if and only if y − x ∈ E+,  such that  E = E+ − E+,  (−E+) ∩ E+ = {0} ,  and  0 ≤ x ≤ y in E implies ∥x∥ ≤ ∥y∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The basic facts concerning the theory of ordered Banach spaces are made available  by the book of Schaefer and Wolff [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See [25] for a short overview centered on  two important particular cases: Rn, the n-dimensional Euclidean space endowed  with the coordinate-wise ordering, and Sym(n, R) the ordered Banach space of all  n × n symmetric matrices with real coefficients endowed with the operator norm  ∥A∥ = sup  ∥x∥≤1  |⟨Ax, x⟩| ,  and the L¨owner (partial) ordering,  A ≤ B if and only if ⟨Ax, x⟩ ≤ ⟨Bx, x⟩ for all x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Here the operator norm can be replaced by any Schatten norm, in particular  with the Frobenius norm,  ∥A∥F =  ��N  i=1  �N  j=1 a2  ij  �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The Frobenius norm is associated to the trace inner product  ⟨A, B⟩ = trace(AB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The positive cone of Rn is the first orthant Rn  +, while the positive cone of  Sym(n, R) is the set Sym+(n, R) consisting of all positive semi-definite matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We  denote by Rn  ++ and Sym++(n, R) respectively the interior of Rn  + and Sym+(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Much of the study of vector-valued convex functions can be reduced to  that of real-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Indeed, in any ordered Banach space E, any inequality  of the form u ≤ v is equivalent to x∗(u) ≤ x∗(v) for all x∗ ∈ E∗  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  As a consequence, a function f : I → E is respectively nondecreasing, convex  or n-convex if and only if x∗ ◦ f has this property whenever x∗ ∈ E∗ is a positive  functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' For E = Rn, this inequality reduces to the components of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Combining Remark 1 with Lemma 1 one obtains the following practical test of  3-convexity for vector-valued differentiable functions:  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Suppose that f is a continuous function defined on an interval I and  taking values in an ordered Banach space E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If f is three times differentiable on  the interior of I and f ′′′ ≥ 0, then f is a 3-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  An example illustrating Theorem 2 is provided by the function  f : R+ → Sym(n, R),  f(t) = −e−tA,  6  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  associated to a positive semi-definite matrix A ∈ Sym(n, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' This function is of  class C∞ and its first three derivatives are given by the formulas  f ′(t) = Ae−tA,  f ′′(t) = −A2e−tA,  f ′′′(t) = A3e−tA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Thus f is nondecreasing, concave and 3-convex (according to the ordering of Sym(n, R)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The matrix A3e−tA is positive semidefinite since the product of commuting positive  semi-definite matrices is also positive semidefinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The functional inequality of Popoviciu  Popoviciu [32] published in 1946 a short note on a functional inequality that we  restate here in a slightly more general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Suppose that E is an ordered Banach space and f : [0, A] → E is a  continuous n-convex function (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then  f(  �n  i=1 xi) −  �  1≤i1<···<in−1≤n f(xi1 + · · · + xin−1)  +  �  1≤i1<···<in−2≤n f(xi1+· · ·+xin−2)−· · ·+(−1)n−1 �n  i=1 f(xi) ≥ (−1)n−1f(0),  for all x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn ≥ 0 with �n  i=1 xi ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Notice that this inequality can be reformulated as  �  ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=',εn∈{0,1}  (−1)n−(ε1+···+εn) f (ε1x1 + · · · + εnxn) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Popoviciu supplied the details only in the case n = 3, for functions f that vanish  at the origin and admit a nondecreasing second order derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The proof of Theorem 3 is by induction, starting with the following instance of  Hardy-Littlewood-P´olya’s majorization inequality (see [26, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 186]):  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If g : [a, b] → R is a continuous convex function and c and d are two  points in [a, b] such that a + b = c + d, then  g(c) + g(d) ≤ g(a) + g(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The case n = 1 is trivial since 1-convexity is equivalent to  the fact that f is nondecreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' For n = 2 the inequality under attention reads as  f(x1) + f(x2) ≤ f(x1 + x2) + f(0),  which follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Suppose thus that the statement of Theorem 3 holds  for all continuous n-convex functions and all x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn ≥ 0 with �n  i=1 xi ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Let f be a continuous (n + 1)-convex function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' According to Remark 1 we may  assume that f is real-valued, while Popoviciu’s approximation theorem (Theorem 1)  allows us to restrict ourselves functions of class C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then f ′ is continuous and n-  convex and the same is true for the function ϕ(x) = f ′(x1 + x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' According to the  induction hypothesis, if x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn, xn+1 ≥ 0 and x1 + · · · + xn+1 ≤ A, we have  f ′(  �n+1  i=1 xi) −  �  2≤i1<···<in−1≤n+1 f ′(x1 + xi1 + · · · + xin−1)  +  �  2≤i1<···<in−2≤n+1 f ′(x1 + xi1 + · · · + xin−2)  − · · · + (−1)n−1 �  2≤j≤n+1 f ′(x1 + xj) ≥ (−1)n−1f ′(x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  7  Similar inequalities occur by permuting the variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Consider x2, x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn+1 fixed in [0, A] and x1 ≥ 0 variable such that x1 + · · · +  xn+1 ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The function F defined by the formula  F (x1) = f(  �n+1  i=1 xi) −  �  1≤i1<···<in≤n+1  f(xi1 + · · · + xin) + · · · +  + (−1)n−1  �  1≤i<j≤n+1  f(xi + xj) + (−1)n �n+1  i=1 f(xi) + (−1)n+1 f (0)  is differentiable and, according to the induction hypothesis,  F ′ (x1) = f ′  ��n+1  i=1 xi  �  −  �  2≤i1<···<in+1≤n+1  f ′(x1 + xi1 + · · · + xin−1)+  + (−1)n−1  �  2≤j≤n+1  f ′(x1 + xj) + (−1)nf ′(x1) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Therefore F is a nondecreasing function, whence F (x1) ≥ F (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' In conclusion  F is a nonnegative function and the proof is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  It is worth noticing that Popoviciu’s inequality can be turned into a characteri-  zation of n-convexity using difference operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The difference operators ∆h (of step size h ≥ 0) can be introduced in a large  category of situations including the case of functions defined on n-dimensional in-  tervals or on convex cones, ordered abelian semigroups, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' They associate to each  such function f, the function ∆hf defined by  (∆hf) (x) = f(x + h) − f(x),  for all x and h such that the right-hand side formula makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Clearly, difference operators are linear and commute with each other,  ∆h1∆h2 = ∆h2∆h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  They also verify the following property of invariance under translation:  ∆h (f ◦ Ta) = (∆hf) ◦ Ta,  where Ta is the translation defined by the formula Ta(x) = x + a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If n is a positive integer, then the following formula holds:  ∆h1∆h2 · · · ∆hnf(x) =  �  ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=',εn∈{0,1}  (−1)n−(ε1+···+εn) f (x + ε1h1 + · · · + εnhn) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The proof is immediate, by mathematical induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The property of convexity of a continuous function f defined on an interval I  can be characterized via the difference operators as follows:  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' A continuous function f : I → R is convex if and only if the following  inequality holds,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1)  ∆a∆bf(x) = f(a + b + x) − f(a + x) − f(b + x) + f(x) ≥ 0  at all interior points x ∈ I and all a, b ≥ 0 for which x + a + b ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  In other words, for continuous functions defined on intervals, convexity is equiv-  alent to the property of having positive differences of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  8  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The fact that convexity implies the inequality ∆a∆bf ≥ 0 is a consequence  of the Hardy-Littlewood-P´olya inequality of majorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' On the  other hand, for u < v arbitrarily fixed in I, choosing a = b = (v − u) /2 and x = u,  we infer from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1) that  f(u) + f(v)  2  ≥ f  �u + v  2  �  ,  which is equivalent to convexity since f was assumed to be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See [26,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='8, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  The following result extends Theorem 4 to the case of higher-order convexity  and originates from an old paper of Boas and Widder [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Suppose that f : [0, A] → R is a continuous function and n ≥ 1 is an  integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then the following conditions are equivalent:  (i) f is n-convex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  (ii) f has positive differences of order n in the sense that  ∆x1∆x2 · · · ∆xnf(t) ≥ 0  for all points t, x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn ≥ 0 such that t + x1 + · · · + xn ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The same works if the interval [0, A] is replaced by R+ and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The implication (i) =⇒ (ii) follows from Theorem 3, when applied to the  n-convex function g(x) = f(x + t) − f(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' An alternative proof is made available by  the paper of Boas and Widder [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The converse implication is immediate and is the objective of [16, Theorem  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 430] in Kuczma’s book.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' A very short proof of the implication (ii) =⇒ (i)  when n = 3 can be found in [2, Proposition 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Suppose that E is an ordered Banach space and f : [0, A] → E is a  continuous function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then f is n-convex if and only if it verifies the inequality  ∆x1∆x2 · · · ∆xnf(x) =  �  ε1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=',εn∈{0,1}  (−1)n−(ε1+···+εn) f (x + ε1x1 + · · · + εnxn) ≥ 0  for all points x, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn ∈ [0, A] such that x + x1 + · · · + xn ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  In particular, a continuous function f : [0, A] → E is 3-convex if and only if it  verifies the inequality  f (x + t) + f (y + t) + f (z + t) + f (x + y + z + t)  ≥ f (x + y + t) + f (y + z + t) + f (z + x + t) + f(t)  for all points x, y, z, t ∈ [0, A] such that x + y + z + t ≤ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Both Theorem 3 and Theorem 5 can be applied successfully to derive a num-  ber of useful inequalities satisfied by the Bernstein functions and the completely  monotonic functions on [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We will come back to this matter in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  In higher dimensions, the equivalence between usual convexity and the property  of having positive differences of second order is no anymore valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Some simple  examples are indicated in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  9  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Consider the case of the infinitely differentiable function  f(x, y) = −2 (xy)1/2 ,  x, y ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  This function is convex, its Hessian being the positive semidefinite matrix  H = 1  2  �  x−3/2y1/2  −x−1/2y−1/2  −x−1/2y−1/2  x1/2y−3/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  However, Φ fails the inequality  ∆A∆Bf(X) ≥ 0  for A, B, X ∈ (0, ∞) × (0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  for example, choose A = (1, 2), B = (2, 1) and X near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The function  M : R2  + → R,  M (x, y) = min {x, y}  is continuous and concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Besides it has positive differences of second order as  min {x + s + u, y + t + v} − min {x + s, y + t}  − min {x + u, y + v} + min {x, y} ≥ 0,  for all s, t, u, v > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The function M proves useful in statistics as the Fr´echet-  Hoeffding upper bound for joint distribution functions of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  See  Nelsen [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Popoviciu [29, 31] introduced the concept of higher order convexity for functions  of several variables using multiple divided differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' To gain some insight, let us  consider the case of a function f = f(x, y) defined on a product I × J of intervals,  and let x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xm be distinct points in I, and y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , yn be distinct points  in J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The divided double differences are defined via the formula  �  x0,  x1,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  , xm  y0,  y1,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  , yn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f  �  = [x0, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' [y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , yn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f((x, ·)]]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2)  = [y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , yn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' [x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f((·, y)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Notice that this formula is invariant under the permutation of variables xk (and  also under the permutation of the variables yk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Drawing a parallel to the one dimensional case, Popoviciu [29, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 78] calls a  function f : I × J → R convex of order (m, n) if the divided differences  � x0,  x1,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  , xm  y0,  y1,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  , yn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' f  �  are nonnegative for all distinct points x0, x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xm ∈ I and y0, y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', yn ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Needless to say, the study of this concept of convexity implies a formidable  formalism, so little progress was made since the times of Popoviciu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The only one  recent contribution is [12] that studies the cases m = n = 1 and m = n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The case of completely monotone functions on cones  The theory of completely monotone functions can be easily extended to the  context of several variables using convex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' In what follows V denotes a  finite-dimensional real vector space and C an open convex cone in V with closure  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Its dual cone is C∗ = {y ∈ E∗ : ⟨y, x⟩ ≥ 0 for all x ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The points in C∗ are  linear functionals that are nonnegative on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  10  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' A function f : C → R+ is called completely monotone if f is C∞ on  C and, for all integers k ≥ 1 and all vectors v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', vk ∈ C, we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1)  (−1)k Dv1 · · · Dvkf(x) ≥ 0  for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Here Dv denotes the directional derivative along the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  A function f : C → R+ is called completely monotone if it is the continuous  extension of a completely monotone function on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  When C = (0, ∞)n, the condition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1) means that  (−1)k  ∂kf  ∂xi1∂xi2 · · · ∂xik  (x) ≥ 0  for all x ∈ (0, ∞)n and all sets of indices 1 ≤ i1 ≤ i2 ≤ · · · ≤ ik ≤ n of arbitrary  length k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  As in the case of completely monotone functions of one real variable, these func-  tions can be obtained as Laplace transforms of Borel measures on the dual cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' (Bernstein-Hausdorff-Widder-Choquet theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Let f be a nonnega-  tive continuous function on the open convex cone C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then f is completely monotone  if and only if it is the Laplace transform of a unique Borel measure µ supported on  the dual cone C∗, that is,  f(x) =  �  C∗ e−⟨y,x⟩dµ(y)  for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  When f admits a continuous extension to C, the last equality works for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  For details, see Choquet [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Finding the positive Borel measure µ that makes the formula of The-  orem 6 working represents a practical way for checking the complete monotonicity  of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' So is the case of Riesz kernels: If α1, α2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', αN > 0, then  x−α1  1  x−α2  2  · · x−αN  N  =  �  RN  ++  e−⟨y,x⟩ x−α1  1  x−α2  2  · · x−αN  N  Γ(α1)Γ(α2) · · · Γ(αN)dy  for all x ∈ RN  ++.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See [15, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' A more subtle case is that of inverse  powers of the determinant  f (X) = (det X)−ρ ,  X ∈ Sym++(N, R),  for which Scott and Sokal [35] have shown that is completely monotone if and only  if ρ ∈ {0, 1/2, 1, 3/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', (N − 1) /2} ∪ ((N − 1) /2, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See also [15, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  It is worth noticing that Siegel established in 1929 the formula  (det A)−ρ =  �  Sym++(N,R)  e− trace AX  (det X)ρ dX  πn(n−1)/4Γ(ρ)Γ (ρ − 1/2) · · · Γ (ρ − (n − 1)/2))  for all A ∈ Sym++(N, R) and ρ ≥ (N + 1) /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See [38, Hilfssatz 37, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 585].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  We next extend (and improve) a result due to Sendov and Zitikis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' see [36, The-  orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Every completely monotone function f : C → R+ satisfies  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2)  �n  i=1 f(xi) −  �  1≤i<j≤n f(xi + xj) + · · · + (−1)n−1f(  �n  i=1 xi) ≥ 0,  11  for every x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn ∈ C and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' When f admits a continuous extension to C,  then then f satisfies the double inequality  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3) f(0) ≥  �n  i=1 f(xi)−  �  1≤i<j≤n f(xi +xj)+· · ·+(−1)n−1f(  �n  i=1 xi) ≥ 0,  for every x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , xn ∈ C and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  For n = 3, the first conclusion of Theorem 7 reads as  �3  i=1 f(xi) −  �  1≤i<j≤3 f(xi + xj) + f(  �3  i=1 xi) ≥ 0,  which is nothing but a Hornich-Hlawka type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The proof of Theorem 7 needs the following auxiliary result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We have  P =  �n  i=1 e−αi −  �  1≤i<j≤n e−(αi+αj) + · · · + (−1)n+1e− �n  i=1 αi ≥ 0  and  Q =  �n  i=1  �  1 − e−αi�  −  �  1≤i<j≤n  �  1 − e−(αi+αj)�  + · · · + (−1)n+1 �  1 − e− �n  i=1 αi�  ≥ 0,  whenever α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , αn ∈ R+ and n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Indeed,  S = 1 −  �n  i=1  �  1 − e−αi�  and Q =  �n  i=1(−1)k+1  �n  k  �  − S = 1 − S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' As per to Theorem 6, f admits the integral representation  f(x) =  �  C∗ e−⟨y,x⟩dµ(y)  for all x ∈ C,  where µ is a Borel measure on C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then, taking into account to the first assertion  of Lemma 5, we have the inequality  0 ≤  �  C∗  �n  i=1 e−⟨xi,y⟩ −  �  1≤i<j≤n e−⟨xi+xj,y⟩  + · · · + (−1)n−1e−⟨�n  i=1 xi,y⟩�  dµ(y)  =  �n  i=1 f(xi) −  �  1≤i<j≤n f(xi + xj) + · · · + (−1)n−1f(  �n  i=1 xi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The case where f is defined on C can be settled in the same manner, using both  assertions of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  Combining Theorem 7 with the aforementioned result of Scott and Sokal (see  Remark 2) one obtains the following result:  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If ρ ∈ {0, 1/2, 1, 3/2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='} ∪ [(N − 1) /2, ∞), then  �n  i=1 det−ρ(Ai) −  �  1≤i<j≤n det−ρ(Ai + Aj)  + · · · + (−1)n−1 det−ρ(  �n  i=1 Ai) ≥ 0,  12  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  for every A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , An ∈ Sym++(N, R) and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Given Corollary 2 one may wonder whether Theorem 7 also specializes to el-  ementary symmetric polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The situation here turns out to be more sub-  tle, and a qualified answer follows from the discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Recall that for any  m = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', N, the m-th elementary symmetric polynomial of x ∈ RN is defined  by the formula  Em,N(x) =  �  1≤i1<···<im≤N xi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='xim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Notice that det and the functions Em,N are hyperbolic polynomials in the sense  of G˚arding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  For details concerning this notion see [26, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Kozhasov,  Michalek and Sturmfels provided a constructive proof for the fact that any elemen-  tary symmetric polynomial Em,n admits a real exponent α′ > 0 such that E−α  m,n is  completely monotone on RN  ++ for all α ≥ α′—see [15, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The restriction of e−x to R+ is a completely monotone function and  thus it verifies the inequality  e−x1 + e−x2 + e−x3 + e−(x1+x2+x3) ≥ e−(x1+x2) + e−(x2+x3) + e−(x3+x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  However there x1, x2, x3 > 0 such that  e−x1 + e−x2 + e−x3 + e−(x1+x2+x3) � e−(x1+x2) + e−(x2+x3) + e−(x3+x1) + e0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  This shows that the inequality  �3  i=1 f(xi) −  �  1≤i<j≤3 f(xi + xj) + f(  �3  i=1 xi) ≥ f(0),  does not characterize the 3-convexity within the class of continuous functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The fact that the Bernstein functions f : R+ → R+ also verify the inequalities  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2) follows from Lemma 5 and the L´evy-Khintchine theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 15  of [34]), which asserts that each such function admits the integral representation  f(x) = a + bx +  � ∞  0  (1 − e−xt)dµ(t)  for some constants a, b ≥ 0 and a positive measure µ on [0, ∞) such that  � ∞  0  min {1, t} dµ(t) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  As far as we know, no attempt was made to extend the theory of Bernstein  functions to the framework of functions defined on cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Functions with positive differences on cones  The difference operators ∆hf : x → ∆hf(x) = f(x + h) − f(x) are well defined  in the case of functions f defined on a convex cone C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Such a function is said to be  a function with positive differences of order n (n ≥ 1) if  ∆h1∆h2 · · · ∆hnf(x) ≥ 0 for all x, h1, h2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', hn ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  For convenience, we say that f has positive differences of order 0 if f ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  In the literature, the concept of n-absolute monotonicity is used with the meaning  that the function under attention has positive differences of order k for all k ∈  {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  13  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If f : C → F+ is a completely monotonic function, then f has positive  differences of any even order k = 0, 2, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' under the same hypothesis, −f has  positive differences of any odd order k = 1, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  This follows easily from the characterization of monotonicity in terms of Gˆateaux  differentiability (as was established by Amann [1], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 184):  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Suppose that E and F are two ordered Banach spaces, C is a convex  subset of E with nonempty interior int C and Φ : C → F is a function, continuous  on C and Gˆateaux differentiable on int C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then Φ is monotone nondecreasing on C  if and only if  DΦ(a)[v] = Φ(a + tv) − Φ(a)  t  ≥ 0  for all points a ∈ int C and all vectors v ∈ E+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Every function of the form  Φ(x) = f (⟨x, w⟩) ,  x ∈ RN  +,  associated to a continuous n-convex function f : R+→ R and a vector w ∈ RN  + has  positive differences of order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The proof is straightforward, as evident from the case n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Indeed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' in this case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  for every x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' t ∈ Rn  + we have  Φ (x + t) + Φ (y + t) + Φ (z + t) + Φ (x + y + z + t)  − Φ (x + y + t) − Φ (y + z + t) − Φ (z + x + t) − Φ(t)  = f (⟨x + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) + f (⟨y + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) + f (⟨z + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) + f (⟨x + y + z + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩)  − f (⟨x + y + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) − f (⟨y + z + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) − f (⟨z + x + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) − f(⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩)  = f (⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩ + ⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) + f (⟨y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩ + ⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) + f (⟨z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩ + ⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩)  + f (⟨x + y + z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩ + ⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) − f (⟨x + y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩ + ⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) − f (⟨y + z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩ + t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩)  − f (⟨z + x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩ + ⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) − f(⟨t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' v⟩) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  A variant of this example is provided by the map  Ψ(A) = f (trace(AW)) ,  A ∈ Sym+(n, R),  associated to a continuous n-convex function f : R+→ R and to an operator W ∈  RN  +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The ambient Hilbert space in this case is Sym(n, R), endowed with the Frobenius  norm and L¨owner ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Linear algebra offers many examples of functions that have positive differences  of any order n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' So is the case of the determinant function det, restricted to  Sym+(N, R) (though det it is not completely monotonic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Clearly, det(X) ≥ 0 and since det is monotonic on the semidefinite matrices,  (∆A det) (X) = det(X + A) − det(X) ≥ 0,  for all A, X ∈ Sym+(N, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Also simple is the fact that  (∆A (∆B det)) (X) = det (A + B + X) − det (A + X) − det(B + X) + det(X) ≥ 0  for all A, B, X ∈ Sym+(N, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' This inequality is mentioned in [42, Problem 36,  pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 215].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' For X = 0 it reduces to the property of superaditivity of the function det,  det(A + B) ≥ det(A) + det(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  14  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  Our next goal is to show the more challenging third-order inequality  ∆A (∆B(∆C det)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  We require some preparatory lemmas before stating our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Let A, B, C ∈ Sym+(N, R), and let ek(X) denote the k-th elementary  symmetric function of a matrix X (0 ≤ k ≤ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then,  ek(A) + ek(B) + ek(C) + ek(A + B + C) ≥ ek(A + B) + ek(B + C) + ek(C + A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Recall that for an N × N matrix X, we have ek(X) = tr(∧kX), where ∧  denotes the anti-symmetric tensor product;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' moreover, ∧k(X) = P ∗  k (⊗kX)Pk for  a suitable projection matrix Pk—see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', [5, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 18] for these facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Using Pk in  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1 in [3], the claimed inequality for ek follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If A, B, C, X ∈ Sym+(N, R), then  det (A + X) + det (B + X) + det (C + X) + det(A + B + C + X)  ≥ det (A + B + X) + det (B + C + X) + det (C + A + X) + det X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  For X = 0 we get the determinantal Hornich-Hlawka inequality  det A + det B + det C + det(A + B + C)  ≥ det (A + B) + det (B + C) + det (C + A) ,  first noticed by Lin [18], who provided a proof based on eigenvalue majorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Without loss of generality we may assume that X is invertible (for example,  replace X by X + εI if necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then  det(A + X) = det(X) det(X−1/2AX−1/2 + I),  where I is the identity matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' the inequality under attention is then equivalent to  det(A + I) + det(B + I) + det(C + I) + det(A + B + C + I)  ≥ det(A + B + I) + det(B + C + I) + det(C + A + I) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Consider the function f(A) := det(A + I) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then the above inequality becomes  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1) f(A) + f(B) + f(C) + f(A + B + C) ≥ f(A + B) + f(B + C) + f(C + A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Now recall the well-known expansion  det(A + I) =  N  �  k=0  ek(A),  where ek(·) denotes the k-th elementary symmetric polynomial (e0 = 1, e1 =  tr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', eN = det)—see [22, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 197] Thus, f(A) = �N  k=1 ek(A),  and inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1) becomes  N  �  k=1  �  ek(A) + ek(B) + ek(C) + ek(A + B + C)  �  ≥  N  �  k=1  �  ek(A + B) + ek(B + C) + ek(C + A)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The proof ends by applying Lemma 7 for each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  15  A similar argument using [3, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4] yields that the function det has  positive differences of any order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We have  ∆A1 (∆A2(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='(∆An det)))(X)  =  n  �  k=0  (−1)k+1  �  1≤i1<i2<···<ik≤n  det(Ai1 + · · · + Aik + X) ≥ 0,  whenever A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' , An, X ∈ Sym+(N, R) and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' In other words, the restriction  of the det function to Sym+(N, R) has positive differences of any order n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Lemma 8 can be extended to a larger class of matrix functions, that of im-  manants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The immanant function dG  χ , associated to a subgroup G of the symmetric  group SN of N letters and to an irreducible character χ of G, is defined via the  formula  dG  χ (A) =  �  σ∈G χ(σ)  �N  i=1 ai σ(i), A ∈ Sym+(N, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  When G = Gm and χ(σ) = sgn σ we have dG  χ (A) = det A, while for χ(σ) ≡ 1 we  obtain the permanent of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The following two inequalities  dG  χ (A + X) − dG  χ (X) ≥ 0  dG  χ (A + B + X) − dG  χ (A + X) − dG  χ (B + X) + dG  χ (X) ≥ 0  occur for all A, B, X ∈ Sym+(N, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See respectively Merris [21], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 228 and Paksoy,  Turkmen and Zhang [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The fact that dG  χ has positive differences of third order  on Sym+(N, R) makes the objective of the following result:  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If A, B, C, X ∈ Sym+(N, R), then  dG  χ (A + X) + dG  χ (B + X) + dG  χ (C + X) + dG  χ (A + B + C + X)  ≥ dG  χ (A + B + X) + dG  χ (B + C + X) + dG  χ (C + A + X) + dG  χ (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Using arguments from multilinear algebra one can show that there exists a matrix  ZG,χ such that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2)  dG  χ (X) = Z∗  G,χ  �  ⊗NX  �  ZG,χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  See [20], p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The representation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2) turns the assertion of Theorem 10 into  an immediate consequence of the following general result:  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' (Operator Hornich-Hlawka inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' If A, B, C, X ∈ Sym+(N, R)  and p ≥ 1 an integer, then  ⊗p(A + X) + ⊗p(B + X) + ⊗p(C + X) + ⊗p(A + B + C + X)  ≥ ⊗p(A + B + X) + ⊗p(B + C + X) + ⊗p(C + A + X) + ⊗pX,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3)  in the L¨owner order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Here ⊗ denotes the usual tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  The proof of Theorem 11 follows the proof structure of [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1], but  due to the additional X term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3) it turns out to be more intricate and requires  some preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We start by introducing the following convenient notation:  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4)  Xj ≡ ⊗jX = X⊗j,  for j ≥ 1,  and X0 = 1 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  16  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  The slight abuse of notation X0 = 1 will be helpful in simplifying the presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Lemma 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Let k, l ≥ 0 be integers and X, Y, Z, V ∈ Sym+(N, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Then,  Xk ⊗ V ⊗ Xl + (X + Y + Z)k ⊗ V ⊗ (X + Y + Z)l  ≥ (X + Y )k ⊗ V ⊗ (X + Y )l + (X + Z)k ⊗ V ⊗ (X + Z)l,  in the sense of L¨owner order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The proof is by induction on k and l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' we provide the argument for k, which  holds essentially unchanged for l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  This approach suffices since for a fixed but  arbitrary l ≥ 0 we prove that the result holds for all k ≥ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' similarly, for an  arbitrarily fixed k ≥ 0 the result holds for all l ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  For the base case of induction, suppose k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Then the inequality under  attention reduces to the following one:  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='5)  V ⊗ Xl + V ⊗ (X + Y + Z)l ≥ V ⊗ (X + Y )l + V ⊗ (X + Z)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  We know from [3, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1] that  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='6)  Xl + (X + Y + Z)l ≥ (X + Y )l + (X + Z)l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Since tensor product preserves inequalities, taking tensor product with V on both  sides of the inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='6) one immediately get (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Assume therefore that  inequality in the the statement of Lemma 9 holds for a fixed l and some k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Then, consider  (X + Y + Z)k+1 ⊗ V ⊗ (X + Y + Z)l  = (X + Y + Z) ⊗  �  (X + Y + Z)k ⊗ V ⊗ (X + Y + Z)l�  ≥ (X + Y + Z) ⊗  �  (X + Y )k ⊗ V ⊗ (X + Y )l  + (X + Z)k ⊗ V ⊗ (X + Z)l − Xk ⊗ V ⊗ Xl�  ,  = (X +Y )k+1 ⊗V ⊗(X +Y )l +(X +Z)k+1 ⊗V ⊗(X +Z)l −Xk+1 ⊗V ⊗Xl +T ,  where the inequality follows from the induction hypothesis and the elementary  monotonicity properties of the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' It remains to show that the term  T = Z⊗(X+Y )k⊗V ⊗(X+Y )l+Y ⊗(X+Z)k⊗V ⊗(X+Z)l−(Y +Z)⊗Xk⊗V ⊗Xl  is a nonnegative operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Since X, Y, Z ≥ 0, it follows that X + Y ≥ X and  X + Z ≥ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Thus, the positive terms in T attached to Y and Z are clearly bigger  than the respective negative terms, whence T ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Inductively, we can conclude  that for fixed l, the inequality in the statement of Lemma 9 holds for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Applying a similar argument for l, we conclude that this inequality works in full  generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  We are now in a position to detail the proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Proof of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Using the auxiliary function  fp(Z) = (Z + X)p − Xp,  the inequality of interest (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3) can be rewritten as  fp(A) + fp(B) + fp(C) + fp(A + B + C) ≥ fp(A + B) + fp(B + C) + fp(C + A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  17  Now introduce the function  gp(Z) = fp(Z)+fp(B)+fp(C)+fp(Z +B+C)−fp(Z +B)−fp(Z +C)−fp(B+C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  We will show that gp is monotonic (as a map from Sym+(N, R) into itself, under  the L¨owner order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Once this monotonicity is established we can conclude that  gp(A) ≥ gp(0) = 0  for every A ∈ Sym+(N, R),  a fact equivalent to the assertion of Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Monotonicity of gp follows from Lemma 6 by considering its derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' To that  end, consider the (directional) derivative of the map Φ(Z) = Zp:  D(Zp)[V ] = V ⊗ Z ⊗ · · · ⊗ Z + Z ⊗ V · · · ⊗ Z + · · · + Z ⊗ · · · ⊗ Z ⊗ V  =  p−1  �  j=0  Zj ⊗ V ⊗ Zp−1−j,  whenever Z ∈ Sym++(N, R) and V ∈ Sym+(N, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' See [6, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='13), pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Indeed, applying this formula to fp(Z) = (Z + X)p − Xp we obtain  Dfp(Z)[V ] =  p−1  �  j=0  (Z + V )j ⊗ V ⊗ (Z + V )p−1−j,  which in turn leads to the identity  Dgp(Z)[V ]  = Dfp(Z)[V ] + Dfp(Z + B + C)[V ] − Dfp(Z + B)[V ] − Dfp(Z + C)[V ]  =  p−1  �  j=0  �  (Z + V )j ⊗ V ⊗ (Z + V )p−1−j +(Z+B+C+V )j⊗V ⊗(Z+B+C+V )p−1−j  −(Z+B+V )j⊗V ⊗(Z+B+V )p−1−j −(Z + C + V )j ⊗ V ⊗ (Z + C + V )p−1−j�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  But this sum evaluates to a positive quantity, which follows from Lemma 9 upon  setting k ← j, l ← p − 1 − j and x ← x + j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Now the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  □  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Further comments and extensions  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Multivariable case for positive operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Given A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', An, X ∈ Sym+(N, R)  and p ∈ N, let us consider the matrices  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='1)  Sp  0 = ⊗pX,  Sp  k =  �  1≤i1<···<ik≤n  ⊗p(Ai1 + · · · + Aik + X) (1 ≤ k ≤ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Then one can prove the following generalization of the operator Hornich-Hlawka  inequality (Theorem 11):  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2)  Sp  n + Sp  n−2 + · · · ≥ Sp  n−1 + Sp  n−3 + · · · ,  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3 of [3] proves inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2) for the special case X = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' One can prove  the general case by a suitable monotonicity argument as in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' The details  are tedious so we omit them, leaving them as an exercise for the interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  18  CONSTANTIN P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' NICULESCU AND SUVRIT SRA  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Two inequalities for determinants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Serre [37] noticed that det1/2 viewed  as a function on Sym+(2, R) verifies the opposite of the Hornich-Hlawka inequality,  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3)  det1/2 A + det1/2 B + det1/2 C + det1/2(A + B + C)  ≤ det1/2 (A + B) + det1/2 (B + C) + det1/2 (C + A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  While in [40], the second named author proved the following Minkowski-like in-  equality for det  1  n , by viewing it as a function on Sym+(n, R):  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4)  det  1  n (A + B) det  1  n (A + C) ≥ det  1  n B det  1  n C + det  1  n A det  1  n (A + B + C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Inequality (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='4) is stronger the a usual log-supermodularity inequality for det, given  the extra det  1  n B det  1  n C term on its right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Vasic-Adamovic inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We close the paper by briefly mentioning the  related class of Vasic-Adamovic inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Inspired by the work of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Smiley  and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Smiley [39] on polygonal inequalities, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Vasi´c and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Adamovi´c [41]  found an inductive scheme for generating inequalities for any function that verifies  the functional Hornich-Hlawka inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' We state here a slightly modified version  of their result [23, Theorem 2, pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 528]:  Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Let S be a commutative additive semigroup with 0 and G be an  ordered abelian group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', an abelian group with an order relation ≤) such that  x, y, z ∈ G and x ≤ y implies x + z ≤ y + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  If ϕ : S → G is a function such that for all x1, x2, x3 ∈ S, we have  �  1≤i<j≤3 ϕ  �  xi + xj  �  ≤  �3  k=1 ϕ (xk) + ϕ  ��3  k=1xk  �  then for each pair {k, n} of integers with 2 ≤ k < n we also have  �  1≤i1<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='<ik≤n  ϕ  � k  �  j=1  xij  �  ≤  �n − 2  k − 1  �  n  �  k=1  ϕ (xk) +  �n − 2  k − 2  �  ϕ  � n  �  k=1  xk  �  ,  whenever x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', xn ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  This claim applies to every function f (defined on a convex cone C and taking  values in an ordered Banach space E) that has positive differences of the third  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Indeed, in their case,  f (x + t) + f (y + t) + f (z + t) + f (x + y + z + t)  ≥ f (x + y + t) + f (y + z + t) + f (z + x + t) + f(t)  for all points x, y, z, t ∈ C and performing the change of function ϕ(v) = f(v + t) −  f(t) we obtain a function that verifies the Hornich-Hlawka inequality  ϕ (x) + ϕ (y) + ϕ (z) + ϕ (x + y + z) ≥ ϕ (x + y) + ϕ (y + z) + ϕ (z + x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  19  References  [1] Amann H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=': Multiple positive fixed points of asymptotically linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 17,  174-213 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Bennett, Some forms of majorization, Houston J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 36 (2010), 1037-1066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [3] Berndt, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', Sra, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=': Hlawka–Popoviciu inequalities on positive definite tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Linear Alge-  bra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 486, 317-327 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [4] Bernstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=': Sur les fonctions absolument monotones, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 52, 1–66 (1929).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [5] Bhatia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=': Matrix Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Springer (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [6] Bhatia, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=': Positive Definite Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Princeton University Press (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [7] Boas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', Widder, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' : Functions with positive differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 7, 496–503  (1940).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Choquet: Deux exemples classiques de repr´esentation int´egrale, Enseign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' 15, 63-75  (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  [9] Del Moral, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=', Niclas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=': A Taylor expansion of the square root matrix function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='  Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
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+page_content='  Department of Mathematics, University of Craiova, Craiova 200585, Romania  Email address: constantin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='niculescu@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='com  Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Email address: suvrit@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE_T4oBgHgl3EQf2xw0/content/2301.08342v1.pdf'}
diff --git a/gNFAT4oBgHgl3EQf8R7j/content/tmp_files/2301.08750v1.pdf.txt b/gNFAT4oBgHgl3EQf8R7j/content/tmp_files/2301.08750v1.pdf.txt
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+DOMAIN-AGNOSTIC AND MULTI-LEVEL EVALUATION OF
+GENERATIVE MODELS
+Girmaw Abebe Tadesse
+IBM Research - Africa
+girmaw.abebe.tadesse@ibm.com
+Jannis Born
+IBM Research - Zurich
+jab@zurich.ibm.com
+Celia Cintas
+IBM Research - Africa
+celia.cintas@ibm.com
+William Ogallo
+IBM Research - Africa
+william.ogallo@ibm.com
+Dmitry Zubarev
+IBM Research - Almaden
+dmitry.zubarev@ibm.com
+Matteo Manica
+IBM Research - Zurich
+tte@zurich.ibm.com
+Komminist Weldemariam
+IBM Research - Yorktown Heights
+kommy@ibm.com
+ABSTRACT
+While the capabilities of generative models heavily improved in different domains (images, text,
+graphs, molecules, etc.), their evaluation metrics largely remain based on simplified quantities
+or manual inspection with limited practicality. To this end, we propose a framework for Multi-
+level Performance Evaluation of Generative mOdels (MPEGO), which could be employed across
+different domains. MPEGO aims to quantify generation performance hierarchically, starting from
+a sub-feature-based low-level evaluation to a global features-based high-level evaluation. MPEGO
+offers great customizability as the employed features are entirely user-driven and can thus be highly
+domain/problem-specific while being arbitrarily complex (e.g., outcomes of experimental procedures).
+We validate MPEGO using multiple generative models across several datasets from the material
+discovery domain. An ablation study is conducted to study the plausibility of intermediate steps
+in MPEGO. Results demonstrate that MPEGO provides a flexible, user-driven, and multi-level
+evaluation framework, with practical insights on the generation quality. The framework, source code,
+and experiments will be available at: https://github.com/GT4SD/mpego.
+Keywords Generative models · Evaluation · Data-centric AI · Foundation models
+1
+Introduction
+Machine Learning (ML) methods, particularly generative models, are effective in addressing critical problems across
+different domains, which includes material sciences. Examples include the design of novel molecules by combining
+data-driven techniques and domain knowledge to efficiently search the space of all plausible molecules and generate
+new and valid ones [1, 2, 3, 4]. Traditional high-throughput wet-lab experiments, physics-based simulations, and
+bioinformatics tools for the molecular design process heavily depend on human expertise. These processes require
+significant resource expenditure to propose, synthesize and test new molecules, thereby limiting the exploration
+space [5, 6, 7].
+For example, generative models have been applied to facilitate the material discovery process by employing inverse
+molecular design problem. This approach transforms the conventional and slow discovery process by mapping the
+desired set of properties to a set of structures. The generative process is then optimized to encourage the generation of
+molecules with those selected properties. Countless approaches have been suggested for such tasks, most prominently
+VAEs with different sampling techniques [8, 9, 10]), GANs [11, 12], diffusion models [13], flow networks [14] and
+Transformers [15].
+arXiv:2301.08750v1  [cs.LG]  20 Jan 2023
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+Though the generation capability has been tremendously improved recently, the quantitative evaluation of these
+generative models in different domains remains a grand challenge [16]. Some of the reasons include the multi-objective
+nature of real discovery problems, the intricacy of evaluating relevant features in-silico, and the lack of widely accepted
+domain- and model-agnostic evaluation frameworks. As a result, existing benchmarks and toolkits in material sciences,
+such as MOSES [17] or GuacaMol [18] include limited metrics, such as validity and uniqueness, that lack the capacity
+to evaluate the complex nature of the generation process (e.g., interactions of multiple properties), thereby less effective
+to provide meaningful insights to subject matter experts (SMEs) to facilitate practical impact.
+In this paper, we introduce a Multi-level Performance Evaluation of Generative mOdels (MPEGO) framework (see
+Fig. 1), which aims to hierarchically characterize and quantify the capability of generative models, using material
+discovery domain as a use case. To that end, MPEGO is a model- and domain-agnostic framework, and its core design
+is derived from two main requirements: representative examples (of training and generated samples) and one or multiple
+features (extracted from these samples). Metrics derived from MPEGO are also interpretable and provide multi-level
+abstractions of the generation process. Specifically, the contributions of this paper are as follows:
+1. We provide a multi-level and domain-agnostic evaluation of generative models, starting with sub-feature- or
+feature-based low-level evaluation to global features based high level-evaluation.
+2. We devise a generation frequency analysis that aims to identify and characterize subsets of samples generated
+with extreme frequencies.
+3. We validate MPEGO framework on multiple generative models (e.g., GCPN [19], GraphAF [20], MolGX [21]
+and Regression Transformer [15], and multiple datasets (e.g., ZINC-250K [22] and MOSES [17] and CIRCA1.
+4. We conduct ablation studies to analyze MPEGO’s sensitivity to design choices.
+2
+Related Work
+The state-of-the-art evaluation approaches for generative models aim to quantify pre-determined requirements, such as
+diversity and validity, using a variety of metrics. Frechet ChemNet Distance (FCD) [23] is one of such metrics, and it
+measures the distance between hidden representations drawn from sets of generated and training samples in the material
+discovery domain, which is limited in providing sub-feature or feature-level evaluation of models. GuacaMol [18] is
+one of the early benchmark platforms for new molecule discovery, which aims to evaluate generative models across
+different tasks, e.g., fidelity and novelty. Molecular Sets (MOSES) [17] is another benchmarking framework, which
+provides training and testing datasets, and a set of metrics to evaluate the quality and diversity of generated structures to
+standardize training and model comparisons.
+Furthermore, automated characterization of subsets of samples, generated with more or less frequency, i.e., generation
+frequency analysis, also still remains challenging as the focus is more on latent- or feature-based evaluation. Overall,
+the challenges associated with evaluating generative models could be summarized as follows. First, multiple evaluation
+metrics are model-dependent. For example, FCD [23] depends on latent representation, and Maximum-mean discrep-
+ancy [24] is more specifically used to evaluate graph-based generative models. State-of-the-art metrics also suffer from
+limited generalizability (across different levels of feature interactions) and interpretability, e.g., by domain experts,
+which is critical to achieve trustworthy AI solutions [25]. In addition, existing evaluation metrics are susceptible to
+potential flaws in predictive models used in goal-oriented or constrained generation. Moreover, existing evaluation
+strategies lack a generic and standalone evaluation metric that combines both distributional metrics (e.g., uniqueness
+and diversity) and property-based metrics that score single property.. The dependency on a single-constraint objective
+lacks a principled approach to incorporate multiple target features. This becomes a significant challenge when a single
+and inaccurate evaluation metric is used, which oversimplifies real discovery problems and hence less practical.
+3
+Proposed: MPEGO Framework
+The proposed MPEGO framework (see Fig. 1) aims to provide an effective and multi-level characterization of generative
+models. The multi-level evaluation of MPEGO (see Fig. 2) starts from sub-feature-based low-level evaluation and
+their step-by-step aggregation to provide high-level evaluations. In this section, we first, formulate the critical research
+questions MPEGO framework is designed to address, followed by the details on its core components.
+2
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+Figure 1: Overview of the MPEGO framework
+3.1
+Problem Statement
+Let G1, G2, · · · , Gk, · · · , GK be datasets comprising samples generated from K black-box generative models
+(Θ1, Θ2, · · · , Θk, · · · , ΘK) trained on a dataset T . Can we evaluate the generation capability of each Θi in a scalable,
+easily interpretable, and multi-objective manner? Specifically, we aim to address two questions.
+Q1: Given a set of features characterizing the samples, how do we quantify the generation capability of each model
+compared to another model or the training data, based on one or more of these features, i.e., at different levels
+of abstractions?
+Q2: What are the characteristics of samples being generated with extreme frequencies (least or most) by each of
+the generative models, compared with an other model or the training data, i.e., generation frequency analysis?
+To address Q1, we propose a Hierarchical Independence Evaluation (HIE) that aims to quantify the performance of
+generative models at different levels of feature interactions hierarchically, starting with a sub-feature level evaluation
+(e.g., a specific range of a feature) to the global aggregation of multiple features. To address Q2, we employ multi-
+dimensional subset scanning (MDSS) [26] that aims to automatically identify and characterize over- and under-generated
+subsets of samples.
+3.2
+Feature Extraction and Pre-processing
+Given representative examples of generated and training samples, MPEGO starts with the extraction of M features from
+these samples, F = {f1, f2, · · · , fm, · · · , fM}. The type of feature values could be binary, continuous, or categorical,
+and a further pre-processing could be applied in the follow up steps. For example, discretization of continuous features
+is required for sub-feature-level performance evaluation, and MPEGO provides different discretization types, e.g., equal
+width, equal frequency or based on k-means.
+3.3
+Hierarchical Independence Evaluation (HIE)
+HIE follows a bottom-up approach (from a sub-feature to global aggregation levels), as shown in Fig. 2, and it evaluates
+the performance generative models at different levels of feature interactions. The lowest level of evaluation in HIE is
+the sub-feature level independence score (SIS), which aims to quantify the performance of generative models across
+different ranges or unique values per feature, e.g., based on a a molecular weight range of 200 − 300 Daltons. The
+second layer in HIE represents feature-level independence score (FIS), which quantifies the generation performance for
+each feature. To this end, FIS could be computed via aggregation of SIS values, thereby providing a weighting strategy
+for SIS scores per feature. FIS values could also be computed directly, without aggregating SIS values, using a different
+choice of objective measure is directly applied on the whole feature.The proposed MPEGO framework is flexible to
+utilize different objective measures (see Appendix D of the Supplementary Material for details). Below we describe the
+computation of SIS and FIS values, using Yule’s Y coefficient [27] as selected objective measure. Note that Gk vs. T
+refers to a case where the comparison is between the kth generative model and the training set T . On the other hand,
+Gk vs. Gj refers to a case when the evaluation between the jth and kth models, where j ̸= k. However, we stick with
+the Gk vs. T comparison below in order to ease readability.
+1https://circa.res.ibm.com/
+3
+
+Hierarchical
+Independence
+Evaluation
+Training
+(HIE)
+Pre-processing
+Features
+Extraction
+Model
+Cm = Kfm}l
+Characterisation and
+m
+Sm=1
+Ranking
+Generation
+Frequency Analysis
+Generated
+(GFA)Domain-agnostic and Multi-level Evaluation of Generative Models
+Figure 2: Details of Multi-level evaluation component of the MPEGO framework that comprises Hierarchical Indepen-
+dence Evaluation (i.e., SIS, FIS, SAFIS and GAFIS) and Generation Frequency Analysis (GFA). A(· · ·) represents
+aggregation operation, and anomalous subset refers to the logical combinations of features that characterize samples
+generated with extreme frequencies.
+Let fm ∈ F is a feature with Cm unique values or ranges, i.e., fm = {f u
+m}, u ∈ [1, 2, · · · , Cm]. Note that Cm is the
+number of unique values for categorical features or the number of bins after discretization of continuous features. SIS
+computation requires the stratification of both the generated (Gk) and training (T ) datasets per each unique value/range
+f u
+m resulting Gu
+km and T u
+m, respectively. The complimentary subsets are then �
+Gu
+km and �
+T u
+m, respectively. Note that
+�
+Gu
+km = Gu
+km|(fm ̸= f u
+m) = Gk − Gu
+km and �
+T u
+m = T u
+m|(fm ̸= f u
+m) = T − T u
+m. Accordingly, a 2 × 2 pivot table is
+generated for each f u
+m as:
+(fm = f u
+m)
+(fm ̸= f u
+m)
+Gkm
+α
+β
+Tm
+δ
+γ
+where α is the number of generated samples in Gu
+km that are characterized by the feature value fm = f u
+m, β is the
+number of generated samples in �
+Gu
+km with fm ̸= f u
+m. Similarly, δ and γ are the numbers of training samples that satisfy
+fm = f u
+m in T u
+m, respectively. Note that α + β are the total numbers of generated samples, i.e., |Gk|. Similarly, δ + γ is
+the number of training samples, i.e., |T |. Then Yule’s Y coefficient is computed from the pivot table as ou
+km ∈ [−1, 1]:
+ou
+km =
+�
+P(Gu
+km)P( �
+T u
+m) −
+�
+P( �
+Gu
+km)P(T u
+m)
+�
+P(Gu
+km)P( �
+T u
+m) +
+�
+P( �
+Gu
+km)P(T u
+m)
+(1)
+ou
+km =
+√αγ − √βδ
+√αγ + √βδ
+(2)
+SIS is then computed from ou
+km value as Iu
+km = 1 − |ou
+km|, where Iu
+km ∈ [0, 1] and higher Iu
+km reflects higher
+independence between Gk and T , i.e., SIS = 1 represents complete independence. Feature-level Independence Score
+(FIS), provides feature-based evaluation, i.e., higher abstraction than SIS. Depending on the objective measure, FIS
+could be computed as 1) via a weighted aggregation of SIS values, i.e., Ikm = �Cm
+u=1 λu
+kmIu
+km where �Cm
+u=1 λu
+km = 1
+and each λu
+km weights the SIS value of f u
+m, or 2) via straightforward computation, without using SIS values, when
+4
+
+GlobalAggregationof
+A(I,...,Im)
+Feature-based Independence
+Score (GAFIS)
+Selective Aggregation of
+A(I1, I2)
+A(Im,.., IMm)
+Feature-based Independence
+Score (SAFiS)
+Feature-based
+I1
+I2
+Im
+IM
+IndependenceScore
+(FIS)
+Sub-feature based
+I1
+Igi
+II
+1
+Im
+IndependenceScore
+(SIS)
+Inputfeatures
+f1
+2
+fm
+Cm = Ifm}l,m E [1,2, ..., M] :
+Anomalous
+Generation Frequency
+subset
+Analysis(GFA)Domain-agnostic and Multi-level Evaluation of Generative Models
+the objective measure is directly applied on each feature without discretization, e.g., using Wasserstein distance. The
+third layer of HIE is Selective Aggregation of Feature-level Independence Score (SAFIS), which aims to aggregate
+FIS values from R < M selected features in F. For example, features including scaffolding, fingerprints, aromaticity,
+and the number of rings could be selected to reflect the structural details of molecules in material discovery domain.
+The last layer of HIE is Global Aggregation of Feature-level Independence Score (GAFIS), which is computed via a
+weighted aggregation of all the FIS values in F. Note that SAFIS and GAFIS are computed as ˆI = �R
+r=1 ηrIkr, where
+�R
+r=1 ηr = 1, and R < M for SAFIS and R = M for GAFIS computation.
+3.4
+Generation Frequency Analysis (GFA)
+Generative models trained on the same dataset will hardly generate samples with exact characteristics. Thus, there
+is a potential over- or under-generation of samples with certain characteristics. To this end, we employ automated
+stratification of samples using multi-dimensional subset scanning (MDSS) [26, 28] to identify subset of samples
+generated with divergent frequencies. Specifically, to identify samples generated with divergent rates by model Θk,
+compared to the training set T , we first merge the corresponding datasets as D = Gk ∪ T , and an outcome label (y)
+is generated, such that yi = 1 for a sample in Gk and yi = 0 for a sample in T . If there are Ng = |Gk| generated
+and Nt = |T | training samples in D, the expectation of generated samples in D is eg =
+Ng
+Ng+Nt . Thus, GFA aims to
+identify a group of samples with extreme deviations in their generation rate compared to eg. The deviation between
+the expectation and observation is evaluated by maximizing a Bernoulli likelihood ratio scoring statistic, Γ(·). The
+null hypothesis assumes that the odds of the generated sample in any subgroup S is similar to the expected, i.e.,
+H0 : odds(S) =
+eg
+1−eg ; while the alternative hypothesis assumes a constant multiplicative increase in the odds of the
+generated samples in S, i.e., H1 : odds(S) = q
+eg
+1−eg where q ̸= 1. Note that q > 1 for over-generated subset, and
+0 < q < 1 for under-generated subset. The divergence score for a subgroup (S) with reference D is formulated as,
+Γ(S, D) and computed as:
+Γ(S, D) = max
+q
+log(q)
+�
+i∈S
+yi − Ns ∗ log(1 − eg + qeg),
+(3)
+where Ns is the number of samples in S. The divergent subset, S, identification is iterated until convergence to a local
+maximum is found, and the global maximum is subsequently optimized using multiple random restarts.
+4
+Experimental Setup
+4.1
+Training Datasets
+We employed three different datasets in material discovery domain to validate our MPEGO framework.
+ZINC-250K
+We utilize the publicly available ZINC-250K2 dataset, which contains 249, 455 small molecules in
+Simplified Molecular-Input Line-Entry System (SMILES) representation. Details on ZINC tool are available in [22].
+MOSES
+The benchmark platform MOSES [17], besides implementing popular molecular generation models and
+metrics, MOSES contains a refined dataset from ZINC3. The dataset has approximately 2M molecules in total, filtered
+by certain parameters such as molecular weight ranges, and number of rotatable bonds, among others.
+CIRCA
+We used IBM’s Chemical Information Resources for Cognitive Analytics (CIRCA) platform4 to construct
+a dataset of organic salts relevant to production of semiconductors via photolitography with chemical amplification.
+Finally, we were able to evaluate environmental and toxicological properties of 866 anions, that comprise the dataset
+used in this study. Steps conducted to obtain final version of the dataset could be found in Appendix A in the
+Supplementary Material.
+4.2
+Generative Models
+We utilized multiple generative models for our validation.
+Particularly, Graph Convolutional Policy Network
+(GCPN) [19] and a Flow-based Autoregressive (GraphAF) [20] were trained separately on ZINC-250K and MOSES
+2https://www.kaggle.com/datasets/basu369victor/zinc250k
+3https://zinc.docking.org/
+4https://circa.res.ibm.com/
+5
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+Figure 3: Visualization of the CIRCA dataset and some of the structural fingerprints relevant to MPEGO analysis (inset).
+The CIRCA dataset is represented as a similarity network, where nodes correspond to anions and links connect nodes if
+Dice similarity of the respective anions is at least 0.5. Node colors encode clustering recovered via modularity analysis
+and node size encodes the number of connections of the node. Chemical structures are included for the most connected
+nodes in the main clusters.
+datasets. Subsequently, 10, 000 valid molecules were generated from each model. We rely on GT4SD [29] for model
+implementations (experimental set-ups shown in Appendix B). Similarly, from the MOSES-trained GCPN and GraphAF,
+14665 and 1680 molecules were generated respectively. For the CIRCA data, we employed MolGX [21] and the
+Regression Transformer [15]) and generated 5000 new anions with MolGX and 32617 new anions with Transformer.
+Note that balanced number of samples were selected across datasets during our experimentation, i.e., 10000, 1680 and
+5000 samples were randomly selected for ZINC-250K-based, MOSES-based, CIRCA-based evaluations.
+4.3
+Feature Extraction and Preprocessing
+We have extracted the following six features from SMILES representations of molecules in ZINK-250K and MOSES
+datasets and their corresponding generated datasets from GCPN and GrpahAF. The following features are selected for
+the evaluation: Aromaticity, ESOL, LogP, Weight, QED, SCScore. On the other hand, we extracted fingerprints (see
+Fig. 3) that reflect structural details from CIRCA dataset and MolGX and Transformer generated datasets. Full details
+of features can be found in the Appendix C of the Supplementary Material. In cases when discretization of continuous
+features is required to compute the SIS values, we employ Yule’s Y coefficient as our default objective measure as it
+satisfies multiple key requirements [30]. We also employ equal frequency discretization type as it better handles outliers
+in the data, with five bins for ZINC and with three bins for CIRCA based evaluations. While aggregating features for
+SAFIS and GAFIS computations using normal averaging. Note that the proposed approach is flexible to utilize different
+discretization types, weighting strategies, and objective measures.
+4.4
+Evaluation Metrics
+Our evaluation metrics include HIE’s SIS, FIS, SAFIS, and GAFIS values that quantify the generation independence of
+models. The independence score is obtained by normalizing the objective measure values to [0, 1]. Note independence
+score = 1.0 represents complete independence. We also utilize the histogram of features to provide a qualitative
+comparison. The characterization of generation frequency analysis involves using the logical combination of feature
+values to describe the identified subgroup, the size of the subgroup Ns, the odds ratio between S and �S = D − S, 95%
+Confidence Interval (CI) and empirical p value. We also reported the divergence score of the identified group from the
+expectation in GFA and elapsed time to identify the group. All the experiments are conducted on a desktop machine,
+2.9 GHz Quad-Core Intel Core i7 (processor), and 16 GB 2133 MHz LPDDR3 (memory).
+6
+
+Fingerprint ID
+Molecular subgraph
+951226070
+3218693969
+3217380708Domain-agnostic and Multi-level Evaluation of Generative Models
+Table 1: HIE scores for GCPN and GraphAF models trained on ZINC-250K dataset
+G vs. T
+G vs. G
+GCPN vs.
+GraphAF vs.
+GCPN vs.
+Level
+ZINC
+ZINC
+GraphAF
+Aromaticity
+1.000
+1.000
+1.000
+ESOL
+0.912
+0.856
+0.853
+LogP
+0.951
+0.942
+0.906
+FIS
+Weight
+0.656
+0.713
+0.809
+QED
+0.946
+0.666
+0.662
+SCScore
+0.825
+0.752
+0.866
+SAFIS
+QED+LogP
+0.949
+0.804
+0.784
+GAFIS
+0.882
+0.821
+0.850
+5
+Results and Discussion
+5.1
+Hierarchical Independence Evaluation
+Table 1 provides extended FIS values across each of the features considered for comparing GCPN and GraphAF models
+with the training ZINC-250K (Gvs.T ) and between each other - head-to-head (Gvs.G). GCPN’s GAFIS value of
+0.882, demonstrating competitive generation independence with GraphAF with GAFIS = 0.821. This is further shown
+with their head-to-head comparison, with Aromaticity (FIS = 1.0) and LogP (FIS = 0.906) features. Divergent
+characteristics are also demonstrated when the two models were evaluated based on QED, Weight and ESOL features,
+achieving FIS of 0.853, 0.809, and 0.662, respectively in their head-to-head comparison. SAFIS values are shown as
+an example of aggregation synthetic metrics, QED and LogP, where GCPN model achieves superiority compared to
+GraphAF. GAFIS values in the bottom row are derived from global aggregation of FIS values above. Overall, the results
+show the flexibility of the MPEGO framework to evaluate models across different levels of feature interactions and
+comparison baselines, i.e., training data or another generated datasets.
+Furthermore, Fig. 4 demonstrates the benefits of sub-feature level evaluation (SIS) of GCPN and GraphAF models
+trained on ZINC-250K, using Molecular Weight as a feature example. Five ranges of the molecular weight in Fig. 4 (a)
+resulted from the discretization of the feature necessary for SIS evaluations. Similarly to Table 1, the comparison is
+performed between generative models and with the training dataset. Results demonstrate that GCPN and GraphAF
+generated molecules with similar weight characteristics but distinctively different at a few particular ranges, i.e., GCPN
+generated molecules similar to ZINC-250K with range [258.1, 308.39) whereas GraphAF showed better resemblance
+with ZINC-250K at ranges [201.63, 258.1) and ≥ 361.47 Daltons. Divergent sub-level generation characteristics is
+also encoded by lower G vs. G independence score at those ranges. The histogram plot in Fig. 4 (b) qualitatively
+compliments the insights from SIS values in Fig. 4 (a) , where GraphAF is shown to generate more molecules with
+extreme weight values. Note that such sub-feature level insights in Fig. 4 are unique to our proposed framework as they
+are currently limited in the state-of-the-art of generative model evaluation.
+The results of comparisons of GCPN and GraphAF trained on MOSES dataset (Table 2) and MolGX and RT trained
+on CIRCA dataset (Table 3) demonstrate MPEGO’s capabilities further. Comparisons between Table 1 with Table 2
+show while MOSES is used as a training, GraphAF achieved more independence than when ZINC-250K is used,
+compared GCPN. Between MolGX and the RT, trained on CIRCA, molecules generated from the RT achieved a higher
+resemblance with CIRCA than MolGX. In their head-to-head comparison, MolGX and RT generated mostly divergent
+structural details, as it is demonstrated by very low FIS values for the majority of the fingerprints in G vs. G column of
+Table 3.
+5.2
+Generation Frequency Analysis
+Table 4 shows the divergence generation frequency of models, compared to the training or other generated set. For
+example, the GCPN model generated molecules with higher LogP (≥ 1.31) and lower weight (< 258.1 Daltons)
+compared to the training ZINC-250K. When we compare the two models head-to-head, GCPN tends to generate
+7
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+G vs. T
+G vs. G
+GCPN vs.
+GraphAF vs.
+GCPN vs.
+Level
+Weight Range
+ZINC
+ZINC
+GraphAF
+<201.63
+0.406
+0.322
+0.859
+[201.63 , 258.1)
+0.631
+0.840
+0.778
+SIS
+[258.1 , 308.39)
+0.986
+0.824
+0.810
+[308.39 , 361.47)
+0.671
+0.637
+0.962
+≥361.47
+0.586
+0.939
+0.638
+(a)
+(b)
+Figure 4: (a) Example of Molecular Weight-based SIS values that provide sub-feature level evaluation of GCPN and
+GraphAF models trained on ZINC-250K; (b) histogram densities to provide qualitative visualization of the SIS values
+in (a)
+molecules with higher QED (≥ 0.64) values compared to GraphAF (< 0.5). When MOSES data is used as validation,
+LogP values become the differentiation factor as GraphAF tends to generate lower values (< 3.91) compared to
+GCPN (≥ 3.91). On CIRCA dataset, MolGX generates samples with a higher occurrence of ’3217380708’ fingerprint
+compared to the RT model and the training CIRCA. The size of the identified group, along with the multiplicative factor
+(q) and the odds ratio values, confirm the significance of the identified divergent generation in our generation frequency
+analysis.
+5.3
+Ablation Study
+We conducted ablation studies to validate the robustness of MPEGO for different design choices of objective measures
+(see Fig. 5) and discretization types (see Fig. 6). Though objective measures could vary in whether they require
+discretization or not, the similar shapes of the plots in Fig. 5 demonstrate similar ranking performance across features.
+For example, in GCPN vs. ZINC comparison (Fig. 5 (a)), Molecular weight achieved the least score among features
+under all the different objective measures employed. The same is true for QED in the GraphAF vs. ZINC validation
+in Fig. 5 (b). We also validated the impact of the discretization types as shown in Fig.6 (a) and (b). In both cases,
+equal width and k-means discretizations provide similar patterns while equal frequency discretization demonstrated
+a slightly different pattern, particularly for features with skewed distribution as Aromaticity. Overall, all three types
+of discretization provide competitive performance scores, thereby validating the stability of our MPEGO framework.
+More ablation results are in Appendix E of the Supplementary Material.
+8
+
+ZINC
+0.008
+GCPN
+GraphAF
+0.006
+0.004
+0.002
+0.000
+0
+200
+400
+600
+800
+Molecular WeightDomain-agnostic and Multi-level Evaluation of Generative Models
+Table 2: HIE Scores for GCPN and GraphAF models trained on MOSES dataset
+G vs. T
+G vs. G
+GCPN vs.
+GraphAF vs.
+GCPN vs.
+Level
+ZINC
+ZINC
+GraphAF
+Aromaticity
+1.000
+1.000
+1.000
+ESOL
+0.419
+0.672
+0.491
+LogP
+0.219
+0.682
+0.319
+FIS
+Weight
+0.550
+0.491
+0.760
+QED
+0.471
+0.576
+0.735
+SCScore
+0.846
+0.865
+0.744
+SAFIS
+QED+LogP
+0.345
+0.629
+0.527
+GAFIS
+0.584
+0.714
+0.675
+Table 3: HIE Scores for MolGX and Regression Transformer (RT) models trained on CIRCA dataset
+G vs. T
+G vs. G
+MolGX vs.
+Transformer vs.
+MolGX vs.
+Level
+Fingerprint
+MOSES
+MOSES
+Transformer
+951226070
+1.000
+1.000
+1.000
+3218693969
+0.599
+0.876
+0.500
+2968968094
+1.000
+1.000
+1.000
+FIS
+882399112
+1.000
+1.000
+1.000
+2245384272
+0.998
+0.847
+0.845
+2246703798
+1.000
+1.000
+1.000
+3217380708
+0.411
+0.661
+0.226
+GAFIS
+0.858
+0.912
+0.796
+6
+Conclusion and Future Work
+We proposed MPEGO - a simple, generalizable, and model-agnostic evaluation framework of generative models
+validated for material discovery domain. MPEGO consists of two main performance evaluation blocks: Hierarchical
+Independence Evaluation (HIE) and Generation Frequency Analysis (GFA). HIE follows a bottom-up approach to
+quantify the generation performance of a model, starting from per sub-feature level (at the bottom) to the global
+aggregation of features (at the top). Thus, HIE provides a flexible performance evaluation of generative models.
+Particularly by evaluating the generation independence of models compared with the training data or other generative
+models using an objective measure set by a user. GFA is applied to detect and characterize divergent generation
+characteristics. Different from the existing evaluation platforms, MPEGO provides interpretable insights that aim to
+facilitate interactions with subject matter experts, which is crucial to develop Trustworthy AI solutions. The proposed
+MPEGO toolkit was validated with multiple datasets (ZINC-250K, MOSES, and CIRCA) and generative models
+trained on these datasets, including GCPN, GraphAF, MolGX, and RT. Conditioned on the training samples in these
+datasets, GCPN, GraphAF, and RT achieved higher generation independence, compared to their counterparts, GraphAF,
+GCPN, and MolGX models, respectively. Future work aims to evaluate generative models from different domains to
+further validate the domain-agnostic nature of the MPEGO framework. We also plan to utilize MPEGO to improve
+the efficiency of latent-based analyses, such as creativity characterization [31] and out-of-distribution detection [32].
+MPEGO will be natively integrated into the GT4SD, the Generative Toolkit for Scientific Discovery [29] and the source
+code and experiments will be available at: https://github.com/GT4SD/mpego.
+9
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+Table 4: Results from generation frequency analysis, validated across three datasets: ZINC250K (with GCPN and
+GraphAF models), MOSES (with GCPN and GraphAF models), and CIRCA (with MolGX and RT). We also compared
+the models headtohead. The q factor represents the multiplicative increase in the odds of the outcome in the identified
+group compared to the null hypothesis. Elapsed time is in seconds.
+Model
+Baseline
+Expected
+Subset
+Group size
+Observed
+q factor
+Odds ratio
+95% CI
+pvalue
+Score
+Elapsed
+GCPN
+ZINC
+0.5
+LogP≥
+1.31
+&
+Weight < 258.1
+5228
+0.888
+7.952
+13.98
+(12.75, 15.33)
+0.0
+1781.619
+9.21
+GraphAF
+0.5
+SCScore < 3.54 &
+QED
+≥
+0.64
+&
+Weight < 362.48
+8040
+0.797
+3.923
+9.14
+(8.54, 9.77)
+0.0
+1493.252
+10.99
+GraphAF
+ZINC
+0.5
+SCScore ≥ 2.62 &
+QED
+<
+0.64
+&
+Weight < 258.1
+4072
+0.946
+17.425
+27.71
+(24.11, 31.85)
+0.0
+1949.670
+8.96
+GCPN
+0.5
+QED < 0.5
+5135
+0.878
+7.164
+12.22
+(11.17, 13.37)
+0.0
+1647.798
+7.73
+GCPN
+MOSES
+0.5
+QED
+<
+0.5
+&
+2.36 ≤LogP< 3.91
+1422
+0.985
+66.714
+397.89
+(253.9, 623.53)
+0.0
+870.290
+4.00
+GraphAF
+0.5
+LogP
+≥
+3.91
+&
+Weight < 298.32
+1284
+0.992
+127.400
+525.01
+(279.16, 987.37)
+0.0
+825.489
+3.43
+GraphAF
+MOSES
+0.5
+Weight < 250.38 &
+−4.75
+≤ ESOL<
+−3.59
+915
+0.991
+113.375
+245.40
+(121.72, 494.77)
+0.0
+584.349
+4.21
+GCPN
+0.5
+LogP
+<
+3.91
+&
+ESOL < −4.75 OR
+ESOL ≥
+−3.59 &
+Weight < 250.38
+1297
+0.976
+40.839
+162.96
+(112.31, 236.46)
+0.0
+744.637
+3.33
+RT
+CIRCA
+0.5
+2245384272 ≥ 1.0
+603
+0.561
+1.275
+1.85
+(1.43, 2.39)
+0.0
+2.430
+1.92
+MolGX
+0.5
+3217380708 < 4.0 &
+3218693969 < 2.0
+353
+0.989
+87.250
+286.60
+(105.19, 780.82)
+0.0
+218.783
+2.27
+MolGX
+CIRCA
+0.5
+3217380708 ≥ 4.0
+484
+0.804
+4.095
+14.94
+(10.99, 20.31)
+0.0
+93.805
+2.30
+RT
+0.5
+3217380708 ≥ 4.0
+416
+0.935
+14.407
+61.39
+(39.48, 95.47)
+0.0
+186.404
+2.11
+References
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+Ceriotti. Machine learning unifies the modeling of materials and molecules. Science advances, 3(12):e1701816,
+2017.
+[2] O Anatole von Lilienfeld and Kieron Burke. Retrospective on a decade of machine learning for chemical discovery.
+Nature communications, 11(1):1–4, 2020.
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+generative models
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+generative models
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+Chanakya Ajit Ekbote, Jie Fu, Tianyu Zhang, Michael Kilgour, Dinghuai Zhang, et al. Biological sequence design
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+GCPN VS ZINC
+1.0
+0.8
+Score
+0.6
+Independence 
+0.4
+Yule-Y
+MI
+0.2
+IMeasure
+0.0
+Gini Index
+Wasserstein
+-0.2
+KS
+-0.4
+ticity
+ESOL
+Neight
+ScScore
+Objective MeasureGraphAF vs ZINC
+1.0
+0.8
+Independence Score
+0.6
+0.4
+Yule-Y
+0.2
+MI
+0.0
+J Measure
+-0.2
+Gini Index
+Wasserstein
+-0.4
+KS
+-0.6
+ticity
+ESOL
+Neight
+Scscore
+Objective MeasureGCPN VS ZINC
+1.0
+Score (Yule-Y)
+0.9
+0.8
+0.7
+Independce s
+0.6
+Equal Width
+0.5
+Equal Frequency
+KMeans
+0.4
+ESOL
+Nelght
+SCScore
+FeaturesGraphAF vs ZINC
+1.0
+0.9
+Independce Score (Yule-Y)
+0.8
+0.7
+0.6
+0.5
+0.4
+Equal Width
+0.3
+Equal Frequency
+0.2
+KMeans
+ESOL
+Neight
+scscore
+FeaturesDomain-agnostic and Multi-level Evaluation of Generative Models
+with gflownets. In International Conference on Machine Learning, pages 9786–9801. PMLR, 2022.
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+outlook. Angewandte Chemie International Edition, 59(52):23414–23436, 2020.
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+Stanislav Belyaev, Rauf Kurbanov, Aleksey Artamonov, Vladimir Aladinskiy, Mark Veselov, et al. Molec-
+ular sets (moses): a benchmarking platform for molecular generation models. Frontiers in pharmacology, 11:1931,
+2020.
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+goal-directed molecular graph generation. Advances in neural information processing systems, 31, 2018.
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+Pitera, Makoto Kogoh, Takumi Hongo, Yenwei Cheng, et al. Molecular inverse-design platform for material
+industries. In Proceedings of the 26th ACM SIGKDD International Conference on Knowledge Discovery & Data
+Mining, pages 2961–2969, 2020.
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+discover chemistry for biology. Journal of chemical information and modeling, 52(7):1757–1768, 2012.
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+detection in dermatology using input perturbation and subset scanning.
+In 2022 IEEE 19th International
+Symposium on Biomedical Imaging (ISBI), pages 1–4, 2022.
+12
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+Appendix A: Steps to prepare CIRCA dataset
+Initially, we carried out a patent search in CIRCA for ionic photoacid generators (PAGs) that are sulfonium or iodonium
+salts. We extracted anionic components of the sulfonium and iodonium salts, finding 493 unique anions. We ran
+additional CIRCA search to enrich the dataset with anions relevant to photolithography using the following query:
+“PAG” OR “photo-acid generator” OR “photo initiator” OR “acid generating agent” OR “photocationic initiator” OR
+“photocationic polymerization initiator” OR “cationic initiator” OR “cationic type initiator” The additional search
+returned ionic compounds with 1398 anions, 189 of these 1398 anions were among the initial 493 anions. We processed
+the set of collected anions, keeping only anions with charge -1 and more than 1 carbon atom, reducing the dataset to
+1086 anions. Out of 1086 anions, we were able to evaluate environmental and toxicological properties of 866 anions,
+that comprise the dataset used in this study
+Appendix B: Experimental Setup for GCPN and GraphAF training
+Table 5: Summary of hyper-parameters set-up to train the two graph-based models: GCPN and GraphAF using
+ZINC-250K dataset.
+Models
+Setting
+GCPN
+GraphAF
+Input Dimension
+18
+9
+Number of relation
+3
+3
+Batch normalization
+False
+True
+Atom types
+[6-9, 15-17, 35, 53]
+[6-9, 15-17, 35, 53]
+Hidden dimensions
+[256, 256, 256, 256]
+[ 256, 256, 256]
+Appendix C: Descriptions of features
+Table 6: Features extracted from samples generated from GCPN and GraphAF models trained on ZINC-250k and
+MOSES datasets.
+Feature
+Description
+QED
+Qualitative Estimate of Drug-likeness
+ESOL
+Estimated SOLubility
+SCScore
+Synthetic Complexity Score
+SAS
+Synthetic Accessibility Score.
+Scaffold
+A molecule is identical to its scaffold
+Ring
+A molecule contains a ring structure
+LargeRing
+A molecule contains a ring structure with more than 6
+atoms
+Aromatic
+A molecule contains an aromatic ring
+SteroCharacter
+SMILES string contains a stereochemistry string.
+Stereocenter
+Whether a molecule contains a stereocenter,
+Heterocycle
+A molecule has at least one ring with at least two different
+atoms
+Lipinski
+A molecule adheres to the Lipinski’s rule of five.
+HBondDonor
+A molecule has not more than 5 hydrogen bond donors
+HBondAcceptor
+A molecule has not more than 10 hydrogen bond acceptors
+MolecularWeight
+The molecular mass in Daltons
+LogP
+Logarithmic partition coefficient
+Multiple fingerprints, informing the presence of particular structural elements in a given molecule, are extracted from
+CIRCA dataset and samples generated from MolGX and RT models. The following fingerprints are selected for
+evaluation in this study due to their higher variance across the datasets.
+13
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+Fingerprint
+Molecular
+Subgraph
+2245384272
+951226070
+3217380708
+2246703798
+882399112
+3218693969
+2968968094
+Table 7: Fingerprints (and their molecular subgraph visualizations) selected for evaluations in CIRCA datasets and its
+associated models: MolGX and RT.
+Appendix D: List of Objective measures
+The MPEGO framework could handle different objective measures to quantify sub-feature or feature-level-based
+association between generated and training datasets or between two generated datasets. The list of objective measures
+implemented in the MPEGO framework includes: Odds ratio, Yule’s Y Coefficients, Yule’s Q coefficients, Mutual Infor-
+mation,J Measure, Gini Index, Wasserstein Distance, Kstest, Support, Confidence, Laplace, Collective Strength, Kappa,
+Conviction, Interest, Cosine, Piatetsky Shapiro, Certainty Factor, Added Value, Jaccard and Kolmogorov–Smirnov.
+14
+
+Domain-agnostic and Multi-level Evaluation of Generative Models
+Appendix E: Ablation results
+Similar ablation studies are applied for head-to-head comparison of models as shown in Fig. 7, where the robustness
+of the MPEGO framework is demonstrated across varieties of objective measures (Fig. 7 (a)) and discretization types
+(Fig. 7 (b)).
+(a)
+(b)
+Figure 7: Results from the ablation study on the variations of (a) objective measures and (b) discretization types on the
+FIS values, validated with a head-to-head comparison of GCPN and GraphAF models, trained on ZINC-250K.
+15
+
+GCPN vs GraphAF
+1.0
+0.8
+Independence Score
+0.6
+0.4
+Yule-Y
+0.2
+MI
+J Measure
+0.0
+Gini Index
+-0.2
+Wasserstein
+-0.4
+KS
+ticity
+ESOL
+Veiont
+Scscore
+Objective MeasureGCPN vs GraphAF
+1.0
+0.9
+0.8
+0.7
+0.6
+0.5
+Equal Width
+Equal Frequency
+0.4
+KMeans
+0.3
+ticity
+ESOL
+Neight
+Scscore
+Features
\ No newline at end of file
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+page_content='com  Jannis Born  IBM Research - Zurich  jab@zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com  Celia Cintas  IBM Research - Africa  celia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='cintas@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com  William Ogallo  IBM Research - Africa  william.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='ogallo@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com  Dmitry Zubarev  IBM Research - Almaden  dmitry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='zubarev@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com  Matteo Manica  IBM Research - Zurich  tte@zurich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com  Komminist Weldemariam  IBM Research - Yorktown Heights  kommy@ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com  ABSTRACT  While the capabilities of generative models heavily improved in different domains (images, text,  graphs, molecules, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' ), their evaluation metrics largely remain based on simplified quantities  or manual inspection with limited practicality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' To this end, we propose a framework for Multi-  level Performance Evaluation of Generative mOdels (MPEGO), which could be employed across  different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' MPEGO aims to quantify generation performance hierarchically, starting from  a sub-feature-based low-level evaluation to a global features-based high-level evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' MPEGO  offers great customizability as the employed features are entirely user-driven and can thus be highly  domain/problem-specific while being arbitrarily complex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', outcomes of experimental procedures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  We validate MPEGO using multiple generative models across several datasets from the material  discovery domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' An ablation study is conducted to study the plausibility of intermediate steps  in MPEGO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Results demonstrate that MPEGO provides a flexible, user-driven, and multi-level  evaluation framework, with practical insights on the generation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The framework, source code,  and experiments will be available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com/GT4SD/mpego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Keywords Generative models · Evaluation · Data-centric AI · Foundation models  1  Introduction  Machine Learning (ML) methods, particularly generative models, are effective in addressing critical problems across  different domains, which includes material sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Examples include the design of novel molecules by combining  data-driven techniques and domain knowledge to efficiently search the space of all plausible molecules and generate  new and valid ones [1, 2, 3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Traditional high-throughput wet-lab experiments, physics-based simulations, and  bioinformatics tools for the molecular design process heavily depend on human expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' These processes require  significant resource expenditure to propose, synthesize and test new molecules, thereby limiting the exploration  space [5, 6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  For example, generative models have been applied to facilitate the material discovery process by employing inverse  molecular design problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' This approach transforms the conventional and slow discovery process by mapping the  desired set of properties to a set of structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The generative process is then optimized to encourage the generation of  molecules with those selected properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Countless approaches have been suggested for such tasks, most prominently  VAEs with different sampling techniques [8, 9, 10]), GANs [11, 12], diffusion models [13], flow networks [14] and  Transformers [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='08750v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='LG]  20 Jan 2023  Domain-agnostic and Multi-level Evaluation of Generative Models  Though the generation capability has been tremendously improved recently, the quantitative evaluation of these  generative models in different domains remains a grand challenge [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Some of the reasons include the multi-objective  nature of real discovery problems, the intricacy of evaluating relevant features in-silico, and the lack of widely accepted  domain- and model-agnostic evaluation frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' As a result, existing benchmarks and toolkits in material sciences,  such as MOSES [17] or GuacaMol [18] include limited metrics, such as validity and uniqueness, that lack the capacity  to evaluate the complex nature of the generation process (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', interactions of multiple properties), thereby less effective  to provide meaningful insights to subject matter experts (SMEs) to facilitate practical impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  In this paper, we introduce a Multi-level Performance Evaluation of Generative mOdels (MPEGO) framework (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 1), which aims to hierarchically characterize and quantify the capability of generative models, using material  discovery domain as a use case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' To that end, MPEGO is a model- and domain-agnostic framework, and its core design  is derived from two main requirements: representative examples (of training and generated samples) and one or multiple  features (extracted from these samples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Metrics derived from MPEGO are also interpretable and provide multi-level  abstractions of the generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Specifically, the contributions of this paper are as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We provide a multi-level and domain-agnostic evaluation of generative models, starting with sub-feature- or  feature-based low-level evaluation to global features based high level-evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We devise a generation frequency analysis that aims to identify and characterize subsets of samples generated  with extreme frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We validate MPEGO framework on multiple generative models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', GCPN [19], GraphAF [20], MolGX [21]  and Regression Transformer [15], and multiple datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', ZINC-250K [22] and MOSES [17] and CIRCA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We conduct ablation studies to analyze MPEGO’s sensitivity to design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  2  Related Work  The state-of-the-art evaluation approaches for generative models aim to quantify pre-determined requirements, such as  diversity and validity, using a variety of metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Frechet ChemNet Distance (FCD) [23] is one of such metrics, and it  measures the distance between hidden representations drawn from sets of generated and training samples in the material  discovery domain, which is limited in providing sub-feature or feature-level evaluation of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' GuacaMol [18] is  one of the early benchmark platforms for new molecule discovery, which aims to evaluate generative models across  different tasks, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', fidelity and novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Molecular Sets (MOSES) [17] is another benchmarking framework, which  provides training and testing datasets, and a set of metrics to evaluate the quality and diversity of generated structures to  standardize training and model comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Furthermore, automated characterization of subsets of samples, generated with more or less frequency, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', generation  frequency analysis, also still remains challenging as the focus is more on latent- or feature-based evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Overall,  the challenges associated with evaluating generative models could be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' First, multiple evaluation  metrics are model-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' For example, FCD [23] depends on latent representation, and Maximum-mean discrep-  ancy [24] is more specifically used to evaluate graph-based generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' State-of-the-art metrics also suffer from  limited generalizability (across different levels of feature interactions) and interpretability, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', by domain experts,  which is critical to achieve trustworthy AI solutions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' In addition, existing evaluation metrics are susceptible to  potential flaws in predictive models used in goal-oriented or constrained generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Moreover, existing evaluation  strategies lack a generic and standalone evaluation metric that combines both distributional metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', uniqueness  and diversity) and property-based metrics that score single property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='. The dependency on a single-constraint objective  lacks a principled approach to incorporate multiple target features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' This becomes a significant challenge when a single  and inaccurate evaluation metric is used, which oversimplifies real discovery problems and hence less practical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  3  Proposed: MPEGO Framework  The proposed MPEGO framework (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 1) aims to provide an effective and multi-level characterization of generative  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The multi-level evaluation of MPEGO (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 2) starts from sub-feature-based low-level evaluation and  their step-by-step aggregation to provide high-level evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' In this section, we first, formulate the critical research  questions MPEGO framework is designed to address, followed by the details on its core components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  2  Domain-agnostic and Multi-level Evaluation of Generative Models  Figure 1: Overview of the MPEGO framework  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1  Problem Statement  Let G1, G2, · · · , Gk, · · · , GK be datasets comprising samples generated from K black-box generative models  (Θ1, Θ2, · · · , Θk, · · · , ΘK) trained on a dataset T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Can we evaluate the generation capability of each Θi in a scalable,  easily interpretable, and multi-objective manner?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Specifically, we aim to address two questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Q1: Given a set of features characterizing the samples, how do we quantify the generation capability of each model  compared to another model or the training data, based on one or more of these features, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', at different levels  of abstractions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Q2: What are the characteristics of samples being generated with extreme frequencies (least or most) by each of  the generative models, compared with an other model or the training data, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', generation frequency analysis?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  To address Q1, we propose a Hierarchical Independence Evaluation (HIE) that aims to quantify the performance of  generative models at different levels of feature interactions hierarchically, starting with a sub-feature level evaluation  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', a specific range of a feature) to the global aggregation of multiple features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' To address Q2, we employ multi-  dimensional subset scanning (MDSS) [26] that aims to automatically identify and characterize over- and under-generated  subsets of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='2  Feature Extraction and Pre-processing  Given representative examples of generated and training samples, MPEGO starts with the extraction of M features from  these samples, F = {f1, f2, · · · , fm, · · · , fM}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The type of feature values could be binary, continuous, or categorical,  and a further pre-processing could be applied in the follow up steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' For example, discretization of continuous features  is required for sub-feature-level performance evaluation, and MPEGO provides different discretization types, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', equal  width, equal frequency or based on k-means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='3  Hierarchical Independence Evaluation (HIE)  HIE follows a bottom-up approach (from a sub-feature to global aggregation levels), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 2, and it evaluates  the performance generative models at different levels of feature interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The lowest level of evaluation in HIE is  the sub-feature level independence score (SIS), which aims to quantify the performance of generative models across  different ranges or unique values per feature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', based on a a molecular weight range of 200 − 300 Daltons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The  second layer in HIE represents feature-level independence score (FIS), which quantifies the generation performance for  each feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' To this end, FIS could be computed via aggregation of SIS values, thereby providing a weighting strategy  for SIS scores per feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' FIS values could also be computed directly, without aggregating SIS values, using a different  choice of objective measure is directly applied on the whole feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='The proposed MPEGO framework is flexible to  utilize different objective measures (see Appendix D of the Supplementary Material for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Below we describe the  computation of SIS and FIS values, using Yule’s Y coefficient [27] as selected objective measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that Gk vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' T  refers to a case where the comparison is between the kth generative model and the training set T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' On the other hand,  Gk vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Gj refers to a case when the evaluation between the jth and kth models, where j ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' However, we stick with  the Gk vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' T comparison below in order to ease readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  1https://circa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com/  3  Hierarchical  Independence  Evaluation  Training  (HIE)  Pre-processing  Features  Extraction  Model  Cm = Kfm}l  Characterisation and  m  Sm=1  Ranking  Generation  Frequency Analysis  Generated  (GFA)Domain-agnostic and Multi-level Evaluation of Generative Models  Figure 2: Details of Multi-level evaluation component of the MPEGO framework that comprises Hierarchical Indepen-  dence Evaluation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', SIS, FIS, SAFIS and GAFIS) and Generation Frequency Analysis (GFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' A(· · ·) represents  aggregation operation, and anomalous subset refers to the logical combinations of features that characterize samples  generated with extreme frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Let fm ∈ F is a feature with Cm unique values or ranges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', fm = {f u  m}, u ∈ [1, 2, · · · , Cm].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that Cm is the  number of unique values for categorical features or the number of bins after discretization of continuous features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' SIS  computation requires the stratification of both the generated (Gk) and training (T ) datasets per each unique value/range  f u  m resulting Gu  km and T u  m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The complimentary subsets are then �  Gu  km and �  T u  m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that  �  Gu  km = Gu  km|(fm ̸= f u  m) = Gk − Gu  km and �  T u  m = T u  m|(fm ̸= f u  m) = T − T u  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Accordingly, a 2 × 2 pivot table is  generated for each f u  m as:  (fm = f u  m)  (fm ̸= f u  m)  Gkm  α  β  Tm  δ  γ  where α is the number of generated samples in Gu  km that are characterized by the feature value fm = f u  m, β is the  number of generated samples in �  Gu  km with fm ̸= f u  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Similarly, δ and γ are the numbers of training samples that satisfy  fm = f u  m in T u  m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that α + β are the total numbers of generated samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', |Gk|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Similarly, δ + γ is  the number of training samples, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', |T |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Then Yule’s Y coefficient is computed from the pivot table as ou  km ∈ [−1, 1]:  ou  km =  �  P(Gu  km)P( �  T u  m) −  �  P( �  Gu  km)P(T u  m)  �  P(Gu  km)P( �  T u  m) +  �  P( �  Gu  km)P(T u  m)  (1)  ou  km =  √αγ − √βδ  √αγ + √βδ  (2)  SIS is then computed from ou  km value as Iu  km = 1 − |ou  km|, where Iu  km ∈ [0, 1] and higher Iu  km reflects higher  independence between Gk and T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', SIS = 1 represents complete independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Feature-level Independence Score  (FIS), provides feature-based evaluation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', higher abstraction than SIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Depending on the objective measure, FIS  could be computed as 1) via a weighted aggregation of SIS values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', Ikm = �Cm  u=1 λu  kmIu  km where �Cm  u=1 λu  km = 1  and each λu  km weights the SIS value of f u  m, or 2) via straightforward computation, without using SIS values, when  4  GlobalAggregationof  A(I,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=',Im)  Feature-based Independence  Score (GAFIS)  Selective Aggregation of  A(I1, I2)  A(Im,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='., IMm)  Feature-based Independence  Score (SAFiS)  Feature-based  I1  I2  Im  IM  IndependenceScore  (FIS)  Sub-feature based  I1  Igi  II  1  Im  IndependenceScore  (SIS)  Inputfeatures  f1  2  fm  Cm = Ifm}l,m E [1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', M] :  Anomalous  Generation Frequency  subset  Analysis(GFA)Domain-agnostic and Multi-level Evaluation of Generative Models  the objective measure is directly applied on each feature without discretization, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', using Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The  third layer of HIE is Selective Aggregation of Feature-level Independence Score (SAFIS), which aims to aggregate  FIS values from R < M selected features in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' For example, features including scaffolding, fingerprints, aromaticity,  and the number of rings could be selected to reflect the structural details of molecules in material discovery domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  The last layer of HIE is Global Aggregation of Feature-level Independence Score (GAFIS), which is computed via a  weighted aggregation of all the FIS values in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that SAFIS and GAFIS are computed as ˆI = �R  r=1 ηrIkr, where  �R  r=1 ηr = 1, and R < M for SAFIS and R = M for GAFIS computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='4  Generation Frequency Analysis (GFA)  Generative models trained on the same dataset will hardly generate samples with exact characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Thus, there  is a potential over- or under-generation of samples with certain characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' To this end, we employ automated  stratification of samples using multi-dimensional subset scanning (MDSS) [26, 28] to identify subset of samples  generated with divergent frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Specifically, to identify samples generated with divergent rates by model Θk,  compared to the training set T , we first merge the corresponding datasets as D = Gk ∪ T , and an outcome label (y)  is generated, such that yi = 1 for a sample in Gk and yi = 0 for a sample in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' If there are Ng = |Gk| generated  and Nt = |T | training samples in D, the expectation of generated samples in D is eg =  Ng  Ng+Nt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Thus, GFA aims to  identify a group of samples with extreme deviations in their generation rate compared to eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The deviation between  the expectation and observation is evaluated by maximizing a Bernoulli likelihood ratio scoring statistic, Γ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The  null hypothesis assumes that the odds of the generated sample in any subgroup S is similar to the expected, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=',  H0 : odds(S) =  eg  1−eg ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' while the alternative hypothesis assumes a constant multiplicative increase in the odds of the  generated samples in S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', H1 : odds(S) = q  eg  1−eg where q ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that q > 1 for over-generated subset, and  0 < q < 1 for under-generated subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The divergence score for a subgroup (S) with reference D is formulated as,  Γ(S, D) and computed as:  Γ(S, D) = max  q  log(q)  �  i∈S  yi − Ns ∗ log(1 − eg + qeg),  (3)  where Ns is the number of samples in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The divergent subset, S, identification is iterated until convergence to a local  maximum is found, and the global maximum is subsequently optimized using multiple random restarts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  4  Experimental Setup  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1  Training Datasets  We employed three different datasets in material discovery domain to validate our MPEGO framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  ZINC-250K  We utilize the publicly available ZINC-250K2 dataset, which contains 249, 455 small molecules in  Simplified Molecular-Input Line-Entry System (SMILES) representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Details on ZINC tool are available in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  MOSES  The benchmark platform MOSES [17], besides implementing popular molecular generation models and  metrics, MOSES contains a refined dataset from ZINC3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The dataset has approximately 2M molecules in total, filtered  by certain parameters such as molecular weight ranges, and number of rotatable bonds, among others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  CIRCA  We used IBM’s Chemical Information Resources for Cognitive Analytics (CIRCA) platform4 to construct  a dataset of organic salts relevant to production of semiconductors via photolitography with chemical amplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Finally, we were able to evaluate environmental and toxicological properties of 866 anions, that comprise the dataset  used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Steps conducted to obtain final version of the dataset could be found in Appendix A in the  Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='2  Generative Models  We utilized multiple generative models for our validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Particularly, Graph Convolutional Policy Network  (GCPN) [19] and a Flow-based Autoregressive (GraphAF) [20] were trained separately on ZINC-250K and MOSES  2https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='kaggle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com/datasets/basu369victor/zinc250k  3https://zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='docking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='org/  4https://circa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='ibm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com/  5  Domain-agnostic and Multi-level Evaluation of Generative Models  Figure 3: Visualization of the CIRCA dataset and some of the structural fingerprints relevant to MPEGO analysis (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  The CIRCA dataset is represented as a similarity network, where nodes correspond to anions and links connect nodes if  Dice similarity of the respective anions is at least 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Node colors encode clustering recovered via modularity analysis  and node size encodes the number of connections of the node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Chemical structures are included for the most connected  nodes in the main clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Subsequently, 10, 000 valid molecules were generated from each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We rely on GT4SD [29] for model  implementations (experimental set-ups shown in Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Similarly, from the MOSES-trained GCPN and GraphAF,  14665 and 1680 molecules were generated respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' For the CIRCA data, we employed MolGX [21] and the  Regression Transformer [15]) and generated 5000 new anions with MolGX and 32617 new anions with Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Note that balanced number of samples were selected across datasets during our experimentation, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', 10000, 1680 and  5000 samples were randomly selected for ZINC-250K-based, MOSES-based, CIRCA-based evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='3  Feature Extraction and Preprocessing  We have extracted the following six features from SMILES representations of molecules in ZINK-250K and MOSES  datasets and their corresponding generated datasets from GCPN and GrpahAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The following features are selected for  the evaluation: Aromaticity, ESOL, LogP, Weight, QED, SCScore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' On the other hand, we extracted fingerprints (see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 3) that reflect structural details from CIRCA dataset and MolGX and Transformer generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Full details  of features can be found in the Appendix C of the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' In cases when discretization of continuous  features is required to compute the SIS values, we employ Yule’s Y coefficient as our default objective measure as it  satisfies multiple key requirements [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We also employ equal frequency discretization type as it better handles outliers  in the data, with five bins for ZINC and with three bins for CIRCA based evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' While aggregating features for  SAFIS and GAFIS computations using normal averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that the proposed approach is flexible to utilize different  discretization types, weighting strategies, and objective measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='4  Evaluation Metrics  Our evaluation metrics include HIE’s SIS, FIS, SAFIS, and GAFIS values that quantify the generation independence of  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The independence score is obtained by normalizing the objective measure values to [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note independence  score = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='0 represents complete independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We also utilize the histogram of features to provide a qualitative  comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The characterization of generation frequency analysis involves using the logical combination of feature  values to describe the identified subgroup, the size of the subgroup Ns, the odds ratio between S and �S = D − S, 95%  Confidence Interval (CI) and empirical p value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We also reported the divergence score of the identified group from the  expectation in GFA and elapsed time to identify the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' All the experiments are conducted on a desktop machine,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='9 GHz Quad-Core Intel Core i7 (processor), and 16 GB 2133 MHz LPDDR3 (memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  6  Fingerprint ID  Molecular subgraph  951226070  3218693969  3217380708Domain-agnostic and Multi-level Evaluation of Generative Models  Table 1: HIE scores for GCPN and GraphAF models trained on ZINC-250K dataset  G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' T  G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' G  GCPN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  GraphAF vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  GCPN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Level  ZINC  ZINC  GraphAF  Aromaticity  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='000  ESOL  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='912  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='856  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='853  LogP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='951  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='942  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='906  FIS  Weight  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='656  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='713  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='809  QED  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='946  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='666  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='662  SCScore  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='825  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='752  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='866  SAFIS  QED+LogP  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='949  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='804  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='784  GAFIS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='882  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='821  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='850  5  Results and Discussion  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1  Hierarchical Independence Evaluation  Table 1 provides extended FIS values across each of the features considered for comparing GCPN and GraphAF models  with the training ZINC-250K (Gvs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='T ) and between each other - head-to-head (Gvs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' GCPN’s GAFIS value of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='882, demonstrating competitive generation independence with GraphAF with GAFIS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' This is further shown  with their head-to-head comparison, with Aromaticity (FIS = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='0) and LogP (FIS = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='906) features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Divergent  characteristics are also demonstrated when the two models were evaluated based on QED, Weight and ESOL features,  achieving FIS of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='853, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='809, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='662, respectively in their head-to-head comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' SAFIS values are shown as  an example of aggregation synthetic metrics, QED and LogP, where GCPN model achieves superiority compared to  GraphAF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' GAFIS values in the bottom row are derived from global aggregation of FIS values above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Overall, the results  show the flexibility of the MPEGO framework to evaluate models across different levels of feature interactions and  comparison baselines, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', training data or another generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Furthermore, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 4 demonstrates the benefits of sub-feature level evaluation (SIS) of GCPN and GraphAF models  trained on ZINC-250K, using Molecular Weight as a feature example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Five ranges of the molecular weight in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 4 (a)  resulted from the discretization of the feature necessary for SIS evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Similarly to Table 1, the comparison is  performed between generative models and with the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Results demonstrate that GCPN and GraphAF  generated molecules with similar weight characteristics but distinctively different at a few particular ranges, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=', GCPN  generated molecules similar to ZINC-250K with range [258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1, 308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='39) whereas GraphAF showed better resemblance  with ZINC-250K at ranges [201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='63, 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1) and ≥ 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='47 Daltons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Divergent sub-level generation characteristics is  also encoded by lower G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' G independence score at those ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The histogram plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 4 (b) qualitatively  compliments the insights from SIS values in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 4 (a) , where GraphAF is shown to generate more molecules with  extreme weight values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Note that such sub-feature level insights in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 4 are unique to our proposed framework as they  are currently limited in the state-of-the-art of generative model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  The results of comparisons of GCPN and GraphAF trained on MOSES dataset (Table 2) and MolGX and RT trained  on CIRCA dataset (Table 3) demonstrate MPEGO’s capabilities further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Comparisons between Table 1 with Table 2  show while MOSES is used as a training, GraphAF achieved more independence than when ZINC-250K is used,  compared GCPN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Between MolGX and the RT, trained on CIRCA, molecules generated from the RT achieved a higher  resemblance with CIRCA than MolGX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' In their head-to-head comparison, MolGX and RT generated mostly divergent  structural details, as it is demonstrated by very low FIS values for the majority of the fingerprints in G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' G column of  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='2  Generation Frequency Analysis  Table 4 shows the divergence generation frequency of models, compared to the training or other generated set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' For  example, the GCPN model generated molecules with higher LogP (≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='31) and lower weight (< 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1 Daltons)  compared to the training ZINC-250K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' When we compare the two models head-to-head, GCPN tends to generate  7  Domain-agnostic and Multi-level Evaluation of Generative Models  G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' T  G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' G  GCPN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  GraphAF vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  GCPN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Level  Weight Range  ZINC  ZINC  GraphAF  <201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='63  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='406  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='322  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='859  [201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='63 , 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='631  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='840  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='778  SIS  [258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='1 , 308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='39)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='986  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='824  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='810  [308.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='39 , 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='47)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='671  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content='962  ≥361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='47  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='586  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='939  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='638  (a)  (b)  Figure 4: (a) Example of Molecular Weight-based SIS values that provide sub-feature level evaluation of GCPN and  GraphAF models trained on ZINC-250K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' (b) histogram densities to provide qualitative visualization of the SIS values  in (a)  molecules with higher QED (≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='64) values compared to GraphAF (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' When MOSES data is used as validation,  LogP values become the differentiation factor as GraphAF tends to generate lower values (< 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='91) compared to  GCPN (≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='91).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' On CIRCA dataset, MolGX generates samples with a higher occurrence of ’3217380708’ fingerprint  compared to the RT model and the training CIRCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The size of the identified group, along with the multiplicative factor  (q) and the odds ratio values, confirm the significance of the identified divergent generation in our generation frequency  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='3  Ablation Study  We conducted ablation studies to validate the robustness of MPEGO for different design choices of objective measures  (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 5) and discretization types (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Though objective measures could vary in whether they require  discretization or not, the similar shapes of the plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 5 demonstrate similar ranking performance across features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  For example, in GCPN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' ZINC comparison (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 5 (a)), Molecular weight achieved the least score among features  under all the different objective measures employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The same is true for QED in the GraphAF vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' ZINC validation  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content=' We also validated the impact of the discretization types as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='6 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' In both cases,  equal width and k-means discretizations provide similar patterns while equal frequency discretization demonstrated  a slightly different pattern, particularly for features with skewed distribution as Aromaticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Overall, all three types  of discretization provide competitive performance scores, thereby validating the stability of our MPEGO framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  More ablation results are in Appendix E of the Supplementary Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content='796  6  Conclusion and Future Work  We proposed MPEGO - a simple, generalizable, and model-agnostic evaluation framework of generative models  validated for material discovery domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' MPEGO consists of two main performance evaluation blocks: Hierarchical  Independence Evaluation (HIE) and Generation Frequency Analysis (GFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' HIE follows a bottom-up approach to  quantify the generation performance of a model, starting from per sub-feature level (at the bottom) to the global  aggregation of features (at the top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Thus, HIE provides a flexible performance evaluation of generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Particularly by evaluating the generation independence of models compared with the training data or other generative  models using an objective measure set by a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' GFA is applied to detect and characterize divergent generation  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Different from the existing evaluation platforms, MPEGO provides interpretable insights that aim to  facilitate interactions with subject matter experts, which is crucial to develop Trustworthy AI solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The proposed  MPEGO toolkit was validated with multiple datasets (ZINC-250K, MOSES, and CIRCA) and generative models  trained on these datasets, including GCPN, GraphAF, MolGX, and RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Conditioned on the training samples in these  datasets, GCPN, GraphAF, and RT achieved higher generation independence, compared to their counterparts, GraphAF,  GCPN, and MolGX models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Future work aims to evaluate generative models from different domains to  further validate the domain-agnostic nature of the MPEGO framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We also plan to utilize MPEGO to improve  the efficiency of latent-based analyses, such as creativity characterization [31] and out-of-distribution detection [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  MPEGO will be natively integrated into the GT4SD, the Generative Toolkit for Scientific Discovery [29] and the source  code and experiments will be available at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='com/GT4SD/mpego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  9  Domain-agnostic and Multi-level Evaluation of Generative Models  Table 4: Results from generation frequency analysis, validated across three datasets: ZINC250K (with GCPN and  GraphAF models), MOSES (with GCPN and GraphAF models), and CIRCA (with MolGX and RT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We also compared  the models headtohead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The q factor represents the multiplicative increase in the odds of the outcome in the identified  group compared to the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Elapsed time is in seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Model  Baseline  Expected  Subset  Group size  Observed  q factor  Odds ratio  95% CI  pvalue  Score  Elapsed  GCPN  ZINC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content=' Advances in Neural Information Processing Systems, 33:4320–4332,  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content=' Data-driven molecular design for discovery  and synthesis of novel ligands: a case study on SARS-CoV-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content='  10  Domain-agnostic and Multi-level Evaluation of Generative Models  (a)  (b)  Figure 5: Different objective measures provided similar ranking per feature-level performance score (FIS) of the  generative models  (a)  (b)  Figure 6: Different discretization types provided similar ranking per feature-level performance score (FIS) of the  generative models  [10] Jannis Born, Tien Huynh, Astrid Stroobants, Wendy D Cornell, and Matteo Manica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Active site sequence  representations of human kinases outperform full sequence representations for affinity prediction and inhibitor  generation: 3d effects in a 1d model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Journal of Chemical Information and Modeling, 62(2):240–257, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content=' De novo generation  of hit-like molecules from gene expression signatures using artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Nature communications,  11(1):1–10, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content='  Mol-cyclegan: a generative model for molecular optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Journal of Cheminformatics, 12(1):1–18, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content=' Geodiff: A geometric diffusion  model for molecular conformation generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content=' Biological sequence design  11  GCPN VS ZINC  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content='4  Equal Width  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content=' Selecting the right objective measure for association  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+page_content='  [31] Celia Cintas, Payel Das, Brian Quanz, Girmaw Abebe Tadesse, Skyler Speakman, and Pin-Yu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Towards  creativity characterization of generative models via group-based subset scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' arXiv preprint arXiv:2203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='00523,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  [32] Hannah Kim, Girmaw Abebe Tadesse, Celia Cintas, Skyler Speakman, and Kush Varshney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Out-of-distribution  detection in dermatology using input perturbation and subset scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  In 2022 IEEE 19th International  Symposium on Biomedical Imaging (ISBI), pages 1–4, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  12  Domain-agnostic and Multi-level Evaluation of Generative Models  Appendix A: Steps to prepare CIRCA dataset  Initially, we carried out a patent search in CIRCA for ionic photoacid generators (PAGs) that are sulfonium or iodonium  salts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We extracted anionic components of the sulfonium and iodonium salts, finding 493 unique anions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We ran  additional CIRCA search to enrich the dataset with anions relevant to photolithography using the following query:  “PAG” OR “photo-acid generator” OR “photo initiator” OR “acid generating agent” OR “photocationic initiator” OR  “photocationic polymerization initiator” OR “cationic initiator” OR “cationic type initiator” The additional search  returned ionic compounds with 1398 anions, 189 of these 1398 anions were among the initial 493 anions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' We processed  the set of collected anions, keeping only anions with charge -1 and more than 1 carbon atom, reducing the dataset to  1086 anions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' Out of 1086 anions, we were able to evaluate environmental and toxicological properties of 866 anions,  that comprise the dataset used in this study  Appendix B: Experimental Setup for GCPN and GraphAF training  Table 5: Summary of hyper-parameters set-up to train the two graph-based models: GCPN and GraphAF using  ZINC-250K dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Models  Setting  GCPN  GraphAF  Input Dimension  18  9  Number of relation  3  3  Batch normalization  False  True  Atom types  [6-9, 15-17, 35, 53]  [6-9, 15-17, 35, 53]  Hidden dimensions  [256, 256, 256, 256]  [ 256, 256, 256]  Appendix C: Descriptions of features  Table 6: Features extracted from samples generated from GCPN and GraphAF models trained on ZINC-250k and  MOSES datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Feature  Description  QED  Qualitative Estimate of Drug-likeness  ESOL  Estimated SOLubility  SCScore  Synthetic Complexity Score  SAS  Synthetic Accessibility Score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Scaffold  A molecule is identical to its scaffold  Ring  A molecule contains a ring structure  LargeRing  A molecule contains a ring structure with more than 6  atoms  Aromatic  A molecule contains an aromatic ring  SteroCharacter  SMILES string contains a stereochemistry string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Stereocenter  Whether a molecule contains a stereocenter,  Heterocycle  A molecule has at least one ring with at least two different  atoms  Lipinski  A molecule adheres to the Lipinski’s rule of five.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  HBondDonor  A molecule has not more than 5 hydrogen bond donors  HBondAcceptor  A molecule has not more than 10 hydrogen bond acceptors  MolecularWeight  The molecular mass in Daltons  LogP  Logarithmic partition coefficient  Multiple fingerprints, informing the presence of particular structural elements in a given molecule, are extracted from  CIRCA dataset and samples generated from MolGX and RT models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The following fingerprints are selected for  evaluation in this study due to their higher variance across the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  13  Domain-agnostic and Multi-level Evaluation of Generative Models  Fingerprint  Molecular  Subgraph  2245384272  951226070  3217380708  2246703798  882399112  3218693969  2968968094  Table 7: Fingerprints (and their molecular subgraph visualizations) selected for evaluations in CIRCA datasets and its  associated models: MolGX and RT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  Appendix D: List of Objective measures  The MPEGO framework could handle different objective measures to quantify sub-feature or feature-level-based  association between generated and training datasets or between two generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' The list of objective measures  implemented in the MPEGO framework includes: Odds ratio, Yule’s Y Coefficients, Yule’s Q coefficients, Mutual Infor-  mation,J Measure, Gini Index, Wasserstein Distance, Kstest, Support, Confidence, Laplace, Collective Strength, Kappa,  Conviction, Interest, Cosine, Piatetsky Shapiro, Certainty Factor, Added Value, Jaccard and Kolmogorov–Smirnov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  14  Domain-agnostic and Multi-level Evaluation of Generative Models  Appendix E: Ablation results  Similar ablation studies are applied for head-to-head comparison of models as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 7, where the robustness  of the MPEGO framework is demonstrated across varieties of objective measures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 7 (a)) and discretization types  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content=' 7 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  (a)  (b)  Figure 7: Results from the ablation study on the variations of (a) objective measures and (b) discretization types on the  FIS values, validated with a head-to-head comparison of GCPN and GraphAF models, trained on ZINC-250K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='  15  GCPN vs GraphAF  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='8  Independence Score  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='4  Yule-Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='2  MI  J Measure  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='0  Gini Index  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='2  Wasserstein  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='4  KS  ticity  ESOL  Veiont  Scscore  Objective MeasureGCPN vs GraphAF  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='5  Equal Width  Equal Frequency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='4  KMeans  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
+page_content='3  ticity  ESOL  Neight  Scscore  Features' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNFAT4oBgHgl3EQf8R7j/content/2301.08750v1.pdf'}
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+Testing High-dimensional Multinomials with Applications to
+Text Analysis
+T. Tony Cai∗, Zheng Tracy Ke†, and Paxton Turner†
+Abstract
+Motivated by applications in text mining and discrete distribution inference, we
+investigate the testing for equality of probability mass functions of K groups of high-
+dimensional multinomial distributions.
+A test statistic, which is shown to have an
+asymptotic standard normal distribution under the null, is proposed. The optimal de-
+tection boundary is established, and the proposed test is shown to achieve this optimal
+detection boundary across the entire parameter space of interest. The proposed method
+is demonstrated in simulation studies and applied to analyze two real-world datasets
+to examine variation among consumer reviews of Amazon movies and diversity of sta-
+tistical paper abstracts.
+Keywords: authorship attribution, closeness testing, consumer reviews, martin-
+gale central limit theorem, minimax optimality, topic model
+1
+Introduction
+Statistical inference for multinomial data has garnered considerable recent interest [Di-
+akonikolas and Kane, 2016, Balakrishnan and Wasserman, 2018]. One important appli-
+cation is in text mining, as it is common to model the word counts in a text document
+by a multinomial distribution [Blei et al., 2003]. We consider a specific example in mar-
+keting, where the study of online customer ratings and reviews has become a trending
+topic [Chevalier and Mayzlin, 2006, Zhu and Zhang, 2010, Leung and Yang, 2020]. Cus-
+tomer reviews are a good proxy to the overall word of mouth (WOM) and can significantly
+influence customers’ decisions [Zhu and Zhang, 2010]. Many research works aim to under-
+stand the patterns in online reviews and their impacts on sales. Classical studies only use
+the numerical ratings but ignore the rich text reviews because of their unstructured na-
+ture. More recent works have revealed the importance of analyzing text reviews [Chevalier
+and Mayzlin, 2006], especially for hedonic products such as books, movies, and hotels. A
+question of great interest is to detect the heterogeneity in reviewers’ response styles. For
+example, Leung and Yang [2020] discovered that younger travelers, women, and travelers
+with less review expertise tend to give more positive reviews and that guests staying in
+∗University of Pennsylvania
+†Harvard University
+1
+arXiv:2301.01381v1  [stat.ME]  3 Jan 2023
+
+high-class hotels tend to have more extreme response styles than those staying in low-class
+hotels. Knowing such differences will offer valuable insights for hotel managers and online
+rating/review sites.
+The aforementioned heterogeneity detection can be cast as a hypothesis test on multi-
+nomial data. Suppose reviews are written on a vocabulary of p distinct words. Let Xi ∈ Rp
+denote the word counts in review i. We model that
+Xi ∼ Multinomial(Ni, Ωi),
+1 ≤ i ≤ n,
+(1.1)
+where Ni is the total length of review i and Ωi ∈ Rp is a probability mass function (PMF)
+containing the population word frequencies. These reviews are divided into K groups by
+reviewer characteristics (e.g., age, gender, new/returning customer), product characteristics
+(e.g., high-class versus low-class hotels), and numeric ratings (e.g., from 1 star to 5 stars),
+where K can be presumably large. We view Ωi as representing the ‘true response’ of review
+i. The “average response” of a group k is defined by a weighted average of the PMFs:
+µk = (nk ¯Nk)−1 �
+i∈Sk
+NiΩi,
+1 ≤ k ≤ K.
+(1.2)
+Here Sk ⊂ {1, 2, . . . , n} is the index set of group k, nk = |Sk| is the total number of reviews
+in group k, and ¯Nk = n−1
+k
+�
+i∈Sk Ni is the average length of reviews in group k. We would
+like to test
+H0 :
+µ1 = µ2 = . . . = µK.
+(1.3)
+When the null hypothesis is rejected, it means there exist statistically significant differences
+among the group-wise “average responses”.
+We call (1.1)-(1.3) the “K-sample testing for equality of average PMFs in multinomials”
+or “K-sample testing for multinomials” for short. Interestingly, as K varies, this problem
+includes several well-defined problems in text mining and discrete distribution inference as
+special cases.
+1. Global testing for topic models. Topic modeling [Blei et al., 2003] is a popular text
+mining tool. In a topic model, each Ωi in (1.1) is a convex combination of M topic
+vectors. Before fitting a topic model to a corpus, it is often desirable to determine
+if the corpus indeed contains multiple topics. This boils down to the global testing
+problem, which tests M = 1 versus M > 1. Under the null hypothesis, Ωi’s are equal
+to each other, and in the alternative hypothesis, Ωi’s can take continuous values in
+a high-dimensional simplex. This is a special case of our problem with K = n and
+nk = 1.
+2. Authorship attribution [Mosteller and Wallace, 1963, Kipnis, 2022]. In these appli-
+cations, the goal is to determine the unknown authorship of an article from other
+articles with known authors. A famous example [Mosteller and Wallace, 2012] is to
+determine the actual authors of a few Federalist Papers written by three authors but
+published under a single pseudonym. It can be formulated [Mosteller and Wallace,
+1963, Kipnis, 2022] as testing the equality of population word frequencies between the
+2
+
+article of interest and the corpus from a known author, a special case of our problem
+with K = 2.
+3. Closeness between discrete distributions [Chan et al., 2014, Bhattacharya and Valiant,
+2015, Balakrishnan and Wasserman, 2019].
+There has been a surge of interest in
+discrete distribution inference. Closeness testing is one of most studied problems.
+The data from two discrete distributions are summarized in two multinomial vectors
+Multinomial(N1, µ) and Multinomial(N2, θ). The goal is to test µ = θ. It is a special
+case of our testing problem with K = 2 and n1 = n2 = 1.
+In this paper, we provide a unified solution to all the aforementioned problems. The
+key to our methodology is a flexible statistic called DELVE (DE-biased and Length-assisted
+Variability Estimator). It provides a general similarity measure for comparing groups of
+discrete distributions such as count vectors associated with text corpora. Similarity mea-
+sures (such as the classical cosine similarity, log-likelihood ratio statistic, and others) are
+fundamental in text mining and have been applied to problems in distribution testing [Kim
+et al., 2022], computational linguistics [Gomaa et al., 2013], econometrics [Hansen et al.,
+2018], and computational biology [Kolodziejczyk et al., 2015]. Our method is a new and
+flexible similarity measure that is potentially useful in these areas.
+We emphasize that our setting does not require that the Xi’s in the same group are
+drawn from the same distribution. Under the null hypothesis (1.3), the group-wise means
+are equal, but the Ωi’s within each group can still be different from each other. As a result,
+the null hypothesis is composite and designing a proper test statistic is non-trivial.
+1.1
+Our results and contributions
+The dimensionality of the testing problem is captured by (n, p, K) and ¯N := n−1 �n
+i=1 Ni.
+We are interested in a high-dimensional setting where
+n ¯N → ∞,
+p → ∞,
+and
+n2 ¯N2/(Kp) → ∞.
+(1.4)
+In most places of this paper, we use a subscript n to indicate asymptotics, but our method
+and theory do apply to the case where n is finite and ¯N → ∞. In text applications, n ¯N is
+the total count of words in the corpus, and a large n ¯N means either there are sufficiently
+many documents, or the documents are sufficiently long. Given that n ¯N → ∞, we further
+allow (p, K) to grow with n at a speed such that Kp ≪ n2 ¯N2. In particular, our settings
+allow K to range from 2 to n, so as to cover all the application examples.
+We propose a test that enjoys the following properties:
+(a) Parameter-free null distribution: We show that the test statistic ψ → N(0, 1) under
+H0. Even under the null hypothesis (1.3), the model contains a large number of free
+parameters because the null hypothesis is only about the equality of “average” PMFs
+but still allows (Ni, Ωi) to differ within each group. As an appealing property, the
+null distribution of ψ does not depend on these individual multinomial parameters;
+hence, we can always conveniently obtain the asymptotic p-value for our proposed
+test.
+3
+
+(b) Minimax optimal detection boundary: We define a quantity ωn := ωn(µ1, µ2, . . . , µK)
+in (3.5) that measures the difference among the K group-wise mean PMF’s. It sat-
+isfies that ωn = 0 if and only if the null hypothesis holds, and it has been properly
+normalized so that ωn is bounded under the alternative hypothesis (provided some
+mild regularity conditions hold). We show that the proposed test has an asymptotic
+full power if ω4
+nn2 ¯N2/(Kp) → ∞. We also provide a matching lower bound by showing
+that the null hypothesis and the alternative hypothesis are asymptotically indistin-
+guishable if ω4
+nn2 ¯N2/(Kp) → 0. Therefore, the proposed test is minimax optimal.
+Furthermore, in the boundary case where ω4
+nn2 ¯N2/(Kp) → c0 for a constant c0 > 0,
+for some special settings, we show that ψ → N(0, 1) under H0, and ψ → N(c1, 1),
+under H1, with the constant c1 being an explicit function of c0.
+To the best of our knowledge, this testing problem for a general K has not been studied
+before. The existing works primarily focused on closeness testing and authorship attribution
+(see Section 1.2), which are special cases with K = 2. In comparison, our test is applicable
+to any value of K, offering a unified solution to multiple applications. Even for K = 2,
+the existing works do not provide a test statistic that has a tractable null distribution.
+They determined the rejection region and calculated p-values using either a (conservative)
+large-deviation bound or a permutation procedure. Our test is the first one equipped with a
+tractable null distribution. Our results about the optimal detection boundary for a general
+K are also new to the literature. By varying K in our theory, we obtain the optimal detec-
+tion boundary for different sub-problems. For some of them (e.g., global testing for topic
+models, authorship attribution with moderate sparsity), the optimal detection boundary
+was not known before; hence, our results help advance the understanding of the statistical
+limits of these problems.
+1.2
+Related literature
+First, we make a connection to discrete distribution inference. Let X ∼ Multinomial(N, Ω)
+represent a size-N sample from a discrete distribution with p categories. The one-sample
+closeness testing aims to test H0 : Ω = µ, for a given PMF µ.
+Existing works focus
+on finding the minimum separation condition in terms of the ℓ1-norm or ℓ2-norm of Ω −
+µ. Balakrishnan and Wasserman [2019] derived the minimum ℓ1-separation condition and
+proposed a truncated chi-square test to achieve it. Valiant and Valiant [2017] studied the
+“local critical radius”, a local separation condition that depends on the “effective sparsity”
+of µ, and they proposed a “2/3rd + tail” test to achieve it. In the two-sample closeness
+testing problem, given X1 ∼ Multinomial(N1, Ω1) and X2 ∼ Multinomial(N2, Ω2), it aims
+to test H0 : Ω1 = Ω2. Again, this literature focuses on finding the minimum separation
+condition in terms of the ℓ1-norm or ℓ2-norm of Ω1 − Ω2. When N1 = N2, Chan et al.
+[2014] derived the minimum ℓ1-separation condition and proposed a weighted chi-square
+test to attain it. Bhattacharya and Valiant [2015] extended their results to the unbalanced
+case where N1 ̸= N2, assuming ∥Ω1 − Ω2∥1 ≥ p−1/12. This assumption was later removed
+by Diakonikolas and Kane [2016], who established the minimum ℓ1-separation condition in
+full generality. Kim et al. [2022] proposed a two-sample kernel U-statistic and showed that
+4
+
+it attains the minimum ℓ2-separation condition.
+Since the two-sample closeness testing is a special case of our problem with K = 2 and
+n1 = n2 = 1, our test is directly applicable. An appealing property of our test is its tractable
+asymptotic null distribution of N(0, 1). In contrast, for the chi-square statistic in Chan et al.
+[2014] or the U-statistic in [Kim et al., 2022], the rejection region is determined by either an
+upper bound from concentration inequalities or a permutation procedure, which may lead
+to a conservative threshold or need additional computational costs. Regarding the testing
+power, we show in Section 4.3 that our test achieves the minimum ℓ2-separation condition,
+i.e., our method is an optimal “ℓ2 testor.” Our test can also be turned into an optimal “ℓ1
+testor” (a test that achieves the minimum ℓ1-separation condition) by re-weighting terms
+in the test statistic (see Section 4.3).
+Next, we make a connection to text mining. In this literature, a multinomial vector
+X ∼ Multinomial(N, Ω) represents the word counts for a document of length N written
+with a dictionary containing p words. In a topic model, each Ωi is a convex combination of
+M “topic vectors”: Ωi = �M
+k=1 wi(k)Ak, where each Ak ∈ Rp is a PMF and the combination
+coefficient vector wi ∈ RK is called the “topic weight” vector for document i. Given a
+collection of documents X1, X2, . . . , Xn, the global testing problem aims to test M = 1
+versus M > 1. Interestingly, the optimal detection boundary for this problem has never
+been rigorously studied. As we have explained, this problem a special case of our testing
+problem with K = n. Our results (a) provide a test statistic that has a tractable null
+distribution and (b) reveal that the optimal detection boundary is ω2
+n ≍ (√n ¯N)−1√p.
+Both (a) and (b) are new results. When comparing our results with those about estimation
+of Ak’s [Ke and Wang, 2022], it suggests that global testing requires a strictly lower signal
+strength than topic estimation.
+For authorship attribution, Kipnis [2022] treats the corpus from a known author as a
+single document and tests the null hypothesis that this combined document and a new
+document have the same population word frequencies. It is a two-sample closeness testing
+problem, except that sparsity is imposed on the difference of two PMFs. Kipnis [2022]
+proposed a test which applies an “exact binomial test” to obtain a p-value for each word
+and combines these p-values using Higher Criticism [Donoho and Jin, 2004]. Donoho and
+Kipnis [2022] analyzed this test when the number of “useful words” is o(√p), and they
+derived a sharp phase diagram (a related one-sample setting was studied in Arias-Castro
+and Wang [2015]). In Section 4.2, we show that our test is applicable to this problem and
+has some nice properties: (a) tractable null distribution; (b) allows for s ≥ c√p, where s
+is the number of useful words; and (c) does not require documents from the known author
+to have identical population word frequencies, making the setting more realistic. On the
+other hand, when s = o(√p), our test is less powerful than the one in Kipnis [2022], Donoho
+and Kipnis [2022], as our test does not utilize sparsity explicitly. We can further improve
+our test in this regime by modifying the DELVE statistic to incorporate sparsity (see the
+remark in Section 4.2).
+5
+
+1.3
+Organization
+The rest of this paper is arranged as follows. In Section 2, we introduce the test statistic and
+explain the rationale behind it. We then present in Section 3 the main theoretical results,
+including the asymptotic null distribution, power analysis, a matching lower bound, the
+study of two special cases (K = n and K = 2), and a discussion of the contiguity regime.
+Section 4 applies our results to text mining and discrete distribution testing. Simulations
+are in Section 5 and real data analysis is in Section 6. The paper is concluded with a
+discussion in Section 7. All proofs are in the appendix.
+2
+The DELVE Test
+Recall that Xi ∼ Multinomial(Ni, Ωi) for 1 ≤ i ≤ n. There is a known partition {1, 2, . . . , n} =
+∪K
+k=1Sk. Write nk = |Sk|, ¯Nk = n−1
+k
+�
+i∈Sk Ni, and ¯N = n−1 �n
+i=1 Ni. In (1.2), we have de-
+fined the group-wise mean PMF µk = (nk ¯Nk)−1 �
+i∈Sk NiΩi. We further define the overall
+mean PMF µ ∈ Rp by
+µ :=
+1
+n ¯N
+K
+�
+k=1
+nk ¯Nkµk =
+1
+n ¯N
+n
+�
+i=1
+NiΩi.
+(2.1)
+We introduce a quantity ρ2 = ρ2(µ1, . . . , µK) by
+ρ2 :=
+K
+�
+k=1
+nk ¯Nk∥µk − µ∥2.
+(2.2)
+This quantity measures the variations across K group-wise mean PMFs. It is true that the
+null hypothesis (1.3) holds if and only if ρ2 = 0. Inspired by this observation, we hope to
+construct an unbiased estimator of ρ2 and develop it to a test statistic.
+We can easily obtain the minimum variance unbiased estimators of µk and µ:
+ˆµk =
+1
+nk ¯Nk
+�
+i∈Sk
+Xi,
+and
+ˆµ =
+1
+n ¯N
+K
+�
+k=1
+nk ¯Nkˆµk =
+1
+n ¯N
+n
+�
+i=1
+Xi.
+(2.3)
+For each 1 ≤ j ≤ p, let µkj, µj, ˆµkj and ˆµj represent the jth entry of µk, µ, ˆµk and ˆµ,
+respectively. A naive estimator of ρ2 is
+�T =
+p
+�
+j=1
+�Tj,
+where
+�Tj =
+K
+�
+k=1
+nk ¯Nk(ˆµkj − ˆµj)2.
+(2.4)
+This estimator is biased. In Section C.1 of the appendix , we show that E[ �Tj] = �K
+k=1
+�
+nk ¯Nk(µkj−
+µj)2 +
+�
+1
+nk ¯
+Nk −
+1
+n ¯
+N
+� �
+i∈Sk NiΩij(1 − Ωij)
+�
+. It motivates us to debias �Tj by using an unbi-
+ased estimate of Ωij(1 − Ωij). By elementary properties of the multinomial distribution,
+E[Xij(Ni − Xij)] = Ni(Ni − 1)Ωij(1 − Ωij).
+We thereby use
+1
+Ni(Ni−1)Xij(Ni − Xij) to
+estimate Ωij(1 − Ωij). This gives rise to an unbiased estimator of ρ2 as
+T =
+p
+�
+j=1
+Tj,
+Tj =
+K
+�
+k=1
+�
+nk ¯Nk(ˆµkj − ˆµj)2 −
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+� �
+i∈Sk
+Xij(Ni − Xij)
+Ni − 1
+�
+.
+(2.5)
+6
+
+Lemma 2.1. Under Models (1.1)-(1.2), the estimator in (2.5) satisfies that E[T] = ρ2.
+To use T for hypothesis testing, we need a proper standardization of this statistic. In
+Sections A.1-A.2 of the appendix , we study V(T), the variance of T. Under mild regularity
+conditions, it can be shown that V(T) = Θn · [1 + o(1)], where
+Θn := 4
+K
+�
+k=1
+p
+�
+j=1
+nk ¯Nk(µkj − µj)2µkj + 2
+K
+�
+k=1
+�
+i∈Sk
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+N3
+i
+Ni − 1Ω2
+ij
+(2.6)
++
+2
+n2 ¯N2
+�
+1≤k̸=ℓ≤K
+�
+i∈Sk
+�
+m∈Sℓ
+p
+�
+j=1
+NiNmΩijΩmj + 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk,
+i̸=m
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩmj.
+In Θn, the first term vanishes under the null, so it suffices to estimate the other three terms
+in Θn. By properties of multinomial distributions, E[XijXmj] = NiNmΩijΩmj, E[X2
+ij] =
+N2
+i Ω2
+ij + NiΩij(1 − Ωij), and E[Xij(Ni − Xij)] = Ni(Ni − 1)Ωij(1 − Ωij). It inspires us to
+estimate ΩijΩmj by XijXmj
+NiNm
+and estimate Ω2
+ij by
+X2
+ij
+N2
+i − Xij(Ni−Xij)
+N2
+i (Ni−1)
+=
+X2
+ij−Xij
+Ni(Ni−1). Define
+V = 2
+K
+�
+k=1
+�
+i∈Sk
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2 X2
+ij − Xij
+Ni(Ni − 1) +
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+p
+�
+j=1
+XijXmj
++ 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk,
+i̸=m
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+XijXmj.
+(2.7)
+The test statistic we propose is as follows (in the rate event V < 0, we simply set ψ = 0):
+ψ = T/
+√
+V .
+(2.8)
+We call ψ the DEbiased and Length-adjusted Variability Estimator (DELVE). In Section 3.1,
+we show that under mild regularity conditions, ψ → N(0, 1) under the null hypothesis. For
+any fixed α ∈ (0, 1), the asymptotic level-α DELVE test rejects H0 if
+ψ > zα,
+where zα is the (1 − α)-quantile of N(0, 1).
+(2.9)
+2.1
+The special cases of K = n and K = 2
+As seen in Section 1, the application examples of K = n and K = 2 are particularly
+intriguing. In these cases, we give more explicit expressions of our test statistic.
+When K = n, we have Sk = {i} and ˆµkj = N−1
+i
+Xij. The null hypothesis becomes
+H0 : Ω1 = Ω2 = . . . = Ωn. The statistic in (2.5) reduces to
+T =
+p
+�
+j=1
+n
+�
+i=1
+�(Xij − Niˆµj)2
+Ni
+−
+�
+1 − Ni
+n ¯N
+�Xij(Ni − Xij)
+Ni(Ni − 1)
+�
+.
+(2.10)
+Moreover, in the variance estimate (2.7), the last term is exactly zero, and it can be shown
+that the third term is negligible compared to the first term. We thereby consider a simpler
+7
+
+variance estimator by only retaining the first term in (2.7):
+V ∗ = 2
+n
+�
+i=1
+p
+�
+j=1
+� 1
+Ni
+−
+1
+n ¯N
+�2 X2
+ij − Xij
+Ni(Ni − 1).
+(2.11)
+The simplified DELVE test statistic is ψ∗ = T/
+√
+V ∗.
+When K = 2, we observe two collections of multinomial vectors, denoted by {Xi}1≤i≤n
+and {Gi}1≤i≤m. We assume for 1 ≤ i ≤ n and 1 ≤ j ≤ m,
+Xi ∼ Multinomial(Ni, Ωi),
+Gj ∼ Multinomial(Mj, Γj).
+(2.12)
+Write ¯N = n−1 �n
+i=1 Ni and ¯
+M = m−1 �m
+i=1 Mi. The null hypothesis becomes
+H0 :
+η = θ,
+where η =
+1
+n ¯N
+n
+�
+i=1
+NiΩi, and θ =
+1
+m ¯
+M
+m
+�
+i=1
+MiΓi,
+(2.13)
+where θ and η are the two group-wise mean PMFs. We estimate them by ˆη = (n ¯N)−1 �n
+i=1 Xi
+and ˆθ = (m ¯
+M)−1 �m
+i=1 Gi. The statistic in (2.5) has an equivalent form as follows:
+T =
+n ¯Nm ¯
+M
+n ¯N + m ¯
+M
+�
+∥ˆη − ˆθ∥2 −
+n
+�
+i=1
+p
+�
+j=1
+Xij(Ni − Xij)
+n2 ¯N2(Ni − 1) −
+m
+�
+i=1
+p
+�
+j=1
+Gij(Mi − Gij)
+m2 ¯
+M2(Mi − 1)
+�
+.
+(2.14)
+The variance estimate (2.7) has an equivalent form as follows:
+V =
+4 �n
+i=1
+�m
+i′=1
+�p
+j=1 XijGi′j
+(n ¯N + m ¯
+M)2
++
+2m2 ¯
+M2� �n
+i=1
+X2
+ij−Xij
+Ni(Ni−1) + �
+1≤i̸=i′≤n XijXi′j
+�
+n2 ¯N2(n ¯N + m ¯
+M)2
++
+2n2 ¯N2� �m
+i=1
+G2
+ij−Gij
+Mi(Mi−1) + �
+1≤i̸=i′≤m GijGi′j
+�
+m2 ¯
+M2(n ¯N + m ¯
+M)2
+.
+(2.15)
+The DELVE test statistic is ψ = T/
+√
+V .
+2.2
+A variant: DELVE+
+We introduce a variant of the DELVE test statistic to better suit real data. Let ˆµ, T and
+V be as in (2.3), (2.5) and (2.7). Define
+ψ+ = T/
+√
+V +,
+where
+V + = V ·
+�
+1 + ∥ˆµ∥2T/
+√
+V
+�
+.
+(2.16)
+We call (2.16) the DELVE+ test statistic. In theory, this modification has little effect on
+the key properties of the test. To see this, we note that ∥ˆµ∥2 = oP(1) in high-dimensional
+settings. Suppose T/
+√
+V → N(0, 1) under H0. Since ∥ˆµ∥2 → 0, it is seen immediately
+that V +/V → 1; hence, the asymptotic normality also holds for ψ+. Suppose T/
+√
+V → ∞
+under the alternative hypothesis. It follows that V + ≤ 2 max{V, ∥ˆµ∥2 · T
+√
+V } and ψ+ ≥
+1
+√
+2 min{T/
+√
+V , ∥ˆµ∥−1
+2 (T/
+√
+V )1/2} → ∞. We have proved the following lemma:
+Lemma 2.2. As n ¯N → ∞, suppose ∥ˆµ∥2 → 0 in probability. Under H0, if T/
+√
+V →
+N(0, 1), then T/
+√
+V + → N(0, 1). Under H1, if T/
+√
+V → ∞, then T/
+√
+V + → ∞.
+8
+
+In practice, this modification avoids extremely small p-values. In some real datasets, V is
+very small and leads to an extremely small p-value in the original DELVE test. In DELVE+,
+as long as T is positive, ψ+ is smaller than ψ, so that the p-value is adjusted.
+In the numerical experiments, we consider both DELVE and DELVE+. For theoretical
+analysis, since these two versions have almost identical theoretical properties, we only focus
+on the original DELVE test statistic.
+3
+Theoretical Properties
+We first present the regularity conditions. For a constant c0 ∈ (0, 1), we assume
+min
+1≤i≤n Ni ≥ 2,
+max
+1≤i≤n ∥Ωi∥∞ ≤ 1 − c0,
+max
+1≤k≤K
+nk ¯Nk
+n ¯N
+≤ 1 − c0.
+(3.1)
+In (3.1), the first condition is mild. The second condition is also mild: note that ∥Ωi∥1 = 1
+for each i; this condition excludes those cases where one of the p categories has an extremely
+dominating probability in the PMF Ωi. In the third condition, nk ¯Nk is the total number of
+counts in all multinomials of group k, and this condition excludes the extremely unbalanced
+case where one group occupies the majority of counts. Note that in the special case of K = 2,
+we relax this condition to allow for severely unbalanced groups (see Section 3.4).
+Recall that µk =
+1
+nk ¯
+Nk
+�
+i∈Sk NiΩi is the mean PMF within group k. We also define a
+‘covariance’ matrix of PMF’s for group k by Σk =
+1
+nk ¯
+Nk
+�
+i∈Sk NiΩiΩ′
+i. Let
+αn := max
+� K
+�
+k=1
+∥µk∥3
+3
+nk ¯Nk
+,
+K
+�
+k=1
+∥µk∥2
+n2
+k ¯N2
+k
+� �� K
+�
+k=1
+∥µk∥2
+�2
+,
+(3.2)
+and
+βn := max
+� K
+�
+k=1
+�
+i∈Sk
+N2
+i
+n2
+k ¯N2
+k
+∥Ωi∥3
+3,
+K
+�
+k=1
+∥Σk∥2
+F
+��
+(K∥µ∥2).
+(3.3)
+We assume that as n ¯N → ∞,
+αn = o(1),
+βn = o(1),
+and
+∥µ∥4
+4
+K∥µ∥4 = o(1).
+(3.4)
+Here αn and βn only depend on group-wise quantities, such as µk, Σk and �
+i∈Sk N2
+i ∥Ωi∥3
+3;
+hence, a small number of ‘outliers’ (i.e., extremely large entries) in Ω has little effect on αn
+and βn. Furthermore, in a simple case where maxk nk ≤ C mink nk, maxk ¯Nk ≤ C mink ¯Nk
+and ∥Ω∥max = O(1/p), it holds that αn = O(max{ 1
+n ¯
+N ,
+Kp
+n2 ¯
+N2 }), βn = O(max{ K2
+n2p, 1
+p}) and
+∥µ∥4
+4
+K∥µ∥4 = O( 1
+Kp). When n ¯N → ∞ and p → ∞, (3.4) reduces to n2 ¯N2/(Kp) → ∞. This
+condition is necessary for successful testing, because our lower bound in Section 3.3 implies
+that the two hypotheses are asymptotically indistinguishable if n2 ¯N2/(Kp) → 0.
+9
+
+3.1
+The asymptotic null distribution
+Under the null hypothesis, the K group-wise mean PMF’s µ1, µ2, . . . , µK, are equal to each
+other, but this hypothesis is still highly composite, as (Ni, Ωi) are not necessarily the same
+within each group. We show that the DELVE test statistic always enjoys a parameter-free
+asymptotic null distribution. Let T, Θn and V be as in (2.5)-(2.7). The next two theorems
+are proved in the appendix.
+Theorem 3.1. Consider Models (1.1)-(1.2), where the null hypothesis (1.3) holds. Suppose
+(3.1) and (3.4) are satisfied. As n ¯N → ∞, T/√Θn → N(0, 1) in distribution.
+Theorem 3.2. Under the conditions of Theorem 3.1, as n ¯N → ∞, V/Θn → 1 in proba-
+bility, and ψ := T/
+√
+V → N(0, 1) in distribution.
+By Theorem 3.2, the asymptotic p-value is computed via 1 − Φ(ψ), where Φ(·) is the
+cumulative distribution function of the standard normal. Moreover, for any fixed α ∈ (0, 1),
+the rejection region of the asymptotic level-α test is as given in (2.9).
+The proofs of Theorems 3.1-3.2 contain two key steps: in the first step, we decompose T
+into the sum of mutually uncorrelated terms. We introduce a set of independent, mean-zero
+random vectors {Zir}1≤i≤n,1≤r≤Ni, where Zir ∼ Multinomial(1, Ωi) − Ωi. By properties of
+multinomial distributions, Xi = NiΩi + �Ni
+r=1 Zir in distribution. We plug it into (2.5) to
+obtain T = T1 +T2 +T3 +T4, where T1 is a linear form of {Zir}, T2, T3 and T4 are quadratic
+forms of {Zir}, and the four terms are uncorrelated with each other (details are contained
+in Section A of the appendix ). In the second step, we construct a martingale for each term
+Tj. This is accomplished by rearranging the double-index sequence Zir to a single-index
+sequence and then successively adding terms in this sequence to Tj. We then apply the
+martingale central limit theorem (CLT) [Hall and Heyde, 2014] to prove the asymptotic
+normality of each Tj. The asymptotic normality of T follows by identifying the dominating
+terms in T1-T4 (as model parameters change, the dominating terms can be different) and
+studying their joint distribution. This step involves extensive calculations to bound the
+conditional variance and to verify the Lindeberg conditions of the martingale CLT, as well
+as numerous subtle uses of the Cauchy-Schwarz inequality to simplify the moment bounds.
+3.2
+Power analysis
+Under the alternative hypothesis, the PMF’s µ1, µ2, . . . , µK are not the same. In Section 2,
+we introduce a quantity ρ2 (see (2.2)) to capture the total variation in µk’s, but this quantity
+is not scale-free. We define a scaled version of ρ2 as
+ωn = ωn(µ1, µ2, . . . , µK) :=
+1
+n ¯N∥µ∥2
+K
+�
+k=1
+nk ¯Nk∥µk − µ∥2.
+(3.5)
+It is seen that ωn ≤ maxk{ ∥µk−µ∥2
+∥µ∥2
+}, which is properly scaled.
+Theorem 3.3. Consider Models (1.1)-(1.2), where (3.1) and (3.4) are satisfied. Then,
+E[T] = n ¯N∥µ∥2ω2
+n, and V(T) = O
+��K
+k=1 ∥µk∥2�
++ E[T] · O
+�
+max1≤k≤K ∥µk∥∞
+�
+.
+10
+
+For the DELVE test to have an asymptotically full power, we need E[T] ≫
+�
+V(T). By
+Theorem 3.3, this is satisfied if E[T] ≫
+��
+k ∥µk∥2 and E[T] ≫ maxk ∥µk∥∞. Between
+these two requirements, the latter one is weaker; hence, we only need E[T] ≫
+��K
+k=1 ∥µk∥2.
+It gives rise to the following theorem:
+Theorem 3.4. Under the conditions of Theorem 3.3, we further assume that under the
+alternative hypothesis, as n ¯N → ∞,
+SNRn :=
+n ¯N∥µ∥2ω2
+n
+��K
+k=1 ∥µk∥2
+→
+∞.
+(3.6)
+The following statements are true. Under the alternative hypothesis, ψ → ∞ in probability.
+For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level of α and an
+asymptotic power of 1. If we choose α = αn such that αn → 0 and 1 − Φ(SNRn) = o(αn),
+where Φ is the CDF of N(0, 1), then the sum of type I and type II errors of the DELVE
+test converges to 0.
+The detection boundary in (3.6) has simpler forms in some special cases. For example,
+if ∥µk∥ ≍ ∥µ∥ for 1 ≤ k ≤ K, then SRNn ≍ n ¯Nω2
+n∥µ∥/
+√
+K. If, furthermore, all entries of
+µ are at the same order, which implies ∥µ∥ ≍ p−1/2, then SRNn ≍ n2 ¯N2ω2
+n/√Kp. In this
+case, the detection boundary simplifies to ω4
+nn2 ¯N2/(Kp) → ∞.
+Remark 1 (The low-dimensional case p = O(1)). Although we are primarily interested in
+the high-dimensional setting p → ∞, it is also worth investigating the case p = O(1). We
+can show the same detection boundary for our test, but the asymptotic normality may not
+hold, because the variance estimator V in (2.7) is not guaranteed to be consistent. To fix
+this issue, we propose a variant of our test by replacing V with a refined variance estimator
+�V , which is consistent for a finite p. The expression of �V is a little complicated. Due to
+space limits, we relegate it to Section E of the appendix.
+3.3
+A matching lower bound
+We have seen that the DELVE test successfully separates two hypotheses if SNRn → ∞,
+where SNRn is as defined in (3.6). We now present a lower bound to show that the two
+hypotheses are asymptotically indistinguishable if SNRn → 0.
+Let ℓi ∈ {1, 2, . . . , K} denote the group label of Xi. Write ξ = {(Ni, Ωi, ℓi)}1≤i≤n. Let
+µk, αn, βn, and ωn be the same as defined in (1.2), (3.2), (3.3), and (3.5), respectively. For
+each given (n, p, K, ¯N), we write µk = µk(ξ) to emphasize its dependence on parameters,
+and similarly for αn, βn, ωn. For any c0 ∈ (0, 1) and sequence ϵn, define
+Qn(c0, ϵn) :=
+�
+ξ = {(Ni, Ωi, ℓi)}n
+i=1 : (3.1) holds for c0, max(αn(ξ), βn(ξ)) ≤ ϵn
+�
+(3.7)
+Furthermore, for any sequence δn, we define a parameter class for the null hypothesis and
+a parameter class for the alternative hypothesis:
+Q∗
+0n(c0, ϵn) = Qn(c0, ϵn) ∩ {ξ : ωn(ξ) = 0} ,
+11
+
+Q∗
+1n(δn; c0, ϵn) = Qn(c0, ϵn) ∩
+�
+�
+�ξ : n ¯N∥µ(ξ)∥2ω2
+n(ξ)
+��K
+k=1 ∥µk(ξ)∥2
+≥ δn
+�
+�
+� .
+(3.8)
+Theorem 3.5. Fix a constant c0 ∈ (0, 1) and two positive sequences ϵn and δn such that
+ϵn → 0 as n → ∞. For any sequence of (n, p, K, ¯N) indexed by n, we consider Models (1.1)-
+(1.2) for Ω ∈ Qn(c0, ϵn). Let Q∗
+0n(c0, ϵn) and Q∗
+1n(δn; c0, ϵn) be as in (3.8). If δn → 0, then
+lim supn→∞ infΨ∈{0,1}
+�
+supξ∈Q∗
+0n(c0,ϵn) Pξ(Ψ = 1) + supξ∈Q∗
+1n(δn;c0,ϵn) Pξ(Ψ = 0)
+�
+= 1.
+By Theorem 3.5, the null and alternative hypotheses are asymptotically indistinguish-
+able if SRNn → 0. Combining it with Theorem 3.4, the DELVE test achieves the minimax
+optimal detection boundary.
+3.4
+The special case of K = 2
+The special case of K = 2 is found in applications such as closeness testing and authorship
+attribution. We study this case more carefully. Given {Xi}1≤i≤n and {Gi}1≤i≤m, we assume
+Xi ∼ Multinomial(Ni, Ωi),
+Gj ∼ Multinomial(Mj, Γj).
+(3.9)
+Write ¯N = n−1 �n
+i=1 Ni and ¯
+M = m−1 �m
+i=1 Mi. The null hypothesis becomes
+H0 :
+η = θ,
+where η =
+1
+n ¯N
+n
+�
+i=1
+NiΩi, and θ =
+1
+m ¯
+M
+m
+�
+i=1
+MiΓi,
+(3.10)
+where θ and η are the two group-wise mean PMFs. In this case, the test statistic ψ has a
+more explicit form as in (2.14)-(2.15).
+In our previous results for a general K, the regularity conditions (e.g., (3.1)) impose
+restrictions on the balance of sample sizes among groups. For K = 2, the severely unbal-
+anced setting is interesting (e.g., in authorship attribution, n = 1 and m can be large). We
+relax the regularity conditions to the following ones:
+Condition 3.1. Let θ and η be as in (3.10) and define two matrices Σ1 =
+1
+n ¯
+N
+�n
+i=1 NiΩiΩ′
+i
+and Σ2 =
+1
+m ¯
+M
+�m
+i=1 MiΓiΓ′
+i. We assume that the following statements are true (a) For
+1 ≤ i ≤ n and 1 ≤ j ≤ m, Ni ≥ 2, ∥Ωi∥∞ ≤ 1 − c0, Mj ≥ 2, and ∥Γj∥∞ ≤ 1 − c0, where
+c0 ∈ (0, 1) is a contant, (b) max
+�� ∥η∥3
+3
+n ¯
+N + ∥θ∥3
+3
+m ¯
+M
+�
+,
+� ∥η∥2
+2
+n2 ¯
+N2 + ∥θ∥2
+2
+m2 ¯
+M2
+2
+�����
+m ¯
+M
+n ¯
+N+m ¯
+M η+
+n ¯
+N
+n ¯
+N+m ¯
+M θ
+��4 =
+o(1), (c) max
+� �
+i
+N2
+i
+n2 ¯
+N2 ∥Ωi∥3
+3, �
+i
+M2
+i
+m2 ¯
+M2 ∥Γi∥3
+3, ∥Σ1∥2
+F + ∥Σ2∥2
+F
+��
+∥µ∥2 = o(1), and (d)
+∥µ∥4
+4/∥µ∥4 = o(1).
+Condition (a) is similar to (3.1), except that we drop the sample size balance requriement.
+Conditions (b)-(d) are equivalent to (3.4) but have more explicit expressions for K = 2.
+Theorem 3.6. In Model (3.9), we test the null hypothesis H0: θ = µ. As min{n ¯N, m ¯
+M} →
+∞, suppose Condition 3.1 is satisfied. Under the alternative hypothesis, we further assume
+∥η − θ∥2
+� 1
+n ¯
+N +
+1
+m ¯
+M
+�
+max{∥η∥, ∥θ∥} → ∞.
+(3.11)
+12
+
+Consider the DELVE test statistic ψ = T/
+√
+V . The following statements are true. Under
+the null hypothesis, ψ → N(0, 1) in distribution. Under the alternative hypothesis, ψ → ∞
+in probability. Moreover for any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic
+level of α and an asymptotic power of 1.
+Compared with the theorems for a general K, first, Theorem 3.6 allows the two groups
+to be severely unbalanced and reveals that the detection boundary depends on the harmonic
+mean of n ¯N and m ¯
+M. Second, the detection boundary is expressed using ∥η − θ∥, which
+is easier to interpret.
+3.5
+The special case of K = n
+The special case of K = n is interesting for two reasons. First, the application example of
+global testing in topic models corresponds to K = n. Second, for any K, when Ωi’s within
+each group are assumed to be the same (e.g., this is the case in closeness testing of discrete
+distributions), it suffices to aggregate the counts in each group, i.e., let Yk = �
+i∈Sk Xi and
+operate on Y1, . . . , YK instead of the original Xi’s; this reduces to the case of K = n.
+When K = n, the null hypothesis has a simpler form:
+H0 :
+Ωi = µ,
+1 ≤ i ≤ n.
+(3.12)
+Moreover, under the alternative hypothesis, the quantity ω2
+n in (3.5) simplifies to
+ωn = ωn(Ω1, Ω2, . . . , Ωn) =
+1
+n ¯N∥µ∥2
+n
+�
+i=1
+Ni∥Ωi − µ∥2.
+(3.13)
+The DELVE test statistic also has a simplified form as in (2.10)-(2.11). We can prove the
+same theoretical results under weaker conditions:
+Condition 3.2. We assume that the following statements are true: (a) For a constant
+c0 ∈ (0, 1), 2 ≤ Ni ≤ (1 − c0)n ¯N and ∥Ωi∥∞ ≤ 1 − c0, 1 ≤ i ≤ n, and
+(b) max
+� �
+i
+∥Ωi∥3
+3
+Ni , �
+i
+∥Ωi∥2
+N2
+i
+��
+(�
+i ∥Ωi∥2)2 = o(1), and (�
+i ∥Ωi∥3
+3)/(n∥µ∥2) = o(1)
+When K = n, Condition (a) is equivalent to (3.1); and Condition (b) is weaker than (3.4),
+as we have dropped the requirement
+∥µ∥4
+4
+K∥µ∥4 = o(1). We obtain weaker conditions for K = n
+because the dominant terms in T differ from those for K < n.
+Theorem 3.7. In Model (1.1), we test the null hypothesis (3.12). As n → ∞, we assume
+that Condition 3.2 is satisfied. Under the alternative, we further assume that
+n ¯N∥µ∥2ω2
+n
+��n
+i=1 ∥Ωi∥2 → ∞.
+(3.14)
+Let T and V ∗ be the same as in (2.10)-(2.11). Consider the simplified DELVE test statistic
+ψ∗ = T/
+√
+V ∗. The following statements are true. Under the null hypothesis, ψ∗ → N(0, 1)
+in distribution. Under the alternative hypothesis, ψ∗ → ∞ in probability. Moreover for any
+fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level of α and an asymptotic
+power of 1.
+13
+
+The detection boundary in (3.14) has a simpler form if �
+i ∥Ωi∥2 ≍ n∥µ∥2. In this case,
+(3.14) is equivalent to √n ¯N∥µ∥ω2
+n → ∞. Additionally, if all entries of µ are at the same
+order, then ∥µ∥ ≍ 1/√p, and (3.14) further reduces to
+�
+n ¯N2/p · ω2
+n → ∞.
+3.6
+A discussion of the contiguity regime
+Our power analysis in Section 3.2 concerns SNRn → ∞, and our lower bound in Section 3.3
+concerns SNRn → 0. We now study the contiguity regime where SNRn tends to a constant.
+For illustration, we consider a special choice of parameters, which allows us to obtain a
+simple expression of the testing risk.
+Suppose K = n and Ni = N for all 1 ≤ i ≤ n. Consider the pair of hypotheses:
+H0 :
+Ωij = p−1,
+v.s.
+H1 :
+Ωij = p−1(1 + βnδij),
+(3.15)
+where {δij}1≤i≤n,1≤j≤p satisfy that |δij| = 1, �p
+j=1 δij = 0 and �n
+i=1 δij = 0. Such δij
+always exist.1 The SNRn in (3.6) satisfies that SNRn ≍ (N√n/√p)β2
+n. We thereby set
+β2
+n =
+√2p
+N√n · a,
+for a constant a > 0.
+(3.16)
+Since K = n here, we consider the simplified DELVE test statistic ψ∗ as in Section 3.5.
+Theorem 3.8. Consider Model (1.1) with Ni = N. For a constant a > 0, let the null
+and alternative hypotheses be specified as in (3.15)-(3.16). As n → ∞, if p = o(N2n), then
+ψ∗ → N(0, 1) under H0 and ψ∗ → N(a, 1) under H1.
+Let Φ be the cumulative distribution function of the standard normal. By Theorem 3.8,
+for any fixed constant t ∈ (0, a), if we reject the null hypothesis when ψ∗ > t, then the sum
+of type I and type II errors converges to [1 − Φ(t)] + [1 − Φ(a − t)].
+4
+Applications
+As mentioned in Section 1, our testing problem includes global testing for topic models,
+authorship attribution, and closeness testing for discrete distributions as special examples.
+In this section, the DELVE test is applied separately to these three problems.
+4.1
+Global testing for topic models
+Topic modeling [Blei et al., 2003] is a popular tool in text mining. It aims to learn a small
+number of “topics” from a large corpus. Given n documents written using a dictionary of p
+words, let Xi ∼ Multinomial(Ni, Ωi) denote the word counts of document i, where Ni is the
+length of this document and Ωi ∈ Rp contains the population word frequencies. In a topic
+model, there exist M topic vectors A1, A2, . . . , AM ∈ Rp, where each Ak is a PMF. Let
+1For example, we can first partition the dictionary into two halves and then partition all the documents
+into two halves; this divides {1, 2, . . . , p} × {1, 2, . . . , n} into four subsets; we construct δij’s freely on one
+subset and then specify the δij’s on the other three subsets by symmetry.
+14
+
+wi ∈ RM be a nonnegative vector whose entries sum up to 1, where wi(k) is the “weight”
+document i puts on topic k. It assumes
+Ωi =
+M
+�
+k=1
+wi(k)Ak,
+1 ≤ i ≤ n.
+(4.1)
+Under (4.1), the matrix Ω = [Ω1, Ω2, . . . , Ωn] admits a low-rank nonnegative factorization.
+Before fitting a topic model, we would like to know whether the corpus indeed involves
+multiple topics. This is the global testing problem: H0 : M = 1 v.s. H1 : M > 1. When
+M = 1, by writing A1 = µ, the topic model reduces to the null hypothesis in (3.12). We
+can apply the DELVE test by treating each Xi as a separate group (i.e., K = n).
+Corollary 4.1. Consider Model (1.1) and define a vector ξ ∈ Rn by ξi = ¯N−1Ni. Suppose
+that Ω = µ1′
+n under the null hypothesis, with µ = n−1Ωξ, and that Ω satisfies (4.1) under
+the alternative hypothesis, with r := rank(Ω) ≥ 2. Suppose ¯N/(mini Ni) = O(1). Denote
+by λ1, λ2, . . . , λr > 0 the singular values of Ω[diag(ξ)]1/2, arranged in the descending order.
+We further assume that under the alternative hypothesis,
+¯N ·
+�r
+k=2 λ2
+k
+��r
+k=1 λ2
+k
+→ ∞.
+(4.2)
+For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level α and an asymptotic
+power 1.
+The least-favorable configuration in the proof of Theorem 3.5 is in fact a topic model
+that follows (4.1) with M = 2. Transferring the argument yields the following lower bound
+that confirms the optimality of DELVE for the global testing of topic models.
+Corollary 4.2. Let Rn,M(ϵn, δn) be the collection of {(Ni, Ωi)}n
+i=1 satisfying the following
+conditions: 1) Ω follows the topic model (4.1) with M topics; 2) Condition 3.2 holds with
+o(1) replaced by ≤ ϵn; 3) ¯N(�r
+k=2 λ2
+k)/(�r
+k=1 λ2
+k)1/2 ≥ δn. If ϵn → 0 and δn → 0, then
+lim supn→∞ infΨ∈{0,1}
+�
+supRn,1(ϵn,0) P(Ψ = 1) + sup∪M≥2Rn,M(δn,δn) P(Ψ = 0)
+�
+= 1.
+The detection boundary (4.2) can be simplified when M = O(1). Following Ke and
+Wang [2022], we define ΣA = A′H−1A and ΣW = n−1WW ′, where A = [A1, A2, . . . , AM],
+W = [w1, w2, . . . , wn] and H = diag(A1M). Ke and Wang [2022] argued that it is reasonable
+to assume that eigenvalues of these two matrices are at the constant order.
+If this is
+true, with some mild additional regularity conditions, each λk is at the order of
+�
+n/p.
+Hence, (4.2) reduces to √n ¯N/√p → ∞. In comparison, Ke and Wang [2022] showed that
+a necessary condition for any estimator ˆA = [ ˆA1, ˆA2, . . . , ˆAM] to achieve
+1
+M
+�M
+k=1 ∥ ˆAk −
+Ak∥1 = o(1) is
+�
+n ¯N/p → ∞. We conclude that consistent estimation of topic vectors
+requires strictly stronger conditions than successful testing.
+4.2
+Authorship attribution
+In authorship attribution, given a corpus from a known author, we want to test whether
+a new document is from the same author.
+It is a special case of our testing problem
+15
+
+with K = 2. We can directly apply the results in Section 3.4. However, the setting in
+Section 3.4 has no sparsity. Kipnis [2022], Donoho and Kipnis [2022] point out that the
+number of words with discriminating power is often much smaller than p. To see how our
+test performs under sparsity, we consider a sparse model. As in Section 3.4, let
+Xi ∼ Multinomial(Ni, Ωi), 1 ≤ i ≤ n,
+and
+Gi ∼ Multinomial(Mi, Γi), 1 ≤ i ≤ m.
+(4.3)
+Let ¯N and ¯
+M be the average of Ni’s and Mi’s, respectively. Write η =
+1
+n ¯
+N
+�n
+i=1 NiΩi and
+θ =
+1
+m ¯
+M
+�m
+i=1 MiΓi. We assume for some βn > 0,
+ηj = θj, for j /∈ S,
+and
+��√ηj −
+�
+θj
+�� ≥ βn, for j ∈ S.
+(4.4)
+Corollary 4.3. Under the model (4.3)-(4.4), consider testing H0 : S = ∅ v.s. H1 : S ̸= ∅,
+where Condition 3.1 is satisfied. Let ηS and θS be the sub-vectors of η and θ restricted to
+the coordinates in S. Suppose that under the alternative hypothesis,
+β2
+n · (∥ηS∥1 + ∥θS∥1)
+� 1
+n ¯
+N +
+1
+m ¯
+M
+�
+max{∥η∥, ∥θ∥} → ∞.
+(4.5)
+As min{n ¯N, m ¯
+M} → ∞, the level-α DELVE test has an asymptotic level α and an asymp-
+totic power 1. Furthermore, if n ¯N ≍ m ¯
+M and minj∈S(ηj +θj) ≥ cp−1 for a constant c > 0,
+then (4.5) reduces to n ¯Nβ2
+n|S|/√p → ∞.
+Donoho and Kipnis [2022] studied a case where N = M, n = m = 1, p → ∞,
+|S| = p1−ϑ,
+and
+βn = c · N−1/2�
+log(p).
+(4.6)
+When ϑ > 1/2 (i.e., |S| = o(√p)), they derived a phase diagram for the aforementioned
+testing problem (under a slightly different setting where the data distributions are Poisson
+instead of multinomial). They showed that when ϑ > 1/2 and c is a properly large constant,
+a Higher-Criticism-based test has an asymptotically full power. Donoho and Kipnis [2022]
+did not study the case of ϑ ≤ 1/2. By Corollary 4.3, when ϑ ≤ 1/2 (i.e., |S| ≥ C√p), the
+DELVE test has asymptotically full power.
+Remark 2. When ϑ > 1/2 in (4.6), the DELVE test is powerless. However, this issue can
+be resolved by borrowing the idea of maximum test or Higher Criticism test [Donoho and
+Jin, 2004] from the classical multiple testing. For example, recalling Tj in (2.5), we can use
+max1≤j≤p{Tj/
+�
+Vj} as the test statistic, where Vj is a proper estimator of the variance of
+Tj. We leave a careful study of this idea to future work.
+4.3
+Closeness testing between discrete distributions
+Two-sample closeness testing is a subject of intensive study in discrete distribution inference
+[Bhattacharya and Valiant, 2015, Chan et al., 2014, Diakonikolas and Kane, 2016, Kim et al.,
+2022]. It is a special case of our problem with K = 2 and n1 = n2 = 1. We thereby apply
+both Theorem 3.6 and Theorem 3.7.
+16
+
+Corollary 4.4. Let Y1 and Y2 be two discrete variables taking values on the same p out-
+comes. Let Ω1 ∈ Rp and Ω2 ∈ Rp be their corresponding PMFs. Suppose we have N1
+samples of Y1 and N2 samples of Y2. The data are summarized in two multinomial vec-
+tors: X1 ∼ Multinomial(N1, Ω1), X2 ∼ Multinomial(N2, Ω2). We test H0 : Ω1 = Ω2. Write
+µ =
+1
+N1+N2 (N1Ω1 + N2Ω2). Suppose min{N1, N2} ≥ 2, max{∥Ω1∥∞, ∥Ω2∥∞} ≤ 1 − c0, for
+a constant c0 ∈ (0, 1). Suppose
+1
+(�2
+k=1 ∥Ωk∥2)2 max
+� �2
+k=1
+∥Ωk∥3
+3
+Nk , �2
+k=1
+∥Ωk∥2
+N2
+k
+�
+= o(1), and
+1
+n∥µ∥2
+�2
+k=1 ∥Ωk∥3
+3 = o(1). We assume that under the alternative hypothesis,
+∥Ω1 − Ω2∥2
+�
+N−1
+1
++ N−1
+2
+�
+max{∥Ω1∥, ∥Ω2∥} → ∞.
+(4.7)
+As min{N1, N2} → ∞, the level-α DELVE test has level α and power 1, asymptotically.
+We notice that (4.7) matches with the minimum ℓ2-separation condition for two-sample
+closeness testing [Kim et al., 2022, Proposition 4.4].
+Therefore, our test is an optimal
+ℓ2-testor. Although other optimal ℓ2-testors have been proposed [Chan et al., 2014, Bhat-
+tacharya and Valiant, 2015, Diakonikolas and Kane, 2016], they are not equipped with
+tractable null distributions.
+Remark 3. We can modify the DELVE test to incorporate frequency-dependent weights.
+Given any nonnegative vector w = (w1, w2, . . . , wp)′, define T(w) := �p
+j=1 wjTj where Tj is
+the same as in (2.5). These weights wj serve to adjust the contributions of different words.
+For example, consider wj =
+�
+max{1/p, ˆµj}
+�−1. This kind of weights have been used in
+discrete distribution inference [Balakrishnan and Wasserman, 2019, Chan et al., 2014] to
+turn an optimal ℓ2 testor to an optimal ℓ1 testor. We can similarly study the power of this
+modified test, except that we need an additional assumption n ¯N ≫ p to guarantee that ˆµj
+is a sufficiently accurate estimator of µj.
+5
+Simulations
+The proposed DELVE test is computationally efficient and easy to implement.
+In this
+section, we investigate its numerical performance in simulation studies. Real data analysis
+will be carried out in Section 6.
+Experiment 1 (Asymptotic normality) . Given (n, p, K, Nmin, Nmax, α), we generate data
+as follows: first, we divide {1, . . . , n} into K equal-size groups. Next, we draw Ωalt
+1 , . . . , Ωalt
+n
+i.i.d. from Dirichlet(p, α1p). Third, we draw Ni
+iid
+∼ Uniform[Nmin, Nmax] and set Ωnull
+i
+= µ,
+where µ :=
+1
+n ¯
+N
+�
+i NiΩalt
+i . Last, we generate X1, . . . , Xn using Model (1.1). We consider
+three sub-experiments. In Experiment 1.1, (n, p, K, Nmin, Nmax, α) = (50, 100, 5, 10, 20, 0.3).
+In Experiment 1.2, α is changed to 1, and the other parameters are the same.
+When
+α = 1, Ωalt
+i
+are drawn from the uniform distribution of the standard probability simplex;
+in comparison, α = 0.3 puts more mass near the boundary of the standard probability
+simplex. In Experiment 1.3, we keep all parameters the same as in Experiment 1.1, except
+that (p, K) are changed to (300, 50).
+For each sub-experiment, we generate 2000 data
+sets under the null hypothesis and plot the histogram of the DELVE test statistic ψ (in
+17
+
+Figure 1: Histograms of the DELVE statistic (top three panels) and the DELVE+ statistic
+(bottom three panels) in Experiments 1.1-1.3. In each plot, the blue and orange histograms
+correspond to the null and alternative hypotheses, respectively; and the green curve is the
+density of N(0, 1).
+blue); similarly, we generate 2000 data sets under the alternative hypothesis and plot the
+histogram of ψ (in orange). The results are contained on the top three panels of Figure 1.
+In Section 2.2, we introduced a variant of DELVE, called DELVE+, in which the variance
+estimator V is replaced by an adjusted one. DELVE+ has similar theoretical properties as
+DELVE but is more suitable for real data. We plot the histograms of the DELVE+ test
+statistics on the bottom three panels of Figure 1.
+We have several observations. In all sub-experiments, when the null hypothesis holds,
+the histograms of both DELVE and DELVE+ fit the standard normal density reasonably
+well. This supports our theory in Section 3.1. Second, when (p, K) increase, the finite
+sample effect becomes slightly more pronounced (c.f., Experiment 1.3 versus Experiment
+1.1). Third, the tests have power in differentiating two hypotheses. As α decreases or K
+increases, the power increases, and the histograms corresponding to two hypotheses become
+further apart. Last, in the alternative hypothesis, DELVE+ has smaller mean and variance
+than DELVE. By Lemma 2.2, these two have similar asymptotic behaviors. The simulation
+results suggest that they have noticeable finite-sample differences.
+Experiment 2 (Power curve). Similarly as before, we divide {1, 2, . . . , n} into K equal-
+size groups and draw Ni ∼ Uniform[Nmin, Nmax]. In this experiment, the PMF’s Ωi are gen-
+erated in a different way. Under the null hypothesis, we generate µ ∼ Dirichlet(p/2, α1p/2)
+and set Ωnull
+i
+= ˜µ, where ˜µj = 1
+2µj for 1 ≤ j ≤ p/2 and ˜µj = 1
+2µp+1−j for p/2 + 1 ≤ j ≤ p.
+Under the alternative hypothesis, we draw z1, . . . , zK, b1, . . . , bp/2
+iid
+∼ Rademacher(1/2) and
+then let Ωalt
+ij = µ(1 + τnzkbj), for all i in group k and 1 ≤ j ≤ p/2, and Ωalt
+ij = µ(1 + τnzkbj)
+for p/2 + 1 ≤ j ≤ p.
+By applying our theory in Section 3.2 together with some cal-
+culations, we obtain that the signal-to-noise ratio is captured by λ := K−1/2n ¯N∥µ∥τn.
+We consider three sub-experiments, Experiment 2.1-2.3, in which the parameter values of
+18
+
+Figure 2:
+Power diagrams (based on 500 repetitions) at level 5%. The x-axis plots the
+SNR λ(ωn) = K−1/2n ¯N∥µ∥ · ωn.
+(n, p, K, Nmin, Nmax, α) are the same as in Experiments 1.1-1.3. For each sub-experiment,
+we consider a grid of 10 equally-spaced values of λ. When λ = 0, it corresponds to the null
+hypothesis; when λ > 0, it corresponds to the alternative hypothesis. For each λ, we gener-
+ate 500 data sets and compute the fraction of rejections of the level-5% DELVE test. This
+gives a power curve for the level-5% DELVE test, in which the first point corresponding to
+λ = 0 is the actual level of the test. The results are contained on the top three panels of
+Figure 2. We repeat the same experiments for the DELVE+ test, which results are on the
+bottom three panels of Figure 2. In all three experiments, the actual level of our proposed
+tests is ≤ 5%, suggesting that our tests perform well at controlling the type-I error. As
+λ increases, the power gradually increased to 1, suggesting that λ is a good metric of the
+signal-to-noise ratio. This supports our theory in Section 3.2.
+6
+Real Data Analysis
+We apply our proposed methods on two real corpora: one consists of abstracts of research
+papers in four statistics journals, and the other consists of movie reviews on Amazon. For
+the analysis of real data, we use DELVE+, which modifies the variance estimator in DELVE
+and reduces the occurrence of extremely small p-values.
+6.1
+Abstracts of statisticians
+We use the data set from Ji and Jin [2016]. It contains the bibtex information of all pub-
+lished papers in four top-tier statistics journals, Annals of Statistics, Biometrika, Journal of
+the American Statistical Association, and Journal of the Royal Statistical Society - Series B,
+from 2003 to the first half of 2012. We pre-process the abstracts of papers by tokenization
+19
+
+(0,0.05)(0,0.03)(0,0.04)(0,0.04)(0,0.02)(0,0.04)Figure 3: (Left) Histogram of nonzero DELVE Z-scores for all authors in the dataset. The
+mean is 4.52 and the standard deviation is 2.94. (Right) Scatter plot of author DELVE
+scores versus the natural log of the number of papers with five statisticians identified.
+Figure 4: Pairwise Z-score plots for Peter Hall (left) and Jianqing Fan (right). In the cell
+(x, y), we compare the corpus of an author’s abstracts from time x with the corpus of that
+author’s abstracts from time y. The heatmap shows the value of DELVE+ with K = 2 for
+each cell.
+and stemming and turn each abstract to a word count vector.
+We conduct two experiments. In the first one, we fix an author and treat the collection
+of his/her co-authored abstracts as a corpus. We apply DELVE+ with K = n, where n is
+the total number of abstracts written by this author. The Z-score measures the “diversity”
+or “variability” of this authors’ abstracts. An author with a high Z-score possesses either
+diverse research interests or a variable writing style. A number of authors have only 1–2
+papers in this data set, and the variance estimator V is often negative; we remove all those
+authors. In Figure 3 (left panel), we plot the histogram of Z-scores of all retained authors.
+The mean is 4.52 and the standard deviation is 2.94. In Figure 3 (right panel), we show
+the scatter plot of Z-score versus logarithm of the number of abstracts written by this
+author, and a few prolific authors who have many papers and a large Z-score are labeled.
+For example, Peter Hall has the most papers in this dataset (82 papers in total). Hall’s
+Z-score is larger than 20, implying a huge diversity in his abstracts. There is also a positive
+association between Z-score and total papers. It suggests that senior authors have more
+diversity in their abstracts, which is as expected.
+20
+
+Peter Hall
+Jianqing Fan
+Raymond J Carrol
+ing Qin
+David DunsonPeter Hall
+2011
+2010
+10
+2009
+8
+2008
+6
+2007
+2006
+4
+2005
+2004
+2003
+TToz T0z 600z 800z L00z 900z 500 F00z E0zJianging Fan
+2007 2008 2009 2011
+10
+8
+6
+4
+2003 2004
+2
+0.
+2003
+2004
+2007
+2008
+2009
+2011Year
+Title
+Journal
+2011
+Nonparametric independence screening in sparse
+ultra-high-dimensional additive models
+JASA
+2011
+Penalized composite quasi-likelihood for ultrahigh
+dimensional variable selection
+JRSS-B
+2011
+Multiple testing via FDRL for large-scale imaging
+data
+Ann. Stat.
+2012
+Vast volatility matrix estimation using high-
+frequency data for portfolio selection
+JASA
+2012
+A road to classification in high dimensional space:
+the regularized optimal affine discriminant
+JRSS-B
+2012
+Variance estimation using refitted cross-validation
+in ultrahigh dimensional regression
+JRSS-B
+Year
+Title
+Journal
+2004
+Low order approximations in deconvolution
+and regression with errors in variables
+JRSS-B
+2004
+Nonparametric inference about service time
+distribution from indirect measurements
+JRSS-B
+2004
+Cross-validation and the estimation of condi-
+tional probability densities
+JASA
+2004
+Nonparametric confidence intervals for re-
+ceiver operating characteristic curves
+Biometrika
+2004
+Bump hunting with non-Gaussian kernels
+Ann. Stat.
+2004
+Attributing a probability to the shape of a
+probability density
+Ann. Stat.
+Figure 5: (Left) Jianqing Fan’s papers in the dataset of Ji and Jin [2016] from 2011 to
+2012. (Right) Peter Hall’s papers in the dataset of Ji and Jin [2016] from 2004.
+In the second experiment, we divide the abstracts of each author into groups by publi-
+cation year. We divide Peter Hall’s abstracts into 9 groups, and each group corresponds to
+one year. We divide Jianqing Fan’s abstracts into 6 groups, with unequal window sizes to
+make all groups have roughly equal numbers of abstracts. Our test can be used to detect
+differences between all groups, but to see more informative results, we do a pairwise com-
+parison: for each pair of groups, we apply DELVE+ with K = 2. This yields a pairwise
+plot of Z-scores. The plot reveals the temporal patterns of this author in abstract writing.
+Figure 4 shows the results for Peter Hall and Jianqing Fan.
+There are interesting temporal patterns. For Jianqing Fan (right panel of Figure 4), the
+group consisting of his 2011-2012 abstracts has comparably large Z-scores in the pairwise
+comparison with other groups. To interpret this , we gathered the titles and abstracts of all
+his papers in the dataset and compared the ones before/after 2011. He published six papers
+in these journals during 2011-2012, whose titles are listed on the left of Figure 5. We see
+that his papers in this period had a strong emphasis on screening and variable selection:
+four out of the six papers mention this subject in their titles and/or abstracts. This shows a
+departure from his previously published topics such as covariance estimation (a focus from
+2007–2009) and semiparametric estimation (a focus before 2010). Though Jianqing Fan had
+previously published papers on variable selection and screening in these journals, he had
+never published so many in such a short time period. For Peter Hall (left panel of Figure 4),
+the group of 2004 abstracts have comparably large Z-scores in the pairwise comparison with
+other groups. We examined the titles and abstracts of his 6 papers published in 2004 in
+this data set. All of his 2004 papers, except the first one, mention bandwidth selection or
+smoothing parameters, and in the last 4 papers, bandwidth selection plays a central role.
+For instance, Bump hunting with non-Gaussian kernels, (Ann. Stat., 2004) studies the
+relationship between the number of modes of a kernel density estimator and its bandwidth
+parameter. Though Peter Hall’s 2014 papers concern many nonparametric statistics topics,
+we find that bandwidth selection is a theme underlying his research in these journals in
+2004.
+21
+
+Rank
+Title
+Z-Score
+Total reviews
+1
+Prometheus
+34.44
+813
+2
+Expelled: No Intelligence Allowed
+34.17
+830
+3
+V for Vendetta
+32.24
+815
+4
+Sin City
+31.72
+828
+5
+No Country for Old Men
+30.57
+819
+...
+...
+...
+...
+16
+John Adams
+20.78
+857
+17
+Cars
+19.98
+902
+18
+Food, Inc.
+17.81
+876
+19
+Jeff Dunham: Arguing with Myself
+4.96
+860
+20
+Jeff Dunham: Spark of Insanity
+4.46
+877
+Figure 6: (Left) Histogram of Z-scores for the 500 most-reviewed movies. The mean is
+19.97 and the standard deviation is 5.07. (Right) Z-scores for the top 20 most reviewed
+movies.
+Figure 7: Pairwise Z-scores for 3 movies. In each cell, we use DELVE+ to compare reviews
+associated to a pair of star ratings. For each movie, the title list the number of reviews of
+each rating from 1–5.
+6.2
+Amazon movie reviews
+We analyze Amazon reviews from the dataset Maurya [2018] that consists of 1,924,471
+reviews of 143,007 visual media products (ie, DVDs, Bluray, or streams).
+We examine
+products with the largest number of reviews. Each product’s review corpus is cleaned and
+stemming is used to group together words with the same root. We obtain word counts
+for each review and a term-document matrix of a product’s review corpus. In the first
+experiment, we fix a movie and apply DELVE+ with K = n to the corpus consisting
+of all reviews of this movie. In Figure 6 (left panel), we plot the histogram of Z-scores
+for the top 500 most reviewed movies. The mean is 19.97 and the standard deviation is
+5.07. Compared with the histogram of Z-scores for statistics paper abstracts, there is much
+larger diversity in movie reviews. In Figure 6 (right panel), we list the 5 movies with the
+highest Z-scores and lowest Z-scores out of the 20 most reviewed movies. Each movie has
+more than 800 reviews, but some have surprisingly low Z-scores. The works by comedian
+Jeff Dunham have the lowest Z-scores, suggesting strong homogeneity among the reviews.
+The 2012 horror film Prometheus has the highest degree of review diversity among the 20
+most reviewed movies. In the second experiment, we divide the reviews of each movie into
+22
+
+Zscores of top 5oo most reviewec mnovies
+80
+70
+60
+50
+40
+30
+20
+10
+0
+5
+10
+15
+20
+25
+30
+355 groups by star rating. We compare each pair of groups using DELVE+ with K = 2,
+resulting in a pairwise Z-score plot. In Figure 7, we plot this for 3 popular movies. We
+see a variety of polarization patterns among the scores. In Harry Potter and the Deathly
+Hallows Part I, DELVE+ signifies that the reviews with ratings in the range 2–4 stars are
+all similar. We see a smooth gradation in how the 1-star reviews differ from those from 2–4
+stars, and similarly for 5-star reviews versus those from 2–4 stars. Twilight Saga: Eclipse
+shows three clusters: 1–2 stars, 3–4 stars, and 5 star, while Night of the living dead shows
+two clusters: 1–2 stars and 3–5 stars.
+7
+Discussions
+We examine the testing for equality of PMFs of K groups of high-dimensional multino-
+mial distributions. The proposed DELVE statistic has a parameter-free limiting null that
+allows for computation of Z-scores and p-values on real data. DELVE achieves the op-
+timal detection boundary over the whole range of parameters (n, p, K, ¯N), including the
+high-dimensional case p → ∞, which is very relevant to applications in text mining.
+This work leads to interesting questions for future study. So far the focus is on testing,
+but one can also consider inference for ρ2 from (2.2), which measures the heterogeneity
+among the group-wise means. Consistent variance estimation under the alternative uses a
+similar strategy, though we omit the calculations in this paper. Establishing asymptotic
+normality of DELVE under the alternative would then lead to asymptotic confidence in-
+tervals for our heterogeneity metric ρ2. Based on the plots in Section 5, it is possible that
+stronger regularity conditions are needed to obtain a pivotal distribution under the alterna-
+tive. As in the two-sample multinomial testing problems described in Kipnis and Donoho
+[2021], Kipnis [2022], such as author attribution, we may also consider an alternative where
+all the group means are the same except for a small set of “giveaway words”. It is interest-
+ing to develop a procedure for identifying these useful words. As discussed in Section 4.2,
+we may modify DELVE by using a version based on the maximum test or higher criticism.
+Another extension is to go beyond ‘bag-of-words’ style analysis and use different types of
+counts besides raw word frequencies. One option is to apply a suitably modified DELVE
+to the counts of multi-grams in the corpus and another is to combine words with similar
+meanings into a ‘superword’ and use superword counts as the basis for DELVE. To do this,
+we can combine words that are close together in some word embedding. We leave these
+interesting tasks for future work.
+Acknowledgments The research of T. Tony Cai was supported in part by NSF Grant
+DMS-2015259 and NIH grant R01-GM129781. The research of Zheng Tracy Ke was sup-
+ported in part by NSF CAREER Grant DMS-1943902.
+23
+
+Notational conventions for the appendix: We write A ≲ B (respectively, A ≳ B) if
+there exists an absolute constant C > 0 such that A ≤ C · B (respectively A ≥ C · B). If
+both A ≲ B and B ≲ A, we write A ≍ B. The implicit constant C may vary from line to
+line. For sequences at, bt indexed by an integer t ∈ N, we write at ≪ bt if bt/at → ∞ as
+t → ∞, and we write at ≫ bt if at/bt → ∞ as t → ∞. We also may write at = o(bt) to
+denote at ≪ bt. In particular, we write at = (1 + o(1))bt if at/bt → 1 as t → ∞.
+A
+Properties of T and V
+We recall that
+Xi ∼ Multinomial(Ni, Ωi),
+1 ≤ i ≤ n.
+(A.1)
+For each 1 ≤ k ≤ K, define
+µk =
+1
+nk ¯Nk
+�
+i∈Sk
+NiΩi ∈ Rp,
+Σk =
+1
+nk ¯Nk
+�
+i∈Sk
+NiΩiΩ′
+i ∈ Rp×p.
+(A.2)
+Moreover, let
+µ =
+1
+n ¯N
+K
+�
+k=1
+nk ¯Nkµk =
+1
+n ¯N
+n
+�
+i=1
+NiΩi ,
+Σ =
+1
+n ¯N
+n
+�
+k=1
+nk ¯NkΣk =
+1
+n ¯N
+�
+i
+NiΩiΩ′
+i
+(A.3)
+The DELVE test statistic is ψ = T/
+√
+V , where T is as in (2.5) and V is as in (2.7). As a
+preparation for the main proofs, in this section, we study T and V separately.
+A.1
+The decomposition of T
+It is well-known that a multinomial with the number of trials equal to N can be equivalently
+written as the sum of N independent multinomials each with the number of trials equal to
+1. This inspires us to introduce a set of independent, mean-zero random vectors:
+{Zir}1≤i≤n,1≤r≤Ni,
+with Zir = Bir − EBir, and Bir ∼ Multinomial(1, Ωi).
+(A.4)
+We use them to get a decomposition of T into mutually uncorrelated terms:
+Lemma A.1. Let {Zir}1≤i≤n,1≤r≤Ni be as in (A.4). For each Zir ∈ Rp, let {Zijr}1≤j≤p
+denote its p coordinates. Recall that ρ2 = �K
+k=1 nk ¯Nk∥µk − µ∥2. For 1 ≤ j ≤ p, define
+U1j
+=
+2
+K
+�
+k=1
+�
+i∈Sk
+Ni
+�
+r=1
+(µkj − µj)Zijr,
+U2j
+=
+K
+�
+k=1
+�
+i∈Sk
+�
+1≤r̸=s≤Ni
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�
+Ni
+Ni − 1ZijrZijs,
+U3j
+=
+− 1
+n ¯N
+�
+1≤k̸=ℓ≤K
+�
+i∈Sk
+�
+m∈Sℓ
+Ni
+�
+r=1
+Nm
+�
+s=1
+ZijrZmjs,
+24
+
+U4j
+=
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i̸=m
+Ni
+�
+r=1
+Nm
+�
+s=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�
+ZijrZmjs.
+Then, T = ρ2 + �4
+κ=1 1′
+pUκ. Moreover, E[Uκ] = 0p and E[UκU ′
+ζ] = 0p×p for 1 ≤ κ ̸= ζ ≤ 4.
+A.2
+The variance of T
+By Lemma A.1, the four terms {1′
+pUκ}1≤κ≤4 are uncorrelated with each other. Therefore,
+Var(T) = Var(1′
+pU1) + Var(1′
+pU2) + Var(1′
+pU3) + Var(1′
+pU4).
+It suffices to study the variance of each of these four terms.
+Lemma A.2. Let U1 be the same as in Lemma A.1. Define
+Θn1 = 4
+K
+�
+k=1
+nk ¯Nk
+��diag(µk)1/2(µk − µ)
+��2
+(A.5)
+Ln = 4
+K
+�
+k=1
+nk ¯Nk
+��Σ1/2
+k
+(µk − µ)
+��2
+(A.6)
+Then Var(1′
+pU1) = Θn1 − Ln. Furthermore, if max1≤k≤K ∥µk∥∞ = o(1), then Var(1′
+pU1) =
+o(ρ2).
+Lemma A.3. Let U2 be the same as in Lemma A.1. Define
+Θn2 = 2
+K
+�
+k=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2 �
+i∈Sk
+N3
+i
+Ni − 1∥Ωi∥2
+(A.7)
+An = 2
+K
+�
+k=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2 �
+i∈Sk
+N3
+i
+Ni − 1∥Ωi∥3
+3
+(A.8)
+Then
+Θn2 − An ≤ Var(1′
+pU2) ≤ Θn2.
+Furthermore, if
+max
+1≤k≤K
+��
+i∈Sk N2
+i ∥Ωi∥3
+3
+�
+i∈Sk N2
+i ∥Ωi∥2
+�
+= o(1),
+(A.9)
+then Var(1′
+pU2) = [1 + o(1)] · Θn2.
+Lemma A.4. Let U3 be the same as in Lemma A.1. Define
+Θn3 =
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j
+NiNmΩijΩmj
+(A.10)
+Bn = 2
+�
+k̸=ℓ
+nknℓ ¯Nk ¯Nℓ
+n2 ¯N2
+1′
+p(Σk ◦ Σℓ)1p
+(A.11)
+Then
+Θn3 − Bn ≤ Var(1′
+pU3) ≤ Θn3 + Bn.
+25
+
+Lemma A.5. Let U4 be the same as in Lemma A.1. Define
+Θn4 = 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i̸=m
+�
+j
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩmj.
+(A.12)
+En = 2
+�
+k
+�
+i∈Sk,m∈Sk,
+i̸=m
+�
+1≤j,j′≤p
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩij′ΩmjΩmj′
+(A.13)
+Then
+Θn4 − En ≤ Var(1′
+pU4) ≤ Θn4 + En
+.
+Using Lemmas A.2-A.5, we derive regularity conditions such that the first term in
+Var(1′
+pUκ) is the dominating term. Observe that Θn = Θn1 + Θn2 + Θn3 + Θn4, where the
+quantity Θn is defined in (2.6). The following intermediate result is useful.
+Lemma A.6. Suppose that (3.1) holds. Then
+Θn2 + Θn3 + Θn4 ≍
+�
+k
+∥µk∥2.
+(A.14)
+Moreover, under the null hypothesis, Θn ≍ K∥µ∥2.
+The next result is useful in proving that our variance estimator V is asymptotically
+unbiased.
+Lemma A.7. Suppose that (3.1) holds, and recall the definition of Θn in (2.6). Define
+βn =
+max
+� �
+k
+�
+i∈Sk
+N2
+i
+n2
+k ¯
+N2
+k ∥Ωi∥3
+3 , �
+k ∥Σk∥2
+F
+�
+K∥µ∥2
+.
+(A.15)
+If βn = o(1), then under the null hypothesis, Var(T) = [1 + o(1)] · Θn.
+We also study the case of K = 2 more explicitly. In the lemmas below we use the
+notation from Section 3.4. First we have an intermediate result analogous to Lemma A.6
+that holds under weaker conditions.
+Lemma A.8. Consider K = 2 and suppose that min Ni ≥ 2, min Mi ≥ 2 Then
+Θn2 + Θn3 + Θn4 ≍
+����
+m ¯
+M
+n ¯N + m ¯
+M η +
+n ¯N
+n ¯N + m ¯
+M θ
+����
+2
+.
+Moreover, under the null hypothesis, Θn ≍ ∥µ∥2.
+The next result is a version of Lemma A.7 for the case K = 2 that holds under weaker
+conditions.
+Lemma A.9. Suppose that mini Ni ≥ 2 and mini Mi ≥ 2. Define
+β(2)
+n
+=
+max
+� �
+i N2
+i ∥Ωi∥3, �
+i M2
+i ∥Γi∥3 , ∥Σ1∥2
+F + ∥Σ2∥2
+F
+�
+∥µ∥2
+.
+(A.16)
+If β(2)
+n
+= o(1), then under the null hypothesis, Var(T) = [1 + o(1)] · Θn.
+26
+
+A.3
+The decomposition of V
+Lemma A.10. Let {Zir}1≤i≤n,1≤r≤Ni be as in (A.4). Recall that
+V = 2
+K
+�
+k=1
+�
+i∈Sk
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2� NiX2
+ij
+Ni − 1 − NiXij(Ni − Xij)
+(Ni − 1)2
+�
+(A.17)
++
+2
+n2 ¯N2
+�
+1≤k̸=ℓ≤K
+�
+i∈Sk
+�
+m∈Sℓ
+p
+�
+j=1
+XijXmj + 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk,
+i̸=m
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+XijXmj.
+Define
+θi =
+�
+1
+nk ¯Nk
+−
+1
+n ¯N )2
+N3
+i
+Ni − 1
+for i ∈ Sk ,
+and let
+αim =
+�
+2
+n2 ¯
+N2
+if i ∈ Sk, m ∈ Sℓ, k ̸= ℓ
+2
+�
+1
+nk ¯
+Nk −
+1
+n ¯
+N )2
+if i, m ∈ Sk
+If we let
+A1 =
+�
+i
+Ni
+�
+r=1
+�
+j
+�4θiΩij
+Ni
++
+�
+m∈[n]\{i}
+2αimNmΩmj
+�
+Zijr,
+(A.18)
+A2 =
+�
+i
+�
+r̸=s∈[Ni]
+2θi
+Ni(Ni − 1)
+� �
+j
+ZijrZijs
+�
+(A.19)
+A3 =
+�
+i̸=m
+Ni
+�
+r=1
+Nm
+�
+s=1
+αim
+� �
+j
+ZijrZmjs
+�
+,
+(A.20)
+then these terms are mean zero, are mutually uncorrelated, and satisfy
+V = A1 + A2 + A3 + Θn2 + Θn3 + Θn4.
+(A.21)
+A.4
+Properties of V
+First we control the variance of V .
+Lemma A.11. Let A1, A2, and A3 be defined as in Lemma A.10. Then
+Var(A1) ≲
+1
+n ¯N ∥µ∥3
+3 +
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+≲
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+Var(A2) ≲
+�
+k
+�
+i∈Sk
+N2
+i ∥Ωi∥2
+2
+n4
+k ¯N4
+k
+≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
+Var(A3) ≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
++
+1
+n2 ¯N2 ∥µ∥2 ≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
+.
+Next we show consistency of V under the null, which is crucial in properly standardizing
+our test statistic and establishing asymptotic normality.
+27
+
+Proposition A.1. Recall the definition of βn in (A.15). Suppose that βn = o(1) and that
+the condition (3.1) holds. If under the null hypothesis we have
+K2∥µ∥4 ≫
+�
+k
+∥µ∥2
+n2
+k ¯N2
+k
+∨
+�
+k
+∥µ∥3
+3
+nk ¯Nk
+,
+(A.22)
+then V/VarT → 1 in probability.
+To later control the type II error, we must also show that V does not dominate the
+true variance under the alternative. We first state an intermediate result that is useful
+throughout.
+Lemma A.12. Suppose that, under either the null or alternative, maxi ∥Ωi∥∞ ≤ 1 − c0
+holds for an absolute constant c0 > 0. Then
+Var(T) ≳ Θn2 + Θn3 + Θn4.
+(A.23)
+Proposition A.2. Suppose that under the alternative (3.1) holds and
+� �
+k
+∥µk∥2�2 ≫
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
+∨
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+.
+(A.24)
+Then V = OP(Var(T)) under the alternative.
+We also require versions of Proposition A.1 and Proposition A.2 that hold under weaker
+conditions in the special case K = 2. We omit the proofs as they are similar. Below we use
+the notation of Section 3.4.
+Proposition A.3. Suppose that K = 2 and recall the definition of β(2)
+n
+in A.16. Suppose
+that β(2)
+n
+= o(1), mini Ni ≥ 2, mini Mi ≥ 2, and maxi ∥Ωi∥∞ ≤ 1 − c0, maxi ∥Γi∥∞ ≤ 1 − c0.
+If under the null hypothesis
+∥µ∥4 ≫ max
+� � ∥µ∥2
+2
+n2 ¯N2 + ∥µ∥2
+2
+m2 ¯
+M2
+2
+�
+,
+�∥µ∥3
+3
+n ¯N + ∥µ∥3
+3
+m ¯
+M
+��
+,
+(A.25)
+then V/Var(T) → 1 in probability.
+Under the alternative we have the following.
+Proposition A.4. Suppose that K = 2, mini Ni ≥ 2, mini Mi ≥ 2, and maxi ∥Ωi∥∞ ≤
+1 − c0, maxi ∥Γi∥∞ ≤ 1 − c0. If under the alternative
+����
+m ¯
+M
+n ¯N + m ¯
+M η +
+n ¯N
+n ¯N + m ¯
+M θ
+����
+4
+≫ max
+� � ∥η∥2
+2
+n2 ¯N2 + ∥θ∥2
+2
+m2 ¯
+M2
+2
+�
+,
+�∥η∥3
+3
+n ¯N + ∥θ∥3
+3
+m ¯
+M
+��
+,
+(A.26)
+then V = OP(Var(T)).
+In the setting of K = n and utilize the variance estimator V ∗. The next results capture
+the behavior of V ∗ under the null and alternative. The proofs are given later in this section.
+28
+
+Proposition A.5. Define
+β(n)
+n
+=
+�
+i ∥Ωi∥3
+n∥µ∥2
+.
+(A.27)
+Suppose that (3.1) holds, β(n)
+n
+= o(1), and
+n2∥µ∥4 ≫
+�
+i
+∥µ∥2
+N2
+i
+∨
+�
+i
+∥µ∥3
+3
+Ni
+.
+(A.28)
+Then V ∗/Var(T) → 1 in probability as n → ∞.
+Proposition A.6. Suppose that under the alternative (3.1) holds and
+� �
+i
+∥Ωi∥2�2 ≫
+�
+i
+∥Ωi∥2
+N2
+i
+∨
+�
+i
+∥Ωi∥3
+3
+Ni
+.
+(A.29)
+Then V ∗ = OP(Var(T)) under the alternative.
+A.5
+Proof of Lemma A.1
+We first show that E[Uκ] = 0p and E[UκU ′
+ζ] = 0p×p for κ ̸= ζ. Note that {Zir}1≤i≤n,1≤r≤Ni
+are independent mean-zero random vectors. It follows that each Uκ is a mean-zero random
+vector. We then compute E[Uκj1Uζj2] for κ ̸= ζ and all 1 ≤ j1, j2 ≤ p. By direct calculations,
+E[U1jU2j2] = 2
+�
+(k,i,r,s)
+�
+(k′,i′,r′)
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�
+(µk′j − µj)
+Ni
+Ni − 1E[Zij2rZij2sZi′j1r′].
+If i′ ̸= i, or if i′ = i and r′ /∈ {r, s}, then Zi′j1r′ is independent of Zij2rZij2s, and it follows
+that E[Zij2rZij2sZi′j1r′] = 0. If i′ = i and r = r′, then E[Zij2rZij2sZi′j1r′] = E[Zij2rZij1r] ·
+E[Zij2s]; since r ̸= s, we also have E[Zij2rZij2sZi′j1r′] = 0. This proves E[U1jU2j∗] = 0.
+Since this holds for all 1 ≤ j1, j2 ≤ p, we immediately have
+E[U1U ′
+2] = 0p×p.
+We can similarly show that E[UκU ′
+ζ] = 0p×p, for other κ ̸= ζ. The proof is omitted.
+It remains to prove the desirable decomposition of T. Recall that T = �p
+j=1 Tj. Write
+ρ2 = �p
+j=1 ρ2
+j, where ρ2
+j = 2 �K
+k=1 nk ¯Nk(µkj − µj)2. It suffices to show that
+Tj = ρ2
+j + U1j + U2j + U3j + U4j,
+for all 1 ≤ j ≤ p.
+(A.30)
+To prove (A.30), we need some preparation. Define
+Yij := Xij
+Ni
+− Ωij = 1
+Ni
+Ni
+�
+r=1
+Zijr,
+Qij := Y 2
+ij − EY 2
+ij = Y 2
+ij − Ωij(1 − Ωij)
+Ni
+.
+(A.31)
+With these notations, Xij = Ni(Ωij + Yij) and NiY 2
+ij = NiQij + Ωij(1 − Ωij). Moreover, we
+can use (A.31) to re-write Qij as a function of {Zijr}1≤r≤Ni as follows:
+Qij =
+1
+N2
+i
+Ni
+�
+r=1
+[Z2
+ijr − Ωij(1 − Ωij)] + 1
+N2
+i
+�
+1≤r̸=s≤Ni
+ZijrZijs.
+29
+
+Note that Zijr = Bijr−Ωij, where Bijr can only take values in {0, 1}. Hence, (Zijr+Ωij)2 =
+(Zijr +Ωij) always holds. Re-arranging the terms gives Z2
+ijr −Ωij(1−Ωij) = (1−2Ωij)Zijr.
+It follows that
+Qij = (1 − 2Ωij)Yij
+Ni
++ 1
+N2
+i
+�
+1≤r̸=s≤Ni
+ZijrZijs.
+(A.32)
+This is a useful equality which we will use in the proof below.
+We now show (A.30). Fix j and write Tj = Rj − Dj, where
+Rj =
+K
+�
+k=1
+nk ¯Nk(ˆµkj − ˆµj)2,
+and
+Dj =
+K
+�
+k=1
+�
+i∈Sk
+ξk
+Xij(Ni − Xij)
+nk ¯Nk(Ni − 1) ,
+with ξk = 1 − nk ¯Nk
+n ¯N
+First, we study Dj. Note that Xij(Nij − Xij) = N2
+i (Ωij + Yij)(1 − Ωij − Yij) = N2
+i Ωij(1 −
+Ωij) − N2
+i Y 2
+ij + N2
+i (1 − 2Ωij)Yij, where Y 2
+ij = Qij + N−1
+i
+Ωij(1 − Ωij). It follows that
+Xij(Nij − Xij)
+Ni(Ni − 1)
+= Ωij(1 − Ωij) − NiQij
+Ni − 1 +
+Ni
+Ni − 1(1 − 2Ωij)Yij.
+We apply (A.32) to get
+Xij(Nij − Xij)
+Ni(Ni − 1)
+= Ωij(1 − Ωij) + (1 − 2Ωij)Yij −
+1
+Ni(Ni − 1)
+�
+1≤r̸=s≤Ni
+ZijrZijs.
+(A.33)
+It follows that
+Dj =
+K
+�
+k=1
+�
+i∈Sk
+ξkNi
+nk ¯Nk
+Ωij(1 − Ωij) +
+K
+�
+k=1
+�
+i∈Sk
+ξkNi
+nk ¯Nk
+(1 − 2Ωij)Yij
+−
+K
+�
+k=1
+�
+i∈Sk
+ξk
+nk ¯Nk(Ni − 1)
+�
+1≤r̸=s≤Ni
+ZijrZijs.
+(A.34)
+Next, we study Rj. Note that nk ¯Nk(ˆµkj − ˆµj) = �
+i∈Sk(Xij − ¯Nkˆµj). It follows that
+Rj =
+K
+�
+k=1
+1
+nk ¯Nk
+��
+i∈Sk
+(Xij − ¯Nkˆµj)
+�2
+.
+Recall that Xij = Ni(Ωij+Yij). By direct calculations, �
+i∈Sk Xij = nk ¯Nkµkj+�
+i∈Sk NiYij,
+and ˆµj = µj + (n ¯N)−1 �n
+m=1 NmYmj. We then have the following decomposition:
+�
+i∈Sk
+(Xij − ¯Nkˆµj) = nk ¯Nk(µkj − µj) +
+�
+i∈Sk
+NiYij − nk ¯Nk
+n ¯N
+�
+n
+�
+m=1
+NmYmj
+�
+.
+Using this decomposition, we can expand [�
+i∈Sk(Xij − ¯Nkˆµj)]2 to a total of 6 terms, where
+3 are quadratic terms and 3 are cross terms. It yields a decomposition of Rj into 6 terms:
+Rj =
+K
+�
+k=1
+nk ¯Nk(µkj − µj)2 +
+K
+�
+k=1
+1
+nk ¯Nk
+��
+i∈Sk
+NiYij
+�2
++
+K
+�
+k=1
+nk ¯Nk
+n2 ¯N2
+�
+n
+�
+m=1
+NmYmj
+�2
+30
+
++ 2
+K
+�
+k=1
+(µkj − µj)
+��
+i∈Sk
+NiYij
+�
+− 2
+K
+�
+k=1
+nk ¯Nk
+n ¯N (µkj − µj)
+�
+n
+�
+m=1
+NmYmj
+�
+−
+2
+n ¯N
+K
+�
+k=1
+��
+i∈Sk
+NiYij
+��
+n
+�
+m=1
+NmYmj
+�
+≡ I1 + I2 + I3 + I4 + I5 + I6.
+(A.35)
+By definition, �K
+k=1 nk ¯Nk = n ¯N and �K
+k=1 nk ¯Nkµkj = n ¯Nµj. It follows that
+I3 =
+1
+n ¯N
+�
+n
+�
+m=1
+NmYmj
+�2
+,
+I5 = 0,
+I6 = − 2
+n ¯N
+�
+n
+�
+m=1
+NmYmj
+�2
+= −2I3.
+It follows that
+Rj = I1 + I2 − I3 + I4.
+(A.36)
+We further simplify I3. Recall that ξk = 1 − (n ¯N)−1nk ¯Nk. By direct calculations,
+I3 =
+1
+n ¯N
+�
+n
+�
+m=1
+NmYmj
+�2
+=
+1
+n ¯N
+� K
+�
+k=1
+��
+i∈Sk
+NiYij
+��2
+=
+1
+n ¯N
+K
+�
+k=1
+��
+i∈Sk
+NiYij
+�2
++
+1
+n ¯N
+�
+1≤k̸=ℓ≤K
+��
+i∈Sk
+NiYij
+�� �
+m∈Sℓ
+NmYmj
+�
+=
+K
+�
+k=1
+(1 − ξk)
+1
+nk ¯Nk
+��
+i∈Sk
+NiYij
+�2
++
+1
+n ¯N
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+NiNmYijYmj
+�
+��
+�
+J1
+= I2 −
+K
+�
+k=1
+�
+i∈Sk
+ξk
+nk ¯Nk
+��
+i∈Sk
+NiYij
+�2
++ J1
+= I2 + J1 −
+K
+�
+k=1
+ξk
+nk ¯Nk
+��
+i∈Sk
+N2
+i Y 2
+ij
+�
+−
+K
+�
+k=1
+ξk
+nk ¯Nk
+�
+i∈Sk,m∈Sk
+i̸=m
+NiNmYijYmj
+�
+��
+�
+J2
+.
+(A.37)
+By (A.31), NiY 2
+ij = NiQi + Ωij(1 − Ωij). We further apply (A.32) to get
+N2
+i Y 2
+ij = Ni(1 − 2Ωij)Yij +
+�
+1≤r̸=s≤Ni
+ZijrZijs + NiΩij(1 − Ωij).
+It follows that
+K
+�
+k=1
+ξk
+nk ¯Nk
+��
+i∈Sk
+N2
+i Y 2
+ij
+�
+=
+K
+�
+k=1
+�
+i∈Sk
+ξkNi
+nk ¯Nk
+(1 − 2Ωij)Yij
+�
+��
+�
+J3
++
+K
+�
+k=1
+�
+i∈Sk
+ξk
+nk ¯Nk
+�
+r̸=s
+ZijrZijs
+�
+��
+�
+J4
++
+K
+�
+k=1
+�
+i∈Sk
+ξkNi
+nk ¯Nk
+Ωij(1 − Ωij)
+�
+��
+�
+J5
+.
+(A.38)
+31
+
+We plug (A.38) into (A.37) to get I3 = I2 + J1 − J2 − J3 − J4 − J5. Further plugging I3
+into the expression of Rj in (A.36), we have
+Rj = I1 + I4 − J1 + J2 + J3 + J4 + J5,
+(A.39)
+where I1 and I4 are defined in (A.35), J1-J2 are defined in (A.37), and J3-J5 are defined in
+(A.38).
+Finally, we combine the expressions of Dj and Rj. By (A.34) and the definitions of
+J1-J5,
+Dj = J5 + J3 −
+K
+�
+k=1
+�
+i∈Sk
+ξk
+nk ¯Nk(Ni − 1)
+�
+r̸=s
+ZijrZijs
+= J5 + J3 + J4 −
+K
+�
+k=1
+�
+i∈Sk
+ξkNi
+nk ¯Nk(Ni − 1)
+�
+r̸=s
+ZijrZijs
+�
+��
+�
+J6
+.
+Combining it with (A.39) gives Tj = Rj − Dj = I1 + I4 − J1 + J2 + J6. We further plug in
+the definition of each term. It follows that
+Tj =
+K
+�
+k=1
+nk ¯Nk(µkj − µj)2 + 2
+K
+�
+k=1
+�
+i∈Sk
+(µkj − µj)NiYij −
+1
+n ¯N
+�
+k̸=ℓ
+�
+i∈Sk,m∈Sℓ
+NiNmYijYmj
++
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i̸=m
+ξk
+nk ¯Nk
+NiNmYijYmj +
+K
+�
+k=1
+�
+i∈Sk
+ξkNi
+nk ¯Nk(Ni − 1)
+�
+r̸=s
+ZijrZijs.
+(A.40)
+We plug in Yij = N−1
+i
+�Ni
+r=1 Zijr and take a sum of 1 ≤ j ≤ p. It gives (A.30) immediately.
+The proof is now complete.
+A.6
+Proof of Lemma A.2
+Recall that {Zir}1≤i≤n,1≤r≤Ni are independent random vectors. Write
+1′
+pU1 = 2
+K
+�
+k=1
+�
+i∈Sk
+Ni
+�
+r=1
+(µk − µ)′Zir.
+The covariance matrix of Zir is diag(Ωi) − ΩiΩ′
+i. It follows that
+Var(1′
+pU1) = 4
+K
+�
+k=1
+�
+i∈Sk
+Ni
+�
+r=1
+(µk − µ)′�
+diag(Ωi) − ΩiΩ′
+i
+�
+(µk − µ)
+= 4
+�
+k
+(µk − µ)′�
+diag
+��
+i∈Sk
+NiΩi
+�
+−
+��
+i∈Sk
+NiΩiΩ′
+i
+��
+(µk − µ)
+= 4
+�
+k
+(µk − µ)′�
+diag(nk ¯Nkµk) − nk ¯NkΣk
+�
+(µk − µ)
+32
+
+= 4
+�
+k
+nk ¯Nk
+��diag(µk)1/2(µk − µ)
+��2 − 4
+�
+k
+nk ¯Nk
+��Σ1/2
+k
+(µk − µ)
+��2.(A.41)
+This proves the first claim. Furthermore, by (A.41),
+Var(1′
+pU1) ≤ 4
+�
+k
+nk ¯Nk
+��diag(µk)1/2(µk − µ)
+��2 ≤ 4
+�
+k
+nk ¯Nk∥diag(µk)∥∥µk − µ∥2.
+Note that ∥diag(µk)∥ = ∥µk∥∞. Therefore, if maxk ∥µk∥∞ = o(1), the right hand side
+above is o(1) · 4 �
+k nk ¯Nk∥µk − µ∥2 = o(ρ2). This proves the second claim.
+A.7
+Proof of Lemma A.3
+For each 1 ≤ k ≤ K, define a set of index triplets: Mk = {(i, r, s) : i ∈ Sk, 1 ≤ r < s ≤ Ni}.
+Let M = ∪K
+k=1Mk. Write for short θi = (
+1
+nk ¯
+Nk −
+1
+n ¯
+N )2 N3
+i
+Ni−1, for i ∈ Sk. It is seen that
+1′
+pU2 = 2
+�
+(i,r,s)∈M
+√θi
+�
+Ni(Ni − 1)
+Wirs,
+with
+Wirs =
+p
+�
+j=1
+ZijrZijs.
+For Wirs and Wi′r′s′, if i ̸= i′, or if i = i′ and {r, s}∩{r′, s′} = ∅, then these two variables are
+independent; if i = i′, r = r′ and s ̸= s′, then E[WirsWirs′] = �
+j,j′ E[ZijrZijsZij′rZij′s′] =
+�
+j,j′ E[ZijrZij′r]·E[Zijs]·E[Zij′s′] = 0. Therefore, {Wirs}(i,r,s)∈M is a collection of mutually
+uncorrelated variables. It follows that
+Var(1′
+pU2) = 4
+�
+(i,r,s)∈M
+θi
+Ni(Ni − 1)Var(Wirs).
+It remains to calculate the variance of each Wirs. By direction calculations,
+Var(Wirs) =
+�
+j
+E[Z2
+ijrZ2
+ijs] + 2
+�
+j<ℓ
+E[ZijrZijsZiℓrZiℓs]
+=
+�
+j
+[Ωij(1 − Ωij)]2 + 2
+�
+j<ℓ
+(−ΩijΩiℓ)2
+=
+�
+j
+Ω2
+ij − 2
+�
+j
+Ω3
+ij +
+��
+j
+Ω2
+ij
+�2
+= ∥Ωi∥2 − 2∥Ωi∥3
+3 + ∥Ωi∥4
+(A.42)
+Since maxij Ωij ≤ 1, we have
+∥Ωi∥2 − ∥Ωi∥3
+3 ≤ Var(Wirs) ≤ ∥Ωi∥2.
+Therefore,
+Var(1′
+pU2) = 4
+K
+�
+k=1
+�
+i∈Sk
+�
+1≤r<s≤Ni
+θi
+Ni(Ni − 1)Var(Wirs)
+33
+
+= 2
+K
+�
+k=1
+�
+i∈Sk
+θiVar(Wirs) ≥ 2
+K
+�
+k=1
+�
+i∈Sk
+θi
+�
+∥Ωi∥2 − ∥Ωi∥3
+3
+�
+= Θn2 − An,
+and similarly Var(1′
+pU2) ≤ Θn2, which proves the first claim. To prove the second claim,
+note that Var(1′
+pU2) = Θn2 + O(An). By (A.9) and the assumption min Ni ≥ 2, we have
+An ≲
+�
+k
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2 �
+i∈Sk
+N2
+i ∥Ωi∥3
+3
+=
+�
+k
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2 · o
+� �
+i∈Sk
+N2
+i ∥Ωi∥2
+�
+= o(Θn2),
+which implies that Var(1pU2) = [1 + o(1)]Θn2, as desired.
+A.8
+Proof of Lemma A.4
+For each 1 ≤ k < ℓ ≤ K, define a set of index quadruples: Jkℓ = {(i, r, m, s) : i ∈ Sk, j ∈
+Sℓ, 1 ≤ r ≤ Ni, 1 ≤ s ≤ Nm}. Let J = ∪(k,ℓ):1≤k<ℓ≤KJkℓ. It is seen that
+1′
+pU3 = − 2
+n ¯N
+�
+(i,r,m,s)∈J
+Virms,
+where Virms =
+p
+�
+j=1
+ZijrZmjs.
+For Virms and Vi′r′m′s′, if {(i, r), (m, s)} ∩ {(i′, r′), (m′, s′)} = ∅, then the two variables are
+independent of each other. If (i, r) = (i′, r′) and (m, s) ̸= (m′, s′), then E[VirmsVirm′s′] =
+�
+j,j′ E[ZijrZmjsZij′rZm′j′s′] = �
+j,j′ E[ZijrZij′r] · E[Zmjs] · E[Zm′js′] = 0. Therefore, the
+only correlated case is when (i, r, m, s) = (i′, r′, m′, s′). This implies that {Virms}(i,r,m,s)∈J
+is a collection of mutually uncorrelated variables. Therefore,
+Var(1′
+pU3) =
+4
+n2 ¯N2
+�
+(i,r,m,s)∈J
+Var(Virms).
+Note that Var(Virms) = E[(�
+j ZijrZmjs)2] = �
+j,j′ E[ZijrZmjsZij′rZmj′s]; also, the covari-
+ance matrix of Zir is diag(Ωi) − ΩiΩ′
+i. It follows that
+Var(Virms) =
+�
+j
+E[Z2
+ijr] · E[Z2
+mjs] +
+�
+j̸=j′
+E[ZijrZij′r] · E[ZmjsZmj′s]
+=
+�
+j
+Ωij(1 − Ωij)Ωmj(1 − Ωmj) +
+�
+j̸=j′
+ΩijΩij′ΩmjΩmj′
+=
+�
+j
+ΩijΩmj − 2
+�
+j
+Ω2
+ijΩ2
+mj +
+�
+j,j′
+ΩijΩij′ΩmjΩmj′.
+(A.43)
+Write for short δim = −2 �
+j Ω2
+ijΩ2
+mj + �
+j,j′ ΩijΩij′ΩmjΩmj′.Combining the above gives
+Var(1′
+pU3) =
+4
+n2 ¯N2
+�
+k<ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+Ni
+�
+r=1
+Nm
+�
+s=1
+��
+j
+ΩijΩmj + δim
+�
+34
+
+=
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j
+NiNmΩijΩmj +
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+NiNmδim. (A.44)
+It is easy to see that |δim| ≤ �
+j,j′ ΩijΩij′ΩmjΩmj′. Also, by the definition of Σk in (A.2),
+we have Σk(j, j′) =
+1
+nk ¯
+Nk
+�
+i∈Sk NiΩijΩij′. Using these results, we immediately have
+���
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+NiNmδim
+��� ≤
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j,j′
+NiNmΩijΩij′ΩmjΩmj′
+=
+2
+n2 ¯N2
+�
+j,j′
+�
+k̸=ℓ
+��
+i∈Sk
+NiΩijΩij′
+�� �
+m∈Sℓ
+NiΩmjΩmj′
+�
+=
+2
+n2 ¯N2
+�
+j,j′
+�
+k̸=ℓ
+nk ¯NkΣk(j, j′) · nℓ ¯NℓΣℓ(j, j′)
+= 2
+�
+k̸=ℓ
+nknℓ ¯Nk ¯Nℓ
+n2 ¯N2
+1′
+p(Σk ◦ Σℓ)1p =: Bn
+(A.45)
+as desired.
+A.9
+Proof of Lemma A.5
+For 1 ≤ k ≤ K, define a set of index quadruples: Qk = {(i, r, m, s) : i ∈ Sk, m ∈ Sk, i <
+m, 1 ≤ r ≤ Ni, 1 ≤ s ≤ Nm}. Let Q = ∪K
+k=1Qk. Write κim = (
+1
+nk ¯
+Nk −
+1
+n ¯
+N )2NiNm, for
+i ∈ Sk and m ∈ Sk. It is seen that
+1′
+pU4 = 2
+�
+(i,r,m,s)∈Q
+√κim
+√NiNm
+Virms,
+where
+Virms =
+p
+�
+j=1
+ZijrZmjs.
+It is not hard to see that Virms and Vi′r′m′s′ are correlated only if (i, r, m, s) = (i′, r′, m′, s′).
+It follows that
+Var(1′
+pU4) = 4
+�
+(i,r,m,s)∈Q
+κim
+NiNm
+Var(Virms).
+In the proof of Lemma A.4, we have studied Var(Virms). In particular, by (A.43), we have
+Var(Virms) =
+�
+j
+ΩijΩmj + δim,
+with
+|δim| ≤
+�
+j,j′
+ΩijΩij′ΩmjΩmj′.
+Thus
+Var(1′
+pU4) = 4
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i<m
+Ni
+�
+i=1
+Nm
+�
+r=1
+κim
+NiNm
+Var(Virms)
+= 4
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i<m
+κim
+��
+j
+ΩijΩmj + δim
+�
+35
+
+= 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i̸=m
+�
+j
+κimΩijΩmj ± 2
+�
+k
+�
+i̸=m∈Sk
+κim
+�
+j,j′
+ΩijΩij′ΩmjΩmj′,
+= Θn3 ± En.
+(A.46)
+which proves the lemma.
+A.10
+Proof of Lemma A.6
+By assumption (3.1), N3
+i /(Ni − 1) ≍ Ni and
+�
+1
+nk ¯
+Nk −
+1
+n ¯
+N
+�2
+≍
+1
+n2
+k ¯
+N2
+k . First, observe that
+Θn2 + Θn4 = 2
+K
+�
+k=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2 �
+i∈Sk
+N3
+i
+Ni − 1∥Ωi∥2
++ 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i̸=m
+�
+j
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩmj
+≍
+�
+k=1
+�
+1
+nk ¯Nk
+�2 �
+j
+�
+i,m∈Sk
+NiΩij · NmΩij =
+�
+k
+∥µk∥2.
+(A.47)
+Recall the definitions of µk and µ in (A.2)-(A.3). By direct calculations, we have
+Θn3 = 2
+�
+j
+�
+k̸=ℓ
+� 1
+n ¯N
+�
+i∈Sk
+NiΩij
+�� 1
+n ¯N
+�
+m∈Sℓ
+NmΩmj
+�
+= 2
+�
+j
+�
+k̸=ℓ
+nk ¯Nk
+n ¯N µkj · nℓ ¯Nℓ
+n ¯N µℓj
+= 2
+�
+k̸=ℓ
+nknℓ ¯Nk ¯Nℓ
+n2 ¯N2
+· µ ′
+k µℓ
+≤ 2
+�
+j
+��
+k
+nk ¯Nk
+n ¯N µkj
+�2
+= 2
+�
+j
+µ2
+j = 2∥µ∥2.
+(A.48)
+By Cauchy–Schwarz,
+∥µ∥2 =
+�
+j
+� �
+k
+(nk ¯
+Nk
+n ¯N )µkj
+�2
+≤
+�
+j
+� �
+k
+(nk ¯
+Nk
+n ¯N )2
+�
+·
+� �
+k
+µ2
+kj
+�
+≤
+�
+j
+� �
+k
+(nk ¯
+Nk
+n ¯N )
+�
+·
+� �
+k
+µ2
+kj
+�
+=
+�
+j
+�
+k
+µ2
+kj =
+�
+k
+∥µk∥2.
+(A.49)
+Combining (A.47), (A.48), and (A.49) yields
+c
+� �
+k
+∥µk∥2�
+≤ Θn2 + Θn3 + Θn4 ≤ C
+� �
+k
+∥µk∥2�
+,
+for absolute constants c, C > 0. This completes the proof.
+36
+
+A.11
+Proof of Lemma A.7
+By (3.1), it holds that
+(
+1
+nk ¯Nk
+−
+1
+n ¯N )2 ≍
+1
+(nk ¯Nk)2 ,
+(A.50)
+and moreover, for all i ∈ {1, 2, . . . , n},
+N3
+i
+Ni − 1 ≍ N2
+i .
+(A.51)
+Recall the definitions of An, Bn, and En in (A.8), (A.11), and (A.13), respectively. Note
+that these are the remainder terms in Lemmas A.3, A.4, and A.5, respectively. Under the
+null hypothesis (recall Θn1 ≡ 0 under the null),
+Var(T) = Θn2 + Θn3 + Θn4 + O(An + Bn + En).
+(A.52)
+It holds that
+An ≤
+K
+�
+k=1
+�
+1
+nk ¯Nk
+�2 �
+i∈Sk
+N2
+i ∥Ωi∥3
+3.
+(A.53)
+Next, by linearity and the definition of Σk, Σ in (A.2), (A.3), respectively,
+Bn ≤ 2
+�
+k,ℓ
+nknℓ ¯Nk ¯Nℓ
+n2 ¯N2
+1′
+p(Σk ◦ Σℓ)1p
+≤ 21′
+p
+� 1
+n ¯N
+�
+k
+nk ¯NkΣk
+�
+◦
+� 1
+n ¯N
+�
+ℓ
+nℓ ¯NℓΣkℓ
+�
+1p
+= 21′
+p(Σ ◦ Σ)1p = 2∥Σ∥2
+F
+By Cauchy–Schwarz,
+Bn ≤ ∥Σ∥2
+F =
+�
+j,j′
+� �
+k
+(nk ¯Nk
+n ¯N Σk(j, j′)
+�2
+≤
+�
+j,j′
+� �
+k
+(nk ¯Nk
+n ¯N )2
+�
+·
+� �
+k
+Σk(j, j′)2
+�
+≤
+�
+j,j′
+� �
+k
+nk ¯Nk
+n ¯N
+�
+·
+� �
+k
+Σk(j, j′)2
+�
+=
+�
+j,j′
+�
+k
+Σk(j, j′)2 =
+�
+k
+∥Σk∥2
+F .
+(A.54)
+Next by the definition of Σk in (A.2), we have Σk(j, j′) =
+1
+nk ¯
+Nk
+�
+i∈Sk NiΩijΩij′. It
+follows that
+En ≤
+�
+k
+�
+j,j′
+�
+1
+nk ¯Nk
+�
+i∈Sk
+NiΩijΩij′
+��
+1
+nk ¯Nk
+�
+m∈Sk
+NmΩmjΩmj′
+�
+=
+�
+k
+�
+j,j′
+Σ2
+k(j, j′) =
+�
+k
+∥Σk∥2
+F .
+(A.55)
+37
+
+Next, Lemma A.6 implies that
+Θn2 + Θn3 + Θn4 ≍
+�
+k
+∥µk∥2 = K∥µ∥2,
+(A.56)
+where we use that the null hypothesis holds. By assumption of the lemma, we have
+βn =
+max
+� �
+k
+�
+i∈Sk
+N2
+i
+n2
+k ¯
+N2
+k ∥Ωi∥3
+3 , �
+k ∥Σk∥2
+F
+�
+K∥µ∥2
+= o(1)
+Combining this with (A.52), (A.53), (A.54), (A.55),and (A.56) completes the proof of the
+first claim. The second claim follows plugging in µk = µ for all k ∈ {1, 2, . . . , K}.
+A.12
+Proof of Lemma A.8
+By assumption, N3
+i /(Ni − 1) ≍ Ni, M3
+i /(Mi − 1) ≍ Mi. By direct calculation,
+Θn2 + Θn4 ≍
+�
+m ¯
+M
+(n ¯N + m ¯
+M)n ¯N
+�2 �
+i,m,j
+NiNmΩijΩmj +
+�
+n ¯N
+(n ¯N + m ¯
+M)m ¯
+M
+�2 �
+i,m
+NiNmΓijΓmj
+=
+1
+(n ¯N + m ¯
+M)2
+�
+(m ¯
+M)2∥η∥2 + n ¯N2∥θ∥2
+�
+.
+(A.57)
+Next
+Θn3 =
+4
+(n ¯N + m ¯
+M)2
+�
+i∈S1
+�
+m∈S2
+�
+j
+NiΩij · NmΓmj
+=
+4
+(n ¯N + m ¯
+M)2 · n ¯Nm ¯
+M⟨θ, η⟩.
+(A.58)
+Combining (A.57) and (A.58) yields
+Θn2 + Θn3 + Θn4 ≍
+1
+(n ¯N + m ¯
+M)2
+�
+(m ¯
+M)2∥η∥2 + 2n ¯Nm ¯
+M⟨θ, η⟩ + n ¯N2∥θ∥2�
+=
+����
+m ¯
+M
+n ¯N + m ¯
+M η +
+n ¯N
+n ¯N + m ¯
+M θ
+����
+2
+,
+which proves the first claim.
+The second follows by plugging in θ = η = µ under the
+null.
+A.13
+Proof of Lemma A.9
+As in (A.52), we have under the null that
+Var(T) = Θn2 + Θn3 + Θn4 + O(An + Bn + En).
+(A.59)
+For general K, observe that the proofs of the bounds
+An ≤
+K
+�
+k=1
+�
+1
+nk ¯Nk
+�2 �
+i∈Sk
+N2
+i ∥Ωi∥3
+3
+38
+
+Bn ≤
+K
+�
+k=1
+∥Σk∥2
+F
+En ≤
+K
+�
+k=1
+∥Σk∥2
+F
+derived in (A.53), (A.54), and (A.55), only use the assumption that Ni, Mi ≥ 2 for all i.
+Translating these bounds to the notation of the K = 2 case, we have
+An ≤
+�
+i
+N2
+i ∥Ωi∥3 +
+�
+i
+M2
+i ∥Γi∥3
+Bn ≤ ∥Σ1∥2
+F + ∥Σ2∥2
+F
+En ≤ ∥Σ1∥2
+F + ∥Σ2∥2
+F .
+(A.60)
+Furthermore, we know that Θn ≥ c∥µ∥2 under the null by Lemma A.8, for an absolute
+constant c > 0. Combining this with (A.59) and (A.60) completes the proof.
+A.14
+Proof of Lemma A.10
+Define
+V1 = 2
+K
+�
+k=1
+�
+i∈Sk
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2� NiX2
+ij
+Ni − 1 − NiXij(Ni − Xij)
+(Ni − 1)2
+�
+V2 =
+2
+n2 ¯N2
+�
+1≤k̸=ℓ≤K
+�
+i∈Sk
+�
+m∈Sℓ
+p
+�
+j=1
+XijXmj
+V3 = 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk,
+i̸=m
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+XijXmj.
+Observe that V1 + V2 + V3 = V . Also define
+A11 =
+�
+i
+Ni
+�
+r=1
+�
+j
+�4θiΩij
+Ni
+�
+Zijr
+(A.61)
+A12 = 2
+�
+i
+Ni
+�
+r=1
+�
+j
+�
+�
+m∈[n]\{i}
+αimNmΩmj
+�
+Zijr
+(A.62)
+and observe that A11 + A12 = A1.
+First, we derive the decomposition of V1. Recall that
+Yij := Xij
+Ni
+− Ωij = 1
+Ni
+Ni
+�
+r=1
+Zijr,
+Qij := Y 2
+ij − EY 2
+ij = Y 2
+ij − Ωij(1 − Ωij)
+Ni
+.
+(A.63)
+With these notations, Xij = Ni(Ωij + Yij) and NiY 2
+ij = NiQij + Ωij(1 − Ωij).
+39
+
+Write
+V1 = 2
+n
+�
+i=1
+n
+�
+i=1
+θi
+Ni
+∆ij,
+where
+∆ij :=
+X2
+ij
+Ni
+− Xij(Ni − Xij)
+Ni(Ni − 1) .
+(A.64)
+Note that Xij = Ni(Ωij + Yij) and Y 2
+ij = Qij + N−1
+i
+Ωij(1 − Ωij). It follows that
+X2
+ij
+Ni
+= NiΩ2
+ij + 2NiΩijYij + NiQij + Ωij(1 − Ωij).
+In (A.32), we have shown that Qij = (1−2Ωij) Yij
+Ni + 1
+N2
+i
+�
+1≤r̸=s≤Ni ZijrZijs. It follows that
+X2
+ij
+Ni
+= NiΩ2
+ij + 2NiΩijYij + (1 − 2Ωij)Yij + 1
+Ni
+�
+1≤r̸=s≤Ni
+ZijrZijs + Ωij(1 − Ωij).
+Additionally, by (A.33),
+Xij(Nij − Xij)
+Ni(Ni − 1)
+= Ωij(1 − Ωij) + (1 − 2Ωij)Yij −
+1
+Ni(Ni − 1)
+�
+1≤r̸=s≤Ni
+ZijrZijs.
+Combining the above gives
+∆ij = NiΩ2
+ij + 2NiΩijYij +
+1
+Ni − 1
+�
+1≤r̸=s≤Ni
+ZijrZijs
+= NiΩ2
+ij + 2Ωij
+Ni
+�
+r=1
+Zijr +
+1
+Ni − 1
+�
+1≤r̸=s≤Ni
+ZijrZijs.
+(A.65)
+Recall the definition of Θn2 in (A.7), A2 in (A.19), and A11 in (A.61). We have
+V1 = 2
+�
+k,i∈Sk
+�
+j
+θi
+Ni
+�
+NiΩ2
+ij + 2Ωij
+Ni
+�
+r=1
+Zijr +
+1
+Ni − 1
+�
+1≤r̸=s≤Ni
+ZijrZijs
+�
+.
+= Θn2 +
+�
+k,i∈Sk
+�
+j
+4θiΩij
+Ni
+Ni
+�
+r=1
+Zijr +
+�
+k,i∈Sk
+�
+j
+2θi
+Ni(Ni − 1)
+�
+1≤r̸=s≤Ni
+ZijrZijs
+= Θn2 + A11 + A2
+(A.66)
+Next, we have
+V2 + V3 =
+�
+i̸=m
+αimNiNm
+�
+j
+�
+(Yij + Ωij)(Ymj + Ωmj)
+�
+=
+�
+i̸=m
+αimNiNm
+�
+j
+YijYmj + 2
+�
+i̸=m
+αimNiNm
+�
+j
+YijΩmj +
+�
+i̸=m
+αimNiNm
+�
+j
+ΩijΩmj
+=
+�
+i̸=m
+Ni
+�
+r=1
+Nm
+�
+s=1
+αim
+� �
+j
+ZijrZmjs
+�
++ 2
+�
+i
+Ni
+�
+r=1
+�
+j
+�
+�
+m∈[n]\{i}
+αimNmΩmj
+�
+Zijr + Θn3 + Θn4
+= A3 + A12 + Θn3 + Θn4.
+40
+
+Hence
+A1 + A2 + A3 + Θn2 + Θn3 + Θn4 = V,
+which verifies (A.21). By inspection, we also see that EAb = 0 for b ∈ {1, 2, 3}. That
+A1, A2, A3 are mutually uncorrelated follows immediately from the linearity of expecta-
+tion and the fact that the random variables {Zijr}i,r ∪ {ZijrZmjs}(i,r)̸=(m,s) are mutually
+uncorrelated.
+A.15
+Proof of Lemma A.11
+Define
+γirj = 4θiΩij
+Ni
++
+�
+m∈[n]\{i}
+2αimNmΩmj
+(A.67)
+and recall that A1 = �
+i
+�
+r∈[Ni]
+�
+j γirjZijr. First we develop a bound on γirj. Suppose
+that i ∈ Sk. Then we have
+γirj ≲ NiΩij
+n2
+k ¯N2
+k
++
+�
+m∈Sk,m̸=i
+NmΩmj
+n2
+k ¯N2
+k
++
+�
+k′∈[K]\{k}
+�
+m∈Sk′
+NmΩmj
+n2 ¯N2
+≲
+µkj
+nk ¯Nk
++ µj
+n ¯N .
+Next using properties of the covariance matrix of a multinomial vector, we have
+Var(A1) =
+�
+i,r∈[Ni]
+Var(γ′
+ir:Zi:r) =
+�
+i,r∈[Ni]
+γ′
+ir:Cov(Zi:r)γir:
+≤
+�
+i,r∈[Ni]
+γ′
+ir:diag(Ωi:)γir: =
+�
+i,r∈[Ni]
+�
+j
+Ωijγ2
+irj
+≲
+�
+k,j
+� µkj
+nk ¯Nk
++ µj
+n ¯N
+�2
+�
+i∈Sk,r∈[Ni]
+Ωij
+≲
+�
+k,j
+� µkj
+nk ¯Nk
+�2nk ¯Nkµkj +
+�
+k,j
+� µj
+n ¯N
+�2nk ¯Nkµkj
+= (
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+) + ∥µ∥3
+3
+n ¯N ≲
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+,
+(A.68)
+which proves the first claim. The last inequality follows because by Jensen’s inequality
+(noting that the function x �→ x3 is convex for x ≥ 0),
+∥µ∥3
+3 =
+�
+j
+� �
+k
+(nk ¯Nk
+n ¯N )µkj
+�3
+≤
+�
+j
+�
+k
+(nk ¯Nk
+n ¯N )µ3
+kj ≤
+�
+k
+∥µk∥3
+3.
+Next observe that
+A2 =
+�
+i
+�
+r̸=s
+2θi
+Ni(Ni − 1)Wirs
+(A.69)
+41
+
+where recall Wirs = �
+j ZijrZijs. Also recall that Wirs and Wi′r′s′ are uncorrelated unless
+i = i′ and {r, s} = {r′, s′}. By (A.42),
+Var(A2) =
+�
+i
+�
+r̸=s
+4θ2
+i
+N2
+i (Ni − 1)2 Var(Wirs)
+≲
+�
+i
+�
+r̸=s
+4θ2
+i
+N2
+i (Ni − 1)2 ∥Ωi∥2
+≲
+�
+k
+�
+i∈Sk
+·(
+1
+nk ¯Nk
+−
+1
+n ¯N )4
+N6
+i
+(Ni − 1)2 ·
+1
+Ni(Ni − 1)∥Ωi∥2
+≲
+�
+k
+�
+i∈Sk
+N2
+i
+n4
+k ¯N4
+k
+∥Ωi∥2
+(A.70)
+Also observe that
+�
+k
+1
+n4
+k ¯N4
+k
+�
+i∈Sk
+N2
+i ∥Ωi∥2
+2 ≤
+�
+k
+1
+n2
+k ¯N2
+k
+�
+i,m∈Sk
+�
+( Ni
+nk ¯Nk
+)Ωi, ( Nm
+nm ¯Nm
+)Ωm
+�
+=
+�
+k
+1
+n2
+k ¯N2
+k
+∥µk∥2.
+This establishes the second claim.
+Last we study A3. Observe that
+A3 =
+�
+i̸=m
+Ni
+�
+r=1
+Nm
+�
+s=1
+αimVirms
+where recall Virms = �
+j ZijrZmjs. Recall that Virms and Vi′r′m′s′ are uncorrelated unless
+(r, s) = (r′, s′) and {i, m} = {i′, m′} .By (A.43),
+Var(A3) ≲
+�
+i̸=m
+α2
+imNiNm
+�
+j
+ΩijΩmj
+≲
+�
+k
+�
+i̸=m∈Sk
+1
+n4
+k ¯N4
+k
+⟨NiΩi, NmΩm⟩ +
+�
+k̸=ℓ
+�
+i∈Sk,m∈Sℓ
+1
+n4 ¯N4 ⟨NiΩi, NmΩm⟩
+≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
++
+�
+k,ℓ
+1
+n4 ¯N4 ⟨nk ¯Nkµk, nℓ ¯Nℓµℓ⟩
+≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
++ ∥µ∥2
+n2 ¯N2 ≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
+.
+(A.71)
+In the last line we use that ∥µ∥2 ≤ 2 � ∥µk∥2 as shown in (A.49). This proves all required
+claims.
+A.16
+Proof of Proposition A.1
+Under the null hypothesis, we have Θn1 ≡ 0. Thus, EV = Θn under the null by Lemma
+A.10. Under (3.1), we have Var(T) = [1 + o(1)]Θn. Therefore,
+EV = [1 + o(1)]Var(T),
+(A.72)
+42
+
+so V is asymptotically unbiased under the null. Furthermore, by Lemma A.6, we have
+Θn ≍ K∥µ∥2.
+(A.73)
+In Lemma A.11, we showed that
+Var(A2) ≲
+�
+k
+�
+i∈Sk
+N2
+i ∥Ωi∥2
+2
+n4
+k ¯N4
+k
+We conclude by Lemma A.11 that under the null
+Var(V ) ≲
+�
+k
+∥µ∥2
+n2
+k ¯N2
+k
+∨
+�
+k
+∥µ∥3
+3
+nk ¯Nk
+.
+(A.74)
+By Chebyshev’s inequality, (A.73), (A.74), and assumption (A.22) of the theorem state-
+ment, we have
+|V − EV |
+Var(T)
+≍ |V − EV |
+K∥µ∥2
+= oP(1).
+Thus by (A.72),
+V
+Var(T) = (V − EV )
+Var(T)
++
+EV
+Var(T) = oP(1) + [1 + o(1)],
+as desired.
+A.17
+Proof of Lemma A.12
+By Lemmas A.1–A.5, we have
+Var(T) =
+4
+�
+a=1
+Var(1′
+pUa) ≥ (
+4
+�
+a=2
+Θna) − (An + Bn + En).
+(A.75)
+Using that maxi ∥Ωi∥∞ ≤ 1 − c0, we have ∥Ωi∥3 ≤ (1 − c0)∥Ωi∥2, which implies that
+An ≤ (1 − c0)Θn2.
+(A.76)
+Again using maxi ∥Ωi∥∞ ≤ 1 − c0, as well as �
+j′ Ωij′ = 1, we have
+Bn =
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j,j′
+NiNmΩijΩij′ΩmjΩmj′
+≤ (1 − c0) ·
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j,j′
+NiNmΩijΩij′Ωmj
+= (1 − c0) ·
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j
+NiNmΩijΩmj
+≤ (1 − c0) · Θn3.
+(A.77)
+43
+
+Similarly to control En, we again use maxi ∥Ωi∥∞ ≤ 1 − c0 and obtain
+En = 2
+�
+k
+�
+i∈Sk,m∈Sk,
+i̸=m
+�
+1≤j,j′≤p
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩij′ΩmjΩmj′
+≤ (1 − c0) · 2
+�
+k
+�
+i∈Sk,m∈Sk,
+i̸=m
+�
+1≤j,j′≤p
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩij′Ωmj
+≤ (1 − c0) · 2
+�
+k
+�
+i∈Sk,m∈Sk,
+i̸=m
+�
+1≤j≤p
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩmj
+≤ (1 − c0) · Θn4.
+(A.78)
+Combining (A.75), (A.76), (A.77), and (A.78) finishes the proof.
+A.18
+Proof of Proposition A.2
+By Lemmas A.6 and A.12,
+Var(T) ≳ Θn2 + Θn3 + Θn4 ≳
+�
+k
+∥µk∥2.
+(A.79)
+By Lemma A.11,
+Var(V ) ≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
+∨
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+(A.80)
+Using a similar argument based on Chebyshev’s inequality as in the proof of Proposition
+A.1 and applying (A.79) and (A.80), we have
+|V − EV |
+Var(T)
+≳ |V − EV |
+�
+k ∥µk∥2 = oP(1).
+(A.81)
+Next, by Lemma A.10 and (A.79),
+EV = Θn2 + Θn3 + Θn4 ≲ Var(T).
+(A.82)
+Combining (A.81) and (A.82) finishes the proof.
+A.19
+Proof of Proposition A.5
+From the proof of Lemma A.10, we have
+V ∗ = V1 = Θn2 + A11 + A2,
+and the terms on the right-hand-side are mutually uncorrelated. From (A.68), we have
+Var(A11) ≲
+�
+i
+∥Ωi∥3
+3
+Ni
+44
+
+Var(A2) ≲
+�
+i
+∥Ωi∥2
+N2
+i
+.
+Hence
+EV ∗ = Θn2
+Var(V ∗) ≲
+�
+i
+∥Ωi∥3
+3
+Ni
+∨
+�
+i
+∥Ωi∥2
+N2
+i
+.
+(A.83)
+Since K = n and the null hypothesis holds, we have Θn1 ≡ Θn4 ≡ 0. Moreover, by
+(A.48), we have
+Θn3 ≲ ∥µ∥2 ≪ Θn2 ≍ n∥µ∥2.
+It follows that
+Var(T) = [1 + o(1)]Θn2 ≍ n∥µ∥2.
+(A.84)
+Thus by (A.83) and Chebyshev’s inequality, we have
+V ∗
+Var(T) = V ∗ − EV ∗
+Var(T)
++
+EV ∗
+Var(T) = oP(1) + 1 + o(1),
+as desired.
+A.20
+Proof of Proposition A.6
+By Lemmas A.6 and A.12,
+Var(T) ≳ Θn2 + Θn3 ≳
+�
+i
+∥Ωi∥2.
+(A.85)
+By (A.83),
+Var(V ∗) ≲
+�
+i
+∥Ωi∥2
+N2
+i
+∨
+�
+i
+∥Ωi∥3
+3
+Ni
+(A.86)
+Using a similar argument based on Chebyshev’s inequality as in the proof of Proposition
+A.1 and applying (A.85) and (A.86), we have
+|V ∗ − EV ∗|
+Var(T)
+≳ |V ∗ − EV ∗|
+�
+i ∥Ωi∥2
+= oP(1).
+(A.87)
+Next, by Lemma A.10 and (A.85),
+EV ∗ = Θn2 ≲ Var(T).
+(A.88)
+Combining (A.81) and (A.88) finishes the proof.
+45
+
+B
+Proofs of asymptotic normality results
+The goal of this section is to prove Theorems 3.1 and 3.2. The argument relies on the
+martingale central limit theorem and the lemmas stated below.
+As a preliminary, we
+describe a martingale decomposition of T under the null.
+Define
+U = 1′
+p(U3 + U4),
+and
+S = 1′
+pU2.
+By Lemma A.1, we have T = U + S under the null hypothesis. It holds that
+U =
+�
+i<i′
+σi,i′
+Ni
+�
+r=1
+Ni′
+�
+s=1
+� �
+j
+ZijrZi′js
+�
+.
+(B.1)
+where we define
+σi,i′ =
+�
+2
+�
+1
+nk ¯
+Nk −
+1
+n ¯
+N
+�
+if i, i′ ∈ Sk for some k
+− 2
+n ¯
+N
+else.
+Define a sequence of random variables
+Dℓ,s =
+�
+i∈[ℓ−1]
+σi,ℓ
+Ni
+�
+r=1
+�
+j
+ZijrZℓjs
+(B.2)
+indexed by (ℓ, s) ∈ {(i, r)}1≤i≤n,1≤r≤Ni, where these tuples are placed in lexicographical
+order. Precisely, we define
+(ℓ1, s1) ≺ (ℓ2, s2)
+if either
+• ℓ1 < ℓ2, or
+• ℓ1 = ℓ2 and s1 < s2.
+Observe that
+�
+ℓ,s
+Dℓ,s = U.
+Next define F≺(ℓ,s) to be the σ-field generated by {Zi:r}(i,r)≺(ℓ,s). Observe that
+E[Dℓ,s|F≺(ℓ,s)] = 0,
+and hence {Dℓ,s} is a martingale difference sequence. Turning to S, we have
+S =
+n
+�
+i=1
+σi
+�
+r<s
+�
+j
+ZijrZijs.
+(B.3)
+where we define
+σi = 2
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�
+Ni
+Ni − 1
+46
+
+if i ∈ Sk. Define
+Eℓ,s = σℓ
+�
+r∈[s−1]
+�
+j
+ZℓjrZℓjs.
+(B.4)
+Note that Eℓ,1 = 0. Order (ℓ, s) lexicographically as above, and recall that F≺(ℓ,s) is the
+σ-field generated by {Zi:r}(i,r)≺(ℓ,s). Observe that
+E[Eℓ,s|F≺(ℓ,s)] = 0,
+and hence {Eℓ,s} is a martingale difference sequence. We have
+�
+(ℓ,s)
+σℓ
+�
+r∈[s−1]
+�
+j
+ZℓjrZℓjs =
+n
+�
+ℓ=1
+Nℓ
+�
+s=1
+σℓ
+�
+r∈[s−1]
+�
+j
+ZℓjrZℓjs = S.
+Define
+Mℓ,s = Dℓ,s + Eℓ,s,
+�
+Mℓ,s =
+Mℓ,s
+�
+Var(T)
+.
+(B.5)
+Thus we obtain the martingale decomposition:
+T = U + S =
+�
+(ℓ,s)
+[Dℓ,s + Eℓ,s] =
+�
+(ℓ,s)
+Mℓ,s.
+(B.6)
+The technical results below are crucial to the proof of Theorem 3.1 given in Section B.1.
+Theorem 3.2 then follows easily from Theorem 3.1 and Theorem A.1.
+Lemma B.1. Let �
+Mℓ,s be defined as in (B.5). It holds that
+E
+� �
+(ℓ,s)
+Var
+� �
+Mℓ,s
+��F≺(ℓ,s)
+��
+= 1.
+Lemma B.2. Suppose that min Ni ≥ 2 and max ∥Ωi∥∞ ≤ 1−c0. Under the null hypothesis,
+it holds that
+Var
+� �
+(ℓ,s)
+Var(Dℓ,s|F≺(ℓ,s))
+�
+≲
+� �
+k
+1
+nk ¯Nk
+�
+∥µ∥3
+3 + K∥µ∥4
+4.
+Lemma B.3. Suppose that min Ni ≥ 2 and max ∥Ωi∥∞ ≤ 1−c0. Under the null hypothesis,
+it holds that
+�
+(ℓ,s)
+ED4
+ℓ,s ≲
+� �
+k
+1
+n2
+k ¯N2
+k
+�
+∥µ∥2 +
+� �
+k
+1
+nk ¯Nk
+�
+∥µ∥3
+3 ,
+Lemma B.4. Suppose that min Ni ≥ 2 and and max ∥Ωi∥∞ ≤ 1 − c0. Then we have
+Var
+� �
+(ℓ,s)
+Var( ˜Eℓ,s|F≺(ℓ,s))
+�
+≲
+�
+k
+�
+i∈Sk
+N3
+i ∥Ωi∥3
+3
+n4
+k ¯N4
+k
+∨
+�
+k
+�
+i∈Sk
+N4
+i ∥Ωi∥4
+4
+n4
+k ¯N4
+k
+(B.7)
+47
+
+Lemma B.5. Suppose that min Ni ≥ 2 and and max ∥Ωi∥∞ ≤ 1 − c0. Then we have
+�
+(ℓ,s)
+E E4
+ℓ,s ≲
+�
+k
+�
+i∈Sk
+N2
+i ∥Ωi∥2
+n4
+k ¯N4
+k
+∨
+�
+k
+�
+i∈Sk
+N3
+i ∥Ωi∥3
+3
+n4
+k ¯N4
+k
+Lemma B.6. Under either the null or alternative, it holds that
+�
+k
+�
+i∈Sk
+N2
+i ∥Ωi∥2
+n4
+k ¯N4
+k
+≤
+�
+k
+1
+n2
+k ¯N2
+k
+∥µk∥2
+�
+k
+�
+i∈Sk
+N3
+i ∥Ωi∥3
+3
+n4
+k ¯N4
+k
+≤
+�
+k
+1
+nk ¯Nk
+∥µk∥3
+3
+�
+k
+�
+i∈Sk
+N4
+i ∥Ωi∥4
+4
+n4
+k ¯N4
+k
+≤
+�
+k
+∥µk∥4
+4
+B.1
+Proof of Theorem 3.1
+By the martingale central limit theorem (see e.g. Hall and Heyde [2014]), we have that
+T/
+�
+Var(T) ⇒ N(0, 1) if the following conditions are satisfied:
+�
+(ℓ,s)
+Var
+� �
+Mℓ,s
+��F≺(ℓ,s)
+� P→ 1
+(B.8)
+�
+(ℓ,s)
+E
+� �
+M2
+ℓ,s1| �
+Mℓ,s|>ε
+��F≺(ℓ,s)
+� P→ 0,
+for any ε > 0.
+(B.9)
+It is known that (B.9), which is a Lindeberg-type condition, is implied by the Lyapunov-type
+condition
+�
+(ℓ,s)
+E �
+M4
+ℓ,s = o(1).
+(B.10)
+See e.g. Jin et al. [2018].
+Since (3.1) holds,
+Var(T) ≳ Θ = Θn2 + Θn3 + Θn4 ≳ K∥µ∥2.
+(B.11)
+Recall that
+�
+Mℓ,s =
+Mℓ,s
+Var(T) = Dℓ,s + Eℓ,s
+Var(T)
+,
+Note that (B.8) holds if
+E
+�
+Var
+� �
+Mℓ,s
+��F≺(ℓ,s)
+��
+→ 1,
+and
+(B.12)
+Var
+�
+Var
+� �
+Mℓ,s
+��F≺(ℓ,s)
+��
+→ 0.
+(B.13)
+Recall that (B.12) holds by Lemma B.1.
+48
+
+Next note that
+E(Dℓ,sEℓ,s|F≺(ℓ,s)) = 0,
+by inspection of the expressions for Dℓ,s and Eℓ,s in (B.2) and (B.4). Therefore
+Var(Mℓ,s|F≺(ℓ,s)) = Var(Dℓ,s|F≺(ℓ,s)) + Var(Eℓ,s|F≺(ℓ,s)).
+Hence by (B.11); Lemmas B.2, B.4 , and B.6; and the assumption (3.4), under the null
+hypothesis, we have
+Var
+�
+Var
+� �
+Mℓ,s
+��F≺(ℓ,s)
+��
+≤
+1
+Var(T)2
+�
+Var
+�
+Var
+�
+Dℓ,s
+��F≺(ℓ,s)
+��
++ Var
+�
+Var
+�
+Eℓ,s
+��F≺(ℓ,s)
+���
+≲
+1
+K2∥µ∥4
+�� �
+k
+1
+nk ¯Nk
+�
+∥µ∥3
+3 + K∥µ∥4
+4
+�
+∥µ∥2
+�
+= o(1).
+This proves (B.13). Thus, (B.12) and (B.13) are established, which proves (B.8).
+Similarly, (B.10) (and thus (B.9)) holds by (B.11); Lemmas (B.3), (B.5), and (B.6), and
+the assumption (3.4). Combining (B.8) and (B.9) verifies the conditions of the martingale
+central limit theorem, so we conclude that T/
+�
+Var(T) ⇒ N(0, 1). Since Var(T) = [1 +
+o(1)]Θn by (3.4) and Lemma A.7, the proof is complete.
+We record a useful proposition that records the weaker conditions under which T/
+�
+Var(T)
+is asymptotically normal.
+Proposition B.1. Recall that αn is defined as
+αn := max
+� K
+�
+k=1
+∥µk∥3
+3
+nk ¯Nk
+,
+K
+�
+k=1
+∥µk∥2
+n2
+k ¯N2
+k
+� �� K
+�
+k=1
+∥µk∥2
+�2
+(B.14)
+in (3.2). If under the null hypothesis,
+αn = max
+� K
+�
+k=1
+∥µk∥3
+3
+nk ¯Nk
+,
+K
+�
+k=1
+∥µk∥2
+n2
+k ¯N2
+k
+� ��
+K∥µ∥2
+�2
+→ 0,
+and
+∥µ∥4
+4
+K∥µ∥4 → 0,
+(B.15)
+then T/
+�
+Var(T) ⇒ N(0, 1).
+B.1.1
+Proof of Theorem 3.2
+By our assumptions, Proposition A.1 holds and V/Var(T) → 1. Thus the variance estimate
+V is consistent under the null. Theorem 3.2 follows immediately from Slutsky’s theorem
+and Theorem 3.1.
+B.2
+Proof of Lemma B.1
+By Lemma A.1, S and U are uncorrelated, and it holds that
+Var(T) = Var(S) + Var(U).
+(B.16)
+49
+
+Next note that
+E(Dℓ,sEℓ,s|F≺(ℓ,s)) = 0,
+by inspection of the expressions for Dℓ,s and Eℓ,s in (B.2) and (B.4). Therefore
+Var(Mℓ,s|F≺(ℓ,s)) = Var(Dℓ,s|F≺(ℓ,s)) + Var(Eℓ,s|F≺(ℓ,s)).
+Observe that
+E
+� �
+(ℓ,s)
+Var(Eℓ,s|F≺(ℓ,s))
+�
+=
+�
+(ℓ,s)
+EE2
+ℓ,s =
+�
+(ℓ,s)
+σ2
+ℓ
+�
+r,r′∈[s−1]
+�
+j,j′
+E[ZℓjrZℓjsZℓj′r′Zℓj′s]
+=
+�
+(ℓ,s)
+σ2
+ℓ
+�
+r∈[s−1]
+�
+j,j′
+E[ZℓjrZℓj′rZℓjsZℓj′s]
+=
+n
+�
+ℓ=1
+σ2
+ℓ
+�
+s∈[Nℓ]
+�
+r∈[s−1]
+E
+� �
+j
+ZℓjrZℓjs
+�2
+= Var(S).
+(B.17)
+The last line is obtained noting that S as defined in (B.3) is a sum of uncorrelated terms
+over (i, r, s).
+Similarly, we have
+E
+� �
+(ℓ,s)
+Var(Dℓ,s|F≺(ℓ,s))
+�
+= E
+� �
+(ℓ,s)
+E
+�
+D2
+ℓ,s|F≺(ℓ,s)
+��
+=
+�
+(ℓ,s)
+E
+�
+D2
+ℓ,s
+�
+=
+�
+(ℓ,s)
+�
+i∈[ℓ−1]
+σ2
+i,ℓVar
+� Ni
+�
+r=1
+�
+j
+ZijrZℓjs
+�
+=
+�
+ℓ
+�
+i∈[ℓ−1]
+σ2
+i,ℓVar
+� Ni
+�
+r=1
+Nℓ
+�
+s=1
+ZijrZℓjs
+�
+= Var(U).
+(B.18)
+The lemma follows by combining (B.16)–(B.18).
+B.3
+Proof of Lemma B.2
+Let Mk = nk ¯Nk and M = n ¯N. Define
+Σ = 1
+M
+�
+k
+MkΣk = 1
+M
+�
+ℓ∈[n]
+NℓΩℓj1Ωℓj2.
+(B.19)
+Our main goal is to control the conditional variance process. Define
+δjj′ℓ = EZℓjrZℓj′r =
+�
+Ωℓj(1 − Ωℓj)
+if j = j′
+−ΩℓjΩℓj′
+else.
+(B.20)
+50
+
+Observe that
+Var(Dℓ,s|F≺(ℓ,s)) = E[
+�
+i,i′∈[ℓ−1]
+�
+r,r′
+�
+j1,j2
+σiℓσi′ℓZij1rZℓj1sZi′j2r′Zℓj2s|F≺(ℓ,s)]
+=
+�
+i,i′∈[ℓ−1]
+�
+r,r′
+�
+j1,j2
+σiℓσi′ℓZij1rZi′j2r′E[Zℓj1sZℓj2s]
+=
+�
+i,i′∈[ℓ−1]
+�
+r,r′
+σiℓσi′ℓ
+�
+j1,j2
+δj1j2ℓZij1rZi′j2r′
+Define
+αii′j1j2 =
+�
+ℓ>i′
+Nℓσiℓσi′ℓδj1j2ℓ.
+(B.21)
+Thus
+�
+(ℓ,s)
+Var(Dℓ,s|F≺(ℓ,s)) =
+�
+ℓ,s
+�
+i,i′∈[ℓ−1]
+Ni
+�
+r=1
+Ni′
+�
+r′=1
+σiℓσi′ℓ
+�
+j1,j2
+δj1j2ℓ Zij1rZi′j2r′
+=
+�
+i
+Ni
+�
+r=1
+Ni
+�
+r′=1
+�
+j1,j2
+� �
+ℓ>i
+Nℓσ2
+iℓδj1j2ℓ
+�
+Zij1rZi′j2r′
++ 2
+�
+i<i′
+Ni
+�
+r=1
+Ni′
+�
+r′=1
+�
+j1,j2
+� �
+ℓ>i′
+Nℓσiℓσi′ℓδj1j2ℓ
+�
+Zij1rZi′j2r′
+=
+�
+i
+Ni
+�
+r=1
+Ni
+�
+r′=1
+�
+j1,j2
+αiij1j2Zij1rZi′j2r′
++ 2
+�
+i<i′
+Ni
+�
+r=1
+Ni′
+�
+r′=1
+�
+j1,j2
+αii′j1j2Zij1rZi′j2r′.
+Define
+ζiri′r′ =
+�
+j1,j2
+αii′j1j2Zij1rZi′j2r′.
+(B.22)
+Then
+�
+(ℓ,s)
+Var(Dℓ,s|F≺(ℓ,s)) =
+�
+i
+�
+r∈[Ni]
+ζirir +
+�
+2
+�
+i
+�
+r<r′∈[Ni]
+ζirir′ + 2
+�
+i<i′
+Ni
+�
+r=1
+Ni′
+�
+r′=1
+ζiri′r′
+�
+=: V1 + V2
+With this decomposition, Lemma B.2 follows directly from Lemmas B.7 and B.8 stated
+below and proved in the next remainder of this subsection.
+Lemma B.7. It holds that
+Var(V1) ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3.
+51
+
+Lemma B.8. It holds that
+Var(V2) ≲ K∥µ∥4
+4
+B.3.1
+Statement and proof of Lemma B.9
+The proofs of Lemmas B.7 and B.8 heavily rely on the following intermediate result that
+bounds the coefficients αii′j1j2 in all cases.
+Lemma B.9. It holds that
+αii′j1j2 ≲
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+Mk µj1
+if i, i′ ∈ Sk, j1 = j2
+1
+Mk Σkj1j2 + 1
+M Σj1j2
+if i, i′ ∈ Sk, j1 ̸= j2
+1
+M µj1
+if i ∈ Sk1, i′ ∈ Sk2, k1 ̸= k2, j1 = j2
+1
+M
+�2
+a=1 Σkaj1j2 + 1
+M Σj1j2
+if i ∈ Sk1, i′ ∈ Sk2, k1 ̸= k2, j1 ̸= j2
+Proof. If j1 = j2 and i, i′ ∈ Sk, we have
+|αii′j1j1| = |
+�
+ℓ>i′
+Nℓσiℓσi′ℓδj1j1ℓ| ≤
+K
+�
+k′=1
+�
+ℓ∈Sk′
+Nℓσiℓσi′ℓδj1j1ℓ
+≲
+1
+Mk
+·
+1
+Mk
+�
+ℓ∈Sk
+NℓΩℓj1 + 1
+M · 1
+M
+�
+ℓ∈[n]
+NℓΩℓj1 ≲
+1
+Mk
+µj1 + 1
+M µj1 ≲
+1
+Mk
+µj1.
+If j1 ̸= j2 and i, i′ ∈ Sk, we have
+|αii′j1j2| = |
+�
+ℓ>i′
+Nℓσiℓσi′ℓδj1j2ℓ| ≤
+�
+ℓ∈[n]
+Nℓ|σiℓσi′ℓ|Ωℓj1Ωℓj2
+≲
+1
+Mk
+·
+1
+Mk
+�
+ℓ∈Sk
+NℓΩℓj1Ωℓj2 + 1
+M · 1
+M
+�
+ℓ∈[n]
+NℓΩℓj1Ωℓj2 ≲
+1
+Mk
+Σkj1j2 + 1
+M Σj1j2.
+If i ̸= i′, j1 = j2, and i ∈ Sk1, i′ ∈ Sk2 where k1 ̸= k2, we have
+|αii′j1j1| = |
+�
+ℓ>i′
+Nℓσiℓσi′ℓδj1j1ℓ| ≤
+�
+ℓ
+Nℓ|σiℓσi′ℓ|Ωℓj1
+≲ 1
+M ·
+2
+�
+a=1
+1
+Mka
+�
+ℓ∈Ska
+NℓΩℓj1 + 1
+M · 1
+M
+�
+ℓ∈[n]
+NℓΩℓj1 = 3
+M µj1.
+If i ̸= i′, j1 ̸= j2, and i ∈ Sk1, i′ ∈ Sk2 where k1 ̸= k2, we have
+|αii′j1j2| = |
+�
+ℓ>i′
+Nℓσiℓσi′ℓδj1j2ℓ| ≲
+�
+ℓ
+Nℓσiℓσi′ℓΩℓj1Ωℓj2
+≲ 1
+M ·
+2
+�
+a=1
+1
+Mka
+�
+ℓ∈Ska
+NℓΩℓj1Ωℓj2 + 1
+M · 1
+M
+�
+ℓ∈[n]
+NℓΩℓj1Ωℓj2
+≤ 1
+M
+2
+�
+a=1
+Σkaj1j2 + 1
+M Σj1j2.
+52
+
+B.3.2
+Proof of Lemma B.7
+We have
+Var(V1) =
+�
+i,r
+Eζ2
+irir.
+Next by symmetry,
+Eζ2
+irir =
+�
+j1,j2,j3,j4
+αiij1j2αiij3j4 EZij1rZij3rZij2rZij4r
+≲
+�
+j1
+α2
+iij1j1Ωij1 +
+�
+j1̸=j4
+αiij1j1αiij1j4 Ωij1Ωij4
++
+�
+j1̸=j3
+αiij1j1αiij3j3 Ωij1Ωij3 +
+�
+j1̸=j2
+α2
+iij1j2 Ωij1Ωij2
++
+�
+j1,j3,j4(dist.)
+αiij1j1αiij3j4 Ωij1Ωij3Ωij4 +
+�
+j1,j2,j4(dist.)
+αiij1j2αiij1j4Ωij1Ωij2Ωij4
++
+�
+j1,j2,j3,j4(dist.)
+αiij1j2αiij3j4Ωij1Ωij2Ωij3Ωij4 =:
+7
+�
+a=1
+Ba,i,r
+Thus
+Var(V1) ≲
+�
+a
+� �
+i,r
+Ba,i,r
+�
+��
+�
+=:Ba
+�
+.
+We analyze B1– B7 separately, bounding the αii′jrjs coefficients using Lemma B.9.
+For B1,
+B1 ≲
+�
+i,r
+�
+j1
+α2
+iij1j2Ωij1 ≲
+k
+�
+k=1
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1
+( 1
+Mk
+µj1)2Ωij1
+≲
+�
+k
+�
+j1
+( 1
+Mk
+µj1)2Mkµj1 ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3.
+(B.23)
+For B2,
+B2 ≲
+�
+i,r
+�
+j1̸=j4
+αiij1j1αiij1j4 Ωij1Ωij4
+≲
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1̸=j4
+1
+Mk
+µj1 ·
+� 1
+Mk
+Σkj1j4 + 1
+M Σj1j4
+�
+· Ωij1Ωij4
+≲
+�
+k
+�
+j1̸=j4
+1
+Mk
+µj1 ·
+� 1
+Mk
+Σkj1j4 + 1
+M Σj1j4
+�
+· MkΣkj1j4
+≲
+�
+k
+1
+Mk
+�
+j1̸=j4
+Σ2
+kj1j4µj1 +
+�
+k
+1
+M
+�
+j1̸=j4
+Σkj1j4Σj1j4µj1
+53
+
+≲
+�
+k
+1′Σ◦2
+k µ
+Mk
++
+�
+k
+1′(Σk ◦ Σ)µ
+M
+=
+�
+k
+1′Σ◦2
+k µ
+Mk
+Next,
+�
+j1̸=j4
+Σ2
+kj1j4µj1 =
+�
+j1̸=j4
+1
+M2
+k
+�
+i,i′∈Sk
+NiNi′Ωij1Ωi′j1Ωij4Ωi′j4 · µj1
+≤
+�
+j1
+1
+M2
+k
+�
+i,i′∈Sk
+NiNi′Ωij1Ωi′j1µj1 ·
+� �
+j4
+Ωij4Ωi′j4
+�
+≤
+�
+j1
+1
+M2
+k
+�
+i,i′∈Sk
+NiNi′Ωij1Ωi′j1 · µj1
+≤
+�
+j1
+µ3
+j1 = ∥µ∥3
+3,
+(B.24)
+and similarly
+�
+j1̸=j4
+Σkj1j4Σj1j4µj1 =
+�
+j1̸=j4
+1
+MkM
+�
+i∈Sk,i′∈[n]
+NiNi′Ωij1Ωi′j1Ωij4Ωi′j4 · µj1
+≤
+�
+j1
+1
+MkM
+�
+i∈Sk,i′∈[n]
+NiNi′Ωij1Ωi′j1µj1
+=
+�
+j1
+µ3
+j1 = ∥µ∥3
+3.
+Thus
+B2 ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3.
+(B.25)
+For B3,
+B3 ≲
+�
+i,r
+�
+j1̸=j3
+αiij1j1αiij3j3 Ωij1Ωij3
+≲
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1̸=j3
+1
+Mk
+µj1 ·
+1
+Mk
+µj3 · Ωij1Ωij3
+≲
+�
+k
+�
+j1̸=j3
+1
+Mk
+µj1 ·
+1
+Mk
+µj3 · MkΣkj1j3 ≲
+�
+k
+µ′Σkµ
+Mk
+.
+We have by Cauchy-Schwarz,
+µ′Σkµ =
+1
+Mk
+�
+i∈Sk
+Niµ′ΩiΩ′
+i′µ
+=
+1
+Mk
+�
+i∈Sk
+Ni
+� �
+j
+µjΩij
+�2
+≤
+1
+Mk
+�
+i∈Sk
+Ni
+� �
+j
+Ωij
+�� �
+j
+µ2
+jΩij
+�
+54
+
+=
+�
+j
+µ3
+j = ∥µ∥3
+3.
+(B.26)
+Thus
+B3 ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3
+(B.27)
+For B4,
+B4 ≲
+�
+i,r
+�
+j1̸=j2
+α2
+iij1j2 Ωij1Ωij2 ≲
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1̸=j2
+� 1
+Mk
+Σkj1j2 + 1
+M Σj1j2
+�2 Ωij1Ωij2
+≲
+�
+k
+�
+j1̸=j2
+� 1
+Mk
+Σkj1j2 + 1
+M Σj1j2
+�2 · MkΣkj1j2 ≲
+�
+k
+1′(Σ◦3
+k )1
+Mk
++
+�
+k
+Mk
+M2 1′(Σk ◦ Σ◦2)1
+≲
+� �
+k
+1′(Σ◦3
+k )1
+Mk
+�
++ 1
+M 1′(Σ◦3)1.
+First,
+1′(Σ◦3
+k )1 =
+1
+M3
+k
+�
+i1,i2,i3∈Sk
+Ni1Ni2Ni3
+� �
+j
+Ωi1jΩi2jΩi3j
+�2
+≤
+1
+M3
+k
+�
+i1,i2,i3∈Sk
+Ni1Ni2Ni3 ·
+�
+j
+Ωi1jΩi2jΩi3j =
+�
+j
+µ3
+j = ∥µ∥3
+3,
+and similarly,
+1′(Σ◦3)1 =
+1
+M3
+�
+i1,i2,i3∈[n]
+Ni1Ni2Ni3
+� �
+j
+Ωi1jΩi2jΩi3j
+�2 ≤ ∥µ∥3
+3.
+Thus
+B4 ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3
+(B.28)
+For B5,
+B5 ≲
+�
+i,r
+�
+j1,j3,j4(dist.)
+αiij1j1αiij3j4 Ωij1Ωij3Ωij4
+≲
+�
+k
+�
+i∈Sk
+Ni
+�
+j1,j3,j4
+1
+Mk
+µj1 · ( 1
+Mk
+Σkj3j4 + 1
+M Σj3j4) · Ωij1Ωij3Ωij4
+≲
+�
+k
+�
+i∈Sk
+�
+j1,j3,j4
+Niµj1Σkj3j4Ωij1Ωij3Ωij4
+M2
+k
++
+�
+k
+�
+i∈Sk
+�
+j1,j3,j4
+Niµj1Σj3j4Ωij1Ωij3Ωij4
+MkM
+=: B51 + B52.
+We have
+B51 =
+�
+k
+1
+M3
+k
+�
+i1,i2∈Sk
+�
+j1,j3,j4
+Ni1Ni2µj1Ωi1j1Ωi1j3Ωi2j3Ωi1j4Ωi2j4
+55
+
+=
+�
+k
+1
+M3
+k
+�
+i1,i2∈Sk
+Ni1Ni2(Ω′
+i1µ) · (Ω′
+i1Ωi2)2
+≤
+�
+k
+1
+M3
+k
+�
+i1,i2∈Sk
+Ni1Ni2 · Ω′
+i1µ · Ω′
+i1Ωi2
+=
+�
+k
+1
+M2
+k
+�
+i1
+Ni1µ′Ωi1Ω′
+i1µ =
+1
+Mk
+µ′Σkµ ≤
+�
+k
+1
+Mk
+∥µ∥3
+3.
+(B.29)
+In the last line we apply (B.26). Similarly,
+B52 =
+�
+k
+1
+MkM2
+�
+i1∈Sk,i2∈[n]
+�
+j1,j3,j4
+Ni1Ni2µj1Ωi1j1Ωi1j3Ωi2j3Ωi1j4Ωi2j4
+≤
+�
+k
+1
+MkM2
+�
+i1∈Sk,i2∈[n]
+Ni1Ni2 · Ω′
+i1µ · Ω′
+i1Ωi2
+≤
+�
+k
+1
+MkM
+�
+i1∈Sk
+Ni1µ′Ωi1Ω′
+i1µ ≤
+�
+k
+1
+M ∥µ∥3
+3.
+(B.30)
+Thus
+B5 ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3.
+(B.31)
+For B6,
+B6 ≲
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j4(dist.)
+� 1
+Mk
+Σkj1j2 + 1
+M Σj1j2
+�� 1
+Mk
+Σkj1j4 + 1
+M Σj1j4
+�
+Ωij1Ωij2Ωij4
+≲
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j4
+Σ2
+kj1j2Ωij1Ωij2Ωij4
+M2
+k
++ 2
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j4
+Σkj1j2Σj1j2Ωij1Ωij2Ωij4
+MkM
++
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j4
+Σ2
+j1j2Ωij1Ωij2Ωij4
+M2
+=: B61 + B62 + B63.
+First,
+B61 ≤
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j4
+Σ2
+kj1j2Ωij1
+M2
+k
+=
+�
+k
+1
+Mk
+1′Σ◦2
+k µ ≤
+�
+k
+1
+Mk
+∥µ∥3
+3,
+where we applied (B.24). Similarly,
+B62 ≲
+�
+k
+1
+Mk
+∥µ∥3
+3, and
+B63 ≲
+�
+k
+1
+Mk
+∥µ∥3
+3.
+Thus
+B6 ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3.
+(B.32)
+56
+
+For B7, we have
+B7 ≲
+�
+j1,j2,j3,j4(dist.)
+� 1
+Mk
+Σkj1j2 + 1
+M Σj1j2
+�� 1
+Mk
+Σkj3j4 + 1
+M Σj3j4
+�
+Ωij1Ωij2Ωij3Ωij4
+≲
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j3,j4
+Σkj1j2Σkj3j4Ωij1Ωij2Ωij3Ωij4
+M2
+k
++ 2
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j3,j4
+Σkj1j2Σj3j4Ωij1Ωij2Ωij3Ωij4
+MkM
++
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j3,j4
+Σj1j2Σj3j4Ωij1Ωij2Ωij3Ωij4
+M2
+=: B71 + B72 + B73.
+Note that
+Σkj1j2 =
+1
+Mk
+�
+i∈Sk
+NiΩij1Ωij2 ≤
+1
+Mk
+�
+i∈Sk
+NiΩij1 = µj1, and
+Σj1j2 = 1
+M
+�
+i∈[n]
+NiΩij1Ωij2 ≤ 1
+M
+�
+i∈[n]
+NiΩij1 = µj1.
+(B.33)
+Thus
+B71 ≤
+�
+k
+�
+i∈Sk
+�
+r∈[Ni]
+�
+j1,j2,j3,j4
+µj1Σkj3j4Ωij1Ωij2Ωij3Ωij4
+M2
+k
+≤
+�
+k
+�
+i∈Sk
+�
+j1,j3,j4
+Niµj1Σkj3j4Ωij1Ωij3Ωij4
+M2
+k
+≤
+�
+k
+1
+Mk
+∥µ∥3
+3
+where we applied (B.29). Similarly,
+B72 ≲
+�
+k
+1
+Mk
+∥µ∥3
+3, and
+B73 ≲
+�
+k
+1
+Mk
+∥µ∥3
+3.
+Thus
+B7 ≲
+� �
+k
+1
+Mk
+�
+∥µ∥3
+3.
+(B.34)
+Combining the results for B1–B7 concludes the proof.
+B.3.3
+Proof of Lemma B.8
+We have
+Var(V2) ≲ 4
+�
+(i,r)̸=(i′,r′)
+Eζ2
+irir′,
+57
+
+where r ∈ [Ni] and r ∈ [Ni′] in the summation above.
+By symmetry, if (i, r) ̸= (i′, r′),
+Eζ2
+iri′r′ =
+�
+j1,j2,j3,j4
+αii′j1j2αii′j3j4 EZij1rZij3r EZi′j2r′Zi′j4r′
+≲
+�
+j1
+α2
+ii′j1j1 Ωij1Ωi′j1 +
+�
+j1̸=j4
+αii′j1j1αii′j1j4 Ωij1Ωi′j1Ωi′j4
++
+�
+j1̸=j3
+αii′j1j1αii′j3j3 Ωij1Ωij3Ωi′j1Ωi′j3 +
+�
+j1̸=j2
+α2
+ii′j1j2 Ωij1Ωi′j2
++
+�
+j1,j3,j4(dist.)
+αii′j1j1αii′j3j4 Ωij1Ωij3Ωi′j1Ωi′j4 +
+�
+j1,j2,j4(dist.)
+αii′j1j2αii′j1j4 Ωij1Ωi′j2Ωi′j4
++
+�
+j1,j2,j3,j4(dist.)
+αii′j1j2αii′j3j4Ωij1Ωij3Ωi′j2Ωi′j4 =:
+7
+�
+a
+Ca,i,r.
+(B.35)
+Thus
+Var(V2) ≲
+7
+�
+a=1
+�
+(i,r)̸=(i′,r′)
+Ca,i,r ≲
+7
+�
+a=1
+�
+i,i′
+NiNi′Ca,i,r
+�
+��
+�
+=:Ca
+.
+Next we analyze C1, . . . , C7, bounding the αii′jrjs coefficients using Lemma B.9.
+For C1,
+C1 ≲
+�
+k
+�
+i,i′∈Sk
+�
+j1
+NiNi′α2
+ii′j1j1Ωij1Ωi′j1 +
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+�
+j1
+NiNi′α2
+ii′j1j1Ωij1Ωi′j1
+≲
+�
+k
+�
+i,i′∈Sk
+�
+j1
+NiNi′( 1
+Mk
+µj1)2Ωij1Ωij1 +
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+�
+j1
+( 1
+M µj1)2Ωij1Ωi′j1
+≲
+�
+k
+�
+j1
+µ4
+j1 +
+�
+k̸=k′
+�
+j1
+MkMk′
+M2
+µ4
+j1 ≲ K∥µ∥4
+4.
+(B.36)
+For C2,
+C2 ≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j4
+αii′j1j1αii′j1j4 Ωij1Ωi′j1Ωi′j4
++
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+j1̸=j4
+αii′j1j1αii′j1j4 Ωij1Ωi′j1Ωi′j4
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j4
+1
+Mk
+µj1 ·
+� 1
+Mk
+Σkj1j4 + 1
+M Σj1j4
+�
+Ωij1Ωi′j1Ωi′j4
++
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+j1̸=j4
+1
+M µj1 ·
+� 1
+M
+�
+a∈{k,k′}
+Σaj1j4 + 1
+M Σj1j4
+�
+Ωij1Ωi′j1Ωi′j4
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j4
+1
+Mk
+µj1 ·
+� 1
+Mk
+µj1 + 1
+M µj1
+�
+Ωij1Ωi′j1Ωi′j4
++
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+j1̸=j4
+1
+M µj1 ·
+� 2
+M µj1 + 1
+M µj1
+�
+Ωij1Ωi′j1Ωi′j4
+58
+
+≲
+�
+k
+�
+j1
+�
+µ4
+j1 + Mk
+M µ4
+j1
+�
++
+�
+k̸=k′
+�
+j1
+MkMk′
+M2
+µ4
+j1 ≲ K∥µ∥4
+4.
+(B.37)
+where we applied (B.33).
+For C3,
+C3 ≲
+� �
+k
+�
+i,i′∈Sk
+NiNi′ +
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+� �
+j1̸=j3
+αii′j1j1αii′j3j3 Ωij1Ωij3Ωi′j1Ωi′j3
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j3
+1
+Mk
+µj1 ·
+1
+Mk
+µj3 · Ωij1Ωij3Ωi′j1Ωi′j3
++
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+j1̸=j3
+1
+M µj1 · 1
+M µj3 Ωij1Ωij3Ωi′j1Ωi′j3
+=
+�
+k
+�
+j1̸=j3
+µj1µj3Σ2
+kj1j3 +
+�
+k̸=k′
+�
+j1̸=j3
+MkMk′
+M2
+µj1µj3Σkj1j3Σk′j1j3
+≤
+� �
+k
+µ′Σ◦2
+k µ
+�
++ µ′Σ◦2µ.
+First, by Cauchy–Schwarz,
+µ′Σ◦2
+k µ =
+1
+M2
+k
+�
+i,i′∈Sk
+NiNi′� �
+j
+µjΩijΩi′j
+�2
+=
+1
+M2
+k
+�
+i,i′∈Sk
+NiNi′� �
+j
+ΩijΩi′j
+� �
+j
+µ2
+jΩijΩi′j
+≤
+1
+M2
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j
+µ2
+jΩijΩi′j =
+�
+j
+µ4
+j = ∥µ∥4
+4.
+(B.38)
+Similarly
+µ′Σ◦2µ ≲ ∥µ∥4
+4.
+(B.39)
+Hence
+C3 ≲ K∥µ∥4
+4.
+(B.40)
+For C4,
+C4 ≲
+� �
+k
+�
+i,i′∈Sk
+NiNi′ +
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+� �
+j1̸=j2
+α2
+ii′j1j2 Ωij1Ωi′j2
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j2
+� 1
+Mk
+Σkj1j2 + 1
+M Σj1j2
+�2 Ωij1Ωi′j2
++
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+j1̸=j2
+� 1
+M
+2
+�
+a∈{k,k′}
+Σaj1j2 + 1
+M Σj1j2
+�2 Ωij1Ωi′j2
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j2
+� 1
+M2
+k
+Σ2
+kj1j2 +
+1
+M2 Σ2
+j1j2
+�
+Ωij1Ωi′j2
+59
+
++
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+j1̸=j2
+� 1
+M2
+2
+�
+a∈{k,k′}
+Σ2
+aj1j2 +
+1
+M2 Σ2
+j1j2
+�
+Ωij1Ωi′j2 =: C41 + C42
+First,
+C41 ≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j2
+1
+M2
+k
+Σ2
+kj1j2Ωij1Ωi′j2 +
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1̸=j2
+1
+M2 Σ2
+j1j2Ωij1Ωi′j2
+≲
+�
+k
+�
+j1̸=j2
+Σ2
+kj1j2µj1µj2 +
+�
+k
+�
+j1̸=j2
+M2
+k
+M2 Σ2
+j1j2µj1µj2 ≤
+�
+k
+µ′Σ◦2
+k µ +
+�
+k
+M2
+k
+M2 µ′Σ◦2µ.
+Similarly,
+C42 ≲
+�
+k̸=k′
+�
+j1̸=j2
+MkMk′
+M2
+Σ2
+kj1j2µj1µj2 +
+�
+k̸=k′
+�
+j1̸=j2
+MkMk′
+M2
+Σ2
+j1j2µj1µj2
+≲
+�
+k̸=k′
+MkMk′
+M2
+�
+µ′Σ◦2
+k µ + µ′Σ◦2µ
+�
+Combining the previous two displays and applying (B.38) and (B.39), we have
+C4 ≲ K∥µ∥4
+4.
+(B.41)
+For C5,
+C5 ≲
+� �
+k
+�
+i,i′∈Sk
+NiNi′ +
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+�
+j1,j3,j4(dist.)
+αii′j1j1αii′j3j4 Ωij1Ωij3Ωi′j1Ωi′j4
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1,j3,j4
+1
+Mk
+µj1 ·
+� 1
+Mk
+Σkj3j4 + 1
+M Σj3j4
+�
+Ωij1Ωij3Ωi′j1Ωi′j4
++
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+j1,j3,j4
+1
+M µj1
+� 1
+M
+2
+�
+a∈{k,k′}
+Σaj3j4 + 1
+M Σj3j4
+�
+Ωij1Ωij3Ωi′j1Ωi′j4
+=
+�
+k
+�
+j1,j3,j4
+µj1Σkj3j4Σkj1j3Σkj1j4 +
+�
+k
+�
+j1,j3,j4
+Mk
+M µj1Σj3j4Σkj1j3Σkj1j4
++ 2
+�
+k̸=k′
+�
+j1,j3,j4
+MkMk′
+M2
+µj1Σkj3j4Σkj1j3Σk′j1j4 +
+�
+k̸=k′
+�
+j1,j3,j4
+MkMk′
+M2
+µj1Σj3j4Σkj1j3Σk′j1j4
+= C51 + C52 + 2C53 + C54
+For C51, we have
+C51 =
+�
+k
+1
+M3
+k
+�
+i1,i2,i3∈Sk
+Ni1Ni2Ni3⟨µ ◦ Ωi1, Ωi2⟩⟨Ωi1, Ωi3⟩⟨Ωi2, Ωi3⟩
+=
+�
+k
+1
+M2
+k
+�
+i1,i2∈Sk
+Ni1Ni2⟨µ ◦ Ωi1, Ωi2⟩ · ⟨Ωi1, ΣkΩi2⟩
+≤
+�
+k
+� 1
+M2
+k
+�
+i1,i2∈Sk
+Ni1Ni2⟨µ ◦ Ωi1, Ωi2⟩2
+�1/2� 1
+M2
+k
+�
+i1,i2∈Sk
+Ni1Ni2⟨Ωi1, ΣkΩi2⟩2
+�1/2
+60
+
+=:
+�
+k
+C1/2
+511k · C1/2
+512k.
+(B.42)
+We have by Cauchy–Schwarz that
+C511k =
+1
+M2
+k
+�
+i1,i2∈Sk
+Ni1Ni2
+� �
+j
+µjΩi1jΩi2j
+�2
+≤
+1
+M2
+k
+�
+i1,i2∈Sk
+Ni1Ni2
+� �
+j
+µ2
+jΩi1jΩi2j
+�� �
+j
+Ωi1jΩi2j
+�
+≤ ∥µ∥4
+4,
+and similarly
+C512k =
+1
+M2
+k
+�
+i1,i2∈Sk
+Ni1Ni2
+� �
+j1,j2
+Ωi1j1Σkj1j2Ωi2j2
+�2
+=
+1
+M2
+k
+�
+i1,i2
+Ni1Ni2
+� �
+j1,j2
+Ωi1j1Σ2
+kj1j2Ωi2j2
+�� �
+j1,j2
+Ωi1j1Ωi2j2
+�
+≤
+1
+M2
+k
+�
+i1,i2
+Ni1Ni2
+� �
+j1,j2
+Ωi1j1Σ2
+kj1j2Ωi2j2
+�
+= µ′ Σ◦2
+k µ
+(B.43)
+Since by Cauchy–Schwarz,
+µ′ Σ◦2
+k µ =
+�
+j1,j2
+µj1µj2
+� 1
+Mk
+�
+i∈Sk
+NiΩij1Ωij2
+�2 =
+1
+M2
+k
+�
+j1,j2
+µj1µj2
+�
+i,i′∈Sk
+NiNi′Ωij1Ωij2Ωi′j1Ωi′j2
+=
+1
+M2
+k
+�
+i,i′∈Sk
+� �
+j
+µjΩijΩi′j
+�2 ≤
+1
+M2
+k
+�
+i,i′∈Sk
+�
+j
+µ2
+jΩijΩi′j ≤ ∥µ∥4
+4
+(B.44)
+we have in total C512k ≲ K∥µ∥4
+4. Combining the result with the bound for C511k implies
+that
+C51 ≲ K∥µ∥4
+4.
+Next we study C52 using a similar argument.
+C52 =
+�
+k
+�
+j1,j3,j4
+Mk
+M µj1Σj3j4Σkj1j3Σkj1j4
+=
+�
+k
+�
+j1,j3,j4
+Mk
+M µj1
+� 1
+M
+�
+i1∈[n]
+Ni1Ωi1j3Ωi1j4
+�� 1
+Mk
+�
+i2∈Sk
+Ni2Ωi2j1Ωi2j3
+�� 1
+Mk
+�
+i3∈Sk
+Ni3Ωi3j1Ωi3j4
+�
+=
+�
+k
+1
+M2Mk
+�
+j1,j2,j3
+�
+i1∈[n]
+i2,i3∈Sk
+Ni1Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi2⟩
+=
+�
+k
+1
+M2
+�
+i2,i3∈[Sk]
+Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi3, ΣΩi2⟩
+≤
+�
+k
+� 1
+M2
+�
+i2,i3∈[Sk]
+Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩2
+�1/2� 1
+M2
+�
+i2,i3∈[Sk]
+Ni2Ni3⟨Ωi3, ΣΩi2⟩
+�1/2
+61
+
+=:
+�
+k
+C1/2
+521kC1/2
+522k.
+(B.45)
+Observe that C521k = C511k, and thus C521 ≲ ∥µ∥4 by (B.43). With a similar argument as
+in (B.44) we obtain C522k ≲ ∥µ∥4
+4. Hence we obtain
+C52 ≤
+�
+k
+C1/2
+521kC1/2
+522k ≲ K∥µ∥4
+4.
+For C53, we have
+C53 =
+�
+k̸=k′
+�
+j1,j3,j4
+MkMk′
+M2
+µj1Σkj3j4Σkj1j3Σk′j1j4
+≤
+�
+k
+�
+j1,j3,j4
+Mk
+M µj1Σkj3j4Σkj1j3Σj1j4
+=
+�
+k
+�
+j1,j3,j4
+Mk
+M µj1
+� 1
+Mk
+�
+i1∈Sk
+Ni1Ωi1j3Ωi1j4
+�� 1
+Mk
+�
+i2∈Sk
+Ni2Ωi2j1Ωi2j3
+�� 1
+M
+�
+i3∈[n]
+Ni3Ωi3j1Ωi3j4
+�
+=
+�
+k
+1
+M2Mk
+�
+i1,i2∈Sk
+i3∈[n]
+Ni1Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi1, Ωi2⟩⟨Ωi1, Ωi3⟩
+=
+�
+k
+1
+M2
+�
+i2∈Sk,i3∈[n]
+Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi2, ΣkΩi3⟩.
+(B.46)
+We then upper bound the last line using a similar strategy as in that we used for C51 and
+C52, respectively. We omit the details and state the final bound:
+C53 ≲ K∥µ∥4
+4
+(B.47)
+Finally for C54, summing over k, k′ we obtain
+C54 ≤
+�
+j1,j3,j4
+µj1Σj3j4Σj1j3Σj1j4 =
+1
+M3
+�
+i1,i2,i3∈[n]
+Ni1Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi1, Ωi2⟩⟨Ωi1, Ωi3⟩.
+(B.48)
+We then proceed as in (B.46) to control the right-hand side. We omit the details and state
+the final bound:
+C54 ≲ K∥µ∥4
+4.
+(B.49)
+Combining the results for C51, . . . , C54, we see that
+C5 ≲ K∥µ∥4.
+For C6, we have
+C6 ≤
+� �
+k
+�
+i,i′∈Sk
+NiNi′ +
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+� �
+j1,j2,j4
+αii′j1j2αii′j1j4 Ωij1Ωi′j2Ωi′j4
+62
+
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1,j2,j4
+� 1
+Mk
+Σkj1j2 + 1
+M Σj1j2
+�� 1
+Mk
+Σkj1j4 + 1
+M Σj1j4
+�
+Ωij1Ωi′j2Ωi′j4
++
+�
+k̸=k′
+i∈Sk,i′∈Sk′
+j1,j2,j4
+NiNi′� 1
+M
+2
+�
+a∈{k,k′}
+Σaj1j2 + 1
+M Σj1j2
+�� 1
+M
+2
+�
+a∈{k,k′}
+Σaj1j4 + 1
+M Σj1j4
+�
+Ωij1Ωi′j2Ωi′j4
+=: C61 + C62.
+For C61, we have
+C61 =
+�
+k
+�
+i′∈Sk
+Ni′
+�
+j1,j2,j4
+1
+Mk
+Σkj1j2Σkj1j4µj1Ωi′j2Ωi′j4
++ 2
+�
+k
+�
+i′∈Sk
+Ni′
+�
+j1,j2,j4
+1
+M Σkj1j2Σj1j4µj1Ωi′j2Ωi′j4
++
+�
+k
+�
+i′∈Sk
+Ni′
+�
+j1,j2,j4
+Mk
+M2 Σj1j2Σj1j4µj1Ωi′j2Ωi′j4 =: C611 + 2C612 + C613.
+Relabeling indices, we see that
+C611 =
+�
+k
+�
+j1,j2,j4
+µj1Σkj1j2Σkj1j4Σkj2j4 = C51
+Hence, C611 ≲ K∥µ∥4
+4. Next,
+C612 ≤
+�
+k
+Mk
+M
+�
+j1,j2,j4
+µj1Σkj1j2Σj1j4Σkj2j4 ≲ K∥µ∥4,
+where we applied (B.46). Similarly,
+C613 =
+�
+k
+M2
+k
+M2
+�
+j1,j2,j4
+µj1Σj1j2Σj1j4Σkj2j4 ≤
+�
+j1,j2,j4
+µj1Σj1j2Σj1j4Σj2j4 ≲ K∥µ∥4,
+where in the final bound we apply (B.48) and (B.49). Combining the results above for
+C611, C612, C613, we obtain
+C61 ≲ K∥µ∥4
+4
+(B.50)
+The argument for C62 is very similar, so we omit proof and state the final bound. We have
+C62 ≲ K∥µ∥4.
+Thus
+C6 ≲ K∥µ∥4
+4
+For C7, we have
+C7 ≲
+� �
+k
+�
+i,i′∈Sk
+NiNi′ +
+�
+k̸=k′
+�
+i∈Sk,i′∈Sk′
+NiNi′
+�
+�
+j1,j2,j3,j4
+αii′j1j2αii′j3j4Ωij1Ωij3Ωi′j2Ωi′j4
+63
+
+≲
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1,j2,j3,j4
+� 1
+Mk
+Σkj1j2 + 1
+M Σj1j2
+�� 1
+Mk
+Σkj3j4 + 1
+M Σj3j4
+�
+Ωij1Ωij3Ωi′j2Ωi′j4
++
+�
+k̸=k′
+�
+j1,j2,j3,j4
+i∈Sk,i′∈Sk′
+NiNi′� 1
+M
+2
+�
+a∈{k,k′}
+Σaj1j2 + 1
+M Σj1j2
+�� 1
+M
+2
+�
+a∈{k,k′}
+Σaj3j4 + 1
+M Σj3j4
+�
+Ωij1Ωij3Ωi′j2Ωi′j4
+=: C71 + C72
+Write
+C71 =
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1,j2,j3,j4
+1
+M2
+k
+Σkj1j2Σkj3j4Ωij1Ωij3Ωi′j2Ωi′j4
++ 2
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1,j2,j3,j4
+1
+MkM Σj1j2Σkj3j4Ωij1Ωij3Ωi′j2Ωi′j4
++
+�
+k
+�
+i,i′∈Sk
+NiNi′
+�
+j1,j2,j3,j4
+1
+M2 Σj1j2Σj3j4Ωij1Ωij3Ωi′j2Ωi′j4 =: C711 + 2C712 + C713.
+For C711, we have
+C711 =
+�
+k
+�
+j1,j2,j3,j4
+Σkj1j2Σkj3j4Σkj1j3Σkj2j4
+=
+�
+k
+1
+M4
+k
+�
+i1,i2,i3,i4∈Sk
+Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩
+=
+1
+M2
+k
+�
+k
+�
+i3,i4
+Ni3Ni4
+�
+Ω′
+i3ΣkΩi4
+�2 =
+�
+k
+1
+M2
+k
+�
+i3,i4
+Ni3Ni4
+� �
+j,j′
+Ω′
+i3jΣkjj′Ωi4j′�2
+≤
+�
+k
+1
+M2
+k
+�
+i3,i4
+Ni3Ni4
+�
+j,j′
+Ω′
+i3jΣ2
+kjj′Ωi4j′ ≤
+�
+k
+�
+j,j′
+µjΣ2
+kjj′µj′ ≲ K∥µ∥4
+4.
+(B.51)
+In the last line we applied Cauchy–Schwarz and (B.44). For C712, we have similarly
+C712 =
+�
+k
+Mk
+M
+�
+j1,j2,j3,j4
+Σj1j2Σkj3j4Σkj1j3Σkj2j4
+=
+�
+k
+1
+M2Mk
+�
+i1∈[n]
+i2,i3,i4∈Sk
+Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩
+=
+�
+k
+Mk
+M2
+�
+i1∈[n],i2∈Sk
+Ni1Ni2⟨Ωi1, ΣkΩi2⟩2 ≤
+�
+k
+Mk
+M2
+�
+i1∈[n],i2∈Sk
+Ni1Ni2
+�
+j,j′
+Ωi1jΣ2
+kjj′Ωi2j′
+≤
+�
+k
+M2
+k
+M2
+�
+j,j′
+µjΣ2
+kjj′µj′ ≲ K∥µ∥4
+4.
+(B.52)
+Next,
+C713 =
+�
+k
+M2
+k
+M2
+�
+j1,j2,j3,j4
+Σj1j2Σj3j4Σkj1j3Σkj2j4
+64
+
+=
+�
+k
+1
+M4
+�
+i1,i2∈[n]
+i3,i4∈Sk
+Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩,
+and applying a similar strategy as in (B.51), (B.52) leads to the bound C713 ≲ K∥µ∥4
+4.
+Thus
+C71 ≲ K∥µ∥4
+4.
+Next , by symmetry and summing over i ∈ Sk, i′ ∈ Sk′, we have
+C72 =
+�
+k̸=k′
+MkMk′
+M2
+�
+j1,j2,j3,j4
+�
+2Σkj1j2Σkj3j4 + 2Σk′j1j2Σkj3j4 + 4Σkj1j2Σj3j4 + Σj1j2Σj3j4
+�
+Σkj1j3Σk′j2j4
+=: 2C721 + 2C722 + 4C723 + C724
+First,
+C721 ≤
+�
+k
+Mk
+M
+�
+j1,j2,j3,j4
+Σkj1j2Σkj3j4Σkj1j3Σj2j4 = C712 ≲ K∥µ∥4
+4
+by (B.52). Next,
+C722 =
+�
+k̸=k′
+MkMk′
+M2
+�
+j1,j2,j3,j4
+Σk′j1j2Σkj3j4Σkj1j3Σk′j2j4
+≤
+�
+k,k′
+1
+M2MkMk′
+�
+i1,i2∈Sk
+i3,i4∈Sk′
+Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩
+=
+�
+k,k′
+Mk
+M2Mk′
+�
+i3,i4∈Sk′
+Ni3Ni4⟨Ωi3, ΣkΩi4⟩2 ≤
+�
+k,k′
+Mk
+M2Mk′
+�
+i3,i4∈Sk′
+Ni3Ni4
+�
+j,j′
+Ωi3jΣ2
+kjj′Ωi4j′
+≤
+�
+k,k′
+MkMk′
+M2
+µ′Σ◦2
+k µ ≤ ∥µ∥4
+4,
+(B.53)
+where we applied Cauchy-Schwarz in the penultimate line and (B.44) in the last line.
+For C723, we have
+C723 =
+�
+k̸=k′
+MkMk′
+M2
+�
+j1,j2,j3,j4
+Σkj1j2Σj3j4Σkj1j3Σk′j2j4 ≤
+�
+k
+Mk
+M
+�
+j1,j2,j3,j4
+Σkj1j2Σj3j4Σkj1j3Σj2j4
+=
+�
+k
+1
+M3Mk
+�
+i1,i3∈Sk
+i2,i4∈[n]
+Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩
+=
+�
+k
+1
+M2
+�
+i3∈Sk,i4∈[n]
+Ni3Ni4⟨Ωi3, ΣkΩi4⟩⟨Ωi3, ΣΩi4⟩
+≤ 1
+2
+�
+k
+1
+M2
+�
+i3∈Sk,i4∈[n]
+Ni3Ni4
+�
+⟨Ωi3, ΣkΩi4⟩2 + ⟨Ωi3, ΣΩi4⟩2�
+65
+
+Using a similar technique as in (B.51)–(B.53) and applying (B.38), (B.39) we obtain
+C723 ≲ ∥µ∥4
+4.
+Finally, for C724 we have
+C724 =
+�
+k̸=k′
+MkMk′
+M2
+�
+j1,j2,j3,j4
+Σj1j2Σj3j4Σkj1j3Σk′j2j4 ≤
+�
+j1,j2,j3,j4
+Σj1j2Σj3j4Σj1j3Σj2j4
+=
+1
+M4
+�
+i1,i2,i3,i4∈[n]
+Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩
+The details are very similar to (B.51)–(B.53), so we omit them and simply state the final
+bound:
+C724 ≲ ∥µ∥4
+4
+Combining the bounds for C721, C722, C723, and C724 yields
+C7 ≲ K∥µ∥4
+4.
+Combining the bounds for C1–C7 proves the result.
+B.4
+Proof of Lemma B.3
+We have
+ED4
+ℓ,s = E
+�� �
+i∈[ℓ−1]
+σi,ℓ
+Ni
+�
+r=1
+�
+j
+ZijrZℓjs
+�4
+�
+=
+�
+i1,i2,i3,i4∈[ℓ−1]
+σi1ℓσi2ℓσi3ℓσi4ℓ
+�
+r1,r2,r3,r4
+j1,j2,j3,j4
+E
+�
+Zi1j1r1Zℓj1sZi2j2r2Zℓj2sZi3j3r3Zℓj3sZi4j4r4Zℓj4s
+�
+=
+�
+i1,i2,i3,i4∈[ℓ−1]
+σi1ℓσi2ℓσi3ℓσi4ℓ
+�
+r1,r2,r3,r4
+j1,j2,j3,j4
+E
+�
+Zi1j1r1Zi2j2r2Zi3j3r3Zi4j4r4
+�
+E
+�
+Zℓj1sZℓj2sZℓj3sZℓj4s
+�
+=
+�
+j1,j2,j3,j4
+E[Zℓj1sZℓj2sZℓj3sZℓj4s]
+�
+i1,i2,i3,i4∈[ℓ−1]
+r1,r2,r3,r4
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zi1j1r1Zi2j2r2Zi3j3r3Zi4j4r4]
+=:
+�
+j1,j2,j3,j4
+E[Zℓj1sZℓj2sZℓj3sZℓj4s]Aj1,j2,j3,j4
+(B.54)
+In the summations above, rt ranges over [Nit].
+Observe that
+|E[Zℓj1sZℓj2sZℓj3sZℓj4s]| ≲
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+Ωℓj1
+if j1 = j2 = j3 = j4
+Ωℓj1Ωℓj4
+if j1 = j2 = j3, j4 ̸= j1
+Ωℓj1Ωℓj3
+if j1 = j2, j3 = j4, j1 ̸= j3
+Ωℓj1Ωℓj3Ωℓj4
+if j1 = j2, j1, j3, j4 dist.
+Ωℓj1Ωℓj2Ωℓj3Ωℓj4
+if j1, j2, j3, j4 dist.
+(B.55)
+66
+
+Up to permutation of the indices j1, . . . , j4, this accounts for all possible cases.
+To proceed we also bound Aj1,j2,j3,j4 by casework on the number of distinct j indices.
+For brevity we define ωt = (it, rt) and slightly abuse notation, letting Zωt,j = Zitjrt. Further
+let Iℓ = {ω = (i, r) : i ∈ [ℓ], 1 ≤ r ≤ Ni}. Our goal is to control
+Aj1,j2,j3,j4 =
+�
+ω1,ω2,ω3,ω4∈Iℓ−1
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j2Zω3j3Zω4j4].
+(B.56)
+To do this, we study (B.56) in five cases that cover all possibilities (up to permutation of
+the indices j1, . . . , j4).
+Case 1: j1 = j2 = j3 = j4. Define j = j1. It holds that
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1jZω2jZω3jZω4j]
+=
+�
+σ4
+i1ℓ EZ4
+ω1j ≲ σ4
+i1ℓ Ωi1j
+if ω1 = ω2 = ω3 = ω4
+σ2
+i1ℓσ2
+i3ℓEZ2
+ω1jEZ2
+ω3j ≲ σ2
+i1ℓσ2
+i3ℓΩi1jΩi3j
+if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3
+(B.57)
+Up to permutation of the indices ω1, . . . , ω4, this accounts for all cases such that (B.57) is
+nonvanishing. To be precise, by symmetry, it also holds that for all permutations π : [4] →
+[4] that if ωπ(1) = ωπ(2), ωπ(3) = ωπ(4), ωπ(1) ̸= ωπ(3), then
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1jZω2jZω3jZω4j] ≲ σ2
+iπ(1)ℓσ2
+iπ(3)ℓΩiπ(1)jΩiπ(3)j.
+In all other cases besides those considered above, we have
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1jZω2jZω3jZω4j] = 0
+by independence.
+Therefore,
+Ajjjj ≲
+�
+ω∈Iℓ−1
+σ4
+iℓΩij +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓΩi1jΩi3j
+(B.58)
+In the remaining Cases 2–6, we follow the same strategy of writing out bounds for
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j2Zω3j3Zω4j4]
+that cover all nonzero cases, up to permutation of the indices ω1, . . . , ω4.
+Case 2: j1 = j2 = j3, j1 ̸= j4. It holds that
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j1Zω3j1Zω4j4]
+=
+�
+σ4
+i1ℓE[Z3
+ω1j1Zω1j4] ≲ σ4
+i1ℓ Ωi1j1Ωi1j4
+if ω1 = ω2 = ω3 = ω4
+σ2
+i1ℓσ2
+i3ℓEZ2
+ω1j1EZω3j1Zω3j4 ≲ σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j1Ωi3j4
+if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3
+(B.59)
+67
+
+Up to permutation of the indices ω1, . . . , ω4, this accounts for all cases such that (B.59) is
+nonvanishing. Thus
+Aj1,j1,j1,j4 ≲
+�
+ω∈Iℓ−1
+σ4
+iℓΩi1j1Ωi1j4 +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j1Ωi3j4
+(B.60)
+Case 3: j1 = j2, j3 = j4, j1 ̸= j3. It holds that
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j1Zω3j3Zω4j3]
+=
+�
+�
+�
+�
+�
+�
+�
+σ4
+i1ℓEZ2
+ω1j1Z2
+ω1j3 ≲ σ4
+i1ℓ Ωi1j1Ωi1j3
+if ω1 = ω2 = ω3 = ω4
+σ2
+i1ℓσ2
+i3ℓEZ2
+ω1j1EZ2
+ω3j3 ≲ σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j3
+if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3
+σ2
+i1ℓσ2
+i3ℓEZω1j1Zω1j3EZω2j1Zω2j3 ≲ σ2
+i1ℓσ2
+i3ℓΩi1j1Ωi1j3Ωi2j1Ωi2j3
+if ω1 = ω3, ω2 = ω4, ω1 ̸= ω2.
+(B.61)
+Up to permutation of the indices ω1, . . . , ω4, this accounts for all cases such that (B.61) is
+nonvanishing. Thus by symmetry,
+Aj1,j1,j3,j3 ≲
+�
+ω∈Iℓ−1
+σ4
+i1ℓ Ωi1j1Ωi1j3 +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j3
+(B.62)
++
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓΩi1j1Ωi1j3Ωi3j1Ωi3j3
+Case 4: j1 = j2 and j1, j3, j4 distinct. We have
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j1Zω3j3Zω4j4]
+=
+�
+�
+�
+�
+�
+�
+�
+σ4
+i1ℓEZ2
+ω1j1Zω1j3Zω1j4 ≲ σ4
+i1ℓ Ωi1j1Ωi1j3Ωi1j4
+if ω1 = ω2 = ω3 = ω4
+σ2
+i1ℓσ2
+i3ℓEZ2
+ω1j1EZω3j3Zω3j4 ≲ σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j3Ωi3j4
+if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3
+σ2
+i1ℓσ2
+i2ℓEZω1j1Zω1j3EZω2j1Zω2j4 ≲ σ2
+i1ℓσ2
+i2ℓ Ωi1j1Ωi1j3Ωi2j1Ωi2j4
+if ω1 = ω3, ω2 = ω4, ω1 ̸= ω2
+(B.63)
+Up to permutation of the indices ω1, . . . , ω4, this accounts for all cases such that (B.63) is
+nonvanishing. Thus
+Aj1,j1,j3,j4 ≲
+�
+ω∈Iℓ−1
+σ4
+i1ℓ Ωi1j1Ωi1j3Ωi1j4 +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j3Ωi3j4
+(B.64)
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i2ℓ Ωi1j1Ωi1j3Ωi3j1Ωi3j4.
+Case 5: j1, j2, j3, j4 distinct. For this final case, it holds that
+σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j2Zω3j3Zω4j4]
+=
+�
+σ4
+i1ℓEZω1j1Zω1j2Zω1j3Zω1j4 ≲ σ4
+i1ℓ Ωi1j1Ωi1j2Ωi1j3Ωi1j4
+if ω1 = ω2 = ω3 = ω4
+σ2
+i1ℓσ2
+i3ℓEZω1j1Zω1j2EZω3j3Zω3j4 ≲ σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi1j2Ωi3j3Ωi3j4
+if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3
+68
+
+The above accounts for all nonzero cases, up to permutation of ω1, ω2, ω3, ω4. Hence
+Aj1,j2,j3,j4 ≲
+�
+ω∈Iℓ−1
+σ4
+i1ℓ Ωi1j1Ωi1j2Ωi1j3Ωi1j4 +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi1j2Ωi3j3Ωi3j4.
+(B.65)
+Finally we control the fourth moment using the casework above. By (B.54) and sym-
+metry,
+ED4
+ℓ,s ≲
+�
+j
+E[ZℓjsZℓjsZℓjsZℓjs]Aj,j,j,j +
+�
+j1̸=j4
+E[Zℓj1sZℓj1sZℓj1sZℓj4s]Aj1,j1,j1,j4
++
+�
+j1̸=j3
+E[Zℓj1sZℓj1sZℓj3sZℓj3s]Aj1,j1,j3,j3 +
+�
+j1,j3,j4 dist.
+E[Zℓj1sZℓj1sZℓj3sZℓj4s]Aj1,j1,j3,j4
++
+�
+j1,j2,j3,j4 dist.
+E[Zℓj1sZℓj2sZℓj3sZℓj4s]Aj1,j2,j3,j4
+=: F1ℓs + F2ℓs + F3ℓs + F4ℓs + F5ℓs
+(B.66)
+By (B.55), (B.58), (B.60) ,(B.62), (B.64), and (B.65),
+F1ℓs ≲
+�
+j
+Ωℓj
+� �
+ω∈Iℓ−1
+σ4
+iℓΩij +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓΩi1jΩi3j
+�
+F2ℓs ≲
+�
+j1̸=j4
+Ωℓj1Ωℓj4
+� �
+ω∈Iℓ−1
+σ4
+iℓΩi1j1Ωi1j4 +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j1Ωi3j4
+�
+F3ℓs ≲
+�
+j1̸=j3
+Ωℓj1Ωℓj3
+� �
+ω∈Iℓ−1
+σ4
+i1ℓ Ωi1j1Ωi1j3 +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j3
++
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓΩi1j1Ωi1j3Ωi3j1Ωi3j3
+�
+F4ℓs ≲
+�
+j1,j3,j4 dist.
+Ωℓj1Ωℓj3Ωℓj4
+� �
+ω∈Iℓ−1
+σ4
+i1ℓ Ωi1j1Ωi1j3Ωi1j4 +
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi3j3Ωi3j4
++
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi1j3Ωi3j1Ωi3j4.
+�
+F5ℓs ≲
+�
+j1,j2,j3,j4 dist.
+Ωℓj1Ωℓj2Ωℓj3Ωℓj4
+� �
+ω∈Iℓ−1
+σ4
+i1ℓ Ωi1j1Ωi1j2Ωi1j3Ωi1j4
++
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ Ωi1j1Ωi1j2Ωi3j3Ωi3j4
+�
+.
+Define
+F11ℓs =
+�
+ω∈Iℓ−1
+σ4
+iℓ
+�
+j
+ΩℓjΩij
+F21ℓs =
+�
+ω∈Iℓ−1
+σ4
+iℓ
+�
+j1̸=j4
+Ωℓj1Ωℓj4Ωi1j1Ωi1j4
+69
+
+F31ℓs =
+�
+ω∈Iℓ−1
+σ4
+i1ℓ
+�
+j1̸=j3
+Ωℓj1Ωℓj3Ωi1j1Ωi1j3
+F41ℓs =
+�
+ω∈Iℓ−1
+σ4
+i1ℓ
+�
+j1,j3,j4 dist.
+Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi1j3Ωi1j4
+F51ℓs =
+�
+ω∈Iℓ−1
+σ4
+i1ℓ
+�
+j1,j2,j3,j4 dist.
+Ωℓj1Ωℓj2Ωℓj3Ωℓj4Ωi1j1Ωi1j2Ωi1j3Ωi1j4
+and
+F12ℓs =
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ
+�
+j
+ΩℓjΩi1jΩi3j
+F22ℓs =
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ
+�
+j1̸=j4
+Ωℓj1Ωℓj4Ωi1j1Ωi3j1Ωi3j4
+F32ℓs =
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ
+�
+j1̸=j3
+�
+Ωℓj1Ωℓj3Ωi1j1Ωi3j3 + Ωℓj1Ωℓj3Ωi1j1Ωi1j3Ωi3j1Ωi3j3
+�
+F42ℓs =
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ
+�
+j1,j3,j4 dist.
+�
+Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi3j3Ωi3j4
++ Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi1j3Ωi3j1Ωi3j4
+�
+F52ℓs =
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ
+�
+j1,j2,j3,j4 dist.
+Ωℓj1Ωℓj2Ωℓj3Ωℓj4Ωi1j1Ωi1j2Ωi3j3Ωi3j4
+Note that �2
+x=1 Ftxℓs = Ftℓs for all t ∈ [5]. Using the fact that �
+j Ωij = 1, we have
+�
+t
+Ft1ℓs ≲ F11ℓs =
+�
+ω∈Iℓ−1
+σ4
+iℓ
+�
+j
+ΩℓjΩij =
+�
+ω∈Iℓ−1
+σ4
+iℓ⟨Ωℓ, Ωi⟩.
+(B.67)
+To control �
+t Ft2ℓs , observe that, since Ωij ≤ 1 for all i, j,
+�
+j
+ΩℓjΩi1j = ⟨Ωℓ, Ωi1 ◦ Ωi3⟩
+�
+j1̸=j4
+Ωℓj1Ωℓj4Ωi1j1Ωi3j1Ωi3j4 ≤ ⟨Ωℓ, Ωi1 ◦ Ωi3⟩ · ⟨Ωℓ, Ωi3⟩
+�
+j1̸=j3
+�
+Ωℓj1Ωℓj3Ωi1j1Ωi3j3 + Ωℓj1Ωℓj3Ωi1j1Ωi1j3Ωi3j1Ωi3j3
+�
+≤ 2⟨Ωℓ, Ωi1⟩ · ⟨Ωℓ, Ωi3⟩
+�
+j1,j3,j4 dist.
+�
+Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi3j3Ωi3j4 + Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi1j3Ωi3j1Ωi3j4
+�
+≤ 2⟨Ωℓ, Ωi1⟩⟨Ωℓ, Ωi3⟩2
+�
+j1,j2,j3,j4 dist.
+Ωℓj1Ωℓj2Ωℓj3Ωℓj4Ωi1j1Ωi1j2Ωi3j3Ωi3j4 ≤ ⟨Ωℓ, Ωi1⟩2⟨Ωℓ, Ωi3⟩2.
+These bounds are relatively sharp, and it is clear that the first and third lines dominate.
+Furthermore as. Hence,
+�
+t
+Ft2ℓs ≲ F12ℓs + F32ℓs ≲
+�
+ω1̸=ω3∈Iℓ−1
+σ2
+i1ℓσ2
+i3ℓ
+�
+⟨Ωℓ, Ωi1 ◦ Ωi3⟩ + ⟨Ωℓ, Ωi1⟩ · ⟨Ωℓ, Ωi3⟩
+�
+.
+(B.68)
+70
+
+Observe that if ℓ ∈ Sk, then
+�
+ω
+σ4
+iℓΩij ≤
+�
+i∈Sk
+1
+n4
+k ¯N4
+k
+NiΩij +
+K
+�
+k′=1
+�
+i∈Sk′
+1
+n4 ¯N4 NiΩij
+(B.69)
+≤
+1
+n3
+k ¯N3
+k
+µkj +
+1
+n3 ¯N3 µj,
+(B.70)
+and
+�
+ω
+σ2
+iℓΩij ≤
+�
+i∈Sk
+1
+n2
+k ¯N2
+k
+NiΩij +
+K
+�
+k′=1
+�
+i∈Sk′
+1
+n ¯N NiΩij
+≤
+1
+nk ¯Nk
+µkj +
+1
+n ¯N µj.
+Next,
+�
+(ℓ,s)
+�
+t
+Ft1ℓs ≲
+�
+(ℓ,s)
+�
+ω∈Iℓ−1
+σ4
+iℓ⟨Ωℓ, Ωi⟩.
+≲
+�
+(ℓ,s)
+�
+j
+Ωℓj
+�
+1
+n3
+k ¯N3
+k
+µkj +
+1
+n3 ¯N3 µj
+�
+≲
+�
+j
+�
+k
+1
+n2
+k ¯N2
+k
+µ2
+kj +
+�
+j
+�
+k
+1
+n2 ¯N2 µ2
+j ≲
+�
+k
+1
+n2
+k ¯N2
+k
+∥µk∥2,
+(B.71)
+where we applied that ∥µ∥2 ≲ �
+k ∥µk∥2 (see (A.49)). Furthermore,
+�
+(ℓ,s)
+�
+t
+Ft2ℓs ≤
+K
+�
+k=1
+�
+ℓ∈Sk
+Nℓ
+�
+ω1,ω3
+σ2
+i1ℓσ2
+i3ℓ
+�
+⟨Ωℓ, Ωi1 ◦ Ωi3⟩ + ⟨Ωℓ, Ωi1⟩ · ⟨Ωℓ, Ωi3⟩
+�
+≲
+�
+k
+�
+ℓ∈Sk
+Nℓ
+� �
+j
+Ωℓj
+�
+1
+nk ¯Nk
+µkj +
+1
+n ¯N µj
+�2 +
+� �
+j
+Ωℓj ·
+�
+1
+nk ¯Nk
+µkj +
+1
+n ¯N µj
+��2�
+≲
+�
+k
+�
+ℓ∈Sk
+Nℓ
+�
+j
+Ωℓj
+�
+1
+nk ¯Nk
+µkj +
+1
+n ¯N µj
+�2
+In the last line we apply Cauchy–Schwarz. Continuing, we have
+�
+(ℓ,s)
+�
+t
+Ft2ℓs ≲
+�
+k
+�
+ℓ∈Sk
+Nℓ
+�
+j
+Ωℓj
+�
+1
+nk ¯Nk
+µkj +
+1
+n ¯N µj
+�2
+≲
+�
+k
+�
+ℓ∈Sk
+Nℓ
+�
+j
+Ωℓj
+�
+1
+nk ¯Nk
+µkj
+�2 +
+�
+k
+�
+ℓ∈Sk
+Nℓ
+�
+j
+Ωℓj
+� 1
+n ¯N µj
+�2
+≲
+�
+k
+∥µk∥3
+3
+nk ¯Nk
++
+�
+k
+∥µ∥3
+3
+n ¯N ≲
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+,
+(B.72)
+where we applied (A.68). Combining (B.66), (B.71) and (B.72), we have
+�
+(ℓ,s)
+ED4
+ℓ,s ≲
+�
+(ℓ,s)
+2
+�
+x=1
+5
+�
+t=1
+Ftxℓs ≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
++
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+,
+as desired.
+71
+
+B.5
+Proof of Lemma B.4
+Var
+� �
+(ℓ,s)
+Var( ˜Eℓ,s|F≺(ℓ,s))
+�
+→ 0
+(B.73)
+Next we study (B.73). We have
+Var(Eℓ,s|F≺(ℓ,s)) = E[E2
+ℓ,s|F≺(ℓ,s)] = σ2
+ℓ
+�
+r,r′∈[s−1]
+�
+j,j′
+E
+�
+ZℓjrZℓjsZℓj′r′Zℓj′s
+��F≺(ℓ,s)
+�
+= σ2
+ℓ
+�
+r,r′∈[s−1]
+�
+j,j′
+ZℓjrZℓj′r′E[ZℓjsZℓj′s]
+= σ2
+ℓ
+�
+r,r′∈[s−1]
+�
+j,j′
+δjj′ℓZℓjrZℓj′r′,
+(B.74)
+where we let
+δjj′ℓ = EZℓjsZℓj′s =
+�
+Ωℓj(1 − Ωℓj)
+if j = j′
+−ΩℓjΩℓj′
+else.
+(B.75)
+Define
+ϕℓrℓr′ =
+�
+j,j′
+δjj′ℓZℓjrZℓj′r′.
+(B.76)
+By (B.74) we have
+�
+(ℓ,s)
+Var(Eℓ,s|F≺(ℓ,s)) =
+n
+�
+ℓ=1
+Nℓ
+�
+s=1
+�
+r,r′∈[s−1]
+σ2
+ℓ ϕℓrℓr′
+=
+n
+�
+ℓ=1
+Nℓ
+�
+s=1
+� �
+r∈[s−1]
+σ2
+ℓ ϕℓrℓr + 2
+�
+r<r′∈[s−1]
+σ2
+ℓ ϕℓrℓr′�
+=
+n
+�
+ℓ=1
+Nℓ
+�
+r=1
+�
+s∈[Nℓ]:s>r
+σ2
+ℓ ϕℓrℓr + 2
+s
+�
+ℓ=1
+�
+r<r′∈[Nℓ]
+�
+s∈[Nℓ]:s>r′
+σ2
+ℓ ϕℓrℓr′
+=
+n
+�
+ℓ=1
+Nℓ
+�
+r=1
+(Nℓ − r)σ2
+ℓ ϕℓrℓr + 2
+s
+�
+ℓ=1
+�
+r<r′∈[Nℓ]
+(Nℓ − r′)σ2
+ℓ ϕℓrℓr′
+≡ S1 + S2.
+Observe that S1 and S2 are uncorrelated. In addition, the terms in the summation defining
+S1 are uncorrelated; the same holds for S2 also.
+First we study S2. Next,
+Eϕ2
+ℓrℓr′ =
+�
+j1,j2,j3,j4
+δj1j2,ℓδj3j4,ℓ EZℓj1rZℓj2r′Zℓj3rZℓj4r′
+=
+�
+j1,j2,j3,j4
+δj1j2ℓδj3j4ℓEZℓj1rZℓj3rEZℓj2r′Zℓj4r′.
+(B.77)
+72
+
+First we study V2. By casework,
+|δj1j2ℓδj3j4ℓEZℓj1rZℓj3rEZℓj2r′Zℓj4r′|
+(B.78)
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+δ2
+jjℓEZ2
+ℓjrEZ2
+ℓjr′ ≲ Ω4
+ℓj
+if j1 = · · · = j4
+δj1j1ℓδj1j4ℓ|EZ2
+ℓj1rEZℓj1r′Zℓj4r′| ≲ Ω4
+ℓj1Ω2
+ℓj4
+if j1 = j2 = j3, j1 ̸= j4
+δj1j1ℓδj3j3ℓEZℓj1rZℓj3rEZℓj1r′Zℓj3r′ ≲ Ω3
+ℓj1Ω3
+ℓj3
+if j1 = j2, j3 = j4, j1 ̸= j3
+δ2
+j1j2ℓEZ2
+ℓj1rEZ2
+ℓj2r′ ≲ Ω3
+ℓj1Ω3
+ℓj2
+if j1 = j3, j2 = j4, j1 ̸= j2
+δj1j1ℓδj3j4ℓEZℓj1rZℓj3rEZℓj1r′Zℓj4r′ ≲ Ω3
+ℓj1Ω2
+ℓj3Ω2
+ℓj4
+if j1 = j2, j1, j3, j4 dist.
+δj1j2ℓδj1j4ℓEZ2
+ℓj1rEZℓj2r′Zℓj4r′ ≲ Ω3
+ℓj1Ω2
+ℓj2Ω2
+ℓj4
+if j1 = j3, j1, j2, j4 dist.
+δj1j2ℓδj3j4ℓEZℓj1rZℓj3rEZℓj2r′Zℓj4r′ ≲ Ω2
+ℓj1Ω2
+ℓj2Ω2
+ℓj3Ω2
+ℓj4
+if j1, j2, j3, j4 dist.
+Up to permutation of the indices j1, . . . , j4, all nonzero terms of (B.77) take one of the
+forms above. By (B.78) and Cauchy–Schwarz, we have
+Eϕ2
+ℓrℓr′ ≲ ∥Ωℓ∥4
+4 + ∥Ωℓ∥4
+4∥Ωℓ∥2 + 2∥Ωℓ∥6
+3 + 2∥Ωℓ∥3
+3∥Ωℓ∥4 + ∥Ωℓ∥8 ≲ ∥Ωℓ∥4
+4.
+(B.79)
+Recalling that {ϕℓrℓr′}ℓ,r<r′∈[Nℓ] are mutually uncorrelated, it follows that
+Var(S2) ≲
+�
+ℓ
+�
+r<r′∈[Nℓ]
+(Nℓ − r′)2σ2
+ℓ Eϕ2
+ℓrℓr′
+≲
+�
+ℓ
+�
+r<r′∈[Nℓ]
+(Nℓ − r′)2σ4
+ℓ ∥Ωℓ∥4
+4
+≲
+�
+k
+�
+ℓ∈Sk
+N4
+ℓ ·
+1
+n4
+k ¯N4
+k
+∥Ωℓ∥4
+4.
+(B.80)
+Next we study S1. We have
+Eϕ2
+ℓrℓr =
+�
+j1,j2,j3,j4
+δj1j2ℓδj3j4ℓEZℓj1rZℓj2rZℓj3rZℓj4r.
+We have the following bounds by casework.
+|δj1j2ℓδj3j4ℓEZℓj1rZℓj2rZℓj3rZℓj4r|
+(B.81)
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+δ2
+jjℓEZ4
+ℓjr ≲ Ω3
+ℓj
+if j1 = · · · = j4
+δj1j1ℓδj1j4ℓ|EZ3
+ℓj1rZℓj4r| ≲ Ω3
+ℓj1Ω2
+ℓj4
+if j1 = j2 = j3, j1 ̸= j4
+δj1j1ℓδj3j3ℓEZ2
+ℓj1rZ2
+ℓj3r ≲ Ω2
+ℓj1Ω2
+ℓj3
+if j1 = j2, j3 = j4, j1 ̸= j3
+δ2
+j1j2ℓEZ2
+ℓj1rZ2
+ℓj2r ≲ Ω3
+ℓj1Ω3
+ℓj2
+if j1 = j3, j2 = j4, j1 ̸= j3
+δj1j1ℓδj3j4ℓ|EZ2
+ℓj1rZℓj3rZℓj4r| ≲ Ω2
+ℓj1Ω2
+ℓj3Ω2
+ℓj4
+if j1 = j2, j1, j3, j4 dist.
+δj1j2ℓδj1j4ℓ|EZ2
+ℓj1rZℓj2rZℓj4| ≲ Ω3
+ℓj1Ω2
+ℓj2Ω2
+ℓj4
+if j1 = j3, j1, j2, j4 dist.
+δj1j2ℓδj3j4ℓ|EZℓj1rZℓj2rZℓj3rZℓj4r| ≲ Ω2
+ℓj1Ω2
+ℓj2Ω2
+ℓj3Ω2
+ℓj4
+if j1, j2, j3, j4 dist.
+Up to symmetry, this accounts for all possible (nonzero) cases. Hence by Cauchy–Schwarz,
+Eϕ2
+ℓrℓr ≲ ∥Ωℓ∥3
+3 + ∥Ωℓ∥3
+3∥Ωℓ∥2 + ∥Ωℓ∥4 + ∥Ωℓ∥6
+3 + ∥Ωℓ∥6 + ∥Ωℓ∥3
+3∥Ωℓ∥4 + ∥Ωℓ∥8 ≲ ∥Ωℓ∥3
+3.
+(B.82)
+73
+
+Recalling that {ϕℓrℓr}ℓ,r∈[Nℓ] is an uncorrelated collection of random variables, we have
+Var(S1) ≲
+�
+ℓ
+�
+r∈[Nℓ]
+(Nℓ − r)2σ4
+ℓ Eϕ2
+ℓrℓr
+≲
+�
+ℓ
+�
+r∈[Nℓ]
+(Nℓ − r)2σ4
+ℓ ∥Ωℓ∥3
+3
+≲
+�
+k
+�
+ℓ∈Sk
+N3
+ℓ ·
+1
+n4
+k ¯N4
+k
+∥Ωℓ∥3
+3.
+(B.83)
+Combining (B.83) and (B.80) proves the result.
+B.6
+Proof of Lemma B.5
+We have
+EE4
+ℓ,s =
+�
+r1,r2,r3,r4∈[s−1]
+σ4
+ℓ
+�
+j1,j2,j3,j4
+EZℓj1r1Zℓj1sZℓj2r2Zℓj2sZℓj3r3Zℓj3sZℓj4r4Zℓj4s
+= σ4
+ℓ
+�
+j1,j2,j3,j4
+�
+E[Zℓj1sZℓj2sZℓj3sZℓj4s] ·
+�
+r1,r2,r3,r4∈[s−1]
+E[Zℓj1r1Zℓj2r2Zℓj3r3Zℓj4r4]
+�
+��
+�
+=:Bℓ,s;j1,j2,j3,j4
+�
+(B.84)
+We have by exhaustive casework that
+|E[Zℓj1r1Zℓj2r2Zℓj3r3Zℓj4r4]|
+(B.85)
+=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+EZ4
+ℓj1r1 ≲ Ωℓj1
+if j1=j2=j3=j4;
+r1=r2=r3=r4
+EZ2
+ℓj1r1EZ2
+ℓj1r3 ≲ Ω2
+ℓj1
+if
+j1=j2=j3=j4;
+r1=r2,r3=r4,r1̸=r3
+|E[Z3
+ℓj1r1Zℓj4r1]| ≲ Ωℓj1Ωℓj4
+if j1=j2=j3,j1̸=j4;
+r1=r2=r3=r4
+|E[Z2
+ℓj1r1EZℓj1r3Zℓj4r3]| ≲ Ω2
+ℓj1Ωℓj4
+if
+j1=j2=j3,j1̸=j4;
+r1=r2,r3=r4,r1̸=r3
+|EZ2
+ℓj1r1Z2
+ℓj3r1| ≲ Ωℓj1Ωℓj3
+if j1=j2,j3=j4,j1̸=j3;
+r1=r2=r3=r4
+|E[Z2
+ℓj1r1Z2
+ℓj3r3]| ≲ Ωℓj1Ωℓj3
+if j1=j2,j3=j4,j1̸=j3;
+r1=r2,r3=r4,r1̸=r3
+|E[Zℓj1r1Zℓj3r1EZℓj1r2Zℓj3r2]| ≲ Ω2
+ℓj1Ω2
+ℓj3
+if j1=j2,j3=j4,j1̸=j3;
+r1=r3,r2=r4,r1̸=r2
+|E[Z2
+ℓj1r1Zℓj3r1Zℓj4r1]| ≲ Ωℓj1Ωℓj3Ωℓj4
+if j1=j2,j1,j3,j4 dist.;
+r1=r2=r3=r4
+|E[Z2
+ℓj1r1EZℓj3r3Zℓj4r3]| ≲ Ωℓj1Ωℓj3Ωℓj4
+if j1=j2,j1,j3,j4 dist.;
+r1=r2,r3=r4,r1̸=r3
+|E[Zℓj1r1Zℓj3r1EZℓj1r2Zℓj4r2]| ≲ Ω2
+ℓj1Ωℓj3Ωℓj4
+if j1=j2,j1,j3,j4 dist.;
+r1=r3,r2=r4,r1̸=r2
+|E[Zℓj1r1Zℓj2r1Zℓj3r1Zℓj4r1]| ≲ Ωℓj1Ωℓj2Ωℓj3Ωℓj4
+if j1,j2,j3,j4 dist;
+r1=r2=r3=r4
+|E[Zℓj1r1Zℓj2r1EZℓj3r3Zℓj4r3]| ≲ Ωℓj1Ωℓj2Ωℓj3Ωℓj4
+if
+j1,j2,j3,j4 dist;
+r1=r2,r3=r4,r1̸=r3
+74
+
+Up to permutation of the indices j1, j2, j3, j4 and r1, r2, r3, r4, this accounts for all possible
+cases such that (B.85) is nonzero. Therefore,
+Bℓ,s;j1,j2,j3,j4 ≲
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+sΩℓj1 + s2Ω2
+ℓj1
+if j1 = j2 = j3 = j4
+sΩℓj1Ωℓj4 + s2Ω2
+ℓj1Ωℓj4
+if j1 = j2 = j3, j1 ̸= j4
+sΩℓj1Ωℓj3 + s2Ωℓj1Ωℓj3
+if j1 = j2, j3 = j4, j1 ̸= j3
+sΩℓj1Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj3Ωℓj4
+if j1 = j2, j1, j3, j4 dist.
+sΩℓj1Ωℓj2Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj2Ωℓj3Ωℓj4
+if j1, j2, j3, j4 dist.
+Up to permutation of j1, j2, j3, j4, this accounts for all possible cases. Returning to (B.84),
+we have by applying (B.55) and the previous display that
+EE4
+ℓ,s ≲ σ4
+ℓ
+� �
+j
+Ωℓj(sΩℓj + s2Ω2
+ℓj) +
+�
+j1̸=j4
+Ωℓj1Ωℓj4(sΩℓj1Ωℓj4 + s2Ω2
+ℓj1Ωℓj4)
++
+�
+j1̸=j3
+Ωℓj1Ωℓj3(sΩℓj1Ωℓj3 + s2Ωℓj1Ωℓj3)
++
+�
+j1,j3,j4(dist.)
+Ωℓj1Ωℓj3Ωℓj4(sΩℓj1Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj3Ωℓj4)
++
+�
+j1,j2,j3,j4 dist.
+Ωℓj1Ωℓj2Ωℓj3Ωℓj4(sΩℓj1Ωℓj2Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj2Ωℓj3Ωℓj4)
+�
+≲ sσ4
+ℓ ∥Ωℓ∥2 + s2σ4
+ℓ ∥Ωℓ∥3
+3.
+In the third line we group the coefficients of s and s2 and use the fact that ∥Ωℓ∥4 ≤ ∥Ωℓ∥3
+3
+by Cauchy–Schwarz. Therefore
+�
+(ℓ,s)
+EE4
+ℓ,s ≲
+�
+(ℓ,s)
+sσ4
+ℓ ∥Ωℓ∥2 +
+�
+(ℓ,s)
+s2σ4
+ℓ ∥Ωℓ∥3
+3
+=
+�
+k
+�
+ℓ∈Sk
+�
+s∈[Nℓ]
+sσ4
+ℓ ∥Ωℓ∥2 +
+�
+k
+�
+ℓ∈Sk
+�
+s∈[Nℓ]
+s2σ4
+ℓ ∥Ωℓ∥3
+3
+≲
+�
+k
+�
+ℓ∈Sk
+N2
+ℓ ·
+1
+n4
+k ¯N4
+k
+∥Ωℓ∥2 +
+�
+k
+�
+ℓ∈Sk
+N3
+ℓ ·
+1
+n4
+k ¯N4
+k
+∥Ωℓ∥3
+3,
+as desired.
+B.7
+Proof of Lemma B.6
+We have
+�
+k
+�
+i∈Sk
+N2
+i ∥Ωi∥2
+n4
+k ¯N4
+k
+≤
+�
+k
+1
+n4
+k ¯N4
+k
+�
+i,m∈Sk
+NiNm⟨Ωi, Ωm⟩
+=
+�
+k
+1
+n2
+k ¯N2
+k
+∥µk∥2,
+which establishes the first claim.
+75
+
+Similarly,
+�
+k
+�
+i∈Sk
+N3
+i ∥Ωi∥3
+3
+n4
+k ¯N4
+k
+≤
+�
+k
+1
+n4
+k ¯N4
+k
+�
+i,m,m′∈Sk
+NiNmNm′
+�
+j
+ΩijΩmjΩm′j
+≤
+�
+k
+1
+nk ¯Nk
+∥µk∥3
+3,
+which proves the second claim.
+The third claim follows similarly and we omit the proof.
+C
+Proofs of other main lemmas and theorems
+C.1
+Proof of Lemma 2.1
+We start from computing E[(ˆµkj−ˆµj)2]. Write Xij = Ni(Ωij+Yij). It follows by elementary
+calculation that
+ˆµkj − ˆµj = µkj − µj +
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+� �
+i∈Sk
+NiYij −
+�
+1≤ℓ≤K:ℓ̸=k
+1
+nℓ ¯Nℓ
+�
+i∈Sℓ
+NiYij.
+For different k, the variables �
+i∈Sk NiYij are independent of each other. It follows that
+E[(ˆµkj − ˆµj)2] = (µkj − µj)2 +
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+E
+���
+i∈Sk
+NiYij
+�2�
++
+�
+ℓ:ℓ̸=k
+1
+n2 ¯N2 E
+���
+i∈Sℓ
+NiYij
+�2�
+= (µkj − µj)2 +
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2 �
+i∈Sk
+NiΩij(1 − Ωij) +
+�
+ℓ:ℓ̸=k
+1
+n2 ¯N2
+�
+i∈Sℓ
+NiΩij(1 − Ωij)
+= (µkj − µj)2 +
+1
+n2
+k ¯N2
+k
+�
+1 − nk ¯Nk
+n ¯N
+� �
+i∈Sk
+NiΩij(1 − Ωij)
++
+1
+n2 ¯N2
+��
+1 − n ¯N
+nk ¯Nk
+� �
+i∈Sk
+NiΩij(1 − Ωij) +
+�
+ℓ:ℓ̸=k
+�
+i∈Sℓ
+NiΩij(1 − Ωij)
+�
+= (µkj − µj)2 +
+1
+n2
+k ¯N2
+k
+�
+1 − nk ¯Nk
+n ¯N
+� �
+i∈Sk
+NiΩij(1 − Ωij)
+−
+1
+n ¯Nnk ¯Nk
+��
+i∈Sk
+NiΩij(1 − Ωij) − nk ¯Nk
+n ¯N
+K
+�
+ℓ=1
+�
+i∈Sℓ
+NiΩij(1 − Ωij)
+�
+�
+��
+�
+δkj
+.
+(C.1)
+Since Xij follows a binomial distribution, it is easy to see that E[Xij] = NiΩij and E[X2
+ij] =
+(E[Xij])2 + Var(Xij) = N2
+i Ω2
+ij + NiΩij(1 − Ωij). Combining them gives
+E[Xij(Ni − Xij)] = Ni(Ni − 1)Ωij(1 − Ωij).
+(C.2)
+76
+
+Define
+ˆζkj = (ˆµkj − ˆµj)2 −
+1
+n2
+k ¯N2
+k
+�
+1 − nk ¯Nk
+n ¯N
+� �
+i∈Sk
+Xij(Ni − Xij)
+Ni − 1
+,
+It follows from (C.1)-(C.2) that
+E[ˆζkj] = (µkj − µj)2 −
+1
+n ¯Nnk ¯Nk
+δkj.
+(C.3)
+We are ready to compute E[T]. By definition, T = �p
+j=1
+�K
+k=1 nk ¯Nk ˆζkj and ρ2 = �
+j,k(µkj−
+µj)2. Consequently,
+E[T] =
+p
+�
+j=1
+K
+�
+k=1
+nk ¯Nk
+�
+(µkj − µj)2 −
+1
+n ¯Nnk ¯Nk
+δkj
+�
+= ρ2 −
+1
+n ¯N
+p
+�
+j=1
+K
+�
+k=1
+δkj.
+(C.4)
+We use the definition of δkj in (C.1). It is seen that for each 1 ≤ j ≤ p,
+K
+�
+k=1
+δkj =
+K
+�
+k=1
+�
+i∈Sk
+NiΩij(1 − Ωij) −
+� K
+�
+k=1
+nk ¯Nk
+n ¯N
+� K
+�
+ℓ=1
+�
+i∈Sℓ
+NiΩij(1 − Ωij) = 0.
+(C.5)
+Combining (C.4)-(C.5) gives E[T] = ρ2. This proves the claim.
+C.2
+Proof of Theorem 3.3
+First we show that
+Var(T) ≲ Θn
+(C.6)
+Recall
+Θn1 = 4
+K
+�
+k=1
+p
+�
+j=1
+nk ¯Nk(µkj − µj)2µkj
+Θn2 = 2
+K
+�
+k=1
+�
+i∈Sk
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+N3
+i
+Ni − 1Ω2
+ij
+Θn3 =
+2
+n2 ¯N2
+�
+1≤k̸=ℓ≤K
+�
+i∈Sk
+�
+m∈Sℓ
+p
+�
+j=1
+NiNmΩijΩmj
+Θn4 = 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk,
+i̸=m
+p
+�
+j=1
+�
+1
+nk ¯Nk
+−
+1
+n ¯N
+�2
+NiNmΩijΩmj.
+and that �4
+a=1 Θna = Θn.
+By Lemma A.2, we immediately have
+Var(1′
+pU1) ≤ Θn1.
+(C.7)
+77
+
+For U2, it is shown in the Proof of Lemma A.3 that
+Var(1′
+pU2) = 4
+K
+�
+k=1
+�
+i∈Sk
+�
+1≤r<s≤Ni
+θi
+Ni(Ni − 1)
+�
+∥Ωi∥2 + O(∥Ωi∥3
+3)].
+Thus
+Var(1′
+pU2) ≲ 4
+K
+�
+k=1
+�
+i∈Sk
+�
+1≤r<s≤Ni
+θi
+Ni(Ni − 1)∥Ωi∥2
+= 2
+K
+�
+k=1
+�
+i∈Sk
+θi∥Ωi∥2 = Θn2
+(C.8)
+Next we study U3. Using that Ωmj′ ≤ 1 and ∥Ωi∥1 = 1, we have
+�
+k̸=ℓ
+nknℓ ¯Nk ¯Nℓ
+n2 ¯N2
+1′
+p(Σk ◦ Σℓ)1p =
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j,j′
+NiNmΩijΩij′ΩmjΩmj′
+≤
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j
+NiNmΩijΩmj
+�
+j′
+Ωij′
+=
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+�
+j
+NiNmΩijΩmj.
+Therefore by Lemma A.4,
+Var(1′
+pU3) ≲
+2
+n2 ¯N2
+�
+1≤k̸=ℓ≤K
+�
+i∈Sk
+�
+m∈Sℓ
+p
+�
+j=1
+NiNmΩijΩmj = Θn3.
+(C.9)
+Similarly for U4, we have by the Proof of Lemma A.5 that
+Var(1′
+pU4) = 4
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i<m
+κim
+��
+j
+ΩijΩmj + δim
+�
+≲
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i<m
+κim
+�
+j
+ΩijΩmj = Θn4.
+(C.10)
+Above we use that |δim| ≤ �
+j ΩijΩmj and recall that κim = (
+1
+nk ¯
+Nk −
+1
+n ¯
+N )2NiNm.
+Observe that by Lemma 2.1,
+Θn1 = 4
+K
+�
+k=1
+p
+�
+j=1
+nk ¯Nk(µkj − µj)2µkj ≲ max
+k
+∥µk∥∞ · ρ2 = max
+k
+∥µk∥∞ · E T.
+(C.11)
+Since (3.1) holds, Lemma A.6 applies and
+Θn2 + Θn3 + Θn4 ≍
+�
+k
+∥µk∥2.
+(C.12)
+Combining (C.6), (C.11), and (C.12) proves the theorem.
+78
+
+C.3
+Proof of Theorem 3.4
+To prove Theorem 3.4, we must prove the following claims:
+(a) Under the alternative hypothesis, ψ → ∞ in probability.
+(b) For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level of α and an
+asymptotic power of 1.
+(c) If we choose α = αn such that αn → 0 and 1−Φ(SNRn) = o(αn), where Φ is the CDF
+of N(0, 1), then the sum of type I and type II errors of the DELVE test converges to
+0.
+We show the first claim, that ψ → ∞, under the alternative hypothesis and the condi-
+tions of Theorem 3.4. In particular, recall we assume that
+ρ2
+��K
+k=1 ∥µk∥2
+=
+n ¯N∥µ∥2ω2
+n
+��K
+k=1 ∥µk∥2
+→ ∞.
+(C.13)
+Our first goal is to show that
+T/
+�
+Var(T) P→ ∞
+(C.14)
+under the alternative. By Chebyshev’s inequality, it suffices to show that
+E T ≫
+�
+Var(T).
+(C.15)
+By Theorem 3.3,
+Var(T) ≲
+�
+k
+∥µk∥2 + max
+k
+∥µk∥∞ · ET =
+�
+k
+∥µk∥2 + max
+k
+∥µk∥∞ · ρ2
+(C.16)
+By (C.13),
+ET = ρ2 ≫
+�
+�
+�
+�
+K
+�
+k=1
+∥µk∥2 ≥ max
+1≤k≤K ∥µk∥∞.
+Therefore,
+�
+max
+1≤k≤K ∥µk∥∞ · ρ ≪ ρ2 = ET.
+(C.17)
+Moreover, by (C.13),
+�
+k
+∥µk∥2 ≪ ρ4 = (ET)2.
+(C.18)
+Combining (C.16), (C.17), and (C.18) implies (C.14).
+Next we show that V > 0 with high probability (i.e., with probability tending to 1 as
+n ¯N → ∞). Recall that by Lemmas A.6, A.10, and A.11,
+EV = Θn2 + Θn3 + Θn4 ≳
+�
+k
+∥µk∥2 > 0, and
+(C.19)
+79
+
+Var(V ) ≲
+�
+k
+∥µk∥2
+n2
+k ¯N2
+k
+∨
+�
+k
+∥µk∥3
+3
+nk ¯Nk
+.
+(C.20)
+Using this, the Markov inequality, and (3.4), we have
+P
+�
+V < E[V ]/2
+�
+≤ P
+�
+|V − E[V ]| ≥ E[V ]/2
+�
+≤ 4Var(V )
+(E[V ])2 = o(1),
+(C.21)
+which implies that V > 0 with high probability.
+To finish the proof of the first claim, note that the assumptions of Proposition A.2 are
+satisfied and we have V/Var(T) = OP(1). By this, (C.14), and (C.21), we have
+ψ = T1V >0
+√
+V
+=
+�
+Var(T)
+√
+V
+·
+T
+�
+Var(T)
+· 1V >0 ≳
+T
+�
+Var(T)
+→ ∞
+in probability.
+The second claim follows directly from the first claim and Theorem 3.2.
+To prove the third claim, by Chebyshev’s inequality and T/
+�
+Var(T) → ∞, it follows
+that T > (1/2)ET = (1/2)ρ2 with high probability as n ¯N → ∞. By a similar Chebyshev
+argument as above, it also holds that V < (3/2)EV with high probability as n ¯N → ∞.
+Recall that EV = Θn2 +Θn3 +Θn4 ≲ �
+k ∥µk∥2 by Lemmas A.6 and A.10. Thus, with high
+probability as n ¯N → ∞, we have
+ψ = T1V >0/
+√
+V ≳ ρ2/
+√
+EV ≳ n ¯N∥µ∥2ω2
+n
+��
+k ∥µk∥2 = SNRn.
+Choosing αn as specified yield the third claim. The proof is complete since all three claims
+are established.
+C.4
+Proof of Theorem 3.5
+Without loss of generality, we assume p is even and write m = p/2.
+Let µ ∈ Rm be
+a nonnegative vector with ∥µ∥1 = 1/2 . Let ˜µ = (µ′, µ′)′ ∈ Rp. We consider the null
+hypothesis:
+H0 :
+Ωi = ˜µ,
+1 ≤ i ≤ n.
+(C.22)
+We pair it with a random alternative hypothesis.
+Let b1, b2, . . . , bm be a collection of
+i.i.d. Rademacher variables. Let z1, z2, . . . , zK denote an independent collection of i.i.d.
+Rademacher random variables conditioned on the event | �
+k zk| ≤ 100
+√
+K. For a properly
+small sequence ωn > 0 of positive numbers, let
+H1 :
+Ωij =
+�
+µj
+�
+1 + ωn(nk ¯Nk)−1� 1
+K
+�
+k∈K nk ¯Nk
+�
+zkbj
+�
+,
+if 1 ≤ j ≤ m, i ∈ Sk
+˜µj
+�
+1 − ωn(nk ¯Nk)−1� 1
+K
+�
+k∈K nk ¯Nk
+�
+zkbj−m
+�
+,
+if m + 1 ≤ j ≤ 2m, i ∈ Sk
+(C.23)
+80
+
+In this section we slightly abuse notation, using ωn to refer to the (deterministic) sequence
+above and reserving ω(Ω) for the random quantity
+ω(Ω) =
+�
+�
+�
+�
+1
+n ¯N∥µ∥2
+K
+�
+k=1
+nk ¯Nk∥µk − µ∥2.
+(C.24)
+As long as
+ωn ≤
+mink nk ¯Nk
+1
+K
+�
+k∈[K] nk ¯Nk
+= mink nk ¯Nk
+n ¯N/K
+,
+then Ωij ≥ 0 for all i ∈ [n], j ∈ [p]. Furthermore, for each 1 ≤ i ≤ n, we have ∥Ωi∥1 =
+2∥µ∥1 = 1. We suppose there exists a constant c ∈ (0, 1) such that
+cK−1n ¯N ≤ nk ¯Nk ≤ c−1K−1n ¯N
+for all k ∈ [K]
+(C.25)
+With (C.25) in hand, we may assume without loss of generality that
+ωn ≤ c/2
+(C.26)
+This assumption implies that (C.23) is well-defined and moreover Ωij ≍ µj.
+Next we characterize the random quantity ω(Ω) in terms of ωn.
+Lemma C.1. Let ω2(Ω) be as in (C.24). When Ω follows Model (C.23), there exists a
+constant c1 ∈ (0, 1) such that c1ω2
+n ≤ ω2(Ω) ≤ c−1
+1 ω2
+n with probability 1.
+The proof of Lemma C.1 is given in Section C.4.1. By Lemma C.1, under the model
+(C.23) it holds with probability 1 that
+n ¯N∥µ∥2ω2(Ω)
+��K
+k=1 ∥µk∥2
+≍ K−1/2n ¯N∥µ∥ω2
+n.
+(C.27)
+Above we use that Ωij ≍ µj , since we assume (C.26)
+We also require Proposition C.1 below, whose proof is given in Section C.4.2.
+Proposition C.1. Suppose that (C.25) and (C.26) hold. Consider the pair of hypotheses
+in (C.22)-(C.23) and let P0, and P1 be the respective probability measures. If
+n ¯N∥µ∥2ω2(Ω)
+��K
+k=1 ∥µk∥2
+≍ K−1/2n ¯N∥µ∥ω2
+n → 0,
+then the chi-square distance between P0 and P1 converges to 0.
+Now we prove Theorem 3.5. Let δn denote an arbitrary sequence tending to 0. Without
+loss of generality, we may assume that δn ≤ c∗ for a small absolute constant c∗ ∈ (0, 1).
+Note that K−1/2n ¯N ≥ 1 since K ≤ n. Thus for appropriate choice of sequences of µ = µn
+and ωn ≤ c/2 in models (C.22), (C.23) and applying (C.27), we obtain
+2δn ≥ n ¯N∥µ∥2ω2(Ω)
+��K
+k=1 ∥µk∥2
+≥ δn.
+(C.28)
+81
+
+Recall the definitions of Q∗
+0n and Q∗
+1n in (3.8). Let Π denote the distribution on ξ =
+{(Ni, Ωi, ℓi)} ∈ Q∗
+1n induced by (C.23).
+Let ξ0 denote the parameter associated to the
+simple null hypothesis in (C.22) associated to our choice of µ and ωn satisfying (C.28). We
+have by standard manipulations,
+R(Q∗
+0n, Q∗
+1n) :=
+inf
+Ψ∈{0,1}
+�
+sup
+ξ∈Q∗
+0n(c0,ϵn)
+Pξ(Ψ = 1) +
+sup
+ξ∈Q∗
+1n(δn;c0,ϵn)
+Pξ(Ψ = 0)
+�
+=
+inf
+Ψ∈{0,1}
+�
+sup
+ξ∈Q∗
+0n(c0,ϵn),ξ′∈Q∗
+1n(δn;c0,ϵn)
+�
+Pξ(Ψ = 1) + Pξ(Ψ = 0)
+�
+≥
+inf
+Ψ∈{0,1}
+�
+sup
+ξ∈Q∗
+0n(c0,ϵn)
+Eξ′∼Π
+�
+Pξ(Ψ = 1) + Pξ′(Ψ = 0)
+��
+≥
+inf
+Ψ∈{0,1}
+�
+Eξ′∼Π
+�
+Pξ0(Ψ = 1) + Pξ′(Ψ = 0)
+��
+=
+inf
+Ψ∈{0,1}
+�
+P0(Ψ = 1) + P1(Ψ = 0)
+�
+.
+In the last line we recall the definition of P0 and P1 in (C.22) and (C.23), noting that for
+all events E,
+P1(E) = Eξ′∼π Pξ′(E).
+Next, by the Neyman–Pearson lemma and the standard inequality TV(P, Q) ≤
+�
+χ2(P, Q)
+(see e.g. Chapter 2 of Tsybakov [2008]),
+R(Q∗
+0n, Q∗
+1n) ≥
+inf
+Ψ∈{0,1}
+�
+P0(Ψ = 1) + P1(Ψ = 0)
+�
+= 1 − TV
+�
+P0, P1
+�
+≥ 1 −
+�
+χ2(P0, P1).
+By Proposition C.1, as δn → 0 we have χ2(P0, P1) → 0 and thus R(Q∗
+0n, Q∗
+1n) → 1, as
+desired.
+C.4.1
+Proof of Proposition C.1
+Next, we perform a change of parameters that preserves the signal strength and chi-squared
+distance. The testing problem (C.22) and (C.23) has parameters Ωij, Ni, ¯Nk, nk, n, and K.
+Let P0 and P1 denote the distributions corresponding to the null and alternative hypotheses,
+respectively. For each k ∈ [K], we combine all documents in sample k to obtain new null
+and alternative distributions ˜P0 and ˜P1 with parameters ˜Ωij, ˜Ni, ¯˜Ni, ˜ni, ˜n, and ˜K such that
+˜K = K = ˜n
+˜Ni = ni ¯Ni
+for i ∈ [ ˜K]
+¯˜Ni ≡ ˜Ni
+for i ∈ [ ˜K]
+˜ni = 1
+for i ∈ [ ˜K] .
+(C.29)
+82
+
+For notational ease, we define ˜N := ¯˜N =
+1
+K
+�
+k∈[K] nk ¯Nk. Furthermore, we have ˜Ωi = µ
+for all i ∈ [˜n] under the null ˜Ωi = µi for all i ∈ [˜n] under the alternative. Explicitly, in the
+reparameterized model, we have the null hypothesis
+H0 :
+Ωi = ˜µ,
+1 ≤ i ≤ n.
+(C.30)
+and alternative hypothesis
+H1 :
+Ωij =
+�
+µj
+�
+1 + ωn ˜N−1
+i
+˜Nzibj
+�
+,
+if 1 ≤ j ≤ m,
+˜µj
+�
+1 − ωn ˜N−1
+i
+˜Nzibj−m
+�
+,
+if m + 1 ≤ j ≤ 2m.
+(C.31)
+for all i ∈ [ ˜K] = [K] = [˜n]. Observe that the likelihood ratio is preserved:
+dP0
+dP1 =
+˜dP0
+d˜P1 and
+also ω(Ω) = ω(�Ω). For simplicity we work with this reparameterized model in this proof.
+If z1, . . . , z˜n are independent Rademacher random variables then with probability at
+least 1/2 it holds that
+|
+�
+i
+zi| ≤ 100
+√
+˜n
+(C.32)
+by Hoeffding’s inequality. Recall that our random model is defined in (C.23) where (i)
+z1, . . . , z˜n are independent Rademacher random variables conditioned on the event | �
+i zi| ≤
+100
+√
+˜n, and (ii) b1, . . . , bm are independent Rademacher random variables.
+Now we study ω2(�Ω). For each 1 ≤ j ≤ m, we have �Ωij = µj(1 + ωn ˜N−1
+i
+˜Nzibj). Define
+ηj = (˜n ˜N)−1 �˜n
+i=1 ˜Ni�Ωij = µj(1 + ωn¯zbj) for 1 ≤ j ≤ m and ηj = (˜n ˜N)−1 �˜n
+i=1 ˜Ni�Ωij =
+˜µj(1 − ωn¯zbj) for m < j ≤ 2m. We have
+˜n
+�
+i=1
+p
+�
+j=1
+˜Ni(�Ωij − ηj)2 = 2
+˜n
+�
+i=1
+m
+�
+j=1
+˜Ni · µ2
+jω2
+n
+˜N2
+˜N2
+i
+(zi − ¯z)2b2
+j
+= 2ω2
+n ˜N2∥µ∥2
+˜n
+�
+i=1
+˜N−1
+i
+(zi − ¯z)2.
+By (C.32), |¯z| ≤ 100
+√
+˜n. Thus |zi − ¯z| ≍ 1. Write ˜N∗ = (˜n−1 �˜n
+i=1 ˜N−1
+i
+). It follows that
+˜n
+�
+i=1
+p
+�
+j=1
+˜Ni(�Ωij − ηj)2 ≍ ω2
+n ˜N2∥µ∥2 · ˜n ˜N−1
+∗ .
+Note that ˜N ≥ ˜N∗. Additionally, by assumption (C.25), ˜Ni ≍ ˜N ≤ c−1 ˜N∗. It follows that
+˜n
+�
+i=1
+p
+�
+j=1
+˜Ni(�Ωij − ηj)2 ≍ ˜n ˜N∥µ∥2ω2
+n.
+(C.33)
+Moreover, ∥η∥2 = �p
+j=1 µ2
+j(1 + ωn¯zbj)2. By our conditioning on the event in (C.32),
+|ωn¯zbj| ≲ ωn˜n−1/2.
+83
+
+Since ωn ≤ 1 and �
+j bj = 0, we have
+∥η∥2 = ∥µ∥2 +
+p
+�
+j=1
+µ2
+jω2
+n¯z2 = ∥µ∥2[1 + O(˜n−1)] ≍ ∥µ∥2.
+(C.34)
+Hence
+ω2(�Ω) = ω2(Ω) ≍ ω2
+n,
+where recall
+ω(�Ω) =
+�˜n
+i=1
+�p
+j=1 ˜Ni(�Ωij − ηj)2
+˜n ˜N∥η∥2
+.
+(C.35)
+This finishes the proof.
+C.4.2
+Proof of Proposition C.1
+In this proof, we continue to employ the reparametrization in (C.29). As discussed there,
+this reparametrization preserves the likelihood ratio and thus the chi-square distance.
+By definition, χ2(P0, P1) =
+�
+( dP1
+dP0 )2dP0 − 1. It suffices to show that
+� �dP1
+dP0
+�2
+dP0 = 1 + o(1).
+(C.36)
+From the density of of multinomial distribution, dP0 = �
+i,j ˜µXij
+j
+, and dP1 = Eb,z[�
+i,j �ΩXij
+ij ].
+It follows that
+dP1
+dP0
+= Eb,z
+� ˜n
+�
+i=1
+p
+�
+j=1
+� �Ωij
+˜µj
+�Xij�
+.
+Let b(0) = (b(0)
+1 , . . . , b(0)
+m )′ and z(0) = (z(0)
+1 , · · · , z(0)
+˜n )′ be independent copies of b and z.
+We construct �Ω(0)
+ij similarly as in (C.31). It is seen that
+� �dP1
+dP0
+�2
+dP0 = EXEb,z,b(0),z(0)
+� ˜n
+�
+i=1
+p
+�
+j=1
+� �Ωij �Ω(0)
+ij
+˜µ2
+j
+�Xij�
+= Eb,z,b(0),z(0)
+� ˜n
+�
+i=1
+EXi
+� p
+�
+j=1
+� �Ωij �Ω(0)
+ij
+˜µ2
+j
+�Xij��
+= Eb,z,b(0),z(0)
+� ˜n
+�
+i=1
+� p
+�
+j=1
+˜µj ·
+�Ωij �Ω(0)
+ij
+˜µ2
+j
+� ˜
+Ni��
+= E[exp(M)],
+with
+M :=
+˜n
+�
+i=1
+˜Ni log
+� p
+�
+j=1
+˜µ−1
+j
+�Ωij �Ω(0)
+ij
+�
+.
+(C.37)
+Here, the third line follows from the moment generating function of a multinomial distri-
+bution. We plug in the expression of �Ωij in (C.23). By direct calculations,
+p
+�
+j=1
+˜µ−1
+j
+�Ωij �Ω(0)
+ij =
+m
+�
+j=1
+µj
+�
+1 + ωn ˜N−1
+i
+˜Nzibj
+��
+1 + ωn ˜N−1
+i
+˜Nz(0)
+i
+b(0)
+j
+�
+84
+
++
+m
+�
+j=1
+µj
+�
+1 − ωn ˜N−1
+i
+˜Nzibj
+��
+1 − ωn ˜N−1
+i
+˜Nz(0)
+i
+b(0)
+j
+�
+= 2∥µ∥1 + 2
+m
+�
+j=1
+µjω2
+n ˜N−2
+i
+˜N2ziz(0)
+i
+bjb(0)
+j
+= 1 + 2
+m
+�
+j=1
+µjω2
+n ˜N−2
+i
+˜N2ziz(0)
+i
+bjb(0)
+j .
+We plug it into M and notice that log(1 + t) ≤ t is always true. It follows that
+M ≤
+˜n
+�
+i=1
+˜Ni · 2
+m
+�
+j=1
+µjω2
+n
+˜N2
+˜N2
+i
+ziz(0)
+i
+bjb(0)
+j
+= 2 ˜Nω2
+n
+� ˜n
+�
+i=1
+˜N
+˜Ni
+ziz(0)
+i
+�� m
+�
+j=1
+µjbjb(0)
+j
+�
+=: M∗.
+(C.38)
+We combine (C.38) with (C.37). It is seen that to show (C.36), it suffices to show that
+E[exp(M∗)] = 1 + o(1).
+(C.39)
+We now show (C.39). Write M1 = �˜n
+i=1( ˜N−1
+i
+˜N)ziz(0)
+i
+and M2 = �p
+j=1 µjbjb(0)
+j .
+Recall that we condition on the event (C.32). By Hoeffding’s inequality, Bayes’s rule,
+and (C.32),
+P(|M1| > t) = P
+�
+|
+�
+i
+˜N
+˜Ni
+ziz(0)
+i
+≥ t
+���� |
+�
+i
+zi| ≤ 100
+√
+˜n, |
+�
+i
+z(0)
+i
+| ≤ 100
+√
+˜n
+�
+=
+P
+�
+| �
+i
+˜
+N
+˜
+Ni ziz(0)
+i
+| ≥ t
+�
+P(| �
+i zi| ≤ 100
+√
+˜n) P(| �
+i z(0)
+i
+| ≤ 100
+√
+˜n)
+≤ 4 · 2 exp
+�
+−
+t2
+8 �˜n
+i=1( ˜N−1
+i
+˜N)2
+�
+= 8 exp
+�
+− t2
+8˜n
+�
+.
+for all t > 0. In the last line, we have used the assumption of ˜Ni ≍ ˜N. By Hoeffding’s
+inequality again, we also have
+P(|M2| > t) ≤ 2 exp
+�
+−
+t2
+8 �p
+j=1 µ2
+j
+�
+= 2 exp
+�
+−
+t2
+8∥µ∥2
+�
+for all t > 0. Write s2
+˜n =
+√
+˜n ˜Nω2
+n∥µ∥. It follows that
+P(M∗ > t) = P
+�
+2 ˜Nω2
+nM1M2 > t
+�
+= P
+�
+M1M2 > t ·
+√
+˜n∥µ∥s−2
+˜n
+�
+≤ P
+�
+M1 >
+√
+t ·
+√
+˜ns−1
+˜n
+�
++ P
+�
+M2 >
+√
+t · ∥µ∥s−1
+˜n
+�
+≤ 8 exp
+�
+− t
+8s2
+˜n
+�
++ 2 exp
+�
+− t
+8s2
+˜n
+�
+≤ 4 exp(−c1t/s2
+˜n),
+(C.40)
+for some constant c1 > 0. Here, in the last line, we have used the assumption of ˜Ni ≍ ˜N.
+85
+
+Let f(x) and F(x) be the density and distribution function of M∗. Write ¯F(x) = 1 −
+F(x). Using integration by part, we have E[exp(M∗)] =
+� ∞
+0 exp(x)f(x)dx = − exp(x) ¯F(x)|∞
+0 +
+� ∞
+0 exp(x) ¯F(x)dx = 1 +
+� ∞
+0 exp(x) ¯F(x)dx, provided that the integral exists. As a result,
+when s˜n = o(1),
+E[exp(M∗)] − 1 =
+� ∞
+0
+exp(t) · P(M∗ > t)
+≤ 4
+� ∞
+0
+exp
+�
+−[c1s−2
+˜n − 1]t
+�
+dt
+≤ 4(c1s−1
+˜n − 1)−1 = 4s˜n/(c1 − s˜n).
+It implies E[exp(M∗)] = 1+o(1), which is exactly (C.39). This completes the proof. because
+s2
+˜n =
+√
+˜n ˜Nω2
+n∥µ∥ = n ¯N∥µ∥ω2
+n
+√
+K
+≍
+n ¯N∥µ∥ω2
+n
+��
+k∈K ∥µk∥2.
+C.5
+Proof of Theorem 3.6
+First we show that
+T/
+�
+Var(T) ⇒ N(0, 1),
+and
+(C.41)
+V/Var(T) → 1.
+(C.42)
+If (C.41) and (C.42) hold, then by mimicking the proof of Theorem 3.2, we see that ψ is
+asymptotically normal and the level-α DELVE test has asymptotic level α. We omit the
+details as they are quite similar.
+Recall the martingale decomposition of T described in Section B. Observe that, under
+our assumptions, Lemmas B.1–B.6 are valid. Moreover, by Lemmas A.8 and A.12
+Var(T) ≳ Θn2 + Θn3 + Θn4 ≳
+����
+m ¯
+M
+n ¯N + m ¯
+M η +
+n ¯N
+n ¯N + m ¯
+M θ
+����
+2
+.
+(C.43)
+Combining (C.43) with Lemmas B.1–B.6 and mimicking the argument in Section B.1 implies
+that T/
+√
+V ⇒ N(0, 1). Thus (C.41) is established.
+Moreover, (C.42) is a direct consequence of our assumptions and Proposition A.3. The
+claims of Theorem 3.6 regarding the null hypothesis follow.
+To prove the claims about the alternative hypothesis, it suffices to show
+T/
+�
+Var(T) → ∞,
+(C.44)
+V > 0 with high probability, and
+(C.45)
+V = OP(Var(T)).
+(C.46)
+Once these claims are established, we prove that ψ = T1V >0/
+√
+V → ∞ under the alterna-
+tive by mimicking the last step of the proof of Theorem 3.4 in Section C.3. We omit the
+details as they are very similar.
+86
+
+Note that (C.46) follows directly from our assumptions and Proposition A.4.
+As in the proof of Theorem 3.4 in Section C.3, to establish (C.44), it suffices to prove
+that
+ET = ρ2 ≫ Var(T).
+(C.47)
+Our main assumption under the alternative when K = 2 is
+∥η − θ∥2
+� 1
+n ¯
+N +
+1
+m ¯
+M
+�
+max{∥η∥, ∥θ∥} → ∞.
+(C.48)
+As shown in Section C.2, we have that
+Var(T) ≲ Θn = Θn1 +
+4
+�
+t=2
+Θnt.
+(C.49)
+Applying (C.17) to the first term and Lemma A.8 to the remaining terms, we have
+Var(T) ≲ max{∥η∥∞, ∥θ∥∞} · ρ2 +
+����
+m ¯
+M
+n ¯N + m ¯
+M η +
+n ¯N
+n ¯N + m ¯
+M θ
+����
+2
+≲ max{∥η∥, ∥θ∥} · ρ2 + max{∥η∥2, ∥θ∥2}
+(C.50)
+Next, note that
+ρ2 = n ¯N∥η − µ∥2 + m ¯
+M∥θ − µ∥2
+= n ¯N
+����η −
+�
+n ¯N
+n ¯N + m ¯
+M η +
+m ¯
+M
+n ¯N + m ¯
+M θ
+�����
+2
++ m ¯
+M
+����θ −
+�
+n ¯N
+n ¯N + m ¯
+M η +
+m ¯
+M
+n ¯N + m ¯
+M θ
+�����
+2
+= n ¯N ·
+�
+m ¯
+M
+n ¯N + m ¯
+M
+�2∥η − θ∥2 + m ¯
+M ·
+�
+n ¯N
+n ¯N + m ¯
+M
+�2∥η − θ∥2
+=
+n ¯Nm ¯
+M
+(n ¯N + m ¯
+M)∥η − θ∥2 =
+� 1
+n ¯N +
+1
+m ¯
+M
+�−1∥η − θ∥2.
+(C.51)
+By (C.48), (C.50), and (C.51), we have
+(ET)2
+Var(T) ≳
+ρ4
+max{∥η∥, ∥θ∥} · ρ2 + max{∥η∥2, ∥θ∥2}
+≳
+∥η − θ∥2
+( 1
+n ¯
+N +
+1
+m ¯
+M ) max{∥η∥, ∥θ∥} +
+�
+∥η − θ∥2
+( 1
+n ¯
+N +
+1
+m ¯
+M ) max{∥η∥, ∥θ∥}
+�2 → ∞,
+which proves (C.47) and thus (C.44).
+To prove (C.45), we mimick the Markov argument in (C.21) and use that under our
+assumptions, Var(V )/(EV )2 = o(1) . We omit the details as they are similar. Since we
+have established (C.44), (C.45), and (C.46), the proof is complete.
+87
+
+C.6
+Proof of Theorem 3.7
+Note that T/
+�
+Var(T) ⇒ N(0, 1) by our assumptions and Proposition B.1. In particular,
+using that n → ∞ and the monotonicity of the ℓp norms we have
+∥µ∥4
+4
+K∥µ∥4 = ∥µ∥4
+4
+n∥µ∥4 ≤ 1
+n · ∥µ∥4
+∥µ∥4 = 1
+n → 0.
+Moreover, V ∗/Var(T) → 1 in probability by Proposition A.5. It follows by Slutsky’s theo-
+rem that ψ∗ = T/
+√
+V ∗ ⇒ N(0, 1) and that the level-α DELVE test has an asymptotic level
+α.
+To conclude the proof, it suffices to show that ψ∗ → ∞ under the alternative. As in the
+proof of Theorem 3.4, this follows immediately if we can show
+T/
+�
+Var(T) → ∞,
+(C.52)
+V ∗ > 0 with high probability, and
+(C.53)
+V ∗ = OP(Var(T)).
+(C.54)
+Note that (C.52) follows from (C.14), and (C.54) is the content of Proposition A.6. Since
+our assumptions imply that EV ∗ ≫
+�
+Var(V ∗), (C.53) follows by a Markov argument as in
+(C.21).
+C.7
+Proof of Theorem 3.8
+We apply Theorem 3.2 to get the asymptotic null distribution. Since Ni = N and µ =
+p−11p, it is easy to see that Condition 3.2 is satisfied under our assumption of p = o(N2n).
+Therefore, by Theorem 3.2, ψ∗ → N(0, 1) under H0.
+We now show the asymptotic alternative distribution. By direct calculations and using
+�n
+i=1 δij = 0 and �p
+j=1 δij = 0, we have
+�
+i,j
+Ni(Ωij − µj)2 = nNβ2
+n
+p
+,
+�
+i,j
+Ni(Ωij − µj)2Ωij = nNβ2
+n
+p2
+,
+�
+i
+∥Ωi∥2 = n(1 + β2
+n)
+p
+.
+We apply Lemmas A.1-A.5 and plug in the above expressions. Let S = 1′
+pU2. It follows
+that
+T = nNβ2
+n
+p
++ S + OP
+�√
+nNβn
+p
++ 1
+√p
+�
+,
+where Var(S) = 2p−1n[1 + o(1)].
+(C.55)
+First, we plug in β2
+n = a√2p/(N√n). It gives p−1nNβ2
+n =
+�
+2n/p. Second, p−1√
+nNβn ≍
+(np)−1/4�
+n/p = o(
+�
+n/p). It follows that
+T = a
+�
+2n/p + S + oP
+��
+n/p
+�
+,
+where Var(S) = (2n/p)[1 + o(1)].
+(C.56)
+Recall the martingale decomposition S = �
+(ℓ,s) Eℓ,s where Eℓ,s is defined in (B.4).
+Observe that Lemmas B.4 and B.5 hold (even under the alternative).
+Define �Eℓ,s =
+88
+
+Eℓ,s/
+�
+Var(S).
+Using Var(S) ≳ n �
+i ∥Ωi∥2 and these lemmas, it is straightforward to
+verify that the following conditions hold:
+�
+(ℓ,s)
+Var
+� �Eℓ,s
+��F≺(ℓ,s)
+� P→ 1
+(C.57)
+�
+(ℓ,s)
+E �E4
+ℓ,s
+P→ 0.
+(C.58)
+As in Section B.1, the martingale CLT applies and we have
+S/
+�
+Var(S) ⇒ N(0, 1).
+By C.55,
+T/
+�
+Var(S)
+→
+N(a, 1).
+(C.59)
+By Lemma A.3 and (A.84),
+Var(S) = [1 + o(1)]Θn2 = [1 + o(1)]Var(T)
+By Proposition A.6, we have that V ∗/Var(T) → 1 in probability. As a result,
+V ∗/Var(S)
+→
+1,
+in probability.
+(C.60)
+We combine (C.59) and (C.60) to conclude that ψ = T/
+√
+V ∗ → N(a, 1).
+D
+Proofs of the corollaries for text analysis
+D.1
+Proof of Corollary 4.1
+Note that Corollary 4.1 follows immediately from the slightly more general result stated
+below.
+Corollary D.1. Consider Model (1.1) and suppose that Ω = µ1′
+n under the null hypothesis
+and that Ω satisfies (4.1) under the alternative hypothesis. Define ξ ∈ Rn by ξi = ¯N−1Ni
+and let �Ω = Ω[diag(ξ)]1/2. Let λ1, . . . , λM > 0 and �λ1, . . . , �λM > 0 denote the singular
+values of Ω and �Ω, respectively, arranged in decreasing order.We further assume that under
+the alternative hypothesis,
+¯N · �M
+k=2 �λ2
+k
+��M
+k=1 λ2
+k
+→ ∞.
+(D.1)
+For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level α and an asymptotic
+power 1. Moreover if Ni ≍ ¯N for all i, we may replace �M
+k=2 �λ2
+k with �M
+k=2 λ2
+k in the
+numerator of (D.1).
+89
+
+Proof of Corollary D.1. This is a special case of our testing problem with K = n. Moreover,
+µ = n−1Ωξ matches with the definition of µ in (1.2). Therefore, we can apply Theorem 3.7
+directly. It remains to verify that the condition
+¯N · �M
+k=2 �λ2
+k
+��M
+k=1 λ2
+k
+→ ∞
+(D.2)
+is sufficient to lead to the condition
+n ¯N∥µ∥2ω2
+n
+��
+i ∥Ωi∥2 → ∞.
+(D.3)
+If we show this then Theorem 3.7 applies directly. We first calculate ω2
+n. Recall ξi = Ni/ ¯N
+for 1 ≤ i ≤ n. Write
+�Ω = Ω[diag(ξ)]1/2,
+�ξ = [diag(ξ)]1/21n.
+For K = n, by (3.13), ω2
+n =
+1
+n ¯
+N∥µ∥2
+�n
+i=1 Ni∥Ωi − µ∥2. It follows that
+ω2
+n =
+1
+n∥µ∥2
+���(Ω − µ1′
+n)[diag(ξ)]1/2���
+2
+F =
+1
+n∥µ∥2
+���Ω − µ�ξ′��2
+F .
+(D.4)
+Recall that �
+λ1, . . . , �λM are the singular values of �Ω. We apply a well-known result in linear
+algebra [Horn and Johnson, 1985], namely Weyl’s inequality: For any rank-1 matrix ∆,
+∥�Ω − ∆∥2
+F ≥ �
+k̸=1 �λ2
+k. In (D.4), µ�ξ′ is a rank-1 matrix. It follows that
+���Ω − µ�ξ′��2
+F ≥
+M
+�
+k=2
+�λ2
+k.
+(D.5)
+Hence
+n ¯N∥µ∥2ω2
+n
+��
+i ∥Ωi∥2 ≥
+¯N · �M
+k=2 �λ2
+k
+∥Ω∥F
+=
+¯N · �M
+k=2 �λ2
+k
+��M
+k=1 λ2
+k
+,
+which implies (D.3) by our assumption. The first claim is proved.
+Next we prove the second claim. Observe that if Ni ≍ ¯N , then by Weyl’s inequality:
+ω2
+n =
+1
+∥µ∥2n ¯N
+�
+i
+Ni∥Ωi − µ|2 ≳
+1
+∥µ∥2
+�
+i
+∥Ωi − µ∥2
+=
+1
+∥µ∥2 ∥Ω − µ1′
+n∥2
+F ≥
+1
+∥µ∥2
+M
+�
+k=2
+λ2
+k.
+Thus
+n ¯N∥µ∥2ω2
+n
+��
+i ∥Ωi∥2 ≥
+¯N · �M
+k=2 λ2
+k
+∥Ω∥F
+=
+¯N · �M
+k=2 λ2
+k
+��M
+k=1 λ2
+k
+.
+We see that the assumption
+¯N · �M
+k=2 λ2
+k
+��M
+k=1 λ2
+k
+→ ∞
+(D.6)
+implies (D.3). The second claim is established and the proof is complete.
+90
+
+D.2
+Proof of Corollary 4.2
+Recall the construction of a simple null and simple (random) alternative model from Section
+C.4.2, specialized below to the case of K = n and Ni ≡ N:
+H0 :
+Ωi = ˜µ,
+1 ≤ i ≤ n.
+(D.7)
+H1 :
+Ωij =
+�
+µj
+�
+1 + ωnzibj
+�
+,
+if 1 ≤ j ≤ m
+˜µj
+�
+1 − ωnzibj−m
+�
+,
+if m + 1 ≤ j ≤ 2m
+(D.8)
+where b1, . . . , bm are i.i.d. Rademacher random variables and z1, . . . , zn are i.i.d Rademacher
+random variables conditioned to satisfy | �
+i zi| ≤ 100√n. Define
+˜b = (b1, . . . , bm, b1, . . . , bm)′.
+To derive the lower bound of Corollary 4.2, we assume without loss of generality that ωn is
+a sufficiently small absolute constant.
+We claim that H1 prescribes a topic model with M = 2 topics. To see this, under the
+alternative,
+Ωi =
+�
+µ ◦ (1p + ωn ˜b)
+if zi = 1
+µ ◦ (1p − ωn ˜b)
+if zi = −1.
+(D.9)
+Moreover, we showed in Section C.4.2 that Ωij ≥ 0 for all i, j and that ∥Ωij∥1 = 1. From
+(D.9), we see that Ω = AW where A ∈ Rp×2 and W ∈ R2×n are defined as follows:
+A:1 = µ ◦ (1p + ωn ˜b),
+A:2 = µ ◦ (1p − ωn ˜b)
+W:i =
+�
+(1, 0)′
+if zi = 1
+(0, 1)′
+if zi = −1.
+Moreover, under the null hypothesis, Ω clearly prescribes a topic model with K = 1.
+Therefore Ω follows the topic model (4.1). Moreover, since Ni ≡ N, we have Ω[diag(ξ)]1/2 =
+Ω.
+By Proposition C.1 specialized to our setting, we know that the χ2 distance between
+the null and alternative goes to zero if
+√nN∥µ∥ω2
+n → 0.
+Thus to prove Corollary 4.2 it suffices to show that
+N �M
+k≥2 λ2
+k
+��M
+k=1 λ2
+k
+=
+Nλ2
+2
+��M
+k=1 λ2
+k
+≳ √nN∥µ∥ω2
+n
+(D.10)
+Accordingly we study the second largest singular value of Ω. First we have some pre-
+liminary calculations. Let U = {i : zi = 1}, and let V = {i : zi = −1}. Define
+u = µ ◦ (1p + ωn ˜b),
+and
+91
+
+v = µ ◦ (1p − ωn ˜b).
+Observe that
+⟨u, v⟩ = ∥µ∥2 − ω2
+n∥µ ◦ ˜b∥2 = ∥µ∥2(1 − ω2
+n).
+Also, since ωn is a sufficiently small absolute constant,
+∥u∥2 = ∥µ∥2 + 2ωn⟨µ, µ ◦ ˜b⟩ + ω2
+n∥µ ◦ ˜b∥2 = (1 + ω2
+n)∥µ∥2 + 2ωn
+�
+j
+µ2
+j ˜bj ≳ ∥µ∥2,
+and
+∥v∥2 = ∥µ∥2 − 2ωn⟨µ, µ ◦ ˜b⟩ + ω2
+n∥µ ◦ ˜b∥2 = (1 + ω2
+n)∥µ∥2 − 2ωn
+�
+j
+µ2
+j ˜bj ≳ ∥µ∥2. (D.11)
+Again, since we assume that ωn is a sufficiently small absolute constant,
+δ2 :=
+⟨u, v⟩2
+∥u∥2∥v∥2 =
+∥µ∥4(1 − ω2
+n)2
+(1 + ω2n)2∥µ∥4 − 4ω2n⟨µ, µ ◦ b⟩2 ≤
+∥µ∥4(1 − ω2
+n)2
+(1 + ω2n)2∥µ∥4 − 4ω2n∥µ∥4
+=
+∥µ∥4(1 − ω2
+n)2
+∥µ∥4(1 + 2ω2n − 3ω4n) =
+(1 − ω2
+n)2
+1 + 2ω2n − 3ω4n
+(D.12)
+Note that
+∥au + bv∥2 = a2∥u∥2 + 2ab⟨u, v⟩ + b2∥v∥2 ≥ a2∥u∥2 + b2∥v∥2 − 2abδ∥u∥∥v∥
+≥ (1 − δ)
+�
+a2∥u∥2 + b2∥v∥2�
++ ∥au − bv∥2 ≥ (1 − δ)
+�
+a2∥u∥2 + b2∥v∥2�
+.
+By (D.12), we have for ωn sufficiently small that
+1 − δ ≥ 1 −
+1 − ω2
+n
+�
+1 + 2ω2n − 3ω4n
+=
+�
+1 + 2ω2n − 3ω4n − 1 + ω2
+n
+�
+1 + 2ω2n − 3ω4n
+≥
+ω2
+n
+�
+1 + 2ω2n − 3ω4n
+≳ ω2
+n.
+Thus
+∥au + bv∥2 ≥ ω2
+n(a2∥u∥2 + b2∥v∥2) ≳ ω2
+n∥µ∥2(a2 + b2)
+(D.13)
+Recall that if M is a rank k matrix, then
+λk(M) =
+sup
+y:∥y∥=1, y∈Ker(M)⊥ ∥My∥ =
+sup
+y:∥y∥=1, y∈Im(M′)
+∥My∥.
+(D.14)
+We have
+ΩΩ′ =
+�
+i∈U
+uu′ +
+�
+i∈V
+vv′ = |U|uu′ + |V |vv′.
+Let y ∈ Rn satisfy ∥y∥ = 1 and y = Ω′x for some x. We have
+Ωy = ΩΩ′x = |U|⟨u, x⟩u + |V |⟨v, x⟩v.
+92
+
+By the previous equation and (D.13),
+∥Ωy∥2 = ∥ΩΩ′x∥2 =
+����|U|⟨u, x⟩u + |V |⟨v, x⟩v
+����
+2
+≳ ω2
+n∥µ∥2�
+|U|2⟨u, x⟩2 + |V |2⟨v, x⟩2�
+.
+By our conditioning on z, we have min(|U|, |V |) ≳ n. Moreover
+1 = ∥y∥2 = ∥Ω′x∥2 = |U|⟨u, x⟩2 + |V |⟨v, x⟩2.
+Applying these facts and (D.14), we obtain
+λ2
+2 ≥ ∥Ωy∥2 = ∥ΩΩ′x∥2 ≳ ω2
+n∥µ∥2n
+�
+|U|⟨u, x⟩2 + |V |⟨v, x⟩2�
+= ω2
+n∥µ∥2n.
+Next,
+M
+�
+k=1
+λ2
+k = ∥Ω∥2
+F =
+�
+i∈U
+∥u∥2 +
+�
+i∈V
+∥v∥2 = |U| · ∥u∥2 + |V | · ∥v∥2 ≍ n∥µ∥2
+(D.15)
+We conclude that
+N �M
+k≥2 λ2
+k
+��M
+k=1 λ2
+k
+=
+Nλ2
+2
+��M
+k=1 λ2
+k
+≳ N · ω2
+n∥µ∥2n
+√n∥µ∥
+= √nN∥µ∥ω2
+n
+which establishes (D.10). The proof is complete.
+D.3
+Proof of Corollary 4.3
+This is a special case of our testing problem with K = 2, we can apply Theorem 3.6 directly.
+It remains to verify that the condition
+β2
+n · (∥ηS∥1 + ∥θS∥1)
+� 1
+n ¯
+N +
+1
+m ¯
+M
+�
+max{∥η∥, ∥θ∥} → ∞
+(D.16)
+is sufficient to yield the condition (3.11) in Theorem 3.6. This is done by calculating ∥η−θ∥2
+directly. By our sparse model (4.4), for j ∈ S, |√ηj −
+�
+θj| ≥ βn. It follows that for j ∈ S,
+|ηj − θj|2 = (√ηj +
+�
+θj)2(√ηj −
+�
+θj)2 ≥ β2
+n(√ηj +
+�
+θj)2 ≥ β2
+n(ηj + θj).
+It follows that
+∥η − θ∥2 ≥ β2
+n
+�
+j∈S
+(ηj + θj) ≥ β2
+n
+�
+∥ηS∥1 + ∥θS∥1
+�
+.
+(D.17)
+We plug it into (3.11) and see immediately that (D.16) implies this condition. The claim
+follows directly from Theorem 3.6.
+93
+
+E
+A modification of DELVE for finite p
+Below we write out the variance of the terms of the raw DELVE statistic under the null,
+using the proofs of Lemmas A.3–A.5.
+Var(1′
+pU2) = 2
+K
+�
+k=1
+�
+i∈Sk
+�
+1≤r<s≤Ni
+(
+1
+nk ¯Nk
+−
+1
+n ¯N )2
+N2
+i
+(Ni − 1)2
+�
+∥Ωi∥2 − 2∥Ωi∥3
+3 + ∥Ωi∥4�
+(E.1)
+Var(1′
+pU3) =
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+NiNm
+��
+j
+ΩijΩmj − 2
+�
+j
+Ω2
+ijΩ2
+mj +
+�
+j,j′
+ΩijΩij′ΩmjΩmj′
+�
+Var(1′
+pU4) = 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i̸=m
+(
+1
+nk ¯Nk
+−
+1
+n ¯N )2NiNm
+��
+j
+ΩijΩmj − 2
+�
+j
+Ω2
+ijΩ2
+mj +
+�
+j,j′
+ΩijΩij′ΩmjΩmj′
+�
+.
+In this section we develop an unbiased estimator for each term above, which leads to an
+unbiased estimator of Var(T) by taking their sum. We require some preliminary results
+proved later in this section. Recall that Lemma E.2 was established in the proof of Lemma
+A.1.
+Lemma E.1. If j ̸= j′, an unbiased estimator of ΩijΩij′ is
+�
+ΩijΩij′ :=
+XijXij′
+Ni(Ni − 1)
+Lemma E.2. An unbiased estimator of Ω2
+ij is
+�
+Ω2
+ij :=
+X2
+ij − Xij
+Ni(Ni − 1).
+(E.2)
+Lemma E.3. If j ̸= j′, an unbiased estimator for Ω2
+ijΩ2
+ij′ is
+�
+Ω2
+ijΩ2
+ij′ =
+(X2
+ij − Xij)(X2
+ij′ − Xij′)
+Ni(Ni − 1)(Ni − 2)(Ni − 3)
+Lemma E.4. An unbiased estimator of Ω3
+ij is
+�
+Ω3
+ij :=
+X3
+ij − 3X2
+ij + 2Xij
+Ni(Ni − 1)(Ni − 2).
+(E.3)
+Lemma E.5. An unbiased estimator of Ω4
+ij is
+�
+Ω4
+ij :=
+X4
+ij − 3X3
+ij − X2
+ij + 3Xij
+Ni(Ni − 1)(Ni − 2)(Ni − 3).
+(E.4)
+Define
+�
+∥Ωi∥2 :=
+�
+j
+�
+Ω2
+ij
+94
+
+�
+∥Ωi∥3
+3 :=
+�
+j
+�
+Ω3
+ij
+�
+∥Ωi∥4 :=
+�
+j
+�
+Ω4
+ij +
+�
+j̸=j′
+�
+Ω2
+ijΩ2
+ij′.
+(E.5)
+Using Lemmas E.1–E.5 and (E.5), we define an unbiased estimator for each term of (E.1).
+Let �
+Ωij = Xij/Ni and define
+�
+Var(1′pU2) = 2
+K
+�
+k=1
+�
+i∈Sk
+�
+1≤r<s≤Ni
+(
+1
+nk ¯Nk
+−
+1
+n ¯N )2
+N2
+i
+(Ni − 1)2
+��
+∥Ωi∥2 − 2�
+∥Ωi∥3
+3 + �
+∥Ωi∥4�
+(E.6)
+�
+Var(1′pU3) =
+2
+n2 ¯N2
+�
+k̸=ℓ
+�
+i∈Sk
+�
+m∈Sℓ
+NiNm
+��
+j
+�
+Ωij �
+Ωmj − 2
+�
+j
+�
+Ω2
+ij �
+Ω2
+mj +
+�
+j,j′
+�
+ΩijΩij′
+�
+ΩmjΩmj′
+�
+�
+Var(1′pU4) = 2
+K
+�
+k=1
+�
+i∈Sk,m∈Sk
+i̸=m
+(
+1
+nk ¯Nk
+−
+1
+n ¯N )2NiNm
+��
+j
+�
+Ωij �
+Ωmj − 2
+�
+j
+�
+Ω2
+ij �
+Ω2
+mj +
+�
+j,j′
+�
+ΩijΩij′
+�
+ΩmjΩmj′
+�
+.
+Define
+�V =
+�
+Var(1′pU2) +
+�
+Var(1′pU3) +
+�
+Var(1′pU4).
+(E.7)
+We define exact DELVE as ˜ψ = T/�V 1/2.
+Combining our results above, we obtain the
+following.
+Proposition E.1. Consider the statistic �V defined in (E.7). Under the null hypothesis, �V
+is an unbiased estimator for Var(T).
+With this result in hand, it is possible to derive consistency of �V as an estimator of
+Var(T) under certain regularity conditions. We omit the details.
+E.1
+Proof of Lemma E.1
+Recall that Bijr is the Bernoulli random variable Bijr = Zijr + Ωij and satisfies Xijr =
+�Ni
+r=1 Bijr. Observe that
+XijXij′ =
+�
+r,s
+BijrBij′s =
+�
+r
+BijrBij′r +
+�
+r̸=s
+BijrBij′s = 0 +
+�
+r̸=s
+BijrBij′s
+Thus
+EXijXij′ = Ni(Ni − 1)ΩijΩij′,
+and we obtain
+�
+ΩijΩij′ =
+XijXij′
+Ni(Ni − 1)
+is an unbiased estimator for ΩijΩij′, as desired.
+95
+
+E.2
+Proof of Lemma E.3
+Note that
+X2
+ijX2
+ij′ =
+� �
+r
+Bijr +
+�
+r̸=s
+BijrBijs
+�� �
+r
+Bij′r +
+�
+r̸=s
+Bij′rBij′s
+�
+=
+�
+r
+BijrBij′r +
+�
+r1̸=r2
+BijrBij′s +
+�
+r1̸=s
+Bijr1Bijs
+�
+r2
+Bij′r2 +
+�
+r1̸=s
+Bij′r1Bij′s
+�
+r2
+Bijr2
++
+� �
+r̸=s
+BijrBijs
+�� �
+r̸=s
+Bij′rBij′s
+�
+=
+�
+r1̸=r2
+BijrBij′s +
+�
+r1̸=s
+Bijr1Bijs
+�
+r2
+Bij′r2 +
+�
+r1̸=s
+Bij′r1Bij′s
+�
+r2
+Bijr2
++
+� �
+r̸=s
+BijrBijs
+�� �
+r̸=s
+Bij′rBij′s
+�
+Since BijrBij′r = 0, note that
+(X2
+ij − Xij)(X2
+ij′ − Xij′) =
+�
+r1̸=s1
+�
+r2̸=s2
+Bijr1Bijs1Bij′r2Bij′s2
+=
+�
+r1,s1,r2,s2 dist.
+Bijr1Bijs1Bij′r2Bij′s2.
+Thus
+E(X2
+ij − Xij)(X2
+ij′ − Xij′) =
+�
+r1,s1,r2,s2 dist.
+E
+�
+Bijr1Bijs1Bij′r2Bij′s2
+�
+= Ni(Ni − 1)(Ni − 2)(Ni − 3) · Ω2
+ijΩ2
+ij′.
+It follows that
+�
+Ω2
+ijΩ2
+ij′ =
+(X2
+ij − Xij)(X2
+ij′ − Xij′)
+Ni(Ni − 1)(Ni − 2)(Ni − 3)
+is an unbiased estimator for Ω2
+ijΩ2
+ij′.
+E.3
+Proof of Lemma E.4
+Recall that Bijr is the Bernoulli random variable Bijr = Zijr + Ωij and satisfies Xijr =
+�Ni
+r=1 Bijr. Observe that
+X3
+ij =
+�
+r
+Bijr + 3
+�
+r1̸=r2
+Bijr1Bijr2 +
+�
+r1̸=r2̸=r3
+Bijr1Bijr2Bijr3.
+Thus
+EX3
+ij = NiΩij + 3Ni(Ni − 1)Ω2
+ij + Ni(Ni − 1)(Ni − 2)Ω3
+ij.
+96
+
+Unbiased estimators for Ωij and Ω2
+ij are
+Xij
+Ni
+X2
+ij
+N2
+i
+− Xij(Ni − Xij)
+N2
+i (Ni − 1)
+=
+1
+Ni(Ni − 1)
+�
+X2
+ij − Xij
+�
+,
+respectively. Hence
+X3
+ij − Xij − 3(X2
+ij − Xij) = X3
+ij − 3X2
+ij + 2Xij
+is an unbiased estimator for Ni(Ni − 1)(Ni − 2)Ω3
+ij, as desired.
+E.4
+Proof of Lemma E.5
+Observe that
+X4
+ij =
+�
+r
+B4
+ijr + 4
+�
+r1̸=r2
+B3
+ijr1Bijr2 + 6
+�
+r1̸=r2
+B2
+ijr1B2
+ijr2
++ 3
+�
+r1̸=r2̸=r3
+B2
+ijr1Bijr2Bijr3 +
+�
+r1̸=r2̸=r3̸=r4
+Bijr1Bijr2Bijr3Bijr4
+=
+�
+r
+Bijr + 10
+�
+r1̸=r2
+Bijr1Bijr2 + 3
+�
+r1̸=r2̸=r3
+Bijr1Bijr2Bijr3
++
+�
+r1̸=r2̸=r3̸=r4
+Bijr1Bijr2Bijr3Bijr4.
+Thus
+EX4
+ij = NiΩij + 10Ni(Ni − 1)Ω2
+ij + 3Ni(Ni − 1)(Ni − 2)Ω3
+ij
++ Ni(Ni − 1)(Ni − 2)(Ni − 3)Ω4
+ij.
+Plugging in unbiased estimators for the first three terms, we have
+X4
+ij − Xij − 10(X2
+ij − Xij) − 3(X3
+ij − 3X2
+ij + 2Xij) = X4
+ij − 3X3
+ij − X2
+ij + 3Xij
+is an unbiased estimator for Ni(Ni − 1)(Ni − 2)(Ni − 3), as desired.
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+
diff --git a/iNAzT4oBgHgl3EQfbPw2/content/tmp_files/load_file.txt b/iNAzT4oBgHgl3EQfbPw2/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..f96d8e9e9ec35a61e1a6b24f79ee979170b6affe
--- /dev/null
+++ b/iNAzT4oBgHgl3EQfbPw2/content/tmp_files/load_file.txt
@@ -0,0 +1,3101 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf,len=3100
+page_content='Testing High-dimensional Multinomials with Applications to  Text Analysis  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Tony Cai∗, Zheng Tracy Ke†, and Paxton Turner†  Abstract  Motivated by applications in text mining and discrete distribution inference, we  investigate the testing for equality of probability mass functions of K groups of high-  dimensional multinomial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A test statistic, which is shown to have an  asymptotic standard normal distribution under the null, is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The optimal de-  tection boundary is established, and the proposed test is shown to achieve this optimal  detection boundary across the entire parameter space of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The proposed method  is demonstrated in simulation studies and applied to analyze two real-world datasets  to examine variation among consumer reviews of Amazon movies and diversity of sta-  tistical paper abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Keywords: authorship attribution, closeness testing, consumer reviews, martin-  gale central limit theorem, minimax optimality, topic model  1  Introduction  Statistical inference for multinomial data has garnered considerable recent interest [Di-  akonikolas and Kane, 2016, Balakrishnan and Wasserman, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' One important appli-  cation is in text mining, as it is common to model the word counts in a text document  by a multinomial distribution [Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2003].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We consider a specific example in mar-  keting, where the study of online customer ratings and reviews has become a trending  topic [Chevalier and Mayzlin, 2006, Zhu and Zhang, 2010, Leung and Yang, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Cus-  tomer reviews are a good proxy to the overall word of mouth (WOM) and can significantly  influence customers’ decisions [Zhu and Zhang, 2010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Many research works aim to under-  stand the patterns in online reviews and their impacts on sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Classical studies only use  the numerical ratings but ignore the rich text reviews because of their unstructured na-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' More recent works have revealed the importance of analyzing text reviews [Chevalier  and Mayzlin, 2006], especially for hedonic products such as books, movies, and hotels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' A  question of great interest is to detect the heterogeneity in reviewers’ response styles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For  example, Leung and Yang [2020] discovered that younger travelers, women, and travelers  with less review expertise tend to give more positive reviews and that guests staying in  ∗University of Pennsylvania  †Harvard University  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='01381v1  [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='ME]  3 Jan 2023  high-class hotels tend to have more extreme response styles than those staying in low-class  hotels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Knowing such differences will offer valuable insights for hotel managers and online  rating/review sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The aforementioned heterogeneity detection can be cast as a hypothesis test on multi-  nomial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose reviews are written on a vocabulary of p distinct words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let Xi ∈ Rp  denote the word counts in review i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We model that  Xi ∼ Multinomial(Ni, Ωi),  1 ≤ i ≤ n,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  where Ni is the total length of review i and Ωi ∈ Rp is a probability mass function (PMF)  containing the population word frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' These reviews are divided into K groups by  reviewer characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', age, gender, new/returning customer), product characteristics  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', high-class versus low-class hotels), and numeric ratings (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', from 1 star to 5 stars),  where K can be presumably large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We view Ωi as representing the ‘true response’ of review  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The “average response” of a group k is defined by a weighted average of the PMFs:  µk = (nk ¯Nk)−1 �  i∈Sk  NiΩi,  1 ≤ k ≤ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  Here Sk ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , n} is the index set of group k, nk = |Sk| is the total number of reviews  in group k, and ¯Nk = n−1  k  �  i∈Sk Ni is the average length of reviews in group k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We would  like to test  H0 :  µ1 = µ2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' = µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  When the null hypothesis is rejected, it means there exist statistically significant differences  among the group-wise “average responses”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We call (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3) the “K-sample testing for equality of average PMFs in multinomials”  or “K-sample testing for multinomials” for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Interestingly, as K varies, this problem  includes several well-defined problems in text mining and discrete distribution inference as  special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Global testing for topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Topic modeling [Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2003] is a popular text  mining tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In a topic model, each Ωi in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) is a convex combination of M topic  vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Before fitting a topic model to a corpus, it is often desirable to determine  if the corpus indeed contains multiple topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This boils down to the global testing  problem, which tests M = 1 versus M > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the null hypothesis, Ωi’s are equal  to each other, and in the alternative hypothesis, Ωi’s can take continuous values in  a high-dimensional simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This is a special case of our problem with K = n and  nk = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Authorship attribution [Mosteller and Wallace, 1963, Kipnis, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In these appli-  cations, the goal is to determine the unknown authorship of an article from other  articles with known authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' A famous example [Mosteller and Wallace, 2012] is to  determine the actual authors of a few Federalist Papers written by three authors but  published under a single pseudonym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It can be formulated [Mosteller and Wallace,  1963, Kipnis, 2022] as testing the equality of population word frequencies between the  2  article of interest and the corpus from a known author, a special case of our problem  with K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Closeness between discrete distributions [Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2014, Bhattacharya and Valiant,  2015, Balakrishnan and Wasserman, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  There has been a surge of interest in  discrete distribution inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Closeness testing is one of most studied problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The data from two discrete distributions are summarized in two multinomial vectors  Multinomial(N1, µ) and Multinomial(N2, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The goal is to test µ = θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is a special  case of our testing problem with K = 2 and n1 = n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In this paper, we provide a unified solution to all the aforementioned problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The  key to our methodology is a flexible statistic called DELVE (DE-biased and Length-assisted  Variability Estimator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It provides a general similarity measure for comparing groups of  discrete distributions such as count vectors associated with text corpora.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Similarity mea-  sures (such as the classical cosine similarity, log-likelihood ratio statistic, and others) are  fundamental in text mining and have been applied to problems in distribution testing [Kim  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2022], computational linguistics [Gomaa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2013], econometrics [Hansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=',  2018], and computational biology [Kolodziejczyk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2015].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Our method is a new and  flexible similarity measure that is potentially useful in these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We emphasize that our setting does not require that the Xi’s in the same group are  drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the null hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3), the group-wise means  are equal, but the Ωi’s within each group can still be different from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As a result,  the null hypothesis is composite and designing a proper test statistic is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Our results and contributions  The dimensionality of the testing problem is captured by (n, p, K) and ¯N := n−1 �n  i=1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We are interested in a high-dimensional setting where  n ¯N → ∞,  p → ∞,  and  n2 ¯N2/(Kp) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  In most places of this paper, we use a subscript n to indicate asymptotics, but our method  and theory do apply to the case where n is finite and ¯N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In text applications, n ¯N is  the total count of words in the corpus, and a large n ¯N means either there are sufficiently  many documents, or the documents are sufficiently long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Given that n ¯N → ∞, we further  allow (p, K) to grow with n at a speed such that Kp ≪ n2 ¯N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In particular, our settings  allow K to range from 2 to n, so as to cover all the application examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We propose a test that enjoys the following properties:  (a) Parameter-free null distribution: We show that the test statistic ψ → N(0, 1) under  H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Even under the null hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3), the model contains a large number of free  parameters because the null hypothesis is only about the equality of “average” PMFs  but still allows (Ni, Ωi) to differ within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As an appealing property, the  null distribution of ψ does not depend on these individual multinomial parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  hence, we can always conveniently obtain the asymptotic p-value for our proposed  test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3  (b) Minimax optimal detection boundary: We define a quantity ωn := ωn(µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , µK)  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) that measures the difference among the K group-wise mean PMF’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It sat-  isfies that ωn = 0 if and only if the null hypothesis holds, and it has been properly  normalized so that ωn is bounded under the alternative hypothesis (provided some  mild regularity conditions hold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We show that the proposed test has an asymptotic  full power if ω4  nn2 ¯N2/(Kp) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We also provide a matching lower bound by showing  that the null hypothesis and the alternative hypothesis are asymptotically indistin-  guishable if ω4  nn2 ¯N2/(Kp) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore, the proposed test is minimax optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Furthermore, in the boundary case where ω4  nn2 ¯N2/(Kp) → c0 for a constant c0 > 0,  for some special settings, we show that ψ → N(0, 1) under H0, and ψ → N(c1, 1),  under H1, with the constant c1 being an explicit function of c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To the best of our knowledge, this testing problem for a general K has not been studied  before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The existing works primarily focused on closeness testing and authorship attribution  (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), which are special cases with K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In comparison, our test is applicable  to any value of K, offering a unified solution to multiple applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Even for K = 2,  the existing works do not provide a test statistic that has a tractable null distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  They determined the rejection region and calculated p-values using either a (conservative)  large-deviation bound or a permutation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Our test is the first one equipped with a  tractable null distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Our results about the optimal detection boundary for a general  K are also new to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By varying K in our theory, we obtain the optimal detec-  tion boundary for different sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For some of them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', global testing for topic  models, authorship attribution with moderate sparsity), the optimal detection boundary  was not known before;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' hence, our results help advance the understanding of the statistical  limits of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Related literature  First, we make a connection to discrete distribution inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let X ∼ Multinomial(N, Ω)  represent a size-N sample from a discrete distribution with p categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The one-sample  closeness testing aims to test H0 : Ω = µ, for a given PMF µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Existing works focus  on finding the minimum separation condition in terms of the ℓ1-norm or ℓ2-norm of Ω −  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Balakrishnan and Wasserman [2019] derived the minimum ℓ1-separation condition and  proposed a truncated chi-square test to achieve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Valiant and Valiant [2017] studied the  “local critical radius”, a local separation condition that depends on the “effective sparsity”  of µ, and they proposed a “2/3rd + tail” test to achieve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the two-sample closeness  testing problem, given X1 ∼ Multinomial(N1, Ω1) and X2 ∼ Multinomial(N2, Ω2), it aims  to test H0 : Ω1 = Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Again, this literature focuses on finding the minimum separation  condition in terms of the ℓ1-norm or ℓ2-norm of Ω1 − Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' When N1 = N2, Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  [2014] derived the minimum ℓ1-separation condition and proposed a weighted chi-square  test to attain it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Bhattacharya and Valiant [2015] extended their results to the unbalanced  case where N1 ̸= N2, assuming ∥Ω1 − Ω2∥1 ≥ p−1/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This assumption was later removed  by Diakonikolas and Kane [2016], who established the minimum ℓ1-separation condition in  full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' [2022] proposed a two-sample kernel U-statistic and showed that  4  it attains the minimum ℓ2-separation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Since the two-sample closeness testing is a special case of our problem with K = 2 and  n1 = n2 = 1, our test is directly applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' An appealing property of our test is its tractable  asymptotic null distribution of N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In contrast, for the chi-square statistic in Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  [2014] or the U-statistic in [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2022], the rejection region is determined by either an  upper bound from concentration inequalities or a permutation procedure, which may lead  to a conservative threshold or need additional computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Regarding the testing  power, we show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3 that our test achieves the minimum ℓ2-separation condition,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', our method is an optimal “ℓ2 testor.” Our test can also be turned into an optimal “ℓ1  testor” (a test that achieves the minimum ℓ1-separation condition) by re-weighting terms  in the test statistic (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next, we make a connection to text mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In this literature, a multinomial vector  X ∼ Multinomial(N, Ω) represents the word counts for a document of length N written  with a dictionary containing p words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In a topic model, each Ωi is a convex combination of  M “topic vectors”: Ωi = �M  k=1 wi(k)Ak, where each Ak ∈ Rp is a PMF and the combination  coefficient vector wi ∈ RK is called the “topic weight” vector for document i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Given a  collection of documents X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , Xn, the global testing problem aims to test M = 1  versus M > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Interestingly, the optimal detection boundary for this problem has never  been rigorously studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As we have explained, this problem a special case of our testing  problem with K = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Our results (a) provide a test statistic that has a tractable null  distribution and (b) reveal that the optimal detection boundary is ω2  n ≍ (√n ¯N)−1√p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Both (a) and (b) are new results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' When comparing our results with those about estimation  of Ak’s [Ke and Wang, 2022], it suggests that global testing requires a strictly lower signal  strength than topic estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For authorship attribution, Kipnis [2022] treats the corpus from a known author as a  single document and tests the null hypothesis that this combined document and a new  document have the same population word frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is a two-sample closeness testing  problem, except that sparsity is imposed on the difference of two PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Kipnis [2022]  proposed a test which applies an “exact binomial test” to obtain a p-value for each word  and combines these p-values using Higher Criticism [Donoho and Jin, 2004].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Donoho and  Kipnis [2022] analyzed this test when the number of “useful words” is o(√p), and they  derived a sharp phase diagram (a related one-sample setting was studied in Arias-Castro  and Wang [2015]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, we show that our test is applicable to this problem and  has some nice properties: (a) tractable null distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' (b) allows for s ≥ c√p, where s  is the number of useful words;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' and (c) does not require documents from the known author  to have identical population word frequencies, making the setting more realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' On the  other hand, when s = o(√p), our test is less powerful than the one in Kipnis [2022], Donoho  and Kipnis [2022], as our test does not utilize sparsity explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We can further improve  our test in this regime by modifying the DELVE statistic to incorporate sparsity (see the  remark in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Organization  The rest of this paper is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Section 2, we introduce the test statistic and  explain the rationale behind it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We then present in Section 3 the main theoretical results,  including the asymptotic null distribution, power analysis, a matching lower bound, the  study of two special cases (K = n and K = 2), and a discussion of the contiguity regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Section 4 applies our results to text mining and discrete distribution testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Simulations  are in Section 5 and real data analysis is in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The paper is concluded with a  discussion in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' All proofs are in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  2  The DELVE Test  Recall that Xi ∼ Multinomial(Ni, Ωi) for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' There is a known partition {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , n} =  ∪K  k=1Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write nk = |Sk|, ¯Nk = n−1  k  �  i∈Sk Ni, and ¯N = n−1 �n  i=1 Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), we have de-  fined the group-wise mean PMF µk = (nk ¯Nk)−1 �  i∈Sk NiΩi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We further define the overall  mean PMF µ ∈ Rp by  µ :=  1  n ¯N  K  �  k=1  nk ¯Nkµk =  1  n ¯N  n  �  i=1  NiΩi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  We introduce a quantity ρ2 = ρ2(µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , µK) by  ρ2 :=  K  �  k=1  nk ¯Nk∥µk − µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  This quantity measures the variations across K group-wise mean PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is true that the  null hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3) holds if and only if ρ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Inspired by this observation, we hope to  construct an unbiased estimator of ρ2 and develop it to a test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We can easily obtain the minimum variance unbiased estimators of µk and µ:  ˆµk =  1  nk ¯Nk  �  i∈Sk  Xi,  and  ˆµ =  1  n ¯N  K  �  k=1  nk ¯Nkˆµk =  1  n ¯N  n  �  i=1  Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  For each 1 ≤ j ≤ p, let µkj, µj, ˆµkj and ˆµj represent the jth entry of µk, µ, ˆµk and ˆµ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' A naive estimator of ρ2 is  �T =  p  �  j=1  �Tj,  where  �Tj =  K  �  k=1  nk ¯Nk(ˆµkj − ˆµj)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  This estimator is biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 of the appendix , we show that E[ �Tj] = �K  k=1  �  nk ¯Nk(µkj−  µj)2 +  �  1  nk ¯  Nk −  1  n ¯  N  � �  i∈Sk NiΩij(1 − Ωij)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It motivates us to debias �Tj by using an unbi-  ased estimate of Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By elementary properties of the multinomial distribution,  E[Xij(Ni − Xij)] = Ni(Ni − 1)Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We thereby use  1  Ni(Ni−1)Xij(Ni − Xij) to  estimate Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This gives rise to an unbiased estimator of ρ2 as  T =  p  �  j=1  Tj,  Tj =  K  �  k=1  �  nk ¯Nk(ˆµkj − ˆµj)2 −  �  1  nk ¯Nk  −  1  n ¯N  � �  i∈Sk  Xij(Ni − Xij)  Ni − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  6  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under Models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), the estimator in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) satisfies that E[T] = ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To use T for hypothesis testing, we need a proper standardization of this statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In  Sections A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 of the appendix , we study V(T), the variance of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under mild regularity  conditions, it can be shown that V(T) = Θn · [1 + o(1)], where  Θn := 4  K  �  k=1  p  �  j=1  nk ¯Nk(µkj − µj)2µkj + 2  K  �  k=1  �  i∈Sk  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2  N3  i  Ni − 1Ω2  ij  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  +  2  n2 ¯N2  �  1≤k̸=ℓ≤K  �  i∈Sk  �  m∈Sℓ  p  �  j=1  NiNmΩijΩmj + 2  K  �  k=1  �  i∈Sk,m∈Sk,  i̸=m  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In Θn, the first term vanishes under the null, so it suffices to estimate the other three terms  in Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By properties of multinomial distributions, E[XijXmj] = NiNmΩijΩmj, E[X2  ij] =  N2  i Ω2  ij + NiΩij(1 − Ωij), and E[Xij(Ni − Xij)] = Ni(Ni − 1)Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It inspires us to  estimate ΩijΩmj by XijXmj  NiNm  and estimate Ω2  ij by  X2  ij  N2  i − Xij(Ni−Xij)  N2  i (Ni−1)  =  X2  ij−Xij  Ni(Ni−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  V = 2  K  �  k=1  �  i∈Sk  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2 X2  ij − Xij  Ni(Ni − 1) +  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  p  �  j=1  XijXmj  + 2  K  �  k=1  �  i∈Sk,m∈Sk,  i̸=m  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2  XijXmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  The test statistic we propose is as follows (in the rate event V < 0, we simply set ψ = 0):  ψ = T/  √  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8)  We call ψ the DEbiased and Length-adjusted Variability Estimator (DELVE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1,  we show that under mild regularity conditions, ψ → N(0, 1) under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For  any fixed α ∈ (0, 1), the asymptotic level-α DELVE test rejects H0 if  ψ > zα,  where zα is the (1 − α)-quantile of N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  The special cases of K = n and K = 2  As seen in Section 1, the application examples of K = n and K = 2 are particularly  intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In these cases, we give more explicit expressions of our test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  When K = n, we have Sk = {i} and ˆµkj = N−1  i  Xij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The null hypothesis becomes  H0 : Ω1 = Ω2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' = Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The statistic in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) reduces to  T =  p  �  j=1  n  �  i=1  �(Xij − Niˆµj)2  Ni  −  �  1 − Ni  n ¯N  �Xij(Ni − Xij)  Ni(Ni − 1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)  Moreover, in the variance estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7), the last term is exactly zero, and it can be shown  that the third term is negligible compared to the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We thereby consider a simpler  7  variance estimator by only retaining the first term in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7):  V ∗ = 2  n  �  i=1  p  �  j=1  � 1  Ni  −  1  n ¯N  �2 X2  ij − Xij  Ni(Ni − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11)  The simplified DELVE test statistic is ψ∗ = T/  √  V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  When K = 2, we observe two collections of multinomial vectors, denoted by {Xi}1≤i≤n  and {Gi}1≤i≤m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We assume for 1 ≤ i ≤ n and 1 ≤ j ≤ m,  Xi ∼ Multinomial(Ni, Ωi),  Gj ∼ Multinomial(Mj, Γj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12)  Write ¯N = n−1 �n  i=1 Ni and ¯  M = m−1 �m  i=1 Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The null hypothesis becomes  H0 :  η = θ,  where η =  1  n ¯N  n  �  i=1  NiΩi, and θ =  1  m ¯  M  m  �  i=1  MiΓi,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13)  where θ and η are the two group-wise mean PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We estimate them by ˆη = (n ¯N)−1 �n  i=1 Xi  and ˆθ = (m ¯  M)−1 �m  i=1 Gi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The statistic in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) has an equivalent form as follows:  T =  n ¯Nm ¯  M  n ¯N + m ¯  M  �  ∥ˆη − ˆθ∥2 −  n  �  i=1  p  �  j=1  Xij(Ni − Xij)  n2 ¯N2(Ni − 1) −  m  �  i=1  p  �  j=1  Gij(Mi − Gij)  m2 ¯  M2(Mi − 1)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14)  The variance estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7) has an equivalent form as follows:  V =  4 �n  i=1  �m  i′=1  �p  j=1 XijGi′j  (n ¯N + m ¯  M)2  +  2m2 ¯  M2� �n  i=1  X2  ij−Xij  Ni(Ni−1) + �  1≤i̸=i′≤n XijXi′j  �  n2 ¯N2(n ¯N + m ¯  M)2  +  2n2 ¯N2� �m  i=1  G2  ij−Gij  Mi(Mi−1) + �  1≤i̸=i′≤m GijGi′j  �  m2 ¯  M2(n ¯N + m ¯  M)2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15)  The DELVE test statistic is ψ = T/  √  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  A variant: DELVE+  We introduce a variant of the DELVE test statistic to better suit real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let ˆµ, T and  V be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  ψ+ = T/  √  V +,  where  V + = V ·  �  1 + ∥ˆµ∥2T/  √  V  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16)  We call (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16) the DELVE+ test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In theory, this modification has little effect on  the key properties of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To see this, we note that ∥ˆµ∥2 = oP(1) in high-dimensional  settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose T/  √  V → N(0, 1) under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Since ∥ˆµ∥2 → 0, it is seen immediately  that V +/V → 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' hence, the asymptotic normality also holds for ψ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose T/  √  V → ∞  under the alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that V + ≤ 2 max{V, ∥ˆµ∥2 · T  √  V } and ψ+ ≥  1  √  2 min{T/  √  V , ∥ˆµ∥−1  2 (T/  √  V )1/2} → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have proved the following lemma:  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As n ¯N → ∞, suppose ∥ˆµ∥2 → 0 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under H0, if T/  √  V →  N(0, 1), then T/  √  V + → N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under H1, if T/  √  V → ∞, then T/  √  V + → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  8  In practice, this modification avoids extremely small p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In some real datasets, V is  very small and leads to an extremely small p-value in the original DELVE test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In DELVE+,  as long as T is positive, ψ+ is smaller than ψ, so that the p-value is adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In the numerical experiments, we consider both DELVE and DELVE+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For theoretical  analysis, since these two versions have almost identical theoretical properties, we only focus  on the original DELVE test statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3  Theoretical Properties  We first present the regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For a constant c0 ∈ (0, 1), we assume  min  1≤i≤n Ni ≥ 2,  max  1≤i≤n ∥Ωi∥∞ ≤ 1 − c0,  max  1≤k≤K  nk ¯Nk  n ¯N  ≤ 1 − c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1), the first condition is mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The second condition is also mild: note that ∥Ωi∥1 = 1  for each i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' this condition excludes those cases where one of the p categories has an extremely  dominating probability in the PMF Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the third condition, nk ¯Nk is the total number of  counts in all multinomials of group k, and this condition excludes the extremely unbalanced  case where one group occupies the majority of counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Note that in the special case of K = 2,  we relax this condition to allow for severely unbalanced groups (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Recall that µk =  1  nk ¯  Nk  �  i∈Sk NiΩi is the mean PMF within group k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We also define a  ‘covariance’ matrix of PMF’s for group k by Σk =  1  nk ¯  Nk  �  i∈Sk NiΩiΩ′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let  αn := max  � K  �  k=1  ∥µk∥3  3  nk ¯Nk  ,  K  �  k=1  ∥µk∥2  n2  k ¯N2  k  � �� K  �  k=1  ∥µk∥2  �2  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  and  βn := max  � K  �  k=1  �  i∈Sk  N2  i  n2  k ¯N2  k  ∥Ωi∥3  3,  K  �  k=1  ∥Σk∥2  F  ��  (K∥µ∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  We assume that as n ¯N → ∞,  αn = o(1),  βn = o(1),  and  ∥µ∥4  4  K∥µ∥4 = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  Here αn and βn only depend on group-wise quantities, such as µk, Σk and �  i∈Sk N2  i ∥Ωi∥3  3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  hence, a small number of ‘outliers’ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', extremely large entries) in Ω has little effect on αn  and βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore, in a simple case where maxk nk ≤ C mink nk, maxk ¯Nk ≤ C mink ¯Nk  and ∥Ω∥max = O(1/p), it holds that αn = O(max{ 1  n ¯  N ,  Kp  n2 ¯  N2 }), βn = O(max{ K2  n2p, 1  p}) and  ∥µ∥4  4  K∥µ∥4 = O( 1  Kp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' When n ¯N → ∞ and p → ∞, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4) reduces to n2 ¯N2/(Kp) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This  condition is necessary for successful testing, because our lower bound in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3 implies  that the two hypotheses are asymptotically indistinguishable if n2 ¯N2/(Kp) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  The asymptotic null distribution  Under the null hypothesis, the K group-wise mean PMF’s µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , µK, are equal to each  other, but this hypothesis is still highly composite, as (Ni, Ωi) are not necessarily the same  within each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We show that the DELVE test statistic always enjoys a parameter-free  asymptotic null distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let T, Θn and V be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The next two theorems  are proved in the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider Models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), where the null hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As n ¯N → ∞, T/√Θn → N(0, 1) in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, as n ¯N → ∞, V/Θn → 1 in proba-  bility, and ψ := T/  √  V → N(0, 1) in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, the asymptotic p-value is computed via 1 − Φ(ψ), where Φ(·) is the  cumulative distribution function of the standard normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover, for any fixed α ∈ (0, 1),  the rejection region of the asymptotic level-α test is as given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The proofs of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 contain two key steps: in the first step, we decompose T  into the sum of mutually uncorrelated terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We introduce a set of independent, mean-zero  random vectors {Zir}1≤i≤n,1≤r≤Ni, where Zir ∼ Multinomial(1, Ωi) − Ωi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By properties of  multinomial distributions, Xi = NiΩi + �Ni  r=1 Zir in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We plug it into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) to  obtain T = T1 +T2 +T3 +T4, where T1 is a linear form of {Zir}, T2, T3 and T4 are quadratic  forms of {Zir}, and the four terms are uncorrelated with each other (details are contained  in Section A of the appendix ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the second step, we construct a martingale for each term  Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This is accomplished by rearranging the double-index sequence Zir to a single-index  sequence and then successively adding terms in this sequence to Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We then apply the  martingale central limit theorem (CLT) [Hall and Heyde, 2014] to prove the asymptotic  normality of each Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The asymptotic normality of T follows by identifying the dominating  terms in T1-T4 (as model parameters change, the dominating terms can be different) and  studying their joint distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This step involves extensive calculations to bound the  conditional variance and to verify the Lindeberg conditions of the martingale CLT, as well  as numerous subtle uses of the Cauchy-Schwarz inequality to simplify the moment bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Power analysis  Under the alternative hypothesis, the PMF’s µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , µK are not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Section 2,  we introduce a quantity ρ2 (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)) to capture the total variation in µk’s, but this quantity  is not scale-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We define a scaled version of ρ2 as  ωn = ωn(µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , µK) :=  1  n ¯N∥µ∥2  K  �  k=1  nk ¯Nk∥µk − µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  It is seen that ωn ≤ maxk{ ∥µk−µ∥2  ∥µ∥2  }, which is properly scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider Models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), where (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Then,  E[T] = n ¯N∥µ∥2ω2  n, and V(T) = O  ��K  k=1 ∥µk∥2�  + E[T] · O  �  max1≤k≤K ∥µk∥∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  10  For the DELVE test to have an asymptotically full power, we need E[T] ≫  �  V(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3, this is satisfied if E[T] ≫  ��  k ∥µk∥2 and E[T] ≫ maxk ∥µk∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Between  these two requirements, the latter one is weaker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' hence, we only need E[T] ≫  ��K  k=1 ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It gives rise to the following theorem:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3, we further assume that under the  alternative hypothesis, as n ¯N → ∞,  SNRn :=  n ¯N∥µ∥2ω2  n  ��K  k=1 ∥µk∥2  →  ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  The following statements are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the alternative hypothesis, ψ → ∞ in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level of α and an  asymptotic power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If we choose α = αn such that αn → 0 and 1 − Φ(SNRn) = o(αn),  where Φ is the CDF of N(0, 1), then the sum of type I and type II errors of the DELVE  test converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The detection boundary in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6) has simpler forms in some special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For example,  if ∥µk∥ ≍ ∥µ∥ for 1 ≤ k ≤ K, then SRNn ≍ n ¯Nω2  n∥µ∥/  √  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If, furthermore, all entries of  µ are at the same order, which implies ∥µ∥ ≍ p−1/2, then SRNn ≍ n2 ¯N2ω2  n/√Kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In this  case, the detection boundary simplifies to ω4  nn2 ¯N2/(Kp) → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Remark 1 (The low-dimensional case p = O(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Although we are primarily interested in  the high-dimensional setting p → ∞, it is also worth investigating the case p = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We  can show the same detection boundary for our test, but the asymptotic normality may not  hold, because the variance estimator V in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7) is not guaranteed to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To fix  this issue, we propose a variant of our test by replacing V with a refined variance estimator  �V , which is consistent for a finite p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The expression of �V is a little complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Due to  space limits, we relegate it to Section E of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  A matching lower bound  We have seen that the DELVE test successfully separates two hypotheses if SNRn → ∞,  where SNRn is as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We now present a lower bound to show that the two  hypotheses are asymptotically indistinguishable if SNRn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let ℓi ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , K} denote the group label of Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write ξ = {(Ni, Ωi, ℓi)}1≤i≤n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let  µk, αn, βn, and ωn be the same as defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For  each given (n, p, K, ¯N), we write µk = µk(ξ) to emphasize its dependence on parameters,  and similarly for αn, βn, ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For any c0 ∈ (0, 1) and sequence ϵn, define  Qn(c0, ϵn) :=  �  ξ = {(Ni, Ωi, ℓi)}n  i=1 : (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds for c0, max(αn(ξ), βn(ξ)) ≤ ϵn  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  Furthermore, for any sequence δn, we define a parameter class for the null hypothesis and  a parameter class for the alternative hypothesis:  Q∗  0n(c0, ϵn) = Qn(c0, ϵn) ∩ {ξ : ωn(ξ) = 0} ,  11  Q∗  1n(δn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' c0, ϵn) = Qn(c0, ϵn) ∩  �  �  �ξ : n ¯N∥µ(ξ)∥2ω2  n(ξ)  ��K  k=1 ∥µk(ξ)∥2  ≥ δn  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8)  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Fix a constant c0 ∈ (0, 1) and two positive sequences ϵn and δn such that  ϵn → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For any sequence of (n, p, K, ¯N) indexed by n, we consider Models (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)-  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2) for Ω ∈ Qn(c0, ϵn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let Q∗  0n(c0, ϵn) and Q∗  1n(δn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' c0, ϵn) be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If δn → 0, then  lim supn→∞ infΨ∈{0,1}  �  supξ∈Q∗  0n(c0,ϵn) Pξ(Ψ = 1) + supξ∈Q∗  1n(δn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='c0,ϵn) Pξ(Ψ = 0)  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5, the null and alternative hypotheses are asymptotically indistinguish-  able if SRNn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Combining it with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4, the DELVE test achieves the minimax  optimal detection boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  The special case of K = 2  The special case of K = 2 is found in applications such as closeness testing and authorship  attribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We study this case more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Given {Xi}1≤i≤n and {Gi}1≤i≤m, we assume  Xi ∼ Multinomial(Ni, Ωi),  Gj ∼ Multinomial(Mj, Γj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9)  Write ¯N = n−1 �n  i=1 Ni and ¯  M = m−1 �m  i=1 Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The null hypothesis becomes  H0 :  η = θ,  where η =  1  n ¯N  n  �  i=1  NiΩi, and θ =  1  m ¯  M  m  �  i=1  MiΓi,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)  where θ and η are the two group-wise mean PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In this case, the test statistic ψ has a  more explicit form as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In our previous results for a general K, the regularity conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)) impose  restrictions on the balance of sample sizes among groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For K = 2, the severely unbal-  anced setting is interesting (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', in authorship attribution, n = 1 and m can be large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We  relax the regularity conditions to the following ones:  Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let θ and η be as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10) and define two matrices Σ1 =  1  n ¯  N  �n  i=1 NiΩiΩ′  i  and Σ2 =  1  m ¯  M  �m  i=1 MiΓiΓ′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We assume that the following statements are true (a) For  1 ≤ i ≤ n and 1 ≤ j ≤ m, Ni ≥ 2, ∥Ωi∥∞ ≤ 1 − c0, Mj ≥ 2, and ∥Γj∥∞ ≤ 1 − c0, where  c0 ∈ (0, 1) is a contant, (b) max  �� ∥η∥3  3  n ¯  N + ∥θ∥3  3  m ¯  M  �  ,  � ∥η∥2  2  n2 ¯  N2 + ∥θ∥2  2  m2 ¯  M2  2  �����  m ¯  M  n ¯  N+m ¯  M η+  n ¯  N  n ¯  N+m ¯  M θ  ��4 =  o(1), (c) max  � �  i  N2  i  n2 ¯  N2 ∥Ωi∥3  3, �  i  M2  i  m2 ¯  M2 ∥Γi∥3  3, ∥Σ1∥2  F + ∥Σ2∥2  F  ��  ∥µ∥2 = o(1), and (d)  ∥µ∥4  4/∥µ∥4 = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Condition (a) is similar to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1), except that we drop the sample size balance requriement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Conditions (b)-(d) are equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4) but have more explicit expressions for K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Model (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9), we test the null hypothesis H0: θ = µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As min{n ¯N, m ¯  M} →  ∞, suppose Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the alternative hypothesis, we further assume  ∥η − θ∥2  � 1  n ¯  N +  1  m ¯  M  �  max{∥η∥, ∥θ∥} → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11)  12  Consider the DELVE test statistic ψ = T/  √  V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The following statements are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under  the null hypothesis, ψ → N(0, 1) in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the alternative hypothesis, ψ → ∞  in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover for any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic  level of α and an asymptotic power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Compared with the theorems for a general K, first, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 allows the two groups  to be severely unbalanced and reveals that the detection boundary depends on the harmonic  mean of n ¯N and m ¯  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Second, the detection boundary is expressed using ∥η − θ∥, which  is easier to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  The special case of K = n  The special case of K = n is interesting for two reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' First, the application example of  global testing in topic models corresponds to K = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Second, for any K, when Ωi’s within  each group are assumed to be the same (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', this is the case in closeness testing of discrete  distributions), it suffices to aggregate the counts in each group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', let Yk = �  i∈Sk Xi and  operate on Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , YK instead of the original Xi’s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' this reduces to the case of K = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  When K = n, the null hypothesis has a simpler form:  H0 :  Ωi = µ,  1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12)  Moreover, under the alternative hypothesis, the quantity ω2  n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) simplifies to  ωn = ωn(Ω1, Ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , Ωn) =  1  n ¯N∥µ∥2  n  �  i=1  Ni∥Ωi − µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13)  The DELVE test statistic also has a simplified form as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We can prove the  same theoretical results under weaker conditions:  Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We assume that the following statements are true: (a) For a constant  c0 ∈ (0, 1), 2 ≤ Ni ≤ (1 − c0)n ¯N and ∥Ωi∥∞ ≤ 1 − c0, 1 ≤ i ≤ n, and  (b) max  � �  i  ∥Ωi∥3  3  Ni , �  i  ∥Ωi∥2  N2  i  ��  (�  i ∥Ωi∥2)2 = o(1), and (�  i ∥Ωi∥3  3)/(n∥µ∥2) = o(1)  When K = n, Condition (a) is equivalent to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' and Condition (b) is weaker than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4),  as we have dropped the requirement  ∥µ∥4  4  K∥µ∥4 = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We obtain weaker conditions for K = n  because the dominant terms in T differ from those for K < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1), we test the null hypothesis (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As n → ∞, we assume  that Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the alternative, we further assume that  n ¯N∥µ∥2ω2  n  ��n  i=1 ∥Ωi∥2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14)  Let T and V ∗ be the same as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider the simplified DELVE test statistic  ψ∗ = T/  √  V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The following statements are true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the null hypothesis, ψ∗ → N(0, 1)  in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the alternative hypothesis, ψ∗ → ∞ in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover for any  fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level of α and an asymptotic  power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  13  The detection boundary in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14) has a simpler form if �  i ∥Ωi∥2 ≍ n∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In this case,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14) is equivalent to √n ¯N∥µ∥ω2  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Additionally, if all entries of µ are at the same  order, then ∥µ∥ ≍ 1/√p, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14) further reduces to  �  n ¯N2/p · ω2  n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  A discussion of the contiguity regime  Our power analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 concerns SNRn → ∞, and our lower bound in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  concerns SNRn → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We now study the contiguity regime where SNRn tends to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For illustration, we consider a special choice of parameters, which allows us to obtain a  simple expression of the testing risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Suppose K = n and Ni = N for all 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider the pair of hypotheses:  H0 :  Ωij = p−1,  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  H1 :  Ωij = p−1(1 + βnδij),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15)  where {δij}1≤i≤n,1≤j≤p satisfy that |δij| = 1, �p  j=1 δij = 0 and �n  i=1 δij = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Such δij  always exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 The SNRn in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6) satisfies that SNRn ≍ (N√n/√p)β2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We thereby set  β2  n =  √2p  N√n · a,  for a constant a > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16)  Since K = n here, we consider the simplified DELVE test statistic ψ∗ as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider Model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) with Ni = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For a constant a > 0, let the null  and alternative hypotheses be specified as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As n → ∞, if p = o(N2n), then  ψ∗ → N(0, 1) under H0 and ψ∗ → N(a, 1) under H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let Φ be the cumulative distribution function of the standard normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8,  for any fixed constant t ∈ (0, a), if we reject the null hypothesis when ψ∗ > t, then the sum  of type I and type II errors converges to [1 − Φ(t)] + [1 − Φ(a − t)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  4  Applications  As mentioned in Section 1, our testing problem includes global testing for topic models,  authorship attribution, and closeness testing for discrete distributions as special examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In this section, the DELVE test is applied separately to these three problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Global testing for topic models  Topic modeling [Blei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2003] is a popular tool in text mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It aims to learn a small  number of “topics” from a large corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Given n documents written using a dictionary of p  words, let Xi ∼ Multinomial(Ni, Ωi) denote the word counts of document i, where Ni is the  length of this document and Ωi ∈ Rp contains the population word frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In a topic  model, there exist M topic vectors A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , AM ∈ Rp, where each Ak is a PMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let  1For example, we can first partition the dictionary into two halves and then partition all the documents  into two halves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' this divides {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , p} × {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , n} into four subsets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' we construct δij’s freely on one  subset and then specify the δij’s on the other three subsets by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  14  wi ∈ RM be a nonnegative vector whose entries sum up to 1, where wi(k) is the “weight”  document i puts on topic k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It assumes  Ωi =  M  �  k=1  wi(k)Ak,  1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  Under (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1), the matrix Ω = [Ω1, Ω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , Ωn] admits a low-rank nonnegative factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Before fitting a topic model, we would like to know whether the corpus indeed involves  multiple topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This is the global testing problem: H0 : M = 1 v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' H1 : M > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' When  M = 1, by writing A1 = µ, the topic model reduces to the null hypothesis in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We  can apply the DELVE test by treating each Xi as a separate group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', K = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider Model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) and define a vector ξ ∈ Rn by ξi = ¯N−1Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose  that Ω = µ1′  n under the null hypothesis, with µ = n−1Ωξ, and that Ω satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) under  the alternative hypothesis, with r := rank(Ω) ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose ¯N/(mini Ni) = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Denote  by λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , λr > 0 the singular values of Ω[diag(ξ)]1/2, arranged in the descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We further assume that under the alternative hypothesis,  ¯N ·  �r  k=2 λ2  k  ��r  k=1 λ2  k  → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level α and an asymptotic  power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The least-favorable configuration in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5 is in fact a topic model  that follows (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) with M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Transferring the argument yields the following lower bound  that confirms the optimality of DELVE for the global testing of topic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let Rn,M(ϵn, δn) be the collection of {(Ni, Ωi)}n  i=1 satisfying the following  conditions: 1) Ω follows the topic model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) with M topics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' 2) Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 holds with  o(1) replaced by ≤ ϵn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' 3) ¯N(�r  k=2 λ2  k)/(�r  k=1 λ2  k)1/2 ≥ δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If ϵn → 0 and δn → 0, then  lim supn→∞ infΨ∈{0,1}  �  supRn,1(ϵn,0) P(Ψ = 1) + sup∪M≥2Rn,M(δn,δn) P(Ψ = 0)  �  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The detection boundary (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2) can be simplified when M = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Following Ke and  Wang [2022], we define ΣA = A′H−1A and ΣW = n−1WW ′, where A = [A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , AM],  W = [w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , wn] and H = diag(A1M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ke and Wang [2022] argued that it is reasonable  to assume that eigenvalues of these two matrices are at the constant order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  If this is  true, with some mild additional regularity conditions, each λk is at the order of  �  n/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Hence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2) reduces to √n ¯N/√p → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In comparison, Ke and Wang [2022] showed that  a necessary condition for any estimator ˆA = [ ˆA1, ˆA2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , ˆAM] to achieve  1  M  �M  k=1 ∥ ˆAk −  Ak∥1 = o(1) is  �  n ¯N/p → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We conclude that consistent estimation of topic vectors  requires strictly stronger conditions than successful testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Authorship attribution  In authorship attribution, given a corpus from a known author, we want to test whether  a new document is from the same author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It is a special case of our testing problem  15  with K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We can directly apply the results in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' However, the setting in  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4 has no sparsity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Kipnis [2022], Donoho and Kipnis [2022] point out that the  number of words with discriminating power is often much smaller than p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To see how our  test performs under sparsity, we consider a sparse model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4, let  Xi ∼ Multinomial(Ni, Ωi), 1 ≤ i ≤ n,  and  Gi ∼ Multinomial(Mi, Γi), 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  Let ¯N and ¯  M be the average of Ni’s and Mi’s, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write η =  1  n ¯  N  �n  i=1 NiΩi and  θ =  1  m ¯  M  �m  i=1 MiΓi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We assume for some βn > 0,  ηj = θj, for j /∈ S,  and  ��√ηj −  �  θj  �� ≥ βn, for j ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4), consider testing H0 : S = ∅ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' H1 : S ̸= ∅,  where Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let ηS and θS be the sub-vectors of η and θ restricted to  the coordinates in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that under the alternative hypothesis,  β2  n · (∥ηS∥1 + ∥θS∥1)  � 1  n ¯  N +  1  m ¯  M  �  max{∥η∥, ∥θ∥} → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  As min{n ¯N, m ¯  M} → ∞, the level-α DELVE test has an asymptotic level α and an asymp-  totic power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore, if n ¯N ≍ m ¯  M and minj∈S(ηj +θj) ≥ cp−1 for a constant c > 0,  then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) reduces to n ¯Nβ2  n|S|/√p → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Donoho and Kipnis [2022] studied a case where N = M, n = m = 1, p → ∞,  |S| = p1−ϑ,  and  βn = c · N−1/2�  log(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  When ϑ > 1/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', |S| = o(√p)), they derived a phase diagram for the aforementioned  testing problem (under a slightly different setting where the data distributions are Poisson  instead of multinomial).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' They showed that when ϑ > 1/2 and c is a properly large constant,  a Higher-Criticism-based test has an asymptotically full power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Donoho and Kipnis [2022]  did not study the case of ϑ ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3, when ϑ ≤ 1/2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', |S| ≥ C√p), the  DELVE test has asymptotically full power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' When ϑ > 1/2 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6), the DELVE test is powerless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' However, this issue can  be resolved by borrowing the idea of maximum test or Higher Criticism test [Donoho and  Jin, 2004] from the classical multiple testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For example, recalling Tj in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5), we can use  max1≤j≤p{Tj/  �  Vj} as the test statistic, where Vj is a proper estimator of the variance of  Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We leave a careful study of this idea to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Closeness testing between discrete distributions  Two-sample closeness testing is a subject of intensive study in discrete distribution inference  [Bhattacharya and Valiant, 2015, Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2014, Diakonikolas and Kane, 2016, Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=',  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is a special case of our problem with K = 2 and n1 = n2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We thereby apply  both Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  16  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let Y1 and Y2 be two discrete variables taking values on the same p out-  comes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let Ω1 ∈ Rp and Ω2 ∈ Rp be their corresponding PMFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose we have N1  samples of Y1 and N2 samples of Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The data are summarized in two multinomial vec-  tors: X1 ∼ Multinomial(N1, Ω1), X2 ∼ Multinomial(N2, Ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We test H0 : Ω1 = Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write  µ =  1  N1+N2 (N1Ω1 + N2Ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose min{N1, N2} ≥ 2, max{∥Ω1∥∞, ∥Ω2∥∞} ≤ 1 − c0, for  a constant c0 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose  1  (�2  k=1 ∥Ωk∥2)2 max  � �2  k=1  ∥Ωk∥3  3  Nk , �2  k=1  ∥Ωk∥2  N2  k  �  = o(1), and  1  n∥µ∥2  �2  k=1 ∥Ωk∥3  3 = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We assume that under the alternative hypothesis,  ∥Ω1 − Ω2∥2  �  N−1  1  + N−1  2  �  max{∥Ω1∥, ∥Ω2∥} → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  As min{N1, N2} → ∞, the level-α DELVE test has level α and power 1, asymptotically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We notice that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7) matches with the minimum ℓ2-separation condition for two-sample  closeness testing [Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2022, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Therefore, our test is an optimal  ℓ2-testor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Although other optimal ℓ2-testors have been proposed [Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2014, Bhat-  tacharya and Valiant, 2015, Diakonikolas and Kane, 2016], they are not equipped with  tractable null distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We can modify the DELVE test to incorporate frequency-dependent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Given any nonnegative vector w = (w1, w2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , wp)′, define T(w) := �p  j=1 wjTj where Tj is  the same as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' These weights wj serve to adjust the contributions of different words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For example, consider wj =  �  max{1/p, ˆµj}  �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This kind of weights have been used in  discrete distribution inference [Balakrishnan and Wasserman, 2019, Chan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2014] to  turn an optimal ℓ2 testor to an optimal ℓ1 testor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We can similarly study the power of this  modified test, except that we need an additional assumption n ¯N ≫ p to guarantee that ˆµj  is a sufficiently accurate estimator of µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  5  Simulations  The proposed DELVE test is computationally efficient and easy to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In this  section, we investigate its numerical performance in simulation studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Real data analysis  will be carried out in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Experiment 1 (Asymptotic normality) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Given (n, p, K, Nmin, Nmax, α), we generate data  as follows: first, we divide {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , n} into K equal-size groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Next, we draw Ωalt  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , Ωalt  n  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' from Dirichlet(p, α1p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Third, we draw Ni  iid  ∼ Uniform[Nmin, Nmax] and set Ωnull  i  = µ,  where µ :=  1  n ¯  N  �  i NiΩalt  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Last, we generate X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , Xn using Model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We consider  three sub-experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, (n, p, K, Nmin, Nmax, α) = (50, 100, 5, 10, 20, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, α is changed to 1, and the other parameters are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  When  α = 1, Ωalt  i  are drawn from the uniform distribution of the standard probability simplex;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  in comparison, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3 puts more mass near the boundary of the standard probability  simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3, we keep all parameters the same as in Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, except  that (p, K) are changed to (300, 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For each sub-experiment, we generate 2000 data  sets under the null hypothesis and plot the histogram of the DELVE test statistic ψ (in  17  Figure 1: Histograms of the DELVE statistic (top three panels) and the DELVE+ statistic  (bottom three panels) in Experiments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In each plot, the blue and orange histograms  correspond to the null and alternative hypotheses, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' and the green curve is the  density of N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  blue);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' similarly, we generate 2000 data sets under the alternative hypothesis and plot the  histogram of ψ (in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The results are contained on the top three panels of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, we introduced a variant of DELVE, called DELVE+, in which the variance  estimator V is replaced by an adjusted one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' DELVE+ has similar theoretical properties as  DELVE but is more suitable for real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We plot the histograms of the DELVE+ test  statistics on the bottom three panels of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We have several observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In all sub-experiments, when the null hypothesis holds,  the histograms of both DELVE and DELVE+ fit the standard normal density reasonably  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This supports our theory in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Second, when (p, K) increase, the finite  sample effect becomes slightly more pronounced (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', Experiment 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3 versus Experiment  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Third, the tests have power in differentiating two hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As α decreases or K  increases, the power increases, and the histograms corresponding to two hypotheses become  further apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Last, in the alternative hypothesis, DELVE+ has smaller mean and variance  than DELVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, these two have similar asymptotic behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The simulation  results suggest that they have noticeable finite-sample differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Experiment 2 (Power curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Similarly as before, we divide {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , n} into K equal-  size groups and draw Ni ∼ Uniform[Nmin, Nmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In this experiment, the PMF’s Ωi are gen-  erated in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the null hypothesis, we generate µ ∼ Dirichlet(p/2, α1p/2)  and set Ωnull  i  = ˜µ, where ˜µj = 1  2µj for 1 ≤ j ≤ p/2 and ˜µj = 1  2µp+1−j for p/2 + 1 ≤ j ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Under the alternative hypothesis, we draw z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , zK, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , bp/2  iid  ∼ Rademacher(1/2) and  then let Ωalt  ij = µ(1 + τnzkbj), for all i in group k and 1 ≤ j ≤ p/2, and Ωalt  ij = µ(1 + τnzkbj)  for p/2 + 1 ≤ j ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By applying our theory in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 together with some cal-  culations, we obtain that the signal-to-noise ratio is captured by λ := K−1/2n ¯N∥µ∥τn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We consider three sub-experiments, Experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3, in which the parameter values of  18  Figure 2:  Power diagrams (based on 500 repetitions) at level 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The x-axis plots the  SNR λ(ωn) = K−1/2n ¯N∥µ∥ · ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (n, p, K, Nmin, Nmax, α) are the same as in Experiments 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For each sub-experiment,  we consider a grid of 10 equally-spaced values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' When λ = 0, it corresponds to the null  hypothesis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' when λ > 0, it corresponds to the alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For each λ, we gener-  ate 500 data sets and compute the fraction of rejections of the level-5% DELVE test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This  gives a power curve for the level-5% DELVE test, in which the first point corresponding to  λ = 0 is the actual level of the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The results are contained on the top three panels of  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We repeat the same experiments for the DELVE+ test, which results are on the  bottom three panels of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In all three experiments, the actual level of our proposed  tests is ≤ 5%, suggesting that our tests perform well at controlling the type-I error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As  λ increases, the power gradually increased to 1, suggesting that λ is a good metric of the  signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This supports our theory in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  6  Real Data Analysis  We apply our proposed methods on two real corpora: one consists of abstracts of research  papers in four statistics journals, and the other consists of movie reviews on Amazon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For  the analysis of real data, we use DELVE+, which modifies the variance estimator in DELVE  and reduces the occurrence of extremely small p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Abstracts of statisticians  We use the data set from Ji and Jin [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It contains the bibtex information of all pub-  lished papers in four top-tier statistics journals, Annals of Statistics, Biometrika, Journal of  the American Statistical Association, and Journal of the Royal Statistical Society - Series B,  from 2003 to the first half of 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We pre-process the abstracts of papers by tokenization  19  (0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='05)(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='03)(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='04)(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='04)(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='02)(0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='04)Figure 3: (Left) Histogram of nonzero DELVE Z-scores for all authors in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The  mean is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52 and the standard deviation is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' (Right) Scatter plot of author DELVE  scores versus the natural log of the number of papers with five statisticians identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Figure 4: Pairwise Z-score plots for Peter Hall (left) and Jianqing Fan (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the cell  (x, y), we compare the corpus of an author’s abstracts from time x with the corpus of that  author’s abstracts from time y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The heatmap shows the value of DELVE+ with K = 2 for  each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  and stemming and turn each abstract to a word count vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We conduct two experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the first one, we fix an author and treat the collection  of his/her co-authored abstracts as a corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We apply DELVE+ with K = n, where n is  the total number of abstracts written by this author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The Z-score measures the “diversity”  or “variability” of this authors’ abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' An author with a high Z-score possesses either  diverse research interests or a variable writing style.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' A number of authors have only 1–2  papers in this data set, and the variance estimator V is often negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' we remove all those  authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Figure 3 (left panel), we plot the histogram of Z-scores of all retained authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The mean is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52 and the standard deviation is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Figure 3 (right panel), we show  the scatter plot of Z-score versus logarithm of the number of abstracts written by this  author, and a few prolific authors who have many papers and a large Z-score are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For example, Peter Hall has the most papers in this dataset (82 papers in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hall’s  Z-score is larger than 20, implying a huge diversity in his abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' There is also a positive  association between Z-score and total papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It suggests that senior authors have more  diversity in their abstracts, which is as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  20  Peter Hall  Jianqing Fan  Raymond J Carrol  ing Qin  David DunsonPeter Hall  2011  2010  10  2009  8  2008  6  2007  2006  4  2005  2004  2003  TToz T0z 600z 800z L00z 900z 500 F00z E0zJianging Fan  2007 2008 2009 2011  10  8  6  4  2003 2004  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  2003  2004  2007  2008  2009  2011Year  Title  Journal  2011  Nonparametric independence screening in sparse  ultra-high-dimensional additive models  JASA  2011  Penalized composite quasi-likelihood for ultrahigh  dimensional variable selection  JRSS-B  2011  Multiple testing via FDRL for large-scale imaging  data  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Vast volatility matrix estimation using high-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='frequency data for portfolio selection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='JASA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='A road to classification in high dimensional space:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='the regularized optimal affine discriminant  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='JRSS-B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2012  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Variance estimation using refitted cross-validation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='in ultrahigh dimensional regression  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='JRSS-B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Year  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Title  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Journal  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2004  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Low order approximations in deconvolution  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='and regression with errors in variables  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='JRSS-B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2004  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Nonparametric inference about service time  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='distribution from indirect measurements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='JRSS-B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2004  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Cross-validation and the estimation of condi-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='tional probability densities  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='JASA  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2004  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Nonparametric confidence intervals for re-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='ceiver operating characteristic curves  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Biometrika  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2004  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Bump hunting with non-Gaussian kernels  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  2004  Attributing a probability to the shape of a  probability density  Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Figure 5: (Left) Jianqing Fan’s papers in the dataset of Ji and Jin [2016] from 2011 to  2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' (Right) Peter Hall’s papers in the dataset of Ji and Jin [2016] from 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In the second experiment, we divide the abstracts of each author into groups by publi-  cation year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We divide Peter Hall’s abstracts into 9 groups, and each group corresponds to  one year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We divide Jianqing Fan’s abstracts into 6 groups, with unequal window sizes to  make all groups have roughly equal numbers of abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Our test can be used to detect  differences between all groups, but to see more informative results, we do a pairwise com-  parison: for each pair of groups, we apply DELVE+ with K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This yields a pairwise  plot of Z-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The plot reveals the temporal patterns of this author in abstract writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Figure 4 shows the results for Peter Hall and Jianqing Fan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  There are interesting temporal patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For Jianqing Fan (right panel of Figure 4), the  group consisting of his 2011-2012 abstracts has comparably large Z-scores in the pairwise  comparison with other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To interpret this , we gathered the titles and abstracts of all  his papers in the dataset and compared the ones before/after 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' He published six papers  in these journals during 2011-2012, whose titles are listed on the left of Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We see  that his papers in this period had a strong emphasis on screening and variable selection:  four out of the six papers mention this subject in their titles and/or abstracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This shows a  departure from his previously published topics such as covariance estimation (a focus from  2007–2009) and semiparametric estimation (a focus before 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Though Jianqing Fan had  previously published papers on variable selection and screening in these journals, he had  never published so many in such a short time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For Peter Hall (left panel of Figure 4),  the group of 2004 abstracts have comparably large Z-scores in the pairwise comparison with  other groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We examined the titles and abstracts of his 6 papers published in 2004 in  this data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' All of his 2004 papers, except the first one, mention bandwidth selection or  smoothing parameters, and in the last 4 papers, bandwidth selection plays a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For instance, Bump hunting with non-Gaussian kernels, (Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', 2004) studies the  relationship between the number of modes of a kernel density estimator and its bandwidth  parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Though Peter Hall’s 2014 papers concern many nonparametric statistics topics,  we find that bandwidth selection is a theme underlying his research in these journals in  2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  21  Rank  Title  Z-Score  Total reviews  1  Prometheus  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44  813  2  Expelled: No Intelligence Allowed  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17  830  3  V for Vendetta  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='24  815  4  Sin City  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='72  828  5  No Country for Old Men  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='57  819  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  16  John Adams  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='78  857  17  Cars  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='98  902  18  Food, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='81  876  19  Jeff Dunham: Arguing with Myself  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='96  860  20  Jeff Dunham: Spark of Insanity  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46  877  Figure 6: (Left) Histogram of Z-scores for the 500 most-reviewed movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The mean is  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='97 and the standard deviation is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' (Right) Z-scores for the top 20 most reviewed  movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Figure 7: Pairwise Z-scores for 3 movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In each cell, we use DELVE+ to compare reviews  associated to a pair of star ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For each movie, the title list the number of reviews of  each rating from 1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Amazon movie reviews  We analyze Amazon reviews from the dataset Maurya [2018] that consists of 1,924,471  reviews of 143,007 visual media products (ie, DVDs, Bluray, or streams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We examine  products with the largest number of reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Each product’s review corpus is cleaned and  stemming is used to group together words with the same root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We obtain word counts  for each review and a term-document matrix of a product’s review corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the first  experiment, we fix a movie and apply DELVE+ with K = n to the corpus consisting  of all reviews of this movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Figure 6 (left panel), we plot the histogram of Z-scores  for the top 500 most reviewed movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The mean is 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='97 and the standard deviation is  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Compared with the histogram of Z-scores for statistics paper abstracts, there is much  larger diversity in movie reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Figure 6 (right panel), we list the 5 movies with the  highest Z-scores and lowest Z-scores out of the 20 most reviewed movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Each movie has  more than 800 reviews, but some have surprisingly low Z-scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The works by comedian  Jeff Dunham have the lowest Z-scores, suggesting strong homogeneity among the reviews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The 2012 horror film Prometheus has the highest degree of review diversity among the 20  most reviewed movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the second experiment, we divide the reviews of each movie into  22  Zscores of top 5oo most reviewec mnovies  80  70  60  50  40  30  20  10  0  5  10  15  20  25  30  355 groups by star rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We compare each pair of groups using DELVE+ with K = 2,  resulting in a pairwise Z-score plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Figure 7, we plot this for 3 popular movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We  see a variety of polarization patterns among the scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In Harry Potter and the Deathly  Hallows Part I, DELVE+ signifies that the reviews with ratings in the range 2–4 stars are  all similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We see a smooth gradation in how the 1-star reviews differ from those from 2–4  stars, and similarly for 5-star reviews versus those from 2–4 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Twilight Saga: Eclipse  shows three clusters: 1–2 stars, 3–4 stars, and 5 star, while Night of the living dead shows  two clusters: 1–2 stars and 3–5 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  7  Discussions  We examine the testing for equality of PMFs of K groups of high-dimensional multino-  mial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The proposed DELVE statistic has a parameter-free limiting null that  allows for computation of Z-scores and p-values on real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' DELVE achieves the op-  timal detection boundary over the whole range of parameters (n, p, K, ¯N), including the  high-dimensional case p → ∞, which is very relevant to applications in text mining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  This work leads to interesting questions for future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' So far the focus is on testing,  but one can also consider inference for ρ2 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), which measures the heterogeneity  among the group-wise means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consistent variance estimation under the alternative uses a  similar strategy, though we omit the calculations in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Establishing asymptotic  normality of DELVE under the alternative would then lead to asymptotic confidence in-  tervals for our heterogeneity metric ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Based on the plots in Section 5, it is possible that  stronger regularity conditions are needed to obtain a pivotal distribution under the alterna-  tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As in the two-sample multinomial testing problems described in Kipnis and Donoho  [2021], Kipnis [2022], such as author attribution, we may also consider an alternative where  all the group means are the same except for a small set of “giveaway words”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is interest-  ing to develop a procedure for identifying these useful words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2,  we may modify DELVE by using a version based on the maximum test or higher criticism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Another extension is to go beyond ‘bag-of-words’ style analysis and use different types of  counts besides raw word frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' One option is to apply a suitably modified DELVE  to the counts of multi-grams in the corpus and another is to combine words with similar  meanings into a ‘superword’ and use superword counts as the basis for DELVE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To do this,  we can combine words that are close together in some word embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We leave these  interesting tasks for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Acknowledgments The research of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Tony Cai was supported in part by NSF Grant  DMS-2015259 and NIH grant R01-GM129781.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The research of Zheng Tracy Ke was sup-  ported in part by NSF CAREER Grant DMS-1943902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  23  Notational conventions for the appendix: We write A ≲ B (respectively, A ≳ B) if  there exists an absolute constant C > 0 such that A ≤ C · B (respectively A ≥ C · B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If  both A ≲ B and B ≲ A, we write A ≍ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The implicit constant C may vary from line to  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For sequences at, bt indexed by an integer t ∈ N, we write at ≪ bt if bt/at → ∞ as  t → ∞, and we write at ≫ bt if at/bt → ∞ as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We also may write at = o(bt) to  denote at ≪ bt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In particular, we write at = (1 + o(1))bt if at/bt → 1 as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A  Properties of T and V  We recall that  Xi ∼ Multinomial(Ni, Ωi),  1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  For each 1 ≤ k ≤ K, define  µk =  1  nk ¯Nk  �  i∈Sk  NiΩi ∈ Rp,  Σk =  1  nk ¯Nk  �  i∈Sk  NiΩiΩ′  i ∈ Rp×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  Moreover, let  µ =  1  n ¯N  K  �  k=1  nk ¯Nkµk =  1  n ¯N  n  �  i=1  NiΩi ,  Σ =  1  n ¯N  n  �  k=1  nk ¯NkΣk =  1  n ¯N  �  i  NiΩiΩ′  i  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  The DELVE test statistic is ψ = T/  √  V , where T is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) and V is as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As a  preparation for the main proofs, in this section, we study T and V separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  The decomposition of T  It is well-known that a multinomial with the number of trials equal to N can be equivalently  written as the sum of N independent multinomials each with the number of trials equal to  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This inspires us to introduce a set of independent, mean-zero random vectors:  {Zir}1≤i≤n,1≤r≤Ni,  with Zir = Bir − EBir, and Bir ∼ Multinomial(1, Ωi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  We use them to get a decomposition of T into mutually uncorrelated terms:  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let {Zir}1≤i≤n,1≤r≤Ni be as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For each Zir ∈ Rp, let {Zijr}1≤j≤p  denote its p coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that ρ2 = �K  k=1 nk ¯Nk∥µk − µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For 1 ≤ j ≤ p, define  U1j  =  2  K  �  k=1  �  i∈Sk  Ni  �  r=1  (µkj − µj)Zijr,  U2j  =  K  �  k=1  �  i∈Sk  �  1≤r̸=s≤Ni  �  1  nk ¯Nk  −  1  n ¯N  �  Ni  Ni − 1ZijrZijs,  U3j  =  − 1  n ¯N  �  1≤k̸=ℓ≤K  �  i∈Sk  �  m∈Sℓ  Ni  �  r=1  Nm  �  s=1  ZijrZmjs,  24  U4j  =  K  �  k=1  �  i∈Sk,m∈Sk  i̸=m  Ni  �  r=1  Nm  �  s=1  �  1  nk ¯Nk  −  1  n ¯N  �  ZijrZmjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Then, T = ρ2 + �4  κ=1 1′  pUκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover, E[Uκ] = 0p and E[UκU ′  ζ] = 0p×p for 1 ≤ κ ̸= ζ ≤ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  The variance of T  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, the four terms {1′  pUκ}1≤κ≤4 are uncorrelated with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore,  Var(T) = Var(1′  pU1) + Var(1′  pU2) + Var(1′  pU3) + Var(1′  pU4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It suffices to study the variance of each of these four terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let U1 be the same as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  Θn1 = 4  K  �  k=1  nk ¯Nk  ��diag(µk)1/2(µk − µ)  ��2  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  Ln = 4  K  �  k=1  nk ¯Nk  ��Σ1/2  k  (µk − µ)  ��2  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  Then Var(1′  pU1) = Θn1 − Ln.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore, if max1≤k≤K ∥µk∥∞ = o(1), then Var(1′  pU1) =  o(ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let U2 be the same as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  Θn2 = 2  K  �  k=1  �  1  nk ¯Nk  −  1  n ¯N  �2 �  i∈Sk  N3  i  Ni − 1∥Ωi∥2  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  An = 2  K  �  k=1  �  1  nk ¯Nk  −  1  n ¯N  �2 �  i∈Sk  N3  i  Ni − 1∥Ωi∥3  3  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8)  Then  Θn2 − An ≤ Var(1′  pU2) ≤ Θn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Furthermore, if  max  1≤k≤K  ��  i∈Sk N2  i ∥Ωi∥3  3  �  i∈Sk N2  i ∥Ωi∥2  �  = o(1),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9)  then Var(1′  pU2) = [1 + o(1)] · Θn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let U3 be the same as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  Θn3 =  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j  NiNmΩijΩmj  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)  Bn = 2  �  k̸=ℓ  nknℓ ¯Nk ¯Nℓ  n2 ¯N2  1′  p(Σk ◦ Σℓ)1p  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11)  Then  Θn3 − Bn ≤ Var(1′  pU3) ≤ Θn3 + Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  25  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let U4 be the same as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  Θn4 = 2  K  �  k=1  �  i∈Sk,m∈Sk  i̸=m  �  j  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12)  En = 2  �  k  �  i∈Sk,m∈Sk,  i̸=m  �  1≤j,j′≤p  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩij′ΩmjΩmj′  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13)  Then  Θn4 − En ≤ Var(1′  pU4) ≤ Θn4 + En  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Using Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5, we derive regularity conditions such that the first term in  Var(1′  pUκ) is the dominating term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that Θn = Θn1 + Θn2 + Θn3 + Θn4, where the  quantity Θn is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The following intermediate result is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Then  Θn2 + Θn3 + Θn4 ≍  �  k  ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14)  Moreover, under the null hypothesis, Θn ≍ K∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The next result is useful in proving that our variance estimator V is asymptotically  unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds, and recall the definition of Θn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  βn =  max  � �  k  �  i∈Sk  N2  i  n2  k ¯  N2  k ∥Ωi∥3  3 , �  k ∥Σk∥2  F  �  K∥µ∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15)  If βn = o(1), then under the null hypothesis, Var(T) = [1 + o(1)] · Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We also study the case of K = 2 more explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the lemmas below we use the  notation from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' First we have an intermediate result analogous to Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  that holds under weaker conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider K = 2 and suppose that min Ni ≥ 2, min Mi ≥ 2 Then  Θn2 + Θn3 + Θn4 ≍  ����  m ¯  M  n ¯N + m ¯  M η +  n ¯N  n ¯N + m ¯  M θ  ����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Moreover, under the null hypothesis, Θn ≍ ∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The next result is a version of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7 for the case K = 2 that holds under weaker  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that mini Ni ≥ 2 and mini Mi ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  β(2)  n  =  max  � �  i N2  i ∥Ωi∥3, �  i M2  i ∥Γi∥3 , ∥Σ1∥2  F + ∥Σ2∥2  F  �  ∥µ∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16)  If β(2)  n  = o(1), then under the null hypothesis, Var(T) = [1 + o(1)] · Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  26  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  The decomposition of V  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let {Zir}1≤i≤n,1≤r≤Ni be as in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that  V = 2  K  �  k=1  �  i∈Sk  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2� NiX2  ij  Ni − 1 − NiXij(Ni − Xij)  (Ni − 1)2  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17)  +  2  n2 ¯N2  �  1≤k̸=ℓ≤K  �  i∈Sk  �  m∈Sℓ  p  �  j=1  XijXmj + 2  K  �  k=1  �  i∈Sk,m∈Sk,  i̸=m  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2  XijXmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define  θi =  �  1  nk ¯Nk  −  1  n ¯N )2  N3  i  Ni − 1  for i ∈ Sk ,  and let  αim =  �  2  n2 ¯  N2  if i ∈ Sk, m ∈ Sℓ, k ̸= ℓ  2  �  1  nk ¯  Nk −  1  n ¯  N )2  if i, m ∈ Sk  If we let  A1 =  �  i  Ni  �  r=1  �  j  �4θiΩij  Ni  +  �  m∈[n]\\{i}  2αimNmΩmj  �  Zijr,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='18)  A2 =  �  i  �  r̸=s∈[Ni]  2θi  Ni(Ni − 1)  � �  j  ZijrZijs  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='19)  A3 =  �  i̸=m  Ni  �  r=1  Nm  �  s=1  αim  � �  j  ZijrZmjs  �  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='20)  then these terms are mean zero, are mutually uncorrelated, and satisfy  V = A1 + A2 + A3 + Θn2 + Θn3 + Θn4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='21)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  Properties of V  First we control the variance of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let A1, A2, and A3 be defined as in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Then  Var(A1) ≲  1  n ¯N ∥µ∥3  3 +  �  k  ∥µk∥3  3  nk ¯Nk  ≲  �  k  ∥µk∥3  3  nk ¯Nk  Var(A2) ≲  �  k  �  i∈Sk  N2  i ∥Ωi∥2  2  n4  k ¯N4  k  ≲  �  k  ∥µk∥2  n2  k ¯N2  k  Var(A3) ≲  �  k  ∥µk∥2  n2  k ¯N2  k  +  1  n2 ¯N2 ∥µ∥2 ≲  �  k  ∥µk∥2  n2  k ¯N2  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next we show consistency of V under the null, which is crucial in properly standardizing  our test statistic and establishing asymptotic normality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  27  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall the definition of βn in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that βn = o(1) and that  the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If under the null hypothesis we have  K2∥µ∥4 ≫  �  k  ∥µ∥2  n2  k ¯N2  k  ∨  �  k  ∥µ∥3  3  nk ¯Nk  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22)  then V/VarT → 1 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To later control the type II error, we must also show that V does not dominate the  true variance under the alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We first state an intermediate result that is useful  throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that, under either the null or alternative, maxi ∥Ωi∥∞ ≤ 1 − c0  holds for an absolute constant c0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Then  Var(T) ≳ Θn2 + Θn3 + Θn4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23)  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that under the alternative (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds and  � �  k  ∥µk∥2�2 ≫  �  k  ∥µk∥2  n2  k ¯N2  k  ∨  �  k  ∥µk∥3  3  nk ¯Nk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='24)  Then V = OP(Var(T)) under the alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We also require versions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 that hold under weaker  conditions in the special case K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We omit the proofs as they are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Below we use  the notation of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that K = 2 and recall the definition of β(2)  n  in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose  that β(2)  n  = o(1), mini Ni ≥ 2, mini Mi ≥ 2, and maxi ∥Ωi∥∞ ≤ 1 − c0, maxi ∥Γi∥∞ ≤ 1 − c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  If under the null hypothesis  ∥µ∥4 ≫ max  � � ∥µ∥2  2  n2 ¯N2 + ∥µ∥2  2  m2 ¯  M2  2  �  ,  �∥µ∥3  3  n ¯N + ∥µ∥3  3  m ¯  M  ��  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='25)  then V/Var(T) → 1 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Under the alternative we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that K = 2, mini Ni ≥ 2, mini Mi ≥ 2, and maxi ∥Ωi∥∞ ≤  1 − c0, maxi ∥Γi∥∞ ≤ 1 − c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If under the alternative  ����  m ¯  M  n ¯N + m ¯  M η +  n ¯N  n ¯N + m ¯  M θ  ����  4  ≫ max  � � ∥η∥2  2  n2 ¯N2 + ∥θ∥2  2  m2 ¯  M2  2  �  ,  �∥η∥3  3  n ¯N + ∥θ∥3  3  m ¯  M  ��  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='26)  then V = OP(Var(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In the setting of K = n and utilize the variance estimator V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The next results capture  the behavior of V ∗ under the null and alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The proofs are given later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  28  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  β(n)  n  =  �  i ∥Ωi∥3  n∥µ∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='27)  Suppose that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds, β(n)  n  = o(1), and  n2∥µ∥4 ≫  �  i  ∥µ∥2  N2  i  ∨  �  i  ∥µ∥3  3  Ni  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='28)  Then V ∗/Var(T) → 1 in probability as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that under the alternative (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds and  � �  i  ∥Ωi∥2�2 ≫  �  i  ∥Ωi∥2  N2  i  ∨  �  i  ∥Ωi∥3  3  Ni  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='29)  Then V ∗ = OP(Var(T)) under the alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  We first show that E[Uκ] = 0p and E[UκU ′  ζ] = 0p×p for κ ̸= ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Note that {Zir}1≤i≤n,1≤r≤Ni  are independent mean-zero random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that each Uκ is a mean-zero random  vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We then compute E[Uκj1Uζj2] for κ ̸= ζ and all 1 ≤ j1, j2 ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direct calculations,  E[U1jU2j2] = 2  �  (k,i,r,s)  �  (k′,i′,r′)  �  1  nk ¯Nk  −  1  n ¯N  �  (µk′j − µj)  Ni  Ni − 1E[Zij2rZij2sZi′j1r′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  If i′ ̸= i, or if i′ = i and r′ /∈ {r, s}, then Zi′j1r′ is independent of Zij2rZij2s, and it follows  that E[Zij2rZij2sZi′j1r′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If i′ = i and r = r′, then E[Zij2rZij2sZi′j1r′] = E[Zij2rZij1r] ·  E[Zij2s];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' since r ̸= s, we also have E[Zij2rZij2sZi′j1r′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This proves E[U1jU2j∗] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Since this holds for all 1 ≤ j1, j2 ≤ p, we immediately have  E[U1U ′  2] = 0p×p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We can similarly show that E[UκU ′  ζ] = 0p×p, for other κ ̸= ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The proof is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It remains to prove the desirable decomposition of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that T = �p  j=1 Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write  ρ2 = �p  j=1 ρ2  j, where ρ2  j = 2 �K  k=1 nk ¯Nk(µkj − µj)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It suffices to show that  Tj = ρ2  j + U1j + U2j + U3j + U4j,  for all 1 ≤ j ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='30)  To prove (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='30), we need some preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  Yij := Xij  Ni  − Ωij = 1  Ni  Ni  �  r=1  Zijr,  Qij := Y 2  ij − EY 2  ij = Y 2  ij − Ωij(1 − Ωij)  Ni  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='31)  With these notations, Xij = Ni(Ωij + Yij) and NiY 2  ij = NiQij + Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover, we  can use (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='31) to re-write Qij as a function of {Zijr}1≤r≤Ni as follows:  Qij =  1  N2  i  Ni  �  r=1  [Z2  ijr − Ωij(1 − Ωij)] + 1  N2  i  �  1≤r̸=s≤Ni  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  29  Note that Zijr = Bijr−Ωij, where Bijr can only take values in {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hence, (Zijr+Ωij)2 =  (Zijr +Ωij) always holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Re-arranging the terms gives Z2  ijr −Ωij(1−Ωij) = (1−2Ωij)Zijr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  Qij = (1 − 2Ωij)Yij  Ni  + 1  N2  i  �  1≤r̸=s≤Ni  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32)  This is a useful equality which we will use in the proof below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We now show (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Fix j and write Tj = Rj − Dj, where  Rj =  K  �  k=1  nk ¯Nk(ˆµkj − ˆµj)2,  and  Dj =  K  �  k=1  �  i∈Sk  ξk  Xij(Ni − Xij)  nk ¯Nk(Ni − 1) ,  with ξk = 1 − nk ¯Nk  n ¯N  First, we study Dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Note that Xij(Nij − Xij) = N2  i (Ωij + Yij)(1 − Ωij − Yij) = N2  i Ωij(1 −  Ωij) − N2  i Y 2  ij + N2  i (1 − 2Ωij)Yij, where Y 2  ij = Qij + N−1  i  Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  Xij(Nij − Xij)  Ni(Ni − 1)  = Ωij(1 − Ωij) − NiQij  Ni − 1 +  Ni  Ni − 1(1 − 2Ωij)Yij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We apply (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32) to get  Xij(Nij − Xij)  Ni(Ni − 1)  = Ωij(1 − Ωij) + (1 − 2Ωij)Yij −  1  Ni(Ni − 1)  �  1≤r̸=s≤Ni  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='33)  It follows that  Dj =  K  �  k=1  �  i∈Sk  ξkNi  nk ¯Nk  Ωij(1 − Ωij) +  K  �  k=1  �  i∈Sk  ξkNi  nk ¯Nk  (1 − 2Ωij)Yij  −  K  �  k=1  �  i∈Sk  ξk  nk ¯Nk(Ni − 1)  �  1≤r̸=s≤Ni  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='34)  Next, we study Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Note that nk ¯Nk(ˆµkj − ˆµj) = �  i∈Sk(Xij − ¯Nkˆµj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  Rj =  K  �  k=1  1  nk ¯Nk  ��  i∈Sk  (Xij − ¯Nkˆµj)  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Recall that Xij = Ni(Ωij+Yij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direct calculations, �  i∈Sk Xij = nk ¯Nkµkj+�  i∈Sk NiYij,  and ˆµj = µj + (n ¯N)−1 �n  m=1 NmYmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We then have the following decomposition:  �  i∈Sk  (Xij − ¯Nkˆµj) = nk ¯Nk(µkj − µj) +  �  i∈Sk  NiYij − nk ¯Nk  n ¯N  �  n  �  m=1  NmYmj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Using this decomposition, we can expand [�  i∈Sk(Xij − ¯Nkˆµj)]2 to a total of 6 terms, where  3 are quadratic terms and 3 are cross terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It yields a decomposition of Rj into 6 terms:  Rj =  K  �  k=1  nk ¯Nk(µkj − µj)2 +  K  �  k=1  1  nk ¯Nk  ��  i∈Sk  NiYij  �2  +  K  �  k=1  nk ¯Nk  n2 ¯N2  �  n  �  m=1  NmYmj  �2  30  + 2  K  �  k=1  (µkj − µj)  ��  i∈Sk  NiYij  �  − 2  K  �  k=1  nk ¯Nk  n ¯N (µkj − µj)  �  n  �  m=1  NmYmj  �  −  2  n ¯N  K  �  k=1  ��  i∈Sk  NiYij  ��  n  �  m=1  NmYmj  �  ≡ I1 + I2 + I3 + I4 + I5 + I6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='35)  By definition, �K  k=1 nk ¯Nk = n ¯N and �K  k=1 nk ¯Nkµkj = n ¯Nµj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  I3 =  1  n ¯N  �  n  �  m=1  NmYmj  �2  ,  I5 = 0,  I6 = − 2  n ¯N  �  n  �  m=1  NmYmj  �2  = −2I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  Rj = I1 + I2 − I3 + I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='36)  We further simplify I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that ξk = 1 − (n ¯N)−1nk ¯Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direct calculations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  I3 =  1  n ¯N  �  n  �  m=1  NmYmj  �2  =  1  n ¯N  � K  �  k=1  ��  i∈Sk  NiYij  ��2  =  1  n ¯N  K  �  k=1  ��  i∈Sk  NiYij  �2  +  1  n ¯N  �  1≤k̸=ℓ≤K  ��  i∈Sk  NiYij  �� �  m∈Sℓ  NmYmj  �  =  K  �  k=1  (1 − ξk)  1  nk ¯Nk  ��  i∈Sk  NiYij  �2  +  1  n ¯N  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  NiNmYijYmj  �  ��  �  J1  = I2 −  K  �  k=1  �  i∈Sk  ξk  nk ¯Nk  ��  i∈Sk  NiYij  �2  + J1  = I2 + J1 −  K  �  k=1  ξk  nk ¯Nk  ��  i∈Sk  N2  i Y 2  ij  �  −  K  �  k=1  ξk  nk ¯Nk  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='m∈Sk  i̸=m  NiNmYijYmj  �  ��  �  J2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='37)  By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='31), NiY 2  ij = NiQi + Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We further apply (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32) to get  N2  i Y 2  ij = Ni(1 − 2Ωij)Yij +  �  1≤r̸=s≤Ni  ZijrZijs + NiΩij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  K  �  k=1  ξk  nk ¯Nk  ��  i∈Sk  N2  i Y 2  ij  �  =  K  �  k=1  �  i∈Sk  ξkNi  nk ¯Nk  (1 − 2Ωij)Yij  �  ��  �  J3  +  K  �  k=1  �  i∈Sk  ξk  nk ¯Nk  �  r̸=s  ZijrZijs  �  ��  �  J4  +  K  �  k=1  �  i∈Sk  ξkNi  nk ¯Nk  Ωij(1 − Ωij)  �  ��  �  J5  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38)  31  We plug (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38) into (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='37) to get I3 = I2 + J1 − J2 − J3 − J4 − J5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Further plugging I3  into the expression of Rj in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='36), we have  Rj = I1 + I4 − J1 + J2 + J3 + J4 + J5,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39)  where I1 and I4 are defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='35), J1-J2 are defined in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='37), and J3-J5 are defined in  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Finally, we combine the expressions of Dj and Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='34) and the definitions of  J1-J5,  Dj = J5 + J3 −  K  �  k=1  �  i∈Sk  ξk  nk ¯Nk(Ni − 1)  �  r̸=s  ZijrZijs  = J5 + J3 + J4 −  K  �  k=1  �  i∈Sk  ξkNi  nk ¯Nk(Ni − 1)  �  r̸=s  ZijrZijs  �  ��  �  J6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Combining it with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39) gives Tj = Rj − Dj = I1 + I4 − J1 + J2 + J6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We further plug in  the definition of each term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  Tj =  K  �  k=1  nk ¯Nk(µkj − µj)2 + 2  K  �  k=1  �  i∈Sk  (µkj − µj)NiYij −  1  n ¯N  �  k̸=ℓ  �  i∈Sk,m∈Sℓ  NiNmYijYmj  +  K  �  k=1  �  i∈Sk,m∈Sk  i̸=m  ξk  nk ¯Nk  NiNmYijYmj +  K  �  k=1  �  i∈Sk  ξkNi  nk ¯Nk(Ni − 1)  �  r̸=s  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='40)  We plug in Yij = N−1  i  �Ni  r=1 Zijr and take a sum of 1 ≤ j ≤ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It gives (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='30) immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Recall that {Zir}1≤i≤n,1≤r≤Ni are independent random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write  1′  pU1 = 2  K  �  k=1  �  i∈Sk  Ni  �  r=1  (µk − µ)′Zir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The covariance matrix of Zir is diag(Ωi) − ΩiΩ′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  Var(1′  pU1) = 4  K  �  k=1  �  i∈Sk  Ni  �  r=1  (µk − µ)′�  diag(Ωi) − ΩiΩ′  i  �  (µk − µ)  = 4  �  k  (µk − µ)′�  diag  ��  i∈Sk  NiΩi  �  −  ��  i∈Sk  NiΩiΩ′  i  ��  (µk − µ)  = 4  �  k  (µk − µ)′�  diag(nk ¯Nkµk) − nk ¯NkΣk  �  (µk − µ)  32  = 4  �  k  nk ¯Nk  ��diag(µk)1/2(µk − µ)  ��2 − 4  �  k  nk ¯Nk  ��Σ1/2  k  (µk − µ)  ��2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='41)  This proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='41),  Var(1′  pU1) ≤ 4  �  k  nk ¯Nk  ��diag(µk)1/2(µk − µ)  ��2 ≤ 4  �  k  nk ¯Nk∥diag(µk)∥∥µk − µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Note that ∥diag(µk)∥ = ∥µk∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore, if maxk ∥µk∥∞ = o(1), the right hand side  above is o(1) · 4 �  k nk ¯Nk∥µk − µ∥2 = o(ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This proves the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  For each 1 ≤ k ≤ K, define a set of index triplets: Mk = {(i, r, s) : i ∈ Sk, 1 ≤ r < s ≤ Ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let M = ∪K  k=1Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write for short θi = (  1  nk ¯  Nk −  1  n ¯  N )2 N3  i  Ni−1, for i ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is seen that  1′  pU2 = 2  �  (i,r,s)∈M  √θi  �  Ni(Ni − 1)  Wirs,  with  Wirs =  p  �  j=1  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For Wirs and Wi′r′s′, if i ̸= i′, or if i = i′ and {r, s}∩{r′, s′} = ∅, then these two variables are  independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' if i = i′, r = r′ and s ̸= s′, then E[WirsWirs′] = �  j,j′ E[ZijrZijsZij′rZij′s′] =  �  j,j′ E[ZijrZij′r]·E[Zijs]·E[Zij′s′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore, {Wirs}(i,r,s)∈M is a collection of mutually  uncorrelated variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  Var(1′  pU2) = 4  �  (i,r,s)∈M  θi  Ni(Ni − 1)Var(Wirs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It remains to calculate the variance of each Wirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direction calculations,  Var(Wirs) =  �  j  E[Z2  ijrZ2  ijs] + 2  �  j<ℓ  E[ZijrZijsZiℓrZiℓs]  =  �  j  [Ωij(1 − Ωij)]2 + 2  �  j<ℓ  (−ΩijΩiℓ)2  =  �  j  Ω2  ij − 2  �  j  Ω3  ij +  ��  j  Ω2  ij  �2  = ∥Ωi∥2 − 2∥Ωi∥3  3 + ∥Ωi∥4  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='42)  Since maxij Ωij ≤ 1, we have  ∥Ωi∥2 − ∥Ωi∥3  3 ≤ Var(Wirs) ≤ ∥Ωi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Therefore,  Var(1′  pU2) = 4  K  �  k=1  �  i∈Sk  �  1≤r<s≤Ni  θi  Ni(Ni − 1)Var(Wirs)  33  = 2  K  �  k=1  �  i∈Sk  θiVar(Wirs) ≥ 2  K  �  k=1  �  i∈Sk  θi  �  ∥Ωi∥2 − ∥Ωi∥3  3  �  = Θn2 − An,  and similarly Var(1′  pU2) ≤ Θn2, which proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To prove the second claim,  note that Var(1′  pU2) = Θn2 + O(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9) and the assumption min Ni ≥ 2, we have  An ≲  �  k  �  1  nk ¯Nk  −  1  n ¯N  �2 �  i∈Sk  N2  i ∥Ωi∥3  3  =  �  k  �  1  nk ¯Nk  −  1  n ¯N  �2 · o  � �  i∈Sk  N2  i ∥Ωi∥2  �  = o(Θn2),  which implies that Var(1pU2) = [1 + o(1)]Θn2, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  For each 1 ≤ k < ℓ ≤ K, define a set of index quadruples: Jkℓ = {(i, r, m, s) : i ∈ Sk, j ∈  Sℓ, 1 ≤ r ≤ Ni, 1 ≤ s ≤ Nm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let J = ∪(k,ℓ):1≤k<ℓ≤KJkℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is seen that  1′  pU3 = − 2  n ¯N  �  (i,r,m,s)∈J  Virms,  where Virms =  p  �  j=1  ZijrZmjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For Virms and Vi′r′m′s′, if {(i, r), (m, s)} ∩ {(i′, r′), (m′, s′)} = ∅, then the two variables are  independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If (i, r) = (i′, r′) and (m, s) ̸= (m′, s′), then E[VirmsVirm′s′] =  �  j,j′ E[ZijrZmjsZij′rZm′j′s′] = �  j,j′ E[ZijrZij′r] · E[Zmjs] · E[Zm′js′] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore, the  only correlated case is when (i, r, m, s) = (i′, r′, m′, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This implies that {Virms}(i,r,m,s)∈J  is a collection of mutually uncorrelated variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore,  Var(1′  pU3) =  4  n2 ¯N2  �  (i,r,m,s)∈J  Var(Virms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Note that Var(Virms) = E[(�  j ZijrZmjs)2] = �  j,j′ E[ZijrZmjsZij′rZmj′s];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' also, the covari-  ance matrix of Zir is diag(Ωi) − ΩiΩ′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  Var(Virms) =  �  j  E[Z2  ijr] · E[Z2  mjs] +  �  j̸=j′  E[ZijrZij′r] · E[ZmjsZmj′s]  =  �  j  Ωij(1 − Ωij)Ωmj(1 − Ωmj) +  �  j̸=j′  ΩijΩij′ΩmjΩmj′  =  �  j  ΩijΩmj − 2  �  j  Ω2  ijΩ2  mj +  �  j,j′  ΩijΩij′ΩmjΩmj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='43)  Write for short δim = −2 �  j Ω2  ijΩ2  mj + �  j,j′ ΩijΩij′ΩmjΩmj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='Combining the above gives  Var(1′  pU3) =  4  n2 ¯N2  �  k<ℓ  �  i∈Sk  �  m∈Sℓ  Ni  �  r=1  Nm  �  s=1  ��  j  ΩijΩmj + δim  �  34  =  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j  NiNmΩijΩmj +  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  NiNmδim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44)  It is easy to see that |δim| ≤ �  j,j′ ΩijΩij′ΩmjΩmj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Also, by the definition of Σk in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2),  we have Σk(j, j′) =  1  nk ¯  Nk  �  i∈Sk NiΩijΩij′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Using these results, we immediately have  ���  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  NiNmδim  ��� ≤  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j,j′  NiNmΩijΩij′ΩmjΩmj′  =  2  n2 ¯N2  �  j,j′  �  k̸=ℓ  ��  i∈Sk  NiΩijΩij′  �� �  m∈Sℓ  NiΩmjΩmj′  �  =  2  n2 ¯N2  �  j,j′  �  k̸=ℓ  nk ¯NkΣk(j, j′) · nℓ ¯NℓΣℓ(j, j′)  = 2  �  k̸=ℓ  nknℓ ¯Nk ¯Nℓ  n2 ¯N2  1′  p(Σk ◦ Σℓ)1p =: Bn  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='45)  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  For 1 ≤ k ≤ K, define a set of index quadruples: Qk = {(i, r, m, s) : i ∈ Sk, m ∈ Sk, i <  m, 1 ≤ r ≤ Ni, 1 ≤ s ≤ Nm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let Q = ∪K  k=1Qk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write κim = (  1  nk ¯  Nk −  1  n ¯  N )2NiNm, for  i ∈ Sk and m ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is seen that  1′  pU4 = 2  �  (i,r,m,s)∈Q  √κim  √NiNm  Virms,  where  Virms =  p  �  j=1  ZijrZmjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It is not hard to see that Virms and Vi′r′m′s′ are correlated only if (i, r, m, s) = (i′, r′, m′, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  Var(1′  pU4) = 4  �  (i,r,m,s)∈Q  κim  NiNm  Var(Virms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4, we have studied Var(Virms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In particular, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='43), we have  Var(Virms) =  �  j  ΩijΩmj + δim,  with  |δim| ≤  �  j,j′  ΩijΩij′ΩmjΩmj′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  Var(1′  pU4) = 4  K  �  k=1  �  i∈Sk,m∈Sk  i<m  Ni  �  i=1  Nm  �  r=1  κim  NiNm  Var(Virms)  = 4  K  �  k=1  �  i∈Sk,m∈Sk  i<m  κim  ��  j  ΩijΩmj + δim  �  35  = 2  K  �  k=1  �  i∈Sk,m∈Sk  i̸=m  �  j  κimΩijΩmj ± 2  �  k  �  i̸=m∈Sk  κim  �  j,j′  ΩijΩij′ΩmjΩmj′,  = Θn3 ± En.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46)  which proves the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  By assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1), N3  i /(Ni − 1) ≍ Ni and  �  1  nk ¯  Nk −  1  n ¯  N  �2  ≍  1  n2  k ¯  N2  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' First, observe that  Θn2 + Θn4 = 2  K  �  k=1  �  1  nk ¯Nk  −  1  n ¯N  �2 �  i∈Sk  N3  i  Ni − 1∥Ωi∥2  + 2  K  �  k=1  �  i∈Sk,m∈Sk  i̸=m  �  j  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩmj  ≍  �  k=1  �  1  nk ¯Nk  �2 �  j  �  i,m∈Sk  NiΩij · NmΩij =  �  k  ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='47)  Recall the definitions of µk and µ in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direct calculations, we have  Θn3 = 2  �  j  �  k̸=ℓ  � 1  n ¯N  �  i∈Sk  NiΩij  �� 1  n ¯N  �  m∈Sℓ  NmΩmj  �  = 2  �  j  �  k̸=ℓ  nk ¯Nk  n ¯N µkj · nℓ ¯Nℓ  n ¯N µℓj  = 2  �  k̸=ℓ  nknℓ ¯Nk ¯Nℓ  n2 ¯N2  µ ′  k µℓ  ≤ 2  �  j  ��  k  nk ¯Nk  n ¯N µkj  �2  = 2  �  j  µ2  j = 2∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='48)  By Cauchy–Schwarz,  ∥µ∥2 =  �  j  � �  k  (nk ¯  Nk  n ¯N )µkj  �2  ≤  �  j  � �  k  (nk ¯  Nk  n ¯N )2  � � �  k  µ2  kj  �  ≤  �  j  � �  k  (nk ¯  Nk  n ¯N )  � � �  k  µ2  kj  �  =  �  j  �  k  µ2  kj =  �  k  ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='49)  Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='47), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='48), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='49) yields  c  � �  k  ∥µk∥2�  ≤ Θn2 + Θn3 + Θn4 ≤ C  � �  k  ∥µk∥2�  ,  for absolute constants c, C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  36  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7  By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1), it holds that  (  1  nk ¯Nk  −  1  n ¯N )2 ≍  1  (nk ¯Nk)2 ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='50)  and moreover, for all i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , n},  N3  i  Ni − 1 ≍ N2  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='51)  Recall the definitions of An, Bn, and En in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Note  that these are the remainder terms in Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the  null hypothesis (recall Θn1 ≡ 0 under the null),  Var(T) = Θn2 + Θn3 + Θn4 + O(An + Bn + En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52)  It holds that  An ≤  K  �  k=1  �  1  nk ¯Nk  �2 �  i∈Sk  N2  i ∥Ωi∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53)  Next, by linearity and the definition of Σk, Σ in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3), respectively,  Bn ≤ 2  �  k,ℓ  nknℓ ¯Nk ¯Nℓ  n2 ¯N2  1′  p(Σk ◦ Σℓ)1p  ≤ 21′  p  � 1  n ¯N  �  k  nk ¯NkΣk  � � 1  n ¯N  �  ℓ  nℓ ¯NℓΣkℓ  �  1p  = 21′  p(Σ ◦ Σ)1p = 2∥Σ∥2  F  By Cauchy–Schwarz,  Bn ≤ ∥Σ∥2  F =  �  j,j′  � �  k  (nk ¯Nk  n ¯N Σk(j, j′)  �2  ≤  �  j,j′  � �  k  (nk ¯Nk  n ¯N )2  � � �  k  Σk(j, j′)2  �  ≤  �  j,j′  � �  k  nk ¯Nk  n ¯N  � � �  k  Σk(j, j′)2  �  =  �  j,j′  �  k  Σk(j, j′)2 =  �  k  ∥Σk∥2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='54)  Next by the definition of Σk in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2), we have Σk(j, j′) =  1  nk ¯  Nk  �  i∈Sk NiΩijΩij′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It  follows that  En ≤  �  k  �  j,j′  �  1  nk ¯Nk  �  i∈Sk  NiΩijΩij′  ��  1  nk ¯Nk  �  m∈Sk  NmΩmjΩmj′  �  =  �  k  �  j,j′  Σ2  k(j, j′) =  �  k  ∥Σk∥2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55)  37  Next, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 implies that  Θn2 + Θn3 + Θn4 ≍  �  k  ∥µk∥2 = K∥µ∥2,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='56)  where we use that the null hypothesis holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By assumption of the lemma, we have  βn =  max  � �  k  �  i∈Sk  N2  i  n2  k ¯  N2  k ∥Ωi∥3  3 , �  k ∥Σk∥2  F  �  K∥µ∥2  = o(1)  Combining this with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='54), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55),and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='56) completes the proof of the  first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The second claim follows plugging in µk = µ for all k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8  By assumption, N3  i /(Ni − 1) ≍ Ni, M3  i /(Mi − 1) ≍ Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direct calculation,  Θn2 + Θn4 ≍  �  m ¯  M  (n ¯N + m ¯  M)n ¯N  �2 �  i,m,j  NiNmΩijΩmj +  �  n ¯N  (n ¯N + m ¯  M)m ¯  M  �2 �  i,m  NiNmΓijΓmj  =  1  (n ¯N + m ¯  M)2  �  (m ¯  M)2∥η∥2 + n ¯N2∥θ∥2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='57)  Next  Θn3 =  4  (n ¯N + m ¯  M)2  �  i∈S1  �  m∈S2  �  j  NiΩij · NmΓmj  =  4  (n ¯N + m ¯  M)2 · n ¯Nm ¯  M⟨θ, η⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='58)  Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='57) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='58) yields  Θn2 + Θn3 + Θn4 ≍  1  (n ¯N + m ¯  M)2  �  (m ¯  M)2∥η∥2 + 2n ¯Nm ¯  M⟨θ, η⟩ + n ¯N2∥θ∥2�  =  ����  m ¯  M  n ¯N + m ¯  M η +  n ¯N  n ¯N + m ¯  M θ  ����  2  ,  which proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The second follows by plugging in θ = η = µ under the  null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9  As in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52), we have under the null that  Var(T) = Θn2 + Θn3 + Θn4 + O(An + Bn + En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='59)  For general K, observe that the proofs of the bounds  An ≤  K  �  k=1  �  1  nk ¯Nk  �2 �  i∈Sk  N2  i ∥Ωi∥3  3  38  Bn ≤  K  �  k=1  ∥Σk∥2  F  En ≤  K  �  k=1  ∥Σk∥2  F  derived in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='54), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55), only use the assumption that Ni, Mi ≥ 2 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Translating these bounds to the notation of the K = 2 case, we have  An ≤  �  i  N2  i ∥Ωi∥3 +  �  i  M2  i ∥Γi∥3  Bn ≤ ∥Σ1∥2  F + ∥Σ2∥2  F  En ≤ ∥Σ1∥2  F + ∥Σ2∥2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='60)  Furthermore, we know that Θn ≥ c∥µ∥2 under the null by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8, for an absolute  constant c > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Combining this with (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='59) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='60) completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10  Define  V1 = 2  K  �  k=1  �  i∈Sk  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2� NiX2  ij  Ni − 1 − NiXij(Ni − Xij)  (Ni − 1)2  �  V2 =  2  n2 ¯N2  �  1≤k̸=ℓ≤K  �  i∈Sk  �  m∈Sℓ  p  �  j=1  XijXmj  V3 = 2  K  �  k=1  �  i∈Sk,m∈Sk,  i̸=m  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2  XijXmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that V1 + V2 + V3 = V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Also define  A11 =  �  i  Ni  �  r=1  �  j  �4θiΩij  Ni  �  Zijr  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='61)  A12 = 2  �  i  Ni  �  r=1  �  j  �  �  m∈[n]\\{i}  αimNmΩmj  �  Zijr  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='62)  and observe that A11 + A12 = A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  First, we derive the decomposition of V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that  Yij := Xij  Ni  − Ωij = 1  Ni  Ni  �  r=1  Zijr,  Qij := Y 2  ij − EY 2  ij = Y 2  ij − Ωij(1 − Ωij)  Ni  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='63)  With these notations, Xij = Ni(Ωij + Yij) and NiY 2  ij = NiQij + Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  39  Write  V1 = 2  n  �  i=1  n  �  i=1  θi  Ni  ∆ij,  where  ∆ij :=  X2  ij  Ni  − Xij(Ni − Xij)  Ni(Ni − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='64)  Note that Xij = Ni(Ωij + Yij) and Y 2  ij = Qij + N−1  i  Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  X2  ij  Ni  = NiΩ2  ij + 2NiΩijYij + NiQij + Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32), we have shown that Qij = (1−2Ωij) Yij  Ni + 1  N2  i  �  1≤r̸=s≤Ni ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  X2  ij  Ni  = NiΩ2  ij + 2NiΩijYij + (1 − 2Ωij)Yij + 1  Ni  �  1≤r̸=s≤Ni  ZijrZijs + Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Additionally, by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='33),  Xij(Nij − Xij)  Ni(Ni − 1)  = Ωij(1 − Ωij) + (1 − 2Ωij)Yij −  1  Ni(Ni − 1)  �  1≤r̸=s≤Ni  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Combining the above gives  ∆ij = NiΩ2  ij + 2NiΩijYij +  1  Ni − 1  �  1≤r̸=s≤Ni  ZijrZijs  = NiΩ2  ij + 2Ωij  Ni  �  r=1  Zijr +  1  Ni − 1  �  1≤r̸=s≤Ni  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='65)  Recall the definition of Θn2 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7), A2 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='19), and A11 in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  V1 = 2  �  k,i∈Sk  �  j  θi  Ni  �  NiΩ2  ij + 2Ωij  Ni  �  r=1  Zijr +  1  Ni − 1  �  1≤r̸=s≤Ni  ZijrZijs  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  = Θn2 +  �  k,i∈Sk  �  j  4θiΩij  Ni  Ni  �  r=1  Zijr +  �  k,i∈Sk  �  j  2θi  Ni(Ni − 1)  �  1≤r̸=s≤Ni  ZijrZijs  = Θn2 + A11 + A2  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='66)  Next, we have  V2 + V3 =  �  i̸=m  αimNiNm  �  j  �  (Yij + Ωij)(Ymj + Ωmj)  �  =  �  i̸=m  αimNiNm  �  j  YijYmj + 2  �  i̸=m  αimNiNm  �  j  YijΩmj +  �  i̸=m  αimNiNm  �  j  ΩijΩmj  =  �  i̸=m  Ni  �  r=1  Nm  �  s=1  αim  � �  j  ZijrZmjs  �  + 2  �  i  Ni  �  r=1  �  j  �  �  m∈[n]\\{i}  αimNmΩmj  �  Zijr + Θn3 + Θn4  = A3 + A12 + Θn3 + Θn4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  40  Hence  A1 + A2 + A3 + Θn2 + Θn3 + Θn4 = V,  which verifies (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By inspection, we also see that EAb = 0 for b ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' That  A1, A2, A3 are mutually uncorrelated follows immediately from the linearity of expecta-  tion and the fact that the random variables {Zijr}i,r ∪ {ZijrZmjs}(i,r)̸=(m,s) are mutually  uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11  Define  γirj = 4θiΩij  Ni  +  �  m∈[n]\\{i}  2αimNmΩmj  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='67)  and recall that A1 = �  i  �  r∈[Ni]  �  j γirjZijr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' First we develop a bound on γirj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose  that i ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Then we have  γirj ≲ NiΩij  n2  k ¯N2  k  +  �  m∈Sk,m̸=i  NmΩmj  n2  k ¯N2  k  +  �  k′∈[K]\\{k}  �  m∈Sk′  NmΩmj  n2 ¯N2  ≲  µkj  nk ¯Nk  + µj  n ¯N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next using properties of the covariance matrix of a multinomial vector, we have  Var(A1) =  �  i,r∈[Ni]  Var(γ′  ir:Zi:r) =  �  i,r∈[Ni]  γ′  ir:Cov(Zi:r)γir:  ≤  �  i,r∈[Ni]  γ′  ir:diag(Ωi:)γir: =  �  i,r∈[Ni]  �  j  Ωijγ2  irj  ≲  �  k,j  � µkj  nk ¯Nk  + µj  n ¯N  �2  �  i∈Sk,r∈[Ni]  Ωij  ≲  �  k,j  � µkj  nk ¯Nk  �2nk ¯Nkµkj +  �  k,j  � µj  n ¯N  �2nk ¯Nkµkj  = (  �  k  ∥µk∥3  3  nk ¯Nk  ) + ∥µ∥3  3  n ¯N ≲  �  k  ∥µk∥3  3  nk ¯Nk  ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='68)  which proves the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The last inequality follows because by Jensen’s inequality  (noting that the function x �→ x3 is convex for x ≥ 0),  ∥µ∥3  3 =  �  j  � �  k  (nk ¯Nk  n ¯N )µkj  �3  ≤  �  j  �  k  (nk ¯Nk  n ¯N )µ3  kj ≤  �  k  ∥µk∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next observe that  A2 =  �  i  �  r̸=s  2θi  Ni(Ni − 1)Wirs  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='69)  41  where recall Wirs = �  j ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Also recall that Wirs and Wi′r′s′ are uncorrelated unless  i = i′ and {r, s} = {r′, s′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='42),  Var(A2) =  �  i  �  r̸=s  4θ2  i  N2  i (Ni − 1)2 Var(Wirs)  ≲  �  i  �  r̸=s  4θ2  i  N2  i (Ni − 1)2 ∥Ωi∥2  ≲  �  k  �  i∈Sk  (  1  nk ¯Nk  −  1  n ¯N )4  N6  i  (Ni − 1)2 ·  1  Ni(Ni − 1)∥Ωi∥2  ≲  �  k  �  i∈Sk  N2  i  n4  k ¯N4  k  ∥Ωi∥2  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='70)  Also observe that  �  k  1  n4  k ¯N4  k  �  i∈Sk  N2  i ∥Ωi∥2  2 ≤  �  k  1  n2  k ¯N2  k  �  i,m∈Sk  �  ( Ni  nk ¯Nk  )Ωi, ( Nm  nm ¯Nm  )Ωm  �  =  �  k  1  n2  k ¯N2  k  ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  This establishes the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Last we study A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that  A3 =  �  i̸=m  Ni  �  r=1  Nm  �  s=1  αimVirms  where recall Virms = �  j ZijrZmjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that Virms and Vi′r′m′s′ are uncorrelated unless  (r, s) = (r′, s′) and {i, m} = {i′, m′} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='43),  Var(A3) ≲  �  i̸=m  α2  imNiNm  �  j  ΩijΩmj  ≲  �  k  �  i̸=m∈Sk  1  n4  k ¯N4  k  ⟨NiΩi, NmΩm⟩ +  �  k̸=ℓ  �  i∈Sk,m∈Sℓ  1  n4 ¯N4 ⟨NiΩi, NmΩm⟩  ≲  �  k  ∥µk∥2  n2  k ¯N2  k  +  �  k,ℓ  1  n4 ¯N4 ⟨nk ¯Nkµk, nℓ ¯Nℓµℓ⟩  ≲  �  k  ∥µk∥2  n2  k ¯N2  k  + ∥µ∥2  n2 ¯N2 ≲  �  k  ∥µk∥2  n2  k ¯N2  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='71)  In the last line we use that ∥µ∥2 ≤ 2 � ∥µk∥2 as shown in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This proves all required  claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16  Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Under the null hypothesis, we have Θn1 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus, EV = Θn under the null by Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1), we have Var(T) = [1 + o(1)]Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore,  EV = [1 + o(1)]Var(T),  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='72)  42  so V is asymptotically unbiased under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6, we have  Θn ≍ K∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='73)  In Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11, we showed that  Var(A2) ≲  �  k  �  i∈Sk  N2  i ∥Ωi∥2  2  n4  k ¯N4  k  We conclude by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11 that under the null  Var(V ) ≲  �  k  ∥µ∥2  n2  k ¯N2  k  ∨  �  k  ∥µ∥3  3  nk ¯Nk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='74)  By Chebyshev’s inequality, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='73), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='74), and assumption (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22) of the theorem state-  ment, we have  |V − EV |  Var(T)  ≍ |V − EV |  K∥µ∥2  = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='72),  V  Var(T) = (V − EV )  Var(T)  +  EV  Var(T) = oP(1) + [1 + o(1)],  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12  By Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5, we have  Var(T) =  4  �  a=1  Var(1′  pUa) ≥ (  4  �  a=2  Θna) − (An + Bn + En).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='75)  Using that maxi ∥Ωi∥∞ ≤ 1 − c0, we have ∥Ωi∥3 ≤ (1 − c0)∥Ωi∥2, which implies that  An ≤ (1 − c0)Θn2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='76)  Again using maxi ∥Ωi∥∞ ≤ 1 − c0, as well as �  j′ Ωij′ = 1, we have  Bn =  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j,j′  NiNmΩijΩij′ΩmjΩmj′  ≤ (1 − c0) ·  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j,j′  NiNmΩijΩij′Ωmj  = (1 − c0) ·  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j  NiNmΩijΩmj  ≤ (1 − c0) · Θn3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='77)  43  Similarly to control En, we again use maxi ∥Ωi∥∞ ≤ 1 − c0 and obtain  En = 2  �  k  �  i∈Sk,m∈Sk,  i̸=m  �  1≤j,j′≤p  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩij′ΩmjΩmj′  ≤ (1 − c0) · 2  �  k  �  i∈Sk,m∈Sk,  i̸=m  �  1≤j,j′≤p  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩij′Ωmj  ≤ (1 − c0) · 2  �  k  �  i∈Sk,m∈Sk,  i̸=m  �  1≤j≤p  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩmj  ≤ (1 − c0) · Θn4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='78)  Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='75), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='76), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='77), and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='78) finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='18  Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  By Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12,  Var(T) ≳ Θn2 + Θn3 + Θn4 ≳  �  k  ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='79)  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11,  Var(V ) ≲  �  k  ∥µk∥2  n2  k ¯N2  k  ∨  �  k  ∥µk∥3  3  nk ¯Nk  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='80)  Using a similar argument based on Chebyshev’s inequality as in the proof of Proposition  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 and applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='79) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='80), we have  |V − EV |  Var(T)  ≳ |V − EV |  �  k ∥µk∥2 = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='81)  Next, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='79),  EV = Θn2 + Θn3 + Θn4 ≲ Var(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='82)  Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='81) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='82) finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='19  Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  From the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10, we have  V ∗ = V1 = Θn2 + A11 + A2,  and the terms on the right-hand-side are mutually uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='68), we have  Var(A11) ≲  �  i  ∥Ωi∥3  3  Ni  44  Var(A2) ≲  �  i  ∥Ωi∥2  N2  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Hence  EV ∗ = Θn2  Var(V ∗) ≲  �  i  ∥Ωi∥3  3  Ni  ∨  �  i  ∥Ωi∥2  N2  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='83)  Since K = n and the null hypothesis holds, we have Θn1 ≡ Θn4 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover, by  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='48), we have  Θn3 ≲ ∥µ∥2 ≪ Θn2 ≍ n∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  Var(T) = [1 + o(1)]Θn2 ≍ n∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='84)  Thus by (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='83) and Chebyshev’s inequality, we have  V ∗  Var(T) = V ∗ − EV ∗  Var(T)  +  EV ∗  Var(T) = oP(1) + 1 + o(1),  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='20  Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  By Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12,  Var(T) ≳ Θn2 + Θn3 ≳  �  i  ∥Ωi∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='85)  By (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='83),  Var(V ∗) ≲  �  i  ∥Ωi∥2  N2  i  ∨  �  i  ∥Ωi∥3  3  Ni  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='86)  Using a similar argument based on Chebyshev’s inequality as in the proof of Proposition  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 and applying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='85) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='86), we have  |V ∗ − EV ∗|  Var(T)  ≳ |V ∗ − EV ∗|  �  i ∥Ωi∥2  = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='87)  Next, by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='85),  EV ∗ = Θn2 ≲ Var(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='88)  Combining (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='81) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='88) finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  45  B  Proofs of asymptotic normality results  The goal of this section is to prove Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The argument relies on the  martingale central limit theorem and the lemmas stated below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  As a preliminary, we  describe a martingale decomposition of T under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define  U = 1′  p(U3 + U4),  and  S = 1′  pU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, we have T = U + S under the null hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  U =  �  i<i′  σi,i′  Ni  �  r=1  Ni′  �  s=1  � �  j  ZijrZi′js  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  where we define  σi,i′ =  �  2  �  1  nk ¯  Nk −  1  n ¯  N  �  if i, i′ ∈ Sk for some k  − 2  n ¯  N  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define a sequence of random variables  Dℓ,s =  �  i∈[ℓ−1]  σi,ℓ  Ni  �  r=1  �  j  ZijrZℓjs  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  indexed by (ℓ, s) ∈ {(i, r)}1≤i≤n,1≤r≤Ni, where these tuples are placed in lexicographical  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Precisely, we define  (ℓ1, s1) ≺ (ℓ2, s2)  if either  ℓ1 < ℓ2, or  ℓ1 = ℓ2 and s1 < s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that  �  ℓ,s  Dℓ,s = U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next define F≺(ℓ,s) to be the σ-field generated by {Zi:r}(i,r)≺(ℓ,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that  E[Dℓ,s|F≺(ℓ,s)] = 0,  and hence {Dℓ,s} is a martingale difference sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Turning to S, we have  S =  n  �  i=1  σi  �  r<s  �  j  ZijrZijs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  where we define  σi = 2  �  1  nk ¯Nk  −  1  n ¯N  �  Ni  Ni − 1  46  if i ∈ Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  Eℓ,s = σℓ  �  r∈[s−1]  �  j  ZℓjrZℓjs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  Note that Eℓ,1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Order (ℓ, s) lexicographically as above, and recall that F≺(ℓ,s) is the  σ-field generated by {Zi:r}(i,r)≺(ℓ,s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that  E[Eℓ,s|F≺(ℓ,s)] = 0,  and hence {Eℓ,s} is a martingale difference sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  �  (ℓ,s)  σℓ  �  r∈[s−1]  �  j  ZℓjrZℓjs =  n  �  ℓ=1  Nℓ  �  s=1  σℓ  �  r∈[s−1]  �  j  ZℓjrZℓjs = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define  Mℓ,s = Dℓ,s + Eℓ,s,  �  Mℓ,s =  Mℓ,s  �  Var(T)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  Thus we obtain the martingale decomposition:  T = U + S =  �  (ℓ,s)  [Dℓ,s + Eℓ,s] =  �  (ℓ,s)  Mℓ,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  The technical results below are crucial to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 given in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 then follows easily from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let �  Mℓ,s be defined as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  E  � �  (ℓ,s)  Var  � �  Mℓ,s  ��F≺(ℓ,s)  ��  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that min Ni ≥ 2 and max ∥Ωi∥∞ ≤ 1−c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the null hypothesis,  it holds that  Var  � �  (ℓ,s)  Var(Dℓ,s|F≺(ℓ,s))  �  ≲  � �  k  1  nk ¯Nk  �  ∥µ∥3  3 + K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that min Ni ≥ 2 and max ∥Ωi∥∞ ≤ 1−c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the null hypothesis,  it holds that  �  (ℓ,s)  ED4  ℓ,s ≲  � �  k  1  n2  k ¯N2  k  �  ∥µ∥2 +  � �  k  1  nk ¯Nk  �  ∥µ∥3  3 ,  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that min Ni ≥ 2 and and max ∥Ωi∥∞ ≤ 1 − c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Then we have  Var  � �  (ℓ,s)  Var( ˜Eℓ,s|F≺(ℓ,s))  �  ≲  �  k  �  i∈Sk  N3  i ∥Ωi∥3  3  n4  k ¯N4  k  ∨  �  k  �  i∈Sk  N4  i ∥Ωi∥4  4  n4  k ¯N4  k  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  47  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that min Ni ≥ 2 and and max ∥Ωi∥∞ ≤ 1 − c0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Then we have  �  (ℓ,s)  E E4  ℓ,s ≲  �  k  �  i∈Sk  N2  i ∥Ωi∥2  n4  k ¯N4  k  ∨  �  k  �  i∈Sk  N3  i ∥Ωi∥3  3  n4  k ¯N4  k  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under either the null or alternative, it holds that  �  k  �  i∈Sk  N2  i ∥Ωi∥2  n4  k ¯N4  k  ≤  �  k  1  n2  k ¯N2  k  ∥µk∥2  �  k  �  i∈Sk  N3  i ∥Ωi∥3  3  n4  k ¯N4  k  ≤  �  k  1  nk ¯Nk  ∥µk∥3  3  �  k  �  i∈Sk  N4  i ∥Ωi∥4  4  n4  k ¯N4  k  ≤  �  k  ∥µk∥4  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  By the martingale central limit theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hall and Heyde [2014]), we have that  T/  �  Var(T) ⇒ N(0, 1) if the following conditions are satisfied:  �  (ℓ,s)  Var  � �  Mℓ,s  ��F≺(ℓ,s)  � P→ 1  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8)  �  (ℓ,s)  E  � �  M2  ℓ,s1| �  Mℓ,s|>ε  ��F≺(ℓ,s)  � P→ 0,  for any ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9)  It is known that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9), which is a Lindeberg-type condition, is implied by the Lyapunov-type  condition  �  (ℓ,s)  E �  M4  ℓ,s = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)  See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds,  Var(T) ≳ Θ = Θn2 + Θn3 + Θn4 ≳ K∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11)  Recall that  �  Mℓ,s =  Mℓ,s  Var(T) = Dℓ,s + Eℓ,s  Var(T)  ,  Note that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8) holds if  E  �  Var  � �  Mℓ,s  ��F≺(ℓ,s)  ��  → 1,  and  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12)  Var  �  Var  � �  Mℓ,s  ��F≺(ℓ,s)  ��  → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13)  Recall that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12) holds by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  48  Next note that  E(Dℓ,sEℓ,s|F≺(ℓ,s)) = 0,  by inspection of the expressions for Dℓ,s and Eℓ,s in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore  Var(Mℓ,s|F≺(ℓ,s)) = Var(Dℓ,s|F≺(ℓ,s)) + Var(Eℓ,s|F≺(ℓ,s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Hence by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4 , and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' and the assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4), under the null  hypothesis, we have  Var  �  Var  � �  Mℓ,s  ��F≺(ℓ,s)  ��  ≤  1  Var(T)2  �  Var  �  Var  �  Dℓ,s  ��F≺(ℓ,s)  ��  + Var  �  Var  �  Eℓ,s  ��F≺(ℓ,s)  ���  ≲  1  K2∥µ∥4  �� �  k  1  nk ¯Nk  �  ∥µ∥3  3 + K∥µ∥4  4  �  ∥µ∥2  �  = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  This proves (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13) are established, which proves (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Similarly, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10) (and thus (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9)) holds by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Lemmas (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6), and  the assumption (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9) verifies the conditions of the martingale  central limit theorem, so we conclude that T/  �  Var(T) ⇒ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Since Var(T) = [1 +  o(1)]Θn by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4) and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We record a useful proposition that records the weaker conditions under which T/  �  Var(T)  is asymptotically normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that αn is defined as  αn := max  � K  �  k=1  ∥µk∥3  3  nk ¯Nk  ,  K  �  k=1  ∥µk∥2  n2  k ¯N2  k  � �� K  �  k=1  ∥µk∥2  �2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14)  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If under the null hypothesis,  αn = max  � K  �  k=1  ∥µk∥3  3  nk ¯Nk  ,  K  �  k=1  ∥µk∥2  n2  k ¯N2  k  � ��  K∥µ∥2  �2  → 0,  and  ∥µ∥4  4  K∥µ∥4 → 0,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15)  then T/  �  Var(T) ⇒ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  By our assumptions, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 holds and V/Var(T) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus the variance estimate  V is consistent under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 follows immediately from Slutsky’s theorem  and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, S and U are uncorrelated, and it holds that  Var(T) = Var(S) + Var(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16)  49  Next note that  E(Dℓ,sEℓ,s|F≺(ℓ,s)) = 0,  by inspection of the expressions for Dℓ,s and Eℓ,s in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore  Var(Mℓ,s|F≺(ℓ,s)) = Var(Dℓ,s|F≺(ℓ,s)) + Var(Eℓ,s|F≺(ℓ,s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that  E  � �  (ℓ,s)  Var(Eℓ,s|F≺(ℓ,s))  �  =  �  (ℓ,s)  EE2  ℓ,s =  �  (ℓ,s)  σ2  ℓ  �  r,r′∈[s−1]  �  j,j′  E[ZℓjrZℓjsZℓj′r′Zℓj′s]  =  �  (ℓ,s)  σ2  ℓ  �  r∈[s−1]  �  j,j′  E[ZℓjrZℓj′rZℓjsZℓj′s]  =  n  �  ℓ=1  σ2  ℓ  �  s∈[Nℓ]  �  r∈[s−1]  E  � �  j  ZℓjrZℓjs  �2  = Var(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17)  The last line is obtained noting that S as defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3) is a sum of uncorrelated terms  over (i, r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Similarly, we have  E  � �  (ℓ,s)  Var(Dℓ,s|F≺(ℓ,s))  �  = E  � �  (ℓ,s)  E  �  D2  ℓ,s|F≺(ℓ,s)  ��  =  �  (ℓ,s)  E  �  D2  ℓ,s  �  =  �  (ℓ,s)  �  i∈[ℓ−1]  σ2  i,ℓVar  � Ni  �  r=1  �  j  ZijrZℓjs  �  =  �  ℓ  �  i∈[ℓ−1]  σ2  i,ℓVar  � Ni  �  r=1  Nℓ  �  s=1  ZijrZℓjs  �  = Var(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='18)  The lemma follows by combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16)–(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Let Mk = nk ¯Nk and M = n ¯N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  Σ = 1  M  �  k  MkΣk = 1  M  �  ℓ∈[n]  NℓΩℓj1Ωℓj2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='19)  Our main goal is to control the conditional variance process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  δjj′ℓ = EZℓjrZℓj′r =  �  Ωℓj(1 − Ωℓj)  if j = j′  −ΩℓjΩℓj′  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='20)  50  Observe that  Var(Dℓ,s|F≺(ℓ,s)) = E[  �  i,i′∈[ℓ−1]  �  r,r′  �  j1,j2  σiℓσi′ℓZij1rZℓj1sZi′j2r′Zℓj2s|F≺(ℓ,s)]  =  �  i,i′∈[ℓ−1]  �  r,r′  �  j1,j2  σiℓσi′ℓZij1rZi′j2r′E[Zℓj1sZℓj2s]  =  �  i,i′∈[ℓ−1]  �  r,r′  σiℓσi′ℓ  �  j1,j2  δj1j2ℓZij1rZi′j2r′  Define  αii′j1j2 =  �  ℓ>i′  Nℓσiℓσi′ℓδj1j2ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='21)  Thus  �  (ℓ,s)  Var(Dℓ,s|F≺(ℓ,s)) =  �  ℓ,s  �  i,i′∈[ℓ−1]  Ni  �  r=1  Ni′  �  r′=1  σiℓσi′ℓ  �  j1,j2  δj1j2ℓ Zij1rZi′j2r′  =  �  i  Ni  �  r=1  Ni  �  r′=1  �  j1,j2  � �  ℓ>i  Nℓσ2  iℓδj1j2ℓ  �  Zij1rZi′j2r′  + 2  �  i<i′  Ni  �  r=1  Ni′  �  r′=1  �  j1,j2  � �  ℓ>i′  Nℓσiℓσi′ℓδj1j2ℓ  �  Zij1rZi′j2r′  =  �  i  Ni  �  r=1  Ni  �  r′=1  �  j1,j2  αiij1j2Zij1rZi′j2r′  + 2  �  i<i′  Ni  �  r=1  Ni′  �  r′=1  �  j1,j2  αii′j1j2Zij1rZi′j2r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define  ζiri′r′ =  �  j1,j2  αii′j1j2Zij1rZi′j2r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22)  Then  �  (ℓ,s)  Var(Dℓ,s|F≺(ℓ,s)) =  �  i  �  r∈[Ni]  ζirir +  �  2  �  i  �  r<r′∈[Ni]  ζirir′ + 2  �  i<i′  Ni  �  r=1  Ni′  �  r′=1  ζiri′r′  �  =: V1 + V2  With this decomposition, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 follows directly from Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8 stated  below and proved in the next remainder of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  Var(V1) ≲  � �  k  1  Mk  �  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  51  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  Var(V2) ≲ K∥µ∥4  4  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Statement and proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9  The proofs of Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8 heavily rely on the following intermediate result that  bounds the coefficients αii′j1j2 in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  αii′j1j2 ≲  �  �  �  �  �  �  �  �  �  �  �  �  �  1  Mk µj1  if i, i′ ∈ Sk, j1 = j2  1  Mk Σkj1j2 + 1  M Σj1j2  if i, i′ ∈ Sk, j1 ̸= j2  1  M µj1  if i ∈ Sk1, i′ ∈ Sk2, k1 ̸= k2, j1 = j2  1  M  �2  a=1 Σkaj1j2 + 1  M Σj1j2  if i ∈ Sk1, i′ ∈ Sk2, k1 ̸= k2, j1 ̸= j2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If j1 = j2 and i, i′ ∈ Sk, we have  |αii′j1j1| = |  �  ℓ>i′  Nℓσiℓσi′ℓδj1j1ℓ| ≤  K  �  k′=1  �  ℓ∈Sk′  Nℓσiℓσi′ℓδj1j1ℓ  ≲  1  Mk 1  Mk  �  ℓ∈Sk  NℓΩℓj1 + 1  M · 1  M  �  ℓ∈[n]  NℓΩℓj1 ≲  1  Mk  µj1 + 1  M µj1 ≲  1  Mk  µj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  If j1 ̸= j2 and i, i′ ∈ Sk, we have  |αii′j1j2| = |  �  ℓ>i′  Nℓσiℓσi′ℓδj1j2ℓ| ≤  �  ℓ∈[n]  Nℓ|σiℓσi′ℓ|Ωℓj1Ωℓj2  ≲  1  Mk 1  Mk  �  ℓ∈Sk  NℓΩℓj1Ωℓj2 + 1  M · 1  M  �  ℓ∈[n]  NℓΩℓj1Ωℓj2 ≲  1  Mk  Σkj1j2 + 1  M Σj1j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  If i ̸= i′, j1 = j2, and i ∈ Sk1, i′ ∈ Sk2 where k1 ̸= k2, we have  |αii′j1j1| = |  �  ℓ>i′  Nℓσiℓσi′ℓδj1j1ℓ| ≤  �  ℓ  Nℓ|σiℓσi′ℓ|Ωℓj1  ≲ 1  M ·  2  �  a=1  1  Mka  �  ℓ∈Ska  NℓΩℓj1 + 1  M · 1  M  �  ℓ∈[n]  NℓΩℓj1 = 3  M µj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  If i ̸= i′, j1 ̸= j2, and i ∈ Sk1, i′ ∈ Sk2 where k1 ̸= k2, we have  |αii′j1j2| = |  �  ℓ>i′  Nℓσiℓσi′ℓδj1j2ℓ| ≲  �  ℓ  Nℓσiℓσi′ℓΩℓj1Ωℓj2  ≲ 1  M ·  2  �  a=1  1  Mka  �  ℓ∈Ska  NℓΩℓj1Ωℓj2 + 1  M · 1  M  �  ℓ∈[n]  NℓΩℓj1Ωℓj2  ≤ 1  M  2  �  a=1  Σkaj1j2 + 1  M Σj1j2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  52  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7  We have  Var(V1) =  �  i,r  Eζ2  irir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next by symmetry,  Eζ2  irir =  �  j1,j2,j3,j4  αiij1j2αiij3j4 EZij1rZij3rZij2rZij4r  ≲  �  j1  α2  iij1j1Ωij1 +  �  j1̸=j4  αiij1j1αiij1j4 Ωij1Ωij4  +  �  j1̸=j3  αiij1j1αiij3j3 Ωij1Ωij3 +  �  j1̸=j2  α2  iij1j2 Ωij1Ωij2  +  �  j1,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αiij1j1αiij3j4 Ωij1Ωij3Ωij4 +  �  j1,j2,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αiij1j2αiij1j4Ωij1Ωij2Ωij4  +  �  j1,j2,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αiij1j2αiij3j4Ωij1Ωij2Ωij3Ωij4 =:  7  �  a=1  Ba,i,r  Thus  Var(V1) ≲  �  a  � �  i,r  Ba,i,r  �  ��  �  =:Ba  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We analyze B1– B7 separately, bounding the αii′jrjs coefficients using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For B1,  B1 ≲  �  i,r  �  j1  α2  iij1j2Ωij1 ≲  k  �  k=1  �  i∈Sk  �  r∈[Ni]  �  j1  ( 1  Mk  µj1)2Ωij1  ≲  �  k  �  j1  ( 1  Mk  µj1)2Mkµj1 ≲  � �  k  1  Mk  �  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23)  For B2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B2 ≲  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r  �  j1̸=j4  αiij1j1αiij1j4 Ωij1Ωij4  ≲  �  k  �  i∈Sk  �  r∈[Ni]  �  j1̸=j4  1  Mk  µj1 ·  � 1  Mk  Σkj1j4 + 1  M Σj1j4  �  Ωij1Ωij4  ≲  �  k  �  j1̸=j4  1  Mk  µj1 ·  � 1  Mk  Σkj1j4 + 1  M Σj1j4  �  MkΣkj1j4  ≲  �  k  1  Mk  �  j1̸=j4  Σ2  kj1j4µj1 +  �  k  1  M  �  j1̸=j4  Σkj1j4Σj1j4µj1  53  ≲  �  k  1′Σ◦2  k µ  Mk  +  �  k  1′(Σk ◦ Σ)µ  M  =  �  k  1′Σ◦2  k µ  Mk  Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  �  j1̸=j4  Σ2  kj1j4µj1 =  �  j1̸=j4  1  M2  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′Ωij1Ωi′j1Ωij4Ωi′j4 · µj1  ≤  �  j1  1  M2  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′Ωij1Ωi′j1µj1 ·  � �  j4  Ωij4Ωi′j4  �  ≤  �  j1  1  M2  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′Ωij1Ωi′j1 · µj1  ≤  �  j1  µ3  j1 = ∥µ∥3  3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='24)  and similarly  �  j1̸=j4  Σkj1j4Σj1j4µj1 =  �  j1̸=j4  1  MkM  �  i∈Sk,i′∈[n]  NiNi′Ωij1Ωi′j1Ωij4Ωi′j4 · µj1  ≤  �  j1  1  MkM  �  i∈Sk,i′∈[n]  NiNi′Ωij1Ωi′j1µj1  =  �  j1  µ3  j1 = ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  B2 ≲  � �  k  1  Mk  �  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='25)  For B3,  B3 ≲  �  i,r  �  j1̸=j3  αiij1j1αiij3j3 Ωij1Ωij3  ≲  �  k  �  i∈Sk  �  r∈[Ni]  �  j1̸=j3  1  Mk  µj1 ·  1  Mk  µj3 · Ωij1Ωij3  ≲  �  k  �  j1̸=j3  1  Mk  µj1 ·  1  Mk  µj3 · MkΣkj1j3 ≲  �  k  µ′Σkµ  Mk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We have by Cauchy-Schwarz,  µ′Σkµ =  1  Mk  �  i∈Sk  Niµ′ΩiΩ′  i′µ  =  1  Mk  �  i∈Sk  Ni  � �  j  µjΩij  �2  ≤  1  Mk  �  i∈Sk  Ni  � �  j  Ωij  �� �  j  µ2  jΩij  �  54  =  �  j  µ3  j = ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='26)  Thus  B3 ≲  � �  k  1  Mk  �  ∥µ∥3  3  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='27)  For B4,  B4 ≲  �  i,r  �  j1̸=j2  α2  iij1j2 Ωij1Ωij2 ≲  �  k  �  i∈Sk  �  r∈[Ni]  �  j1̸=j2  � 1  Mk  Σkj1j2 + 1  M Σj1j2  �2 Ωij1Ωij2  ≲  �  k  �  j1̸=j2  � 1  Mk  Σkj1j2 + 1  M Σj1j2  �2 · MkΣkj1j2 ≲  �  k  1′(Σ◦3  k )1  Mk  +  �  k  Mk  M2 1′(Σk ◦ Σ◦2)1  ≲  � �  k  1′(Σ◦3  k )1  Mk  �  + 1  M 1′(Σ◦3)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  First,  1′(Σ◦3  k )1 =  1  M3  k  �  i1,i2,i3∈Sk  Ni1Ni2Ni3  � �  j  Ωi1jΩi2jΩi3j  �2  ≤  1  M3  k  �  i1,i2,i3∈Sk  Ni1Ni2Ni3 ·  �  j  Ωi1jΩi2jΩi3j =  �  j  µ3  j = ∥µ∥3  3,  and similarly,  1′(Σ◦3)1 =  1  M3  �  i1,i2,i3∈[n]  Ni1Ni2Ni3  � �  j  Ωi1jΩi2jΩi3j  �2 ≤ ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  B4 ≲  � �  k  1  Mk  �  ∥µ∥3  3  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='28)  For B5,  B5 ≲  �  i,r  �  j1,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αiij1j1αiij3j4 Ωij1Ωij3Ωij4  ≲  �  k  �  i∈Sk  Ni  �  j1,j3,j4  1  Mk  µj1 · ( 1  Mk  Σkj3j4 + 1  M Σj3j4) · Ωij1Ωij3Ωij4  ≲  �  k  �  i∈Sk  �  j1,j3,j4  Niµj1Σkj3j4Ωij1Ωij3Ωij4  M2  k  +  �  k  �  i∈Sk  �  j1,j3,j4  Niµj1Σj3j4Ωij1Ωij3Ωij4  MkM  =: B51 + B52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We have  B51 =  �  k  1  M3  k  �  i1,i2∈Sk  �  j1,j3,j4  Ni1Ni2µj1Ωi1j1Ωi1j3Ωi2j3Ωi1j4Ωi2j4  55  =  �  k  1  M3  k  �  i1,i2∈Sk  Ni1Ni2(Ω′  i1µ) · (Ω′  i1Ωi2)2  ≤  �  k  1  M3  k  �  i1,i2∈Sk  Ni1Ni2 · Ω′  i1µ · Ω′  i1Ωi2  =  �  k  1  M2  k  �  i1  Ni1µ′Ωi1Ω′  i1µ =  1  Mk  µ′Σkµ ≤  �  k  1  Mk  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='29)  In the last line we apply (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Similarly,  B52 =  �  k  1  MkM2  �  i1∈Sk,i2∈[n]  �  j1,j3,j4  Ni1Ni2µj1Ωi1j1Ωi1j3Ωi2j3Ωi1j4Ωi2j4  ≤  �  k  1  MkM2  �  i1∈Sk,i2∈[n]  Ni1Ni2 · Ω′  i1µ · Ω′  i1Ωi2  ≤  �  k  1  MkM  �  i1∈Sk  Ni1µ′Ωi1Ω′  i1µ ≤  �  k  1  M ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='30)  Thus  B5 ≲  � �  k  1  Mk  �  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='31)  For B6,  B6 ≲  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  � 1  Mk  Σkj1j2 + 1  M Σj1j2  �� 1  Mk  Σkj1j4 + 1  M Σj1j4  �  Ωij1Ωij2Ωij4  ≲  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j4  Σ2  kj1j2Ωij1Ωij2Ωij4  M2  k  + 2  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j4  Σkj1j2Σj1j2Ωij1Ωij2Ωij4  MkM  +  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j4  Σ2  j1j2Ωij1Ωij2Ωij4  M2  =: B61 + B62 + B63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  First,  B61 ≤  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j4  Σ2  kj1j2Ωij1  M2  k  =  �  k  1  Mk  1′Σ◦2  k µ ≤  �  k  1  Mk  ∥µ∥3  3,  where we applied (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Similarly,  B62 ≲  �  k  1  Mk  ∥µ∥3  3, and  B63 ≲  �  k  1  Mk  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  B6 ≲  � �  k  1  Mk  �  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32)  56  For B7, we have  B7 ≲  �  j1,j2,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  � 1  Mk  Σkj1j2 + 1  M Σj1j2  �� 1  Mk  Σkj3j4 + 1  M Σj3j4  �  Ωij1Ωij2Ωij3Ωij4  ≲  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j3,j4  Σkj1j2Σkj3j4Ωij1Ωij2Ωij3Ωij4  M2  k  + 2  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j3,j4  Σkj1j2Σj3j4Ωij1Ωij2Ωij3Ωij4  MkM  +  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j3,j4  Σj1j2Σj3j4Ωij1Ωij2Ωij3Ωij4  M2  =: B71 + B72 + B73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Note that  Σkj1j2 =  1  Mk  �  i∈Sk  NiΩij1Ωij2 ≤  1  Mk  �  i∈Sk  NiΩij1 = µj1, and  Σj1j2 = 1  M  �  i∈[n]  NiΩij1Ωij2 ≤ 1  M  �  i∈[n]  NiΩij1 = µj1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='33)  Thus  B71 ≤  �  k  �  i∈Sk  �  r∈[Ni]  �  j1,j2,j3,j4  µj1Σkj3j4Ωij1Ωij2Ωij3Ωij4  M2  k  ≤  �  k  �  i∈Sk  �  j1,j3,j4  Niµj1Σkj3j4Ωij1Ωij3Ωij4  M2  k  ≤  �  k  1  Mk  ∥µ∥3  3  where we applied (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Similarly,  B72 ≲  �  k  1  Mk  ∥µ∥3  3, and  B73 ≲  �  k  1  Mk  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  B7 ≲  � �  k  1  Mk  �  ∥µ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='34)  Combining the results for B1–B7 concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8  We have  Var(V2) ≲ 4  �  (i,r)̸=(i′,r′)  Eζ2  irir′,  57  where r ∈ [Ni] and r ∈ [Ni′] in the summation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By symmetry, if (i, r) ̸= (i′, r′),  Eζ2  iri′r′ =  �  j1,j2,j3,j4  αii′j1j2αii′j3j4 EZij1rZij3r EZi′j2r′Zi′j4r′  ≲  �  j1  α2  ii′j1j1 Ωij1Ωi′j1 +  �  j1̸=j4  αii′j1j1αii′j1j4 Ωij1Ωi′j1Ωi′j4  +  �  j1̸=j3  αii′j1j1αii′j3j3 Ωij1Ωij3Ωi′j1Ωi′j3 +  �  j1̸=j2  α2  ii′j1j2 Ωij1Ωi′j2  +  �  j1,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αii′j1j1αii′j3j4 Ωij1Ωij3Ωi′j1Ωi′j4 +  �  j1,j2,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αii′j1j2αii′j1j4 Ωij1Ωi′j2Ωi′j4  +  �  j1,j2,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αii′j1j2αii′j3j4Ωij1Ωij3Ωi′j2Ωi′j4 =:  7  �  a  Ca,i,r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='35)  Thus  Var(V2) ≲  7  �  a=1  �  (i,r)̸=(i′,r′)  Ca,i,r ≲  7  �  a=1  �  i,i′  NiNi′Ca,i,r  �  ��  �  =:Ca  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next we analyze C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , C7, bounding the αii′jrjs coefficients using Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For C1,  C1 ≲  �  k  �  i,i′∈Sk  �  j1  NiNi′α2  ii′j1j1Ωij1Ωi′j1 +  �  k̸=k′  �  i∈Sk,i′∈Sk′  �  j1  NiNi′α2  ii′j1j1Ωij1Ωi′j1  ≲  �  k  �  i,i′∈Sk  �  j1  NiNi′( 1  Mk  µj1)2Ωij1Ωij1 +  �  k̸=k′  �  i∈Sk,i′∈Sk′  �  j1  ( 1  M µj1)2Ωij1Ωi′j1  ≲  �  k  �  j1  µ4  j1 +  �  k̸=k′  �  j1  MkMk′  M2  µ4  j1 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='36)  For C2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C2 ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1̸=j4  αii′j1j1αii′j1j4 Ωij1Ωi′j1Ωi′j4  +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  �  j1̸=j4  αii′j1j1αii′j1j4 Ωij1Ωi′j1Ωi′j4  ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1̸=j4  1  Mk  µj1 ·  � 1  Mk  Σkj1j4 + 1  M Σj1j4  �  Ωij1Ωi′j1Ωi′j4  +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  �  j1̸=j4  1  M µj1 ·  � 1  M  �  a∈{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='k′}  Σaj1j4 + 1  M Σj1j4  �  Ωij1Ωi′j1Ωi′j4  ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1̸=j4  1  Mk  µj1 ·  � 1  Mk  µj1 + 1  M µj1  �  Ωij1Ωi′j1Ωi′j4  +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  �  j1̸=j4  1  M µj1 ·  � 2  M µj1 + 1  M µj1  �  Ωij1Ωi′j1Ωi′j4  58  ≲  �  k  �  j1  �  µ4  j1 + Mk  M µ4  j1  �  +  �  k̸=k′  �  j1  MkMk′  M2  µ4  j1 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='37)  where we applied (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For C3,  C3 ≲  � �  k  �  i,i′∈Sk  NiNi′ +  �  k̸=k′  �  i∈Sk,i′∈Sk′  NiNi′  � �  j1̸=j3  αii′j1j1αii′j3j3 Ωij1Ωij3Ωi′j1Ωi′j3  ≲  �  k  �  i,i′∈Sk  NiNi′  �  j1̸=j3  1  Mk  µj1 ·  1  Mk  µj3 · Ωij1Ωij3Ωi′j1Ωi′j3  +  �  k̸=k′  �  i∈Sk,i′∈Sk′  NiNi′  �  j1̸=j3  1  M µj1 · 1  M µj3 Ωij1Ωij3Ωi′j1Ωi′j3  =  �  k  �  j1̸=j3  µj1µj3Σ2  kj1j3 +  �  k̸=k′  �  j1̸=j3  MkMk′  M2  µj1µj3Σkj1j3Σk′j1j3  ≤  � �  k  µ′Σ◦2  k µ  �  + µ′Σ◦2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  First, by Cauchy–Schwarz,  µ′Σ◦2  k µ =  1  M2  k  �  i,i′∈Sk  NiNi′� �  j  µjΩijΩi′j  �2  =  1  M2  k  �  i,i′∈Sk  NiNi′� �  j  ΩijΩi′j  � �  j  µ2  jΩijΩi′j  ≤  1  M2  k  �  i,i′∈Sk  NiNi′  �  j  µ2  jΩijΩi′j =  �  j  µ4  j = ∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38)  Similarly  µ′Σ◦2µ ≲ ∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39)  Hence  C3 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='40)  For C4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C4 ≲  � �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′ +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  � �  j1̸=j2  α2  ii′j1j2 Ωij1Ωi′j2  ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1̸=j2  � 1  Mk  Σkj1j2 + 1  M Σj1j2  �2 Ωij1Ωi′j2  +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  �  j1̸=j2  � 1  M  2  �  a∈{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='k′}  Σaj1j2 + 1  M Σj1j2  �2 Ωij1Ωi′j2  ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1̸=j2  � 1  M2  k  Σ2  kj1j2 +  1  M2 Σ2  j1j2  �  Ωij1Ωi′j2  59  +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  �  j1̸=j2  � 1  M2  2  �  a∈{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='k′}  Σ2  aj1j2 +  1  M2 Σ2  j1j2  �  Ωij1Ωi′j2 =: C41 + C42  First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C41 ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1̸=j2  1  M2  k  Σ2  kj1j2Ωij1Ωi′j2 +  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1̸=j2  1  M2 Σ2  j1j2Ωij1Ωi′j2  ≲  �  k  �  j1̸=j2  Σ2  kj1j2µj1µj2 +  �  k  �  j1̸=j2  M2  k  M2 Σ2  j1j2µj1µj2 ≤  �  k  µ′Σ◦2  k µ +  �  k  M2  k  M2 µ′Σ◦2µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Similarly,  C42 ≲  �  k̸=k′  �  j1̸=j2  MkMk′  M2  Σ2  kj1j2µj1µj2 +  �  k̸=k′  �  j1̸=j2  MkMk′  M2  Σ2  j1j2µj1µj2  ≲  �  k̸=k′  MkMk′  M2  �  µ′Σ◦2  k µ + µ′Σ◦2µ  �  Combining the previous two displays and applying (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39), we have  C4 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='41)  For C5,  C5 ≲  � �  k  �  i,i′∈Sk  NiNi′ +  �  k̸=k′  �  i∈Sk,i′∈Sk′  NiNi′  �  �  j1,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  αii′j1j1αii′j3j4 Ωij1Ωij3Ωi′j1Ωi′j4  ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  1  Mk  µj1 ·  � 1  Mk  Σkj3j4 + 1  M Σj3j4  �  Ωij1Ωij3Ωi′j1Ωi′j4  +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  1  M µj1  � 1  M  2  �  a∈{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='k′}  Σaj3j4 + 1  M Σj3j4  �  Ωij1Ωij3Ωi′j1Ωi′j4  =  �  k  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  µj1Σkj3j4Σkj1j3Σkj1j4 +  �  k  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  Mk  M µj1Σj3j4Σkj1j3Σkj1j4  + 2  �  k̸=k′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  MkMk′  M2  µj1Σkj3j4Σkj1j3Σk′j1j4 +  �  k̸=k′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  MkMk′  M2  µj1Σj3j4Σkj1j3Σk′j1j4  = C51 + C52 + 2C53 + C54  For C51,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' we have  C51 =  �  k  1  M3  k  �  i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3∈Sk  Ni1Ni2Ni3⟨µ ◦ Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi2⟩⟨Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi3⟩⟨Ωi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi3⟩  =  �  k  1  M2  k  �  i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i2∈Sk  Ni1Ni2⟨µ ◦ Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi2⟩ · ⟨Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ΣkΩi2⟩  ≤  �  k  � 1  M2  k  �  i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i2∈Sk  Ni1Ni2⟨µ ◦ Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi2⟩2  �1/2� 1  M2  k  �  i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i2∈Sk  Ni1Ni2⟨Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ΣkΩi2⟩2  �1/2  60  =:  �  k  C1/2  511k · C1/2  512k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='42)  We have by Cauchy–Schwarz that  C511k =  1  M2  k  �  i1,i2∈Sk  Ni1Ni2  � �  j  µjΩi1jΩi2j  �2  ≤  1  M2  k  �  i1,i2∈Sk  Ni1Ni2  � �  j  µ2  jΩi1jΩi2j  �� �  j  Ωi1jΩi2j  �  ≤ ∥µ∥4  4,  and similarly  C512k =  1  M2  k  �  i1,i2∈Sk  Ni1Ni2  � �  j1,j2  Ωi1j1Σkj1j2Ωi2j2  �2  =  1  M2  k  �  i1,i2  Ni1Ni2  � �  j1,j2  Ωi1j1Σ2  kj1j2Ωi2j2  �� �  j1,j2  Ωi1j1Ωi2j2  �  ≤  1  M2  k  �  i1,i2  Ni1Ni2  � �  j1,j2  Ωi1j1Σ2  kj1j2Ωi2j2  �  = µ′ Σ◦2  k µ  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='43)  Since by Cauchy–Schwarz,  µ′ Σ◦2  k µ =  �  j1,j2  µj1µj2  � 1  Mk  �  i∈Sk  NiΩij1Ωij2  �2 =  1  M2  k  �  j1,j2  µj1µj2  �  i,i′∈Sk  NiNi′Ωij1Ωij2Ωi′j1Ωi′j2  =  1  M2  k  �  i,i′∈Sk  � �  j  µjΩijΩi′j  �2 ≤  1  M2  k  �  i,i′∈Sk  �  j  µ2  jΩijΩi′j ≤ ∥µ∥4  4  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44)  we have in total C512k ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Combining the result with the bound for C511k implies  that  C51 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next we study C52 using a similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C52 =  �  k  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  Mk  M µj1Σj3j4Σkj1j3Σkj1j4  =  �  k  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  Mk  M µj1  � 1  M  �  i1∈[n]  Ni1Ωi1j3Ωi1j4  �� 1  Mk  �  i2∈Sk  Ni2Ωi2j1Ωi2j3  �� 1  Mk  �  i3∈Sk  Ni3Ωi3j1Ωi3j4  �  =  �  k  1  M2Mk  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3  �  i1∈[n]  i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3∈Sk  Ni1Ni2Ni3⟨µ ◦ Ωi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi3⟩⟨Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi3⟩⟨Ωi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi2⟩  =  �  k  1  M2  �  i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3∈[Sk]  Ni2Ni3⟨µ ◦ Ωi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi3⟩⟨Ωi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ΣΩi2⟩  ≤  �  k  � 1  M2  �  i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3∈[Sk]  Ni2Ni3⟨µ ◦ Ωi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Ωi3⟩2  �1/2� 1  M2  �  i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3∈[Sk]  Ni2Ni3⟨Ωi3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ΣΩi2⟩  �1/2  61  =:  �  k  C1/2  521kC1/2  522k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='45)  Observe that C521k = C511k, and thus C521 ≲ ∥µ∥4 by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' With a similar argument as  in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44) we obtain C522k ≲ ∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hence we obtain  C52 ≤  �  k  C1/2  521kC1/2  522k ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For C53, we have  C53 =  �  k̸=k′  �  j1,j3,j4  MkMk′  M2  µj1Σkj3j4Σkj1j3Σk′j1j4  ≤  �  k  �  j1,j3,j4  Mk  M µj1Σkj3j4Σkj1j3Σj1j4  =  �  k  �  j1,j3,j4  Mk  M µj1  � 1  Mk  �  i1∈Sk  Ni1Ωi1j3Ωi1j4  �� 1  Mk  �  i2∈Sk  Ni2Ωi2j1Ωi2j3  �� 1  M  �  i3∈[n]  Ni3Ωi3j1Ωi3j4  �  =  �  k  1  M2Mk  �  i1,i2∈Sk  i3∈[n]  Ni1Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi1, Ωi2⟩⟨Ωi1, Ωi3⟩  =  �  k  1  M2  �  i2∈Sk,i3∈[n]  Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi2, ΣkΩi3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46)  We then upper bound the last line using a similar strategy as in that we used for C51 and  C52, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We omit the details and state the final bound:  C53 ≲ K∥µ∥4  4  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='47)  Finally for C54, summing over k, k′ we obtain  C54 ≤  �  j1,j3,j4  µj1Σj3j4Σj1j3Σj1j4 =  1  M3  �  i1,i2,i3∈[n]  Ni1Ni2Ni3⟨µ ◦ Ωi2, Ωi3⟩⟨Ωi1, Ωi2⟩⟨Ωi1, Ωi3⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='48)  We then proceed as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46) to control the right-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We omit the details and state  the final bound:  C54 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='49)  Combining the results for C51, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , C54, we see that  C5 ≲ K∥µ∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For C6, we have  C6 ≤  � �  k  �  i,i′∈Sk  NiNi′ +  �  k̸=k′  �  i∈Sk,i′∈Sk′  NiNi′  � �  j1,j2,j4  αii′j1j2αii′j1j4 Ωij1Ωi′j2Ωi′j4  62  ≲  �  k  �  i,i′∈Sk  NiNi′  �  j1,j2,j4  � 1  Mk  Σkj1j2 + 1  M Σj1j2  �� 1  Mk  Σkj1j4 + 1  M Σj1j4  �  Ωij1Ωi′j2Ωi′j4  +  �  k̸=k′  i∈Sk,i′∈Sk′  j1,j2,j4  NiNi′� 1  M  2  �  a∈{k,k′}  Σaj1j2 + 1  M Σj1j2  �� 1  M  2  �  a∈{k,k′}  Σaj1j4 + 1  M Σj1j4  �  Ωij1Ωi′j2Ωi′j4  =: C61 + C62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For C61, we have  C61 =  �  k  �  i′∈Sk  Ni′  �  j1,j2,j4  1  Mk  Σkj1j2Σkj1j4µj1Ωi′j2Ωi′j4  + 2  �  k  �  i′∈Sk  Ni′  �  j1,j2,j4  1  M Σkj1j2Σj1j4µj1Ωi′j2Ωi′j4  +  �  k  �  i′∈Sk  Ni′  �  j1,j2,j4  Mk  M2 Σj1j2Σj1j4µj1Ωi′j2Ωi′j4 =: C611 + 2C612 + C613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Relabeling indices, we see that  C611 =  �  k  �  j1,j2,j4  µj1Σkj1j2Σkj1j4Σkj2j4 = C51  Hence, C611 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Next,  C612 ≤  �  k  Mk  M  �  j1,j2,j4  µj1Σkj1j2Σj1j4Σkj2j4 ≲ K∥µ∥4,  where we applied (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Similarly,  C613 =  �  k  M2  k  M2  �  j1,j2,j4  µj1Σj1j2Σj1j4Σkj2j4 ≤  �  j1,j2,j4  µj1Σj1j2Σj1j4Σj2j4 ≲ K∥µ∥4,  where in the final bound we apply (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='48) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Combining the results above for  C611, C612, C613, we obtain  C61 ≲ K∥µ∥4  4  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='50)  The argument for C62 is very similar, so we omit proof and state the final bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  C62 ≲ K∥µ∥4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  C6 ≲ K∥µ∥4  4  For C7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' we have  C7 ≲  � �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′ +  �  k̸=k′  �  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′  �  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  αii′j1j2αii′j3j4Ωij1Ωij3Ωi′j2Ωi′j4  63  ≲  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  � 1  Mk  Σkj1j2 + 1  M Σj1j2  �� 1  Mk  Σkj3j4 + 1  M Σj3j4  �  Ωij1Ωij3Ωi′j2Ωi′j4  +  �  k̸=k′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  i∈Sk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk′  NiNi′� 1  M  2  �  a∈{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='k′}  Σaj1j2 + 1  M Σj1j2  �� 1  M  2  �  a∈{k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='k′}  Σaj3j4 + 1  M Σj3j4  �  Ωij1Ωij3Ωi′j2Ωi′j4  =: C71 + C72  Write  C71 =  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  1  M2  k  Σkj1j2Σkj3j4Ωij1Ωij3Ωi′j2Ωi′j4  + 2  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  1  MkM Σj1j2Σkj3j4Ωij1Ωij3Ωi′j2Ωi′j4  +  �  k  �  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i′∈Sk  NiNi′  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  1  M2 Σj1j2Σj3j4Ωij1Ωij3Ωi′j2Ωi′j4 =: C711 + 2C712 + C713.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For C711, we have  C711 =  �  k  �  j1,j2,j3,j4  Σkj1j2Σkj3j4Σkj1j3Σkj2j4  =  �  k  1  M4  k  �  i1,i2,i3,i4∈Sk  Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩  =  1  M2  k  �  k  �  i3,i4  Ni3Ni4  �  Ω′  i3ΣkΩi4  �2 =  �  k  1  M2  k  �  i3,i4  Ni3Ni4  � �  j,j′  Ω′  i3jΣkjj′Ωi4j′�2  ≤  �  k  1  M2  k  �  i3,i4  Ni3Ni4  �  j,j′  Ω′  i3jΣ2  kjj′Ωi4j′ ≤  �  k  �  j,j′  µjΣ2  kjj′µj′ ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='51)  In the last line we applied Cauchy–Schwarz and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For C712, we have similarly  C712 =  �  k  Mk  M  �  j1,j2,j3,j4  Σj1j2Σkj3j4Σkj1j3Σkj2j4  =  �  k  1  M2Mk  �  i1∈[n]  i2,i3,i4∈Sk  Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩  =  �  k  Mk  M2  �  i1∈[n],i2∈Sk  Ni1Ni2⟨Ωi1, ΣkΩi2⟩2 ≤  �  k  Mk  M2  �  i1∈[n],i2∈Sk  Ni1Ni2  �  j,j′  Ωi1jΣ2  kjj′Ωi2j′  ≤  �  k  M2  k  M2  �  j,j′  µjΣ2  kjj′µj′ ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52)  Next,  C713 =  �  k  M2  k  M2  �  j1,j2,j3,j4  Σj1j2Σj3j4Σkj1j3Σkj2j4  64  =  �  k  1  M4  �  i1,i2∈[n]  i3,i4∈Sk  Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩,  and applying a similar strategy as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='51), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52) leads to the bound C713 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  C71 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next , by symmetry and summing over i ∈ Sk, i′ ∈ Sk′, we have  C72 =  �  k̸=k′  MkMk′  M2  �  j1,j2,j3,j4  �  2Σkj1j2Σkj3j4 + 2Σk′j1j2Σkj3j4 + 4Σkj1j2Σj3j4 + Σj1j2Σj3j4  �  Σkj1j3Σk′j2j4  =: 2C721 + 2C722 + 4C723 + C724  First,  C721 ≤  �  k  Mk  M  �  j1,j2,j3,j4  Σkj1j2Σkj3j4Σkj1j3Σj2j4 = C712 ≲ K∥µ∥4  4  by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Next,  C722 =  �  k̸=k′  MkMk′  M2  �  j1,j2,j3,j4  Σk′j1j2Σkj3j4Σkj1j3Σk′j2j4  ≤  �  k,k′  1  M2MkMk′  �  i1,i2∈Sk  i3,i4∈Sk′  Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩  =  �  k,k′  Mk  M2Mk′  �  i3,i4∈Sk′  Ni3Ni4⟨Ωi3, ΣkΩi4⟩2 ≤  �  k,k′  Mk  M2Mk′  �  i3,i4∈Sk′  Ni3Ni4  �  j,j′  Ωi3jΣ2  kjj′Ωi4j′  ≤  �  k,k′  MkMk′  M2  µ′Σ◦2  k µ ≤ ∥µ∥4  4,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53)  where we applied Cauchy-Schwarz in the penultimate line and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44) in the last line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For C723, we have  C723 =  �  k̸=k′  MkMk′  M2  �  j1,j2,j3,j4  Σkj1j2Σj3j4Σkj1j3Σk′j2j4 ≤  �  k  Mk  M  �  j1,j2,j3,j4  Σkj1j2Σj3j4Σkj1j3Σj2j4  =  �  k  1  M3Mk  �  i1,i3∈Sk  i2,i4∈[n]  Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩  =  �  k  1  M2  �  i3∈Sk,i4∈[n]  Ni3Ni4⟨Ωi3, ΣkΩi4⟩⟨Ωi3, ΣΩi4⟩  ≤ 1  2  �  k  1  M2  �  i3∈Sk,i4∈[n]  Ni3Ni4  �  ⟨Ωi3, ΣkΩi4⟩2 + ⟨Ωi3, ΣΩi4⟩2�  65  Using a similar technique as in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='51)–(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53) and applying (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39) we obtain  C723 ≲ ∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Finally, for C724 we have  C724 =  �  k̸=k′  MkMk′  M2  �  j1,j2,j3,j4  Σj1j2Σj3j4Σkj1j3Σk′j2j4 ≤  �  j1,j2,j3,j4  Σj1j2Σj3j4Σj1j3Σj2j4  =  1  M4  �  i1,i2,i3,i4∈[n]  Ni1Ni2Ni3Ni4⟨Ωi1, Ωi3⟩⟨Ωi1, Ωi4⟩⟨Ωi2, Ωi3⟩⟨Ωi2, Ωi4⟩  The details are very similar to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='51)–(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53), so we omit them and simply state the final  bound:  C724 ≲ ∥µ∥4  4  Combining the bounds for C721, C722, C723, and C724 yields  C7 ≲ K∥µ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Combining the bounds for C1–C7 proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  We have  ED4  ℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='s = E  �� �  i∈[ℓ−1]  σi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='ℓ  Ni  �  r=1  �  j  ZijrZℓjs  �4  �  =  �  i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i4∈[ℓ−1]  σi1ℓσi2ℓσi3ℓσi4ℓ  �  r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r4  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  E  �  Zi1j1r1Zℓj1sZi2j2r2Zℓj2sZi3j3r3Zℓj3sZi4j4r4Zℓj4s  �  =  �  i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i4∈[ℓ−1]  σi1ℓσi2ℓσi3ℓσi4ℓ  �  r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r4  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  E  �  Zi1j1r1Zi2j2r2Zi3j3r3Zi4j4r4  �  E  �  Zℓj1sZℓj2sZℓj3sZℓj4s  �  =  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  E[Zℓj1sZℓj2sZℓj3sZℓj4s]  �  i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i4∈[ℓ−1]  r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r4  σi1ℓσi2ℓσi3ℓσi4ℓE[Zi1j1r1Zi2j2r2Zi3j3r3Zi4j4r4]  =:  �  j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  E[Zℓj1sZℓj2sZℓj3sZℓj4s]Aj1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j4  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='54)  In the summations above, rt ranges over [Nit].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that  |E[Zℓj1sZℓj2sZℓj3sZℓj4s]| ≲  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  Ωℓj1  if j1 = j2 = j3 = j4  Ωℓj1Ωℓj4  if j1 = j2 = j3, j4 ̸= j1  Ωℓj1Ωℓj3  if j1 = j2, j3 = j4, j1 ̸= j3  Ωℓj1Ωℓj3Ωℓj4  if j1 = j2, j1, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj2Ωℓj3Ωℓj4  if j1, j2, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55)  66  Up to permutation of the indices j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , j4, this accounts for all possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To proceed we also bound Aj1,j2,j3,j4 by casework on the number of distinct j indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For brevity we define ωt = (it, rt) and slightly abuse notation, letting Zωt,j = Zitjrt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Further  let Iℓ = {ω = (i, r) : i ∈ [ℓ], 1 ≤ r ≤ Ni}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Our goal is to control  Aj1,j2,j3,j4 =  �  ω1,ω2,ω3,ω4∈Iℓ−1  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j2Zω3j3Zω4j4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='56)  To do this, we study (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='56) in five cases that cover all possibilities (up to permutation of  the indices j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , j4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Case 1: j1 = j2 = j3 = j4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define j = j1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1jZω2jZω3jZω4j]  =  �  σ4  i1ℓ EZ4  ω1j ≲ σ4  i1ℓ Ωi1j  if ω1 = ω2 = ω3 = ω4  σ2  i1ℓσ2  i3ℓEZ2  ω1jEZ2  ω3j ≲ σ2  i1ℓσ2  i3ℓΩi1jΩi3j  if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='57)  Up to permutation of the indices ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , ω4, this accounts for all cases such that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='57) is  nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To be precise, by symmetry, it also holds that for all permutations π : [4] →  [4] that if ωπ(1) = ωπ(2), ωπ(3) = ωπ(4), ωπ(1) ̸= ωπ(3), then  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1jZω2jZω3jZω4j] ≲ σ2  iπ(1)ℓσ2  iπ(3)ℓΩiπ(1)jΩiπ(3)j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In all other cases besides those considered above, we have  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1jZω2jZω3jZω4j] = 0  by independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Therefore,  Ajjjj ≲  �  ω∈Iℓ−1  σ4  iℓΩij +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓΩi1jΩi3j  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='58)  In the remaining Cases 2–6, we follow the same strategy of writing out bounds for  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j2Zω3j3Zω4j4]  that cover all nonzero cases, up to permutation of the indices ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , ω4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Case 2: j1 = j2 = j3, j1 ̸= j4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j1Zω3j1Zω4j4]  =  �  σ4  i1ℓE[Z3  ω1j1Zω1j4] ≲ σ4  i1ℓ Ωi1j1Ωi1j4  if ω1 = ω2 = ω3 = ω4  σ2  i1ℓσ2  i3ℓEZ2  ω1j1EZω3j1Zω3j4 ≲ σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j1Ωi3j4  if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='59)  67  Up to permutation of the indices ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , ω4, this accounts for all cases such that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='59) is  nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus  Aj1,j1,j1,j4 ≲  �  ω∈Iℓ−1  σ4  iℓΩi1j1Ωi1j4 +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j1Ωi3j4  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='60)  Case 3: j1 = j2, j3 = j4, j1 ̸= j3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It holds that  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j1Zω3j3Zω4j3]  =  �  �  �  �  �  �  �  σ4  i1ℓEZ2  ω1j1Z2  ω1j3 ≲ σ4  i1ℓ Ωi1j1Ωi1j3  if ω1 = ω2 = ω3 = ω4  σ2  i1ℓσ2  i3ℓEZ2  ω1j1EZ2  ω3j3 ≲ σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j3  if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3  σ2  i1ℓσ2  i3ℓEZω1j1Zω1j3EZω2j1Zω2j3 ≲ σ2  i1ℓσ2  i3ℓΩi1j1Ωi1j3Ωi2j1Ωi2j3  if ω1 = ω3, ω2 = ω4, ω1 ̸= ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='61)  Up to permutation of the indices ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , ω4, this accounts for all cases such that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='61) is  nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus by symmetry,  Aj1,j1,j3,j3 ≲  �  ω∈Iℓ−1  σ4  i1ℓ Ωi1j1Ωi1j3 +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j3  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='62)  +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓΩi1j1Ωi1j3Ωi3j1Ωi3j3  Case 4: j1 = j2 and j1, j3, j4 distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j1Zω3j3Zω4j4]  =  �  �  �  �  �  �  �  σ4  i1ℓEZ2  ω1j1Zω1j3Zω1j4 ≲ σ4  i1ℓ Ωi1j1Ωi1j3Ωi1j4  if ω1 = ω2 = ω3 = ω4  σ2  i1ℓσ2  i3ℓEZ2  ω1j1EZω3j3Zω3j4 ≲ σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j3Ωi3j4  if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3  σ2  i1ℓσ2  i2ℓEZω1j1Zω1j3EZω2j1Zω2j4 ≲ σ2  i1ℓσ2  i2ℓ Ωi1j1Ωi1j3Ωi2j1Ωi2j4  if ω1 = ω3, ω2 = ω4, ω1 ̸= ω2  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='63)  Up to permutation of the indices ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , ω4, this accounts for all cases such that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='63) is  nonvanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus  Aj1,j1,j3,j4 ≲  �  ω∈Iℓ−1  σ4  i1ℓ Ωi1j1Ωi1j3Ωi1j4 +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j3Ωi3j4  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='64)  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i2ℓ Ωi1j1Ωi1j3Ωi3j1Ωi3j4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Case 5: j1, j2, j3, j4 distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For this final case, it holds that  σi1ℓσi2ℓσi3ℓσi4ℓE[Zω1j1Zω2j2Zω3j3Zω4j4]  =  �  σ4  i1ℓEZω1j1Zω1j2Zω1j3Zω1j4 ≲ σ4  i1ℓ Ωi1j1Ωi1j2Ωi1j3Ωi1j4  if ω1 = ω2 = ω3 = ω4  σ2  i1ℓσ2  i3ℓEZω1j1Zω1j2EZω3j3Zω3j4 ≲ σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi1j2Ωi3j3Ωi3j4  if ω1 = ω2, ω3 = ω4, ω1 ̸= ω3  68  The above accounts for all nonzero cases, up to permutation of ω1, ω2, ω3, ω4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hence  Aj1,j2,j3,j4 ≲  �  ω∈Iℓ−1  σ4  i1ℓ Ωi1j1Ωi1j2Ωi1j3Ωi1j4 +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi1j2Ωi3j3Ωi3j4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='65)  Finally we control the fourth moment using the casework above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='54) and sym-  metry,  ED4  ℓ,s ≲  �  j  E[ZℓjsZℓjsZℓjsZℓjs]Aj,j,j,j +  �  j1̸=j4  E[Zℓj1sZℓj1sZℓj1sZℓj4s]Aj1,j1,j1,j4  +  �  j1̸=j3  E[Zℓj1sZℓj1sZℓj3sZℓj3s]Aj1,j1,j3,j3 +  �  j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  E[Zℓj1sZℓj1sZℓj3sZℓj4s]Aj1,j1,j3,j4  +  �  j1,j2,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  E[Zℓj1sZℓj2sZℓj3sZℓj4s]Aj1,j2,j3,j4  =: F1ℓs + F2ℓs + F3ℓs + F4ℓs + F5ℓs  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='66)  By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='58), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='60) ,(B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='62), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='64), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='65),  F1ℓs ≲  �  j  Ωℓj  � �  ω∈Iℓ−1  σ4  iℓΩij +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓΩi1jΩi3j  �  F2ℓs ≲  �  j1̸=j4  Ωℓj1Ωℓj4  � �  ω∈Iℓ−1  σ4  iℓΩi1j1Ωi1j4 +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j1Ωi3j4  �  F3ℓs ≲  �  j1̸=j3  Ωℓj1Ωℓj3  � �  ω∈Iℓ−1  σ4  i1ℓ Ωi1j1Ωi1j3 +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j3  +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓΩi1j1Ωi1j3Ωi3j1Ωi3j3  �  F4ℓs ≲  �  j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj3Ωℓj4  � �  ω∈Iℓ−1  σ4  i1ℓ Ωi1j1Ωi1j3Ωi1j4 +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi3j3Ωi3j4  +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi1j3Ωi3j1Ωi3j4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  �  F5ℓs ≲  �  j1,j2,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj2Ωℓj3Ωℓj4  � �  ω∈Iℓ−1  σ4  i1ℓ Ωi1j1Ωi1j2Ωi1j3Ωi1j4  +  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ Ωi1j1Ωi1j2Ωi3j3Ωi3j4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define  F11ℓs =  �  ω∈Iℓ−1  σ4  iℓ  �  j  ΩℓjΩij  F21ℓs =  �  ω∈Iℓ−1  σ4  iℓ  �  j1̸=j4  Ωℓj1Ωℓj4Ωi1j1Ωi1j4  69  F31ℓs =  �  ω∈Iℓ−1  σ4  i1ℓ  �  j1̸=j3  Ωℓj1Ωℓj3Ωi1j1Ωi1j3  F41ℓs =  �  ω∈Iℓ−1  σ4  i1ℓ  �  j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi1j3Ωi1j4  F51ℓs =  �  ω∈Iℓ−1  σ4  i1ℓ  �  j1,j2,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj2Ωℓj3Ωℓj4Ωi1j1Ωi1j2Ωi1j3Ωi1j4  and  F12ℓs =  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ  �  j  ΩℓjΩi1jΩi3j  F22ℓs =  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ  �  j1̸=j4  Ωℓj1Ωℓj4Ωi1j1Ωi3j1Ωi3j4  F32ℓs =  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ  �  j1̸=j3  �  Ωℓj1Ωℓj3Ωi1j1Ωi3j3 + Ωℓj1Ωℓj3Ωi1j1Ωi1j3Ωi3j1Ωi3j3  �  F42ℓs =  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ  �  j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  �  Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi3j3Ωi3j4  + Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi1j3Ωi3j1Ωi3j4  �  F52ℓs =  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ  �  j1,j2,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj2Ωℓj3Ωℓj4Ωi1j1Ωi1j2Ωi3j3Ωi3j4  Note that �2  x=1 Ftxℓs = Ftℓs for all t ∈ [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Using the fact that �  j Ωij = 1, we have  �  t  Ft1ℓs ≲ F11ℓs =  �  ω∈Iℓ−1  σ4  iℓ  �  j  ΩℓjΩij =  �  ω∈Iℓ−1  σ4  iℓ⟨Ωℓ, Ωi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='67)  To control �  t Ft2ℓs , observe that, since Ωij ≤ 1 for all i, j,  �  j  ΩℓjΩi1j = ⟨Ωℓ, Ωi1 ◦ Ωi3⟩  �  j1̸=j4  Ωℓj1Ωℓj4Ωi1j1Ωi3j1Ωi3j4 ≤ ⟨Ωℓ, Ωi1 ◦ Ωi3⟩ · ⟨Ωℓ, Ωi3⟩  �  j1̸=j3  �  Ωℓj1Ωℓj3Ωi1j1Ωi3j3 + Ωℓj1Ωℓj3Ωi1j1Ωi1j3Ωi3j1Ωi3j3  �  ≤ 2⟨Ωℓ, Ωi1⟩ · ⟨Ωℓ, Ωi3⟩  �  j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  �  Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi3j3Ωi3j4 + Ωℓj1Ωℓj3Ωℓj4Ωi1j1Ωi1j3Ωi3j1Ωi3j4  �  ≤ 2⟨Ωℓ, Ωi1⟩⟨Ωℓ, Ωi3⟩2  �  j1,j2,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj2Ωℓj3Ωℓj4Ωi1j1Ωi1j2Ωi3j3Ωi3j4 ≤ ⟨Ωℓ, Ωi1⟩2⟨Ωℓ, Ωi3⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  These bounds are relatively sharp, and it is clear that the first and third lines dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Furthermore as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hence,  �  t  Ft2ℓs ≲ F12ℓs + F32ℓs ≲  �  ω1̸=ω3∈Iℓ−1  σ2  i1ℓσ2  i3ℓ  �  ⟨Ωℓ, Ωi1 ◦ Ωi3⟩ + ⟨Ωℓ, Ωi1⟩ · ⟨Ωℓ, Ωi3⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='68)  70  Observe that if ℓ ∈ Sk, then  �  ω  σ4  iℓΩij ≤  �  i∈Sk  1  n4  k ¯N4  k  NiΩij +  K  �  k′=1  �  i∈Sk′  1  n4 ¯N4 NiΩij  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='69)  ≤  1  n3  k ¯N3  k  µkj +  1  n3 ¯N3 µj,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='70)  and  �  ω  σ2  iℓΩij ≤  �  i∈Sk  1  n2  k ¯N2  k  NiΩij +  K  �  k′=1  �  i∈Sk′  1  n ¯N NiΩij  ≤  1  nk ¯Nk  µkj +  1  n ¯N µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next,  �  (ℓ,s)  �  t  Ft1ℓs ≲  �  (ℓ,s)  �  ω∈Iℓ−1  σ4  iℓ⟨Ωℓ, Ωi⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  ≲  �  (ℓ,s)  �  j  Ωℓj  �  1  n3  k ¯N3  k  µkj +  1  n3 ¯N3 µj  �  ≲  �  j  �  k  1  n2  k ¯N2  k  µ2  kj +  �  j  �  k  1  n2 ¯N2 µ2  j ≲  �  k  1  n2  k ¯N2  k  ∥µk∥2,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='71)  where we applied that ∥µ∥2 ≲ �  k ∥µk∥2 (see (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='49)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore,  �  (ℓ,s)  �  t  Ft2ℓs ≤  K  �  k=1  �  ℓ∈Sk  Nℓ  �  ω1,ω3  σ2  i1ℓσ2  i3ℓ  �  ⟨Ωℓ, Ωi1 ◦ Ωi3⟩ + ⟨Ωℓ, Ωi1⟩ · ⟨Ωℓ, Ωi3⟩  �  ≲  �  k  �  ℓ∈Sk  Nℓ  � �  j  Ωℓj  �  1  nk ¯Nk  µkj +  1  n ¯N µj  �2 +  � �  j  Ωℓj ·  �  1  nk ¯Nk  µkj +  1  n ¯N µj  ��2�  ≲  �  k  �  ℓ∈Sk  Nℓ  �  j  Ωℓj  �  1  nk ¯Nk  µkj +  1  n ¯N µj  �2  In the last line we apply Cauchy–Schwarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Continuing, we have  �  (ℓ,s)  �  t  Ft2ℓs ≲  �  k  �  ℓ∈Sk  Nℓ  �  j  Ωℓj  �  1  nk ¯Nk  µkj +  1  n ¯N µj  �2  ≲  �  k  �  ℓ∈Sk  Nℓ  �  j  Ωℓj  �  1  nk ¯Nk  µkj  �2 +  �  k  �  ℓ∈Sk  Nℓ  �  j  Ωℓj  � 1  n ¯N µj  �2  ≲  �  k  ∥µk∥3  3  nk ¯Nk  +  �  k  ∥µ∥3  3  n ¯N ≲  �  k  ∥µk∥3  3  nk ¯Nk  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='72)  where we applied (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='66), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='71) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='72), we have  �  (ℓ,s)  ED4  ℓ,s ≲  �  (ℓ,s)  2  �  x=1  5  �  t=1  Ftxℓs ≲  �  k  ∥µk∥2  n2  k ¯N2  k  +  �  k  ∥µk∥3  3  nk ¯Nk  ,  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  71  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  Var  � �  (ℓ,s)  Var( ˜Eℓ,s|F≺(ℓ,s))  �  → 0  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='73)  Next we study (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  Var(Eℓ,s|F≺(ℓ,s)) = E[E2  ℓ,s|F≺(ℓ,s)] = σ2  ℓ  �  r,r′∈[s−1]  �  j,j′  E  �  ZℓjrZℓjsZℓj′r′Zℓj′s  ��F≺(ℓ,s)  �  = σ2  ℓ  �  r,r′∈[s−1]  �  j,j′  ZℓjrZℓj′r′E[ZℓjsZℓj′s]  = σ2  ℓ  �  r,r′∈[s−1]  �  j,j′  δjj′ℓZℓjrZℓj′r′,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='74)  where we let  δjj′ℓ = EZℓjsZℓj′s =  �  Ωℓj(1 − Ωℓj)  if j = j′  −ΩℓjΩℓj′  else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='75)  Define  ϕℓrℓr′ =  �  j,j′  δjj′ℓZℓjrZℓj′r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='76)  By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='74) we have  �  (ℓ,s)  Var(Eℓ,s|F≺(ℓ,s)) =  n  �  ℓ=1  Nℓ  �  s=1  �  r,r′∈[s−1]  σ2  ℓ ϕℓrℓr′  =  n  �  ℓ=1  Nℓ  �  s=1  � �  r∈[s−1]  σ2  ℓ ϕℓrℓr + 2  �  r<r′∈[s−1]  σ2  ℓ ϕℓrℓr′�  =  n  �  ℓ=1  Nℓ  �  r=1  �  s∈[Nℓ]:s>r  σ2  ℓ ϕℓrℓr + 2  s  �  ℓ=1  �  r<r′∈[Nℓ]  �  s∈[Nℓ]:s>r′  σ2  ℓ ϕℓrℓr′  =  n  �  ℓ=1  Nℓ  �  r=1  (Nℓ − r)σ2  ℓ ϕℓrℓr + 2  s  �  ℓ=1  �  r<r′∈[Nℓ]  (Nℓ − r′)σ2  ℓ ϕℓrℓr′  ≡ S1 + S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that S1 and S2 are uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In addition, the terms in the summation defining  S1 are uncorrelated;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' the same holds for S2 also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  First we study S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Next,  Eϕ2  ℓrℓr′ =  �  j1,j2,j3,j4  δj1j2,ℓδj3j4,ℓ EZℓj1rZℓj2r′Zℓj3rZℓj4r′  =  �  j1,j2,j3,j4  δj1j2ℓδj3j4ℓEZℓj1rZℓj3rEZℓj2r′Zℓj4r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='77)  72  First we study V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By casework,  |δj1j2ℓδj3j4ℓEZℓj1rZℓj3rEZℓj2r′Zℓj4r′|  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='78)  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  δ2  jjℓEZ2  ℓjrEZ2  ℓjr′ ≲ Ω4  ℓj  if j1 = · · · = j4  δj1j1ℓδj1j4ℓ|EZ2  ℓj1rEZℓj1r′Zℓj4r′| ≲ Ω4  ℓj1Ω2  ℓj4  if j1 = j2 = j3, j1 ̸= j4  δj1j1ℓδj3j3ℓEZℓj1rZℓj3rEZℓj1r′Zℓj3r′ ≲ Ω3  ℓj1Ω3  ℓj3  if j1 = j2, j3 = j4, j1 ̸= j3  δ2  j1j2ℓEZ2  ℓj1rEZ2  ℓj2r′ ≲ Ω3  ℓj1Ω3  ℓj2  if j1 = j3, j2 = j4, j1 ̸= j2  δj1j1ℓδj3j4ℓEZℓj1rZℓj3rEZℓj1r′Zℓj4r′ ≲ Ω3  ℓj1Ω2  ℓj3Ω2  ℓj4  if j1 = j2, j1, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  δj1j2ℓδj1j4ℓEZ2  ℓj1rEZℓj2r′Zℓj4r′ ≲ Ω3  ℓj1Ω2  ℓj2Ω2  ℓj4  if j1 = j3, j1, j2, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  δj1j2ℓδj3j4ℓEZℓj1rZℓj3rEZℓj2r′Zℓj4r′ ≲ Ω2  ℓj1Ω2  ℓj2Ω2  ℓj3Ω2  ℓj4  if j1, j2, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Up to permutation of the indices j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , j4, all nonzero terms of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='77) take one of the  forms above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='78) and Cauchy–Schwarz, we have  Eϕ2  ℓrℓr′ ≲ ∥Ωℓ∥4  4 + ∥Ωℓ∥4  4∥Ωℓ∥2 + 2∥Ωℓ∥6  3 + 2∥Ωℓ∥3  3∥Ωℓ∥4 + ∥Ωℓ∥8 ≲ ∥Ωℓ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='79)  Recalling that {ϕℓrℓr′}ℓ,r<r′∈[Nℓ] are mutually uncorrelated, it follows that  Var(S2) ≲  �  ℓ  �  r<r′∈[Nℓ]  (Nℓ − r′)2σ2  ℓ Eϕ2  ℓrℓr′  ≲  �  ℓ  �  r<r′∈[Nℓ]  (Nℓ − r′)2σ4  ℓ ∥Ωℓ∥4  4  ≲  �  k  �  ℓ∈Sk  N4  ℓ ·  1  n4  k ¯N4  k  ∥Ωℓ∥4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='80)  Next we study S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  Eϕ2  ℓrℓr =  �  j1,j2,j3,j4  δj1j2ℓδj3j4ℓEZℓj1rZℓj2rZℓj3rZℓj4r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We have the following bounds by casework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  |δj1j2ℓδj3j4ℓEZℓj1rZℓj2rZℓj3rZℓj4r|  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='81)  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  δ2  jjℓEZ4  ℓjr ≲ Ω3  ℓj  if j1 = · · · = j4  δj1j1ℓδj1j4ℓ|EZ3  ℓj1rZℓj4r| ≲ Ω3  ℓj1Ω2  ℓj4  if j1 = j2 = j3, j1 ̸= j4  δj1j1ℓδj3j3ℓEZ2  ℓj1rZ2  ℓj3r ≲ Ω2  ℓj1Ω2  ℓj3  if j1 = j2, j3 = j4, j1 ̸= j3  δ2  j1j2ℓEZ2  ℓj1rZ2  ℓj2r ≲ Ω3  ℓj1Ω3  ℓj2  if j1 = j3, j2 = j4, j1 ̸= j3  δj1j1ℓδj3j4ℓ|EZ2  ℓj1rZℓj3rZℓj4r| ≲ Ω2  ℓj1Ω2  ℓj3Ω2  ℓj4  if j1 = j2, j1, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  δj1j2ℓδj1j4ℓ|EZ2  ℓj1rZℓj2rZℓj4| ≲ Ω3  ℓj1Ω2  ℓj2Ω2  ℓj4  if j1 = j3, j1, j2, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  δj1j2ℓδj3j4ℓ|EZℓj1rZℓj2rZℓj3rZℓj4r| ≲ Ω2  ℓj1Ω2  ℓj2Ω2  ℓj3Ω2  ℓj4  if j1, j2, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Up to symmetry, this accounts for all possible (nonzero) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hence by Cauchy–Schwarz,  Eϕ2  ℓrℓr ≲ ∥Ωℓ∥3  3 + ∥Ωℓ∥3  3∥Ωℓ∥2 + ∥Ωℓ∥4 + ∥Ωℓ∥6  3 + ∥Ωℓ∥6 + ∥Ωℓ∥3  3∥Ωℓ∥4 + ∥Ωℓ∥8 ≲ ∥Ωℓ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='82)  73  Recalling that {ϕℓrℓr}ℓ,r∈[Nℓ] is an uncorrelated collection of random variables, we have  Var(S1) ≲  �  ℓ  �  r∈[Nℓ]  (Nℓ − r)2σ4  ℓ Eϕ2  ℓrℓr  ≲  �  ℓ  �  r∈[Nℓ]  (Nℓ − r)2σ4  ℓ ∥Ωℓ∥3  3  ≲  �  k  �  ℓ∈Sk  N3  ℓ ·  1  n4  k ¯N4  k  ∥Ωℓ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='83)  Combining (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='83) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='80) proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  We have  EE4  ℓ,s =  �  r1,r2,r3,r4∈[s−1]  σ4  ℓ  �  j1,j2,j3,j4  EZℓj1r1Zℓj1sZℓj2r2Zℓj2sZℓj3r3Zℓj3sZℓj4r4Zℓj4s  = σ4  ℓ  �  j1,j2,j3,j4  �  E[Zℓj1sZℓj2sZℓj3sZℓj4s] ·  �  r1,r2,r3,r4∈[s−1]  E[Zℓj1r1Zℓj2r2Zℓj3r3Zℓj4r4]  �  ��  �  =:Bℓ,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j1,j2,j3,j4  �  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='84)  We have by exhaustive casework that  |E[Zℓj1r1Zℓj2r2Zℓj3r3Zℓj4r4]|  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='85)  =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  EZ4  ℓj1r1 ≲ Ωℓj1  if j1=j2=j3=j4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2=r3=r4  EZ2  ℓj1r1EZ2  ℓj1r3 ≲ Ω2  ℓj1  if  j1=j2=j3=j4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2,r3=r4,r1̸=r3  |E[Z3  ℓj1r1Zℓj4r1]| ≲ Ωℓj1Ωℓj4  if j1=j2=j3,j1̸=j4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2=r3=r4  |E[Z2  ℓj1r1EZℓj1r3Zℓj4r3]| ≲ Ω2  ℓj1Ωℓj4  if  j1=j2=j3,j1̸=j4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2,r3=r4,r1̸=r3  |EZ2  ℓj1r1Z2  ℓj3r1| ≲ Ωℓj1Ωℓj3  if j1=j2,j3=j4,j1̸=j3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2=r3=r4  |E[Z2  ℓj1r1Z2  ℓj3r3]| ≲ Ωℓj1Ωℓj3  if j1=j2,j3=j4,j1̸=j3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2,r3=r4,r1̸=r3  |E[Zℓj1r1Zℓj3r1EZℓj1r2Zℓj3r2]| ≲ Ω2  ℓj1Ω2  ℓj3  if j1=j2,j3=j4,j1̸=j3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r3,r2=r4,r1̸=r2  |E[Z2  ℓj1r1Zℓj3r1Zℓj4r1]| ≲ Ωℓj1Ωℓj3Ωℓj4  if j1=j2,j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2=r3=r4  |E[Z2  ℓj1r1EZℓj3r3Zℓj4r3]| ≲ Ωℓj1Ωℓj3Ωℓj4  if j1=j2,j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2,r3=r4,r1̸=r3  |E[Zℓj1r1Zℓj3r1EZℓj1r2Zℓj4r2]| ≲ Ω2  ℓj1Ωℓj3Ωℓj4  if j1=j2,j1,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r3,r2=r4,r1̸=r2  |E[Zℓj1r1Zℓj2r1Zℓj3r1Zℓj4r1]| ≲ Ωℓj1Ωℓj2Ωℓj3Ωℓj4  if j1,j2,j3,j4 dist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2=r3=r4  |E[Zℓj1r1Zℓj2r1EZℓj3r3Zℓj4r3]| ≲ Ωℓj1Ωℓj2Ωℓj3Ωℓj4  if  j1,j2,j3,j4 dist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  r1=r2,r3=r4,r1̸=r3  74  Up to permutation of the indices j1, j2, j3, j4 and r1, r2, r3, r4, this accounts for all possible  cases such that (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='85) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore,  Bℓ,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='j1,j2,j3,j4 ≲  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  sΩℓj1 + s2Ω2  ℓj1  if j1 = j2 = j3 = j4  sΩℓj1Ωℓj4 + s2Ω2  ℓj1Ωℓj4  if j1 = j2 = j3, j1 ̸= j4  sΩℓj1Ωℓj3 + s2Ωℓj1Ωℓj3  if j1 = j2, j3 = j4, j1 ̸= j3  sΩℓj1Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj3Ωℓj4  if j1 = j2, j1, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  sΩℓj1Ωℓj2Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj2Ωℓj3Ωℓj4  if j1, j2, j3, j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Up to permutation of j1, j2, j3, j4, this accounts for all possible cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Returning to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='84),  we have by applying (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55) and the previous display that  EE4  ℓ,s ≲ σ4  ℓ  � �  j  Ωℓj(sΩℓj + s2Ω2  ℓj) +  �  j1̸=j4  Ωℓj1Ωℓj4(sΩℓj1Ωℓj4 + s2Ω2  ℓj1Ωℓj4)  +  �  j1̸=j3  Ωℓj1Ωℓj3(sΩℓj1Ωℓj3 + s2Ωℓj1Ωℓj3)  +  �  j1,j3,j4(dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=')  Ωℓj1Ωℓj3Ωℓj4(sΩℓj1Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj3Ωℓj4)  +  �  j1,j2,j3,j4 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Ωℓj1Ωℓj2Ωℓj3Ωℓj4(sΩℓj1Ωℓj2Ωℓj3Ωℓj4 + s2Ωℓj1Ωℓj2Ωℓj3Ωℓj4)  �  ≲ sσ4  ℓ ∥Ωℓ∥2 + s2σ4  ℓ ∥Ωℓ∥3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In the third line we group the coefficients of s and s2 and use the fact that ∥Ωℓ∥4 ≤ ∥Ωℓ∥3  3  by Cauchy–Schwarz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore  �  (ℓ,s)  EE4  ℓ,s ≲  �  (ℓ,s)  sσ4  ℓ ∥Ωℓ∥2 +  �  (ℓ,s)  s2σ4  ℓ ∥Ωℓ∥3  3  =  �  k  �  ℓ∈Sk  �  s∈[Nℓ]  sσ4  ℓ ∥Ωℓ∥2 +  �  k  �  ℓ∈Sk  �  s∈[Nℓ]  s2σ4  ℓ ∥Ωℓ∥3  3  ≲  �  k  �  ℓ∈Sk  N2  ℓ ·  1  n4  k ¯N4  k  ∥Ωℓ∥2 +  �  k  �  ℓ∈Sk  N3  ℓ ·  1  n4  k ¯N4  k  ∥Ωℓ∥3  3,  as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7  Proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  We have  �  k  �  i∈Sk  N2  i ∥Ωi∥2  n4  k ¯N4  k  ≤  �  k  1  n4  k ¯N4  k  �  i,m∈Sk  NiNm⟨Ωi, Ωm⟩  =  �  k  1  n2  k ¯N2  k  ∥µk∥2,  which establishes the first claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  75  Similarly,  �  k  �  i∈Sk  N3  i ∥Ωi∥3  3  n4  k ¯N4  k  ≤  �  k  1  n4  k ¯N4  k  �  i,m,m′∈Sk  NiNmNm′  �  j  ΩijΩmjΩm′j  ≤  �  k  1  nk ¯Nk  ∥µk∥3  3,  which proves the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The third claim follows similarly and we omit the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C  Proofs of other main lemmas and theorems  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  We start from computing E[(ˆµkj−ˆµj)2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write Xij = Ni(Ωij+Yij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows by elementary  calculation that  ˆµkj − ˆµj = µkj − µj +  �  1  nk ¯Nk  −  1  n ¯N  � �  i∈Sk  NiYij −  �  1≤ℓ≤K:ℓ̸=k  1  nℓ ¯Nℓ  �  i∈Sℓ  NiYij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For different k, the variables �  i∈Sk NiYij are independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='E[(ˆµkj − ˆµj)2] = (µkj − µj)2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='nk ¯Nk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='n ¯N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='NiYij  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='ℓ:ℓ̸=k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='n2 ¯N2 E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='���  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='NiYij  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�2�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='= (µkj − µj)2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='nk ¯Nk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='n ¯N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�2 �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='NiΩij(1 − Ωij) +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='ℓ:ℓ̸=k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='n2 ¯N2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content='+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content='1 − n ¯N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='nk ¯Nk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='ℓ:ℓ̸=k  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='NiΩij(1 − Ωij)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='= (µkj − µj)2 +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 − nk ¯Nk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content='� �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='NiΩij(1 − Ωij)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='n ¯Nnk ¯Nk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='NiΩij(1 − Ωij) − nk ¯Nk  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='n ¯N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='K  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='ℓ=1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i∈Sℓ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='NiΩij(1 − Ωij)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='��  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='δkj  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  Since Xij follows a binomial distribution, it is easy to see that E[Xij] = NiΩij and E[X2  ij] =  (E[Xij])2 + Var(Xij) = N2  i Ω2  ij + NiΩij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Combining them gives  E[Xij(Ni − Xij)] = Ni(Ni − 1)Ωij(1 − Ωij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  76  Define  ˆζkj = (ˆµkj − ˆµj)2 −  1  n2  k ¯N2  k  �  1 − nk ¯Nk  n ¯N  � �  i∈Sk  Xij(Ni − Xij)  Ni − 1  ,  It follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)-(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2) that  E[ˆζkj] = (µkj − µj)2 −  1  n ¯Nnk ¯Nk  δkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  We are ready to compute E[T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By definition, T = �p  j=1  �K  k=1 nk ¯Nk ˆζkj and ρ2 = �  j,k(µkj−  µj)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consequently,  E[T] =  p  �  j=1  K  �  k=1  nk ¯Nk  �  (µkj − µj)2 −  1  n ¯Nnk ¯Nk  δkj  �  = ρ2 −  1  n ¯N  p  �  j=1  K  �  k=1  δkj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  We use the definition of δkj in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is seen that for each 1 ≤ j ≤ p,  K  �  k=1  δkj =  K  �  k=1  �  i∈Sk  NiΩij(1 − Ωij) −  � K  �  k=1  nk ¯Nk  n ¯N  � K  �  ℓ=1  �  i∈Sℓ  NiΩij(1 − Ωij) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  Combining (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)-(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5) gives E[T] = ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  First we show that  Var(T) ≲ Θn  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  Recall  Θn1 = 4  K  �  k=1  p  �  j=1  nk ¯Nk(µkj − µj)2µkj  Θn2 = 2  K  �  k=1  �  i∈Sk  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2  N3  i  Ni − 1Ω2  ij  Θn3 =  2  n2 ¯N2  �  1≤k̸=ℓ≤K  �  i∈Sk  �  m∈Sℓ  p  �  j=1  NiNmΩijΩmj  Θn4 = 2  K  �  k=1  �  i∈Sk,m∈Sk,  i̸=m  p  �  j=1  �  1  nk ¯Nk  −  1  n ¯N  �2  NiNmΩijΩmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  and that �4  a=1 Θna = Θn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, we immediately have  Var(1′  pU1) ≤ Θn1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  77  For U2, it is shown in the Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3 that  Var(1′  pU2) = 4  K  �  k=1  �  i∈Sk  �  1≤r<s≤Ni  θi  Ni(Ni − 1)  �  ∥Ωi∥2 + O(∥Ωi∥3  3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  Var(1′  pU2) ≲ 4  K  �  k=1  �  i∈Sk  �  1≤r<s≤Ni  θi  Ni(Ni − 1)∥Ωi∥2  = 2  K  �  k=1  �  i∈Sk  θi∥Ωi∥2 = Θn2  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8)  Next we study U3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Using that Ωmj′ ≤ 1 and ∥Ωi∥1 = 1, we have  �  k̸=ℓ  nknℓ ¯Nk ¯Nℓ  n2 ¯N2  1′  p(Σk ◦ Σℓ)1p =  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j,j′  NiNmΩijΩij′ΩmjΩmj′  ≤  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j  NiNmΩijΩmj  �  j′  Ωij′  =  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  �  j  NiNmΩijΩmj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Therefore by Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4,  Var(1′  pU3) ≲  2  n2 ¯N2  �  1≤k̸=ℓ≤K  �  i∈Sk  �  m∈Sℓ  p  �  j=1  NiNmΩijΩmj = Θn3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9)  Similarly for U4, we have by the Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5 that  Var(1′  pU4) = 4  K  �  k=1  �  i∈Sk,m∈Sk  i<m  κim  ��  j  ΩijΩmj + δim  �  ≲  K  �  k=1  �  i∈Sk,m∈Sk  i<m  κim  �  j  ΩijΩmj = Θn4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)  Above we use that |δim| ≤ �  j ΩijΩmj and recall that κim = (  1  nk ¯  Nk −  1  n ¯  N )2NiNm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1,  Θn1 = 4  K  �  k=1  p  �  j=1  nk ¯Nk(µkj − µj)2µkj ≲ max  k  ∥µk∥∞ · ρ2 = max  k  ∥µk∥∞ · E T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11)  Since (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) holds, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 applies and  Θn2 + Θn3 + Θn4 ≍  �  k  ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12)  Combining (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12) proves the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  78  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4, we must prove the following claims:  (a) Under the alternative hypothesis, ψ → ∞ in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (b) For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level of α and an  asymptotic power of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (c) If we choose α = αn such that αn → 0 and 1−Φ(SNRn) = o(αn), where Φ is the CDF  of N(0, 1), then the sum of type I and type II errors of the DELVE test converges to  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We show the first claim, that ψ → ∞, under the alternative hypothesis and the condi-  tions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In particular, recall we assume that  ρ2  ��K  k=1 ∥µk∥2  =  n ¯N∥µ∥2ω2  n  ��K  k=1 ∥µk∥2  → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13)  Our first goal is to show that  T/  �  Var(T) P→ ∞  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14)  under the alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By Chebyshev’s inequality, it suffices to show that  E T ≫  �  Var(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15)  By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3,  Var(T) ≲  �  k  ∥µk∥2 + max  k  ∥µk∥∞ · ET =  �  k  ∥µk∥2 + max  k  ∥µk∥∞ · ρ2  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16)  By (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13),  ET = ρ2 ≫  �  �  �  �  K  �  k=1  ∥µk∥2 ≥ max  1≤k≤K ∥µk∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Therefore,  �  max  1≤k≤K ∥µk∥∞ · ρ ≪ ρ2 = ET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17)  Moreover, by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13),  �  k  ∥µk∥2 ≪ ρ4 = (ET)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='18)  Combining (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='18) implies (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next we show that V > 0 with high probability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=', with probability tending to 1 as  n ¯N → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that by Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11,  EV = Θn2 + Θn3 + Θn4 ≳  �  k  ∥µk∥2 > 0, and  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='19)  79  Var(V ) ≲  �  k  ∥µk∥2  n2  k ¯N2  k  ∨  �  k  ∥µk∥3  3  nk ¯Nk  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='20)  Using this, the Markov inequality, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4), we have  P  �  V < E[V ]/2  �  ≤ P  �  |V − E[V ]| ≥ E[V ]/2  �  ≤ 4Var(V )  (E[V ])2 = o(1),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='21)  which implies that V > 0 with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To finish the proof of the first claim, note that the assumptions of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 are  satisfied and we have V/Var(T) = OP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By this, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='21), we have  ψ = T1V >0  √  V  =  �  Var(T)  √  V T  �  Var(T)  1V >0 ≳  T  �  Var(T)  → ∞  in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The second claim follows directly from the first claim and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To prove the third claim, by Chebyshev’s inequality and T/  �  Var(T) → ∞, it follows  that T > (1/2)ET = (1/2)ρ2 with high probability as n ¯N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By a similar Chebyshev  argument as above, it also holds that V < (3/2)EV with high probability as n ¯N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Recall that EV = Θn2 +Θn3 +Θn4 ≲ �  k ∥µk∥2 by Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus, with high  probability as n ¯N → ∞, we have  ψ = T1V >0/  √  V ≳ ρ2/  √  EV ≳ n ¯N∥µ∥2ω2  n  ��  k ∥µk∥2 = SNRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Choosing αn as specified yield the third claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The proof is complete since all three claims  are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  Without loss of generality, we assume p is even and write m = p/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let µ ∈ Rm be  a nonnegative vector with ∥µ∥1 = 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let ˜µ = (µ′, µ′)′ ∈ Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We consider the null  hypothesis:  H0 :  Ωi = ˜µ,  1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22)  We pair it with a random alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let b1, b2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , bm be a collection of  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Rademacher variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let z1, z2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , zK denote an independent collection of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Rademacher random variables conditioned on the event | �  k zk| ≤ 100  √  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For a properly  small sequence ωn > 0 of positive numbers, let  H1 :  Ωij =  �  µj  �  1 + ωn(nk ¯Nk)−1� 1  K  �  k∈K nk ¯Nk  �  zkbj  �  ,  if 1 ≤ j ≤ m, i ∈ Sk  ˜µj  �  1 − ωn(nk ¯Nk)−1� 1  K  �  k∈K nk ¯Nk  �  zkbj−m  �  ,  if m + 1 ≤ j ≤ 2m, i ∈ Sk  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23)  80  In this section we slightly abuse notation, using ωn to refer to the (deterministic) sequence  above and reserving ω(Ω) for the random quantity  ω(Ω) =  �  �  �  �  1  n ¯N∥µ∥2  K  �  k=1  nk ¯Nk∥µk − µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='24)  As long as  ωn ≤  mink nk ¯Nk  1  K  �  k∈[K] nk ¯Nk  = mink nk ¯Nk  n ¯N/K  ,  then Ωij ≥ 0 for all i ∈ [n], j ∈ [p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore, for each 1 ≤ i ≤ n, we have ∥Ωi∥1 =  2∥µ∥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We suppose there exists a constant c ∈ (0, 1) such that  cK−1n ¯N ≤ nk ¯Nk ≤ c−1K−1n ¯N  for all k ∈ [K]  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='25)  With (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='25) in hand, we may assume without loss of generality that  ωn ≤ c/2  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='26)  This assumption implies that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23) is well-defined and moreover Ωij ≍ µj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next we characterize the random quantity ω(Ω) in terms of ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let ω2(Ω) be as in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' When Ω follows Model (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23), there exists a  constant c1 ∈ (0, 1) such that c1ω2  n ≤ ω2(Ω) ≤ c−1  1 ω2  n with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  The proof of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 is given in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, under the model  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23) it holds with probability 1 that  n ¯N∥µ∥2ω2(Ω)  ��K  k=1 ∥µk∥2  ≍ K−1/2n ¯N∥µ∥ω2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='27)  Above we use that Ωij ≍ µj , since we assume (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='26)  We also require Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 below, whose proof is given in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Suppose that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='25) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='26) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider the pair of hypotheses  in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22)-(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23) and let P0, and P1 be the respective probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If  n ¯N∥µ∥2ω2(Ω)  ��K  k=1 ∥µk∥2  ≍ K−1/2n ¯N∥µ∥ω2  n → 0,  then the chi-square distance between P0 and P1 converges to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Now we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let δn denote an arbitrary sequence tending to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Without  loss of generality, we may assume that δn ≤ c∗ for a small absolute constant c∗ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Note that K−1/2n ¯N ≥ 1 since K ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus for appropriate choice of sequences of µ = µn  and ωn ≤ c/2 in models (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23) and applying (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='27), we obtain  2δn ≥ n ¯N∥µ∥2ω2(Ω)  ��K  k=1 ∥µk∥2  ≥ δn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='28)  81  Recall the definitions of Q∗  0n and Q∗  1n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let Π denote the distribution on ξ =  {(Ni, Ωi, ℓi)} ∈ Q∗  1n induced by (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let ξ0 denote the parameter associated to the  simple null hypothesis in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22) associated to our choice of µ and ωn satisfying (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We  have by standard manipulations,  R(Q∗  0n, Q∗  1n) :=  inf  Ψ∈{0,1}  �  sup  ξ∈Q∗  0n(c0,ϵn)  Pξ(Ψ = 1) +  sup  ξ∈Q∗  1n(δn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='c0,ϵn)  Pξ(Ψ = 0)  �  =  inf  Ψ∈{0,1}  �  sup  ξ∈Q∗  0n(c0,ϵn),ξ′∈Q∗  1n(δn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='c0,ϵn)  �  Pξ(Ψ = 1) + Pξ(Ψ = 0)  �  ≥  inf  Ψ∈{0,1}  �  sup  ξ∈Q∗  0n(c0,ϵn)  Eξ′∼Π  �  Pξ(Ψ = 1) + Pξ′(Ψ = 0)  ��  ≥  inf  Ψ∈{0,1}  �  Eξ′∼Π  �  Pξ0(Ψ = 1) + Pξ′(Ψ = 0)  ��  =  inf  Ψ∈{0,1}  �  P0(Ψ = 1) + P1(Ψ = 0)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In the last line we recall the definition of P0 and P1 in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23), noting that for  all events E,  P1(E) = Eξ′∼π Pξ′(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next, by the Neyman–Pearson lemma and the standard inequality TV(P, Q) ≤  �  χ2(P, Q)  (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Chapter 2 of Tsybakov [2008]),  R(Q∗  0n, Q∗  1n) ≥  inf  Ψ∈{0,1}  �  P0(Ψ = 1) + P1(Ψ = 0)  �  = 1 − TV  �  P0, P1  �  ≥ 1 −  �  χ2(P0, P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, as δn → 0 we have χ2(P0, P1) → 0 and thus R(Q∗  0n, Q∗  1n) → 1, as  desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Proof of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Next, we perform a change of parameters that preserves the signal strength and chi-squared  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The testing problem (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='22) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23) has parameters Ωij, Ni, ¯Nk, nk, n, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let P0 and P1 denote the distributions corresponding to the null and alternative hypotheses,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For each k ∈ [K], we combine all documents in sample k to obtain new null  and alternative distributions ˜P0 and ˜P1 with parameters ˜Ωij, ˜Ni, ¯˜Ni, ˜ni, ˜n, and ˜K such that  ˜K = K = ˜n  ˜Ni = ni ¯Ni  for i ∈ [ ˜K]  ¯˜Ni ≡ ˜Ni  for i ∈ [ ˜K]  ˜ni = 1  for i ∈ [ ˜K] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='29)  82  For notational ease, we define ˜N := ¯˜N =  1  K  �  k∈[K] nk ¯Nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Furthermore, we have ˜Ωi = µ  for all i ∈ [˜n] under the null ˜Ωi = µi for all i ∈ [˜n] under the alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Explicitly, in the  reparameterized model, we have the null hypothesis  H0 :  Ωi = ˜µ,  1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='30)  and alternative hypothesis  H1 :  Ωij =  �  µj  �  1 + ωn ˜N−1  i  ˜Nzibj  �  ,  if 1 ≤ j ≤ m,  ˜µj  �  1 − ωn ˜N−1  i  ˜Nzibj−m  �  ,  if m + 1 ≤ j ≤ 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='31)  for all i ∈ [ ˜K] = [K] = [˜n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that the likelihood ratio is preserved:  dP0  dP1 =  ˜dP0  d˜P1 and  also ω(Ω) = ω(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For simplicity we work with this reparameterized model in this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  If z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , z˜n are independent Rademacher random variables then with probability at  least 1/2 it holds that  |  �  i  zi| ≤ 100  √  ˜n  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32)  by Hoeffding’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that our random model is defined in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23) where (i)  z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , z˜n are independent Rademacher random variables conditioned on the event | �  i zi| ≤  100  √  ˜n, and (ii) b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , bm are independent Rademacher random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Now we study ω2(�Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' For each 1 ≤ j ≤ m, we have �Ωij = µj(1 + ωn ˜N−1  i  ˜Nzibj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  ηj = (˜n ˜N)−1 �˜n  i=1 ˜Ni�Ωij = µj(1 + ωn¯zbj) for 1 ≤ j ≤ m and ηj = (˜n ˜N)−1 �˜n  i=1 ˜Ni�Ωij =  ˜µj(1 − ωn¯zbj) for m < j ≤ 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  ˜n  �  i=1  p  �  j=1  ˜Ni(�Ωij − ηj)2 = 2  ˜n  �  i=1  m  �  j=1  ˜Ni · µ2  jω2  n  ˜N2  ˜N2  i  (zi − ¯z)2b2  j  = 2ω2  n ˜N2∥µ∥2  ˜n  �  i=1  ˜N−1  i  (zi − ¯z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32), |¯z| ≤ 100  √  ˜n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus |zi − ¯z| ≍ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write ˜N∗ = (˜n−1 �˜n  i=1 ˜N−1  i  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  ˜n  �  i=1  p  �  j=1  ˜Ni(�Ωij − ηj)2 ≍ ω2  n ˜N2∥µ∥2 · ˜n ˜N−1  ∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Note that ˜N ≥ ˜N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Additionally, by assumption (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='25), ˜Ni ≍ ˜N ≤ c−1 ˜N∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  ˜n  �  i=1  p  �  j=1  ˜Ni(�Ωij − ηj)2 ≍ ˜n ˜N∥µ∥2ω2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='33)  Moreover, ∥η∥2 = �p  j=1 µ2  j(1 + ωn¯zbj)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By our conditioning on the event in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32),  |ωn¯zbj| ≲ ωn˜n−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  83  Since ωn ≤ 1 and �  j bj = 0, we have  ∥η∥2 = ∥µ∥2 +  p  �  j=1  µ2  jω2  n¯z2 = ∥µ∥2[1 + O(˜n−1)] ≍ ∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='34)  Hence  ω2(�Ω) = ω2(Ω) ≍ ω2  n,  where recall  ω(�Ω) =  �˜n  i=1  �p  j=1 ˜Ni(�Ωij − ηj)2  ˜n ˜N∥η∥2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='35)  This finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Proof of Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  In this proof, we continue to employ the reparametrization in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As discussed there,  this reparametrization preserves the likelihood ratio and thus the chi-square distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By definition, χ2(P0, P1) =  �  ( dP1  dP0 )2dP0 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It suffices to show that  � �dP1  dP0  �2  dP0 = 1 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='36)  From the density of of multinomial distribution, dP0 = �  i,j ˜µXij  j  , and dP1 = Eb,z[�  i,j �ΩXij  ij ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  dP1  dP0  = Eb,z  � ˜n  �  i=1  p  �  j=1  � �Ωij  ˜µj  �Xij�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let b(0) = (b(0)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , b(0)  m )′ and z(0) = (z(0)  1 , · · · , z(0)  ˜n )′ be independent copies of b and z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We construct �Ω(0)  ij similarly as in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is seen that  � �dP1  dP0  �2  dP0 = EXEb,z,b(0),z(0)  � ˜n  �  i=1  p  �  j=1  � �Ωij �Ω(0)  ij  ˜µ2  j  �Xij�  = Eb,z,b(0),z(0)  � ˜n  �  i=1  EXi  � p  �  j=1  � �Ωij �Ω(0)  ij  ˜µ2  j  �Xij��  = Eb,z,b(0),z(0)  � ˜n  �  i=1  � p  �  j=1  ˜µj ·  �Ωij �Ω(0)  ij  ˜µ2  j  � ˜  Ni��  = E[exp(M)],  with  M :=  ˜n  �  i=1  ˜Ni log  � p  �  j=1  ˜µ−1  j  �Ωij �Ω(0)  ij  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='37)  Here, the third line follows from the moment generating function of a multinomial distri-  bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We plug in the expression of �Ωij in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direct calculations,  p  �  j=1  ˜µ−1  j  �Ωij �Ω(0)  ij =  m  �  j=1  µj  �  1 + ωn ˜N−1  i  ˜Nzibj  ��  1 + ωn ˜N−1  i  ˜Nz(0)  i  b(0)  j  �  84  +  m  �  j=1  µj  �  1 − ωn ˜N−1  i  ˜Nzibj  ��  1 − ωn ˜N−1  i  ˜Nz(0)  i  b(0)  j  �  = 2∥µ∥1 + 2  m  �  j=1  µjω2  n ˜N−2  i  ˜N2ziz(0)  i  bjb(0)  j  = 1 + 2  m  �  j=1  µjω2  n ˜N−2  i  ˜N2ziz(0)  i  bjb(0)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We plug it into M and notice that log(1 + t) ≤ t is always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  M ≤  ˜n  �  i=1  ˜Ni · 2  m  �  j=1  µjω2  n  ˜N2  ˜N2  i  ziz(0)  i  bjb(0)  j  = 2 ˜Nω2  n  � ˜n  �  i=1  ˜N  ˜Ni  ziz(0)  i  �� m  �  j=1  µjbjb(0)  j  �  =: M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38)  We combine (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='38) with (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It is seen that to show (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='36), it suffices to show that  E[exp(M∗)] = 1 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39)  We now show (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write M1 = �˜n  i=1( ˜N−1  i  ˜N)ziz(0)  i  and M2 = �p  j=1 µjbjb(0)  j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Recall that we condition on the event (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By Hoeffding’s inequality, Bayes’s rule,  and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='32),  P(|M1| > t) = P  �  |  �  i  ˜N  ˜Ni  ziz(0)  i  ≥ t  ���� |  �  i  zi| ≤ 100  √  ˜n, |  �  i  z(0)  i  | ≤ 100  √  ˜n  �  =  P  �  | �  i  ˜  N  ˜  Ni ziz(0)  i  | ≥ t  �  P(| �  i zi| ≤ 100  √  ˜n) P(| �  i z(0)  i  | ≤ 100  √  ˜n)  ≤ 4 · 2 exp  �  −  t2  8 �˜n  i=1( ˜N−1  i  ˜N)2  �  = 8 exp  �  − t2  8˜n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In the last line, we have used the assumption of ˜Ni ≍ ˜N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By Hoeffding’s  inequality again, we also have  P(|M2| > t) ≤ 2 exp  �  −  t2  8 �p  j=1 µ2  j  �  = 2 exp  �  −  t2  8∥µ∥2  �  for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write s2  ˜n =  √  ˜n ˜Nω2  n∥µ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  P(M∗ > t) = P  �  2 ˜Nω2  nM1M2 > t  �  = P  �  M1M2 > t ·  √  ˜n∥µ∥s−2  ˜n  �  ≤ P  �  M1 >  √  t ·  √  ˜ns−1  ˜n  �  + P  �  M2 >  √  t · ∥µ∥s−1  ˜n  �  ≤ 8 exp  �  − t  8s2  ˜n  �  + 2 exp  �  − t  8s2  ˜n  �  ≤ 4 exp(−c1t/s2  ˜n),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='40)  for some constant c1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Here, in the last line, we have used the assumption of ˜Ni ≍ ˜N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  85  Let f(x) and F(x) be the density and distribution function of M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write ¯F(x) = 1 −  F(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Using integration by part, we have E[exp(M∗)] =  � ∞  0 exp(x)f(x)dx = − exp(x) ¯F(x)|∞  0 +  � ∞  0 exp(x) ¯F(x)dx = 1 +  � ∞  0 exp(x) ¯F(x)dx, provided that the integral exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As a result,  when s˜n = o(1),  E[exp(M∗)] − 1 =  � ∞  0  exp(t) · P(M∗ > t)  ≤ 4  � ∞  0  exp  �  −[c1s−2  ˜n − 1]t  �  dt  ≤ 4(c1s−1  ˜n − 1)−1 = 4s˜n/(c1 − s˜n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It implies E[exp(M∗)] = 1+o(1), which is exactly (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' because  s2  ˜n =  √  ˜n ˜Nω2  n∥µ∥ = n ¯N∥µ∥ω2  n  √  K  ≍  n ¯N∥µ∥ω2  n  ��  k∈K ∥µk∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  First we show that  T/  �  Var(T) ⇒ N(0, 1),  and  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='41)  V/Var(T) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='42)  If (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='41) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='42) hold, then by mimicking the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, we see that ψ is  asymptotically normal and the level-α DELVE test has asymptotic level α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We omit the  details as they are quite similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Recall the martingale decomposition of T described in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that, under  our assumptions, Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1–B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover, by Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12  Var(T) ≳ Θn2 + Θn3 + Θn4 ≳  ����  m ¯  M  n ¯N + m ¯  M η +  n ¯N  n ¯N + m ¯  M θ  ����  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='43)  Combining (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='43) with Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1–B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 and mimicking the argument in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 implies  that T/  √  V ⇒ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Thus (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='41) is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Moreover, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='42) is a direct consequence of our assumptions and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The  claims of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 regarding the null hypothesis follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To prove the claims about the alternative hypothesis, it suffices to show  T/  �  Var(T) → ∞,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44)  V > 0 with high probability, and  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='45)  V = OP(Var(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46)  Once these claims are established, we prove that ψ = T1V >0/  √  V → ∞ under the alterna-  tive by mimicking the last step of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4 in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We omit the  details as they are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  86  Note that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46) follows directly from our assumptions and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  As in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4 in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3, to establish (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44), it suffices to prove  that  ET = ρ2 ≫ Var(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='47)  Our main assumption under the alternative when K = 2 is  ∥η − θ∥2  � 1  n ¯  N +  1  m ¯  M  �  max{∥η∥, ∥θ∥} → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='48)  As shown in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, we have that  Var(T) ≲ Θn = Θn1 +  4  �  t=2  Θnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='49)  Applying (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17) to the first term and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8 to the remaining terms, we have  Var(T) ≲ max{∥η∥∞, ∥θ∥∞} · ρ2 +  ����  m ¯  M  n ¯N + m ¯  M η +  n ¯N  n ¯N + m ¯  M θ  ����  2  ≲ max{∥η∥, ∥θ∥} · ρ2 + max{∥η∥2, ∥θ∥2}  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='50)  Next, note that  ρ2 = n ¯N∥η − µ∥2 + m ¯  M∥θ − µ∥2  = n ¯N  ����η −  �  n ¯N  n ¯N + m ¯  M η +  m ¯  M  n ¯N + m ¯  M θ  �����  2  + m ¯  M  ����θ −  �  n ¯N  n ¯N + m ¯  M η +  m ¯  M  n ¯N + m ¯  M θ  �����  2  = n ¯N ·  �  m ¯  M  n ¯N + m ¯  M  �2∥η − θ∥2 + m ¯  M ·  �  n ¯N  n ¯N + m ¯  M  �2∥η − θ∥2  =  n ¯Nm ¯  M  (n ¯N + m ¯  M)∥η − θ∥2 =  � 1  n ¯N +  1  m ¯  M  �−1∥η − θ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='51)  By (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='48), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='50), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='51), we have  (ET)2  Var(T) ≳  ρ4  max{∥η∥, ∥θ∥} · ρ2 + max{∥η∥2, ∥θ∥2}  ≳  ∥η − θ∥2  ( 1  n ¯  N +  1  m ¯  M ) max{∥η∥, ∥θ∥} +  �  ∥η − θ∥2  ( 1  n ¯  N +  1  m ¯  M ) max{∥η∥, ∥θ∥}  �2 → ∞,  which proves (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='47) and thus (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To prove (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='45), we mimick the Markov argument in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='21) and use that under our  assumptions, Var(V )/(EV )2 = o(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We omit the details as they are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Since we  have established (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='44), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='45), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='46), the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  87  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7  Note that T/  �  Var(T) ⇒ N(0, 1) by our assumptions and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In particular,  using that n → ∞ and the monotonicity of the ℓp norms we have  ∥µ∥4  4  K∥µ∥4 = ∥µ∥4  4  n∥µ∥4 ≤ 1  n · ∥µ∥4  ∥µ∥4 = 1  n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Moreover, V ∗/Var(T) → 1 in probability by Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows by Slutsky’s theo-  rem that ψ∗ = T/  √  V ∗ ⇒ N(0, 1) and that the level-α DELVE test has an asymptotic level  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To conclude the proof, it suffices to show that ψ∗ → ∞ under the alternative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As in the  proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4, this follows immediately if we can show  T/  �  Var(T) → ∞,  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52)  V ∗ > 0 with high probability, and  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53)  V ∗ = OP(Var(T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='54)  Note that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='52) follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14), and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='54) is the content of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Since  our assumptions imply that EV ∗ ≫  �  Var(V ∗), (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='53) follows by a Markov argument as in  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8  We apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 to get the asymptotic null distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Since Ni = N and µ =  p−11p, it is easy to see that Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 is satisfied under our assumption of p = o(N2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Therefore, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, ψ∗ → N(0, 1) under H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We now show the asymptotic alternative distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By direct calculations and using  �n  i=1 δij = 0 and �p  j=1 δij = 0, we have  �  i,j  Ni(Ωij − µj)2 = nNβ2  n  p  ,  �  i,j  Ni(Ωij − µj)2Ωij = nNβ2  n  p2  ,  �  i  ∥Ωi∥2 = n(1 + β2  n)  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We apply Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5 and plug in the above expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let S = 1′  pU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows  that  T = nNβ2  n  p  + S + OP  �√  nNβn  p  + 1  √p  �  ,  where Var(S) = 2p−1n[1 + o(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55)  First, we plug in β2  n = a√2p/(N√n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It gives p−1nNβ2  n =  �  2n/p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Second, p−1√  nNβn ≍  (np)−1/4�  n/p = o(  �  n/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  T = a  �  2n/p + S + oP  ��  n/p  �  ,  where Var(S) = (2n/p)[1 + o(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='56)  Recall the martingale decomposition S = �  (ℓ,s) Eℓ,s where Eℓ,s is defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that Lemmas B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5 hold (even under the alternative).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define �Eℓ,s =  88  Eℓ,s/  �  Var(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Using Var(S) ≳ n �  i ∥Ωi∥2 and these lemmas, it is straightforward to  verify that the following conditions hold:  �  (ℓ,s)  Var  � �Eℓ,s  ��F≺(ℓ,s)  � P→ 1  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='57)  �  (ℓ,s)  E �E4  ℓ,s  P→ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='58)  As in Section B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1, the martingale CLT applies and we have  S/  �  Var(S) ⇒ N(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='55,  T/  �  Var(S)  →  N(a, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='59)  By Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3 and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='84),  Var(S) = [1 + o(1)]Θn2 = [1 + o(1)]Var(T)  By Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6, we have that V ∗/Var(T) → 1 in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' As a result,  V ∗/Var(S)  →  1,  in probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='60)  We combine (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='59) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='60) to conclude that ψ = T/  √  V ∗ → N(a, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  D  Proofs of the corollaries for text analysis  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Note that Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 follows immediately from the slightly more general result stated  below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider Model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) and suppose that Ω = µ1′  n under the null hypothesis  and that Ω satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1) under the alternative hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define ξ ∈ Rn by ξi = ¯N−1Ni  and let �Ω = Ω[diag(ξ)]1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , λM > 0 and �λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , �λM > 0 denote the singular  values of Ω and �Ω, respectively, arranged in decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='We further assume that under  the alternative hypothesis,  ¯N · �M  k=2 �λ2  k  ��M  k=1 λ2  k  → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  For any fixed α ∈ (0, 1), the level-α DELVE test has an asymptotic level α and an asymptotic  power 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover if Ni ≍ ¯N for all i, we may replace �M  k=2 �λ2  k with �M  k=2 λ2  k in the  numerator of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  89  Proof of Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This is a special case of our testing problem with K = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover,  µ = n−1Ωξ matches with the definition of µ in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Therefore, we can apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It remains to verify that the condition  ¯N · �M  k=2 �λ2  k  ��M  k=1 λ2  k  → ∞  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  is sufficient to lead to the condition  n ¯N∥µ∥2ω2  n  ��  i ∥Ωi∥2 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  If we show this then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7 applies directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We first calculate ω2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall ξi = Ni/ ¯N  for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Write  �Ω = Ω[diag(ξ)]1/2,  �ξ = [diag(ξ)]1/21n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  For K = n, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13), ω2  n =  1  n ¯  N∥µ∥2  �n  i=1 Ni∥Ωi − µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  ω2  n =  1  n∥µ∥2  ���(Ω − µ1′  n)[diag(ξ)]1/2���  2  F =  1  n∥µ∥2  ���Ω − µ�ξ′��2  F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  Recall that �  λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , �λM are the singular values of �Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We apply a well-known result in linear  algebra [Horn and Johnson, 1985], namely Weyl’s inequality: For any rank-1 matrix ∆,  ∥�Ω − ∆∥2  F ≥ �  k̸=1 �λ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' In (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4), µ�ξ′ is a rank-1 matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that  ���Ω − µ�ξ′��2  F ≥  M  �  k=2  �λ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  Hence  n ¯N∥µ∥2ω2  n  ��  i ∥Ωi∥2 ≥  ¯N · �M  k=2 �λ2  k  ∥Ω∥F  =  ¯N · �M  k=2 �λ2  k  ��M  k=1 λ2  k  ,  which implies (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3) by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The first claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next we prove the second claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that if Ni ≍ ¯N , then by Weyl’s inequality:  ω2  n =  1  ∥µ∥2n ¯N  �  i  Ni∥Ωi − µ|2 ≳  1  ∥µ∥2  �  i  ∥Ωi − µ∥2  =  1  ∥µ∥2 ∥Ω − µ1′  n∥2  F ≥  1  ∥µ∥2  M  �  k=2  λ2  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  n ¯N∥µ∥2ω2  n  ��  i ∥Ωi∥2 ≥  ¯N · �M  k=2 λ2  k  ∥Ω∥F  =  ¯N · �M  k=2 λ2  k  ��M  k=1 λ2  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We see that the assumption  ¯N · �M  k=2 λ2  k  ��M  k=1 λ2  k  → ∞  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  implies (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The second claim is established and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  90  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Recall the construction of a simple null and simple (random) alternative model from Section  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, specialized below to the case of K = n and Ni ≡ N:  H0 :  Ωi = ˜µ,  1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  H1 :  Ωij =  �  µj  �  1 + ωnzibj  �  ,  if 1 ≤ j ≤ m  ˜µj  �  1 − ωnzibj−m  �  ,  if m + 1 ≤ j ≤ 2m  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='8)  where b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , bm are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Rademacher random variables and z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , zn are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='d Rademacher  random variables conditioned to satisfy | �  i zi| ≤ 100√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  ˜b = (b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , bm, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' , bm)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  To derive the lower bound of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2, we assume without loss of generality that ωn is  a sufficiently small absolute constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  We claim that H1 prescribes a topic model with M = 2 topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' To see this, under the  alternative,  Ωi =  �  µ ◦ (1p + ωn ˜b)  if zi = 1  µ ◦ (1p − ωn ˜b)  if zi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9)  Moreover, we showed in Section C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 that Ωij ≥ 0 for all i, j and that ∥Ωij∥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' From  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='9), we see that Ω = AW where A ∈ Rp×2 and W ∈ R2×n are defined as follows:  A:1 = µ ◦ (1p + ωn ˜b),  A:2 = µ ◦ (1p − ωn ˜b)  W:i =  �  (1, 0)′  if zi = 1  (0, 1)′  if zi = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Moreover, under the null hypothesis, Ω clearly prescribes a topic model with K = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Therefore Ω follows the topic model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover, since Ni ≡ N, we have Ω[diag(ξ)]1/2 =  Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1 specialized to our setting, we know that the χ2 distance between  the null and alternative goes to zero if  √nN∥µ∥ω2  n → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus to prove Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 it suffices to show that  N �M  k≥2 λ2  k  ��M  k=1 λ2  k  =  Nλ2  2  ��M  k=1 λ2  k  ≳ √nN∥µ∥ω2  n  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10)  Accordingly we study the second largest singular value of Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' First we have some pre-  liminary calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Let U = {i : zi = 1}, and let V = {i : zi = −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Define  u = µ ◦ (1p + ωn ˜b),  and  91  v = µ ◦ (1p − ωn ˜b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Observe that  ⟨u, v⟩ = ∥µ∥2 − ω2  n∥µ ◦ ˜b∥2 = ∥µ∥2(1 − ω2  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Also, since ωn is a sufficiently small absolute constant,  ∥u∥2 = ∥µ∥2 + 2ωn⟨µ, µ ◦ ˜b⟩ + ω2  n∥µ ◦ ˜b∥2 = (1 + ω2  n)∥µ∥2 + 2ωn  �  j  µ2  j ˜bj ≳ ∥µ∥2,  and  ∥v∥2 = ∥µ∥2 − 2ωn⟨µ, µ ◦ ˜b⟩ + ω2  n∥µ ◦ ˜b∥2 = (1 + ω2  n)∥µ∥2 − 2ωn  �  j  µ2  j ˜bj ≳ ∥µ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11)  Again, since we assume that ωn is a sufficiently small absolute constant,  δ2 :=  ⟨u, v⟩2  ∥u∥2∥v∥2 =  ∥µ∥4(1 − ω2  n)2  (1 + ω2n)2∥µ∥4 − 4ω2n⟨µ, µ ◦ b⟩2 ≤  ∥µ∥4(1 − ω2  n)2  (1 + ω2n)2∥µ∥4 − 4ω2n∥µ∥4  =  ∥µ∥4(1 − ω2  n)2  ∥µ∥4(1 + 2ω2n − 3ω4n) =  (1 − ω2  n)2  1 + 2ω2n − 3ω4n  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12)  Note that  ∥au + bv∥2 = a2∥u∥2 + 2ab⟨u, v⟩ + b2∥v∥2 ≥ a2∥u∥2 + b2∥v∥2 − 2abδ∥u∥∥v∥  ≥ (1 − δ)  �  a2∥u∥2 + b2∥v∥2�  + ∥au − bv∥2 ≥ (1 − δ)  �  a2∥u∥2 + b2∥v∥2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='12), we have for ωn sufficiently small that  1 − δ ≥ 1 −  1 − ω2  n  �  1 + 2ω2n − 3ω4n  =  �  1 + 2ω2n − 3ω4n − 1 + ω2  n  �  1 + 2ω2n − 3ω4n  ≥  ω2  n  �  1 + 2ω2n − 3ω4n  ≳ ω2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  ∥au + bv∥2 ≥ ω2  n(a2∥u∥2 + b2∥v∥2) ≳ ω2  n∥µ∥2(a2 + b2)  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13)  Recall that if M is a rank k matrix, then  λk(M) =  sup  y:∥y∥=1, y∈Ker(M)⊥ ∥My∥ =  sup  y:∥y∥=1, y∈Im(M′)  ∥My∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14)  We have  ΩΩ′ =  �  i∈U  uu′ +  �  i∈V  vv′ = |U|uu′ + |V |vv′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let y ∈ Rn satisfy ∥y∥ = 1 and y = Ω′x for some x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We have  Ωy = ΩΩ′x = |U|⟨u, x⟩u + |V |⟨v, x⟩v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  92  By the previous equation and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='13),  ∥Ωy∥2 = ∥ΩΩ′x∥2 =  ����|U|⟨u, x⟩u + |V |⟨v, x⟩v  ����  2  ≳ ω2  n∥µ∥2�  |U|2⟨u, x⟩2 + |V |2⟨v, x⟩2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  By our conditioning on z, we have min(|U|, |V |) ≳ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Moreover  1 = ∥y∥2 = ∥Ω′x∥2 = |U|⟨u, x⟩2 + |V |⟨v, x⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Applying these facts and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='14), we obtain  λ2  2 ≥ ∥Ωy∥2 = ∥ΩΩ′x∥2 ≳ ω2  n∥µ∥2n  �  |U|⟨u, x⟩2 + |V |⟨v, x⟩2�  = ω2  n∥µ∥2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Next,  M  �  k=1  λ2  k = ∥Ω∥2  F =  �  i∈U  ∥u∥2 +  �  i∈V  ∥v∥2 = |U| · ∥u∥2 + |V | · ∥v∥2 ≍ n∥µ∥2  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='15)  We conclude that  N �M  k≥2 λ2  k  ��M  k=1 λ2  k  =  Nλ2  2  ��M  k=1 λ2  k  ≳ N · ω2  n∥µ∥2n  √n∥µ∥  = √nN∥µ∥ω2  n  which establishes (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  This is a special case of our testing problem with K = 2, we can apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6 directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It remains to verify that the condition  β2  n · (∥ηS∥1 + ∥θS∥1)  � 1  n ¯  N +  1  m ¯  M  �  max{∥η∥, ∥θ∥} → ∞  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16)  is sufficient to yield the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11) in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' This is done by calculating ∥η−θ∥2  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' By our sparse model (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4), for j ∈ S, |√ηj −  �  θj| ≥ βn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' It follows that for j ∈ S,  |ηj − θj|2 = (√ηj +  �  θj)2(√ηj −  �  θj)2 ≥ β2  n(√ηj +  �  θj)2 ≥ β2  n(ηj + θj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  ∥η − θ∥2 ≥ β2  n  �  j∈S  (ηj + θj) ≥ β2  n  �  ∥ηS∥1 + ∥θS∥1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='17)  We plug it into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='11) and see immediately that (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='16) implies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' The claim  follows directly from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  93  E  A modification of DELVE for finite p  Below we write out the variance of the terms of the raw DELVE statistic under the null,  using the proofs of Lemmas A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3–A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Var(1′  pU2) = 2  K  �  k=1  �  i∈Sk  �  1≤r<s≤Ni  (  1  nk ¯Nk  −  1  n ¯N )2  N2  i  (Ni − 1)2  �  ∥Ωi∥2 − 2∥Ωi∥3  3 + ∥Ωi∥4�  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1)  Var(1′  pU3) =  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  NiNm  ��  j  ΩijΩmj − 2  �  j  Ω2  ijΩ2  mj +  �  j,j′  ΩijΩij′ΩmjΩmj′  �  Var(1′  pU4) = 2  K  �  k=1  �  i∈Sk,m∈Sk  i̸=m  (  1  nk ¯Nk  −  1  n ¯N )2NiNm  ��  j  ΩijΩmj − 2  �  j  Ω2  ijΩ2  mj +  �  j,j′  ΩijΩij′ΩmjΩmj′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  In this section we develop an unbiased estimator for each term above, which leads to an  unbiased estimator of Var(T) by taking their sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We require some preliminary results  proved later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Recall that Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2 was established in the proof of Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If j ̸= j′, an unbiased estimator of ΩijΩij′ is  �  ΩijΩij′ :=  XijXij′  Ni(Ni − 1)  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' An unbiased estimator of Ω2  ij is  �  Ω2  ij :=  X2  ij − Xij  Ni(Ni − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2)  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' If j ̸= j′, an unbiased estimator for Ω2  ijΩ2  ij′ is  �  Ω2  ijΩ2  ij′ =  (X2  ij − Xij)(X2  ij′ − Xij′)  Ni(Ni − 1)(Ni − 2)(Ni − 3)  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' An unbiased estimator of Ω3  ij is  �  Ω3  ij :=  X3  ij − 3X2  ij + 2Xij  Ni(Ni − 1)(Ni − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3)  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' An unbiased estimator of Ω4  ij is  �  Ω4  ij :=  X4  ij − 3X3  ij − X2  ij + 3Xij  Ni(Ni − 1)(Ni − 2)(Ni − 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4)  Define  �  ∥Ωi∥2 :=  �  j  �  Ω2  ij  94  �  ∥Ωi∥3  3 :=  �  j  �  Ω3  ij  �  ∥Ωi∥4 :=  �  j  �  Ω4  ij +  �  j̸=j′  �  Ω2  ijΩ2  ij′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5)  Using Lemmas E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1–E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5 and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5), we define an unbiased estimator for each term of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Let �  Ωij = Xij/Ni and define  �  Var(1′pU2) = 2  K  �  k=1  �  i∈Sk  �  1≤r<s≤Ni  (  1  nk ¯Nk  −  1  n ¯N )2  N2  i  (Ni − 1)2  ��  ∥Ωi∥2 − 2�  ∥Ωi∥3  3 + �  ∥Ωi∥4�  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='6)  �  Var(1′pU3) =  2  n2 ¯N2  �  k̸=ℓ  �  i∈Sk  �  m∈Sℓ  NiNm  ��  j  �  Ωij �  Ωmj − 2  �  j  �  Ω2  ij �  Ω2  mj +  �  j,j′  �  ΩijΩij′  �  ΩmjΩmj′  �  �  Var(1′pU4) = 2  K  �  k=1  �  i∈Sk,m∈Sk  i̸=m  (  1  nk ¯Nk  −  1  n ¯N )2NiNm  ��  j  �  Ωij �  Ωmj − 2  �  j  �  Ω2  ij �  Ω2  mj +  �  j,j′  �  ΩijΩij′  �  ΩmjΩmj′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Define  �V =  �  Var(1′pU2) +  �  Var(1′pU3) +  �  Var(1′pU4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7)  We define exact DELVE as ˜ψ = T/�V 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Combining our results above, we obtain the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Proposition E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Consider the statistic �V defined in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Under the null hypothesis, �V  is an unbiased estimator for Var(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  With this result in hand, it is possible to derive consistency of �V as an estimator of  Var(T) under certain regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Proof of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='1  Recall that Bijr is the Bernoulli random variable Bijr = Zijr + Ωij and satisfies Xijr =  �Ni  r=1 Bijr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that  XijXij′ =  �  r,s  BijrBij′s =  �  r  BijrBij′r +  �  r̸=s  BijrBij′s = 0 +  �  r̸=s  BijrBij′s  Thus  EXijXij′ = Ni(Ni − 1)ΩijΩij′,  and we obtain  �  ΩijΩij′ =  XijXij′  Ni(Ni − 1)  is an unbiased estimator for ΩijΩij′, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  95  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='2  Proof of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Note that  X2  ijX2  ij′ =  � �  r  Bijr +  �  r̸=s  BijrBijs  �� �  r  Bij′r +  �  r̸=s  Bij′rBij′s  �  =  �  r  BijrBij′r +  �  r1̸=r2  BijrBij′s +  �  r1̸=s  Bijr1Bijs  �  r2  Bij′r2 +  �  r1̸=s  Bij′r1Bij′s  �  r2  Bijr2  +  � �  r̸=s  BijrBijs  �� �  r̸=s  Bij′rBij′s  �  =  �  r1̸=r2  BijrBij′s +  �  r1̸=s  Bijr1Bijs  �  r2  Bij′r2 +  �  r1̸=s  Bij′r1Bij′s  �  r2  Bijr2  +  � �  r̸=s  BijrBijs  �� �  r̸=s  Bij′rBij′s  �  Since BijrBij′r = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' note that  (X2  ij − Xij)(X2  ij′ − Xij′) =  �  r1̸=s1  �  r2̸=s2  Bijr1Bijs1Bij′r2Bij′s2  =  �  r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='s1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='s2 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Bijr1Bijs1Bij′r2Bij′s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  E(X2  ij − Xij)(X2  ij′ − Xij′) =  �  r1,s1,r2,s2 dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  E  �  Bijr1Bijs1Bij′r2Bij′s2  �  = Ni(Ni − 1)(Ni − 2)(Ni − 3) · Ω2  ijΩ2  ij′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  It follows that  �  Ω2  ijΩ2  ij′ =  (X2  ij − Xij)(X2  ij′ − Xij′)  Ni(Ni − 1)(Ni − 2)(Ni − 3)  is an unbiased estimator for Ω2  ijΩ2  ij′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='3  Proof of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  Recall that Bijr is the Bernoulli random variable Bijr = Zijr + Ωij and satisfies Xijr =  �Ni  r=1 Bijr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Observe that  X3  ij =  �  r  Bijr + 3  �  r1̸=r2  Bijr1Bijr2 +  �  r1̸=r2̸=r3  Bijr1Bijr2Bijr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  EX3  ij = NiΩij + 3Ni(Ni − 1)Ω2  ij + Ni(Ni − 1)(Ni − 2)Ω3  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  96  Unbiased estimators for Ωij and Ω2  ij are  Xij  Ni  X2  ij  N2  i  − Xij(Ni − Xij)  N2  i (Ni − 1)  =  1  Ni(Ni − 1)  �  X2  ij − Xij  �  ,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' Hence  X3  ij − Xij − 3(X2  ij − Xij) = X3  ij − 3X2  ij + 2Xij  is an unbiased estimator for Ni(Ni − 1)(Ni − 2)Ω3  ij, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='4  Proof of Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='5  Observe that  X4  ij =  �  r  B4  ijr + 4  �  r1̸=r2  B3  ijr1Bijr2 + 6  �  r1̸=r2  B2  ijr1B2  ijr2  + 3  �  r1̸=r2̸=r3  B2  ijr1Bijr2Bijr3 +  �  r1̸=r2̸=r3̸=r4  Bijr1Bijr2Bijr3Bijr4  =  �  r  Bijr + 10  �  r1̸=r2  Bijr1Bijr2 + 3  �  r1̸=r2̸=r3  Bijr1Bijr2Bijr3  +  �  r1̸=r2̸=r3̸=r4  Bijr1Bijr2Bijr3Bijr4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Thus  EX4  ij = NiΩij + 10Ni(Ni − 1)Ω2  ij + 3Ni(Ni − 1)(Ni − 2)Ω3  ij  + Ni(Ni − 1)(Ni − 2)(Ni − 3)Ω4  ij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content='  Plugging in unbiased estimators for the first three terms, we have  X4  ij − Xij − 10(X2  ij − Xij) − 3(X3  ij − 3X2  ij + 2Xij) = X4  ij − 3X3  ij − X2  ij + 3Xij  is an unbiased estimator for Ni(Ni − 1)(Ni − 2)(Ni − 3), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content='  Springer New York, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
+page_content=' ISBN 9780387790527.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content='  Gregory Valiant and Paul Valiant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+page_content=' Journal of Marketing, 74(2):133–148,  2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/iNAzT4oBgHgl3EQfbPw2/content/2301.01381v1.pdf'}
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+Coverability in 2-VASS with One Unary Counter is in NP
+Filip Mazowiecki ∗
+Henry Sinclair-Banks †
+Karol Węgrzycki ‡
+Abstract
+Coverability in Petri nets fnds applications in verifcation of safety properties of reactive
+systems. We study coverability in the equivalent model: Vector Addition Systems with
+States (VASS).
+A k-VASS can be seen as k counters and a fnite automaton whose transitions are labelled
+with k integers. Counter values are updated by adding the respective transition labels.
+A confguration in this system consists of a state and k counter values. Importantly, the
+counters are never allowed to take negative values. The coverability problem asks whether
+one can traverse the k-VASS from the initial confguration to a confguration with at least
+the counter values of the target.
+In a well-established line of work on k-VASS, coverability in 2-VASS is already PSPACE-
+hard when the integer updates are encoded in binary. This lower bound limits the practical-
+ity of applications, so it is natural to focus on restrictions. In this paper we initiate the study
+of 2-VASS with one unary counter. Here, one counter receives binary encoded updates and
+the other receives unary encoded updates. Our main result is that coverability in 2-VASS
+with one unary counter is in NP. This improves upon the inherited state-of-the-art PSPACE
+upper bound. Our main technical contribution is that one only needs to consider runs in a
+certain compressed linear form.
+1
+Introduction
+Vector Addition Systems with States (VASS) are a well-studied class of infnite-state systems
+(see the survey [37]). These are fnite automata with counters that can be updated, but are never
+allowed to take negative values. Thus, a confguration consists of a state and a vector over the
+natural numbers. The central decision problems are the reachability and coverability problems.
+The reachability problem asks whether from a given start confguration one can reach the tar-
+get confguration. The coverability problem is the same except that the target confguration
+need not be reached exactly, counter values are allowed to be greater. Both problems are not
+only mathematically elegant, but they have interesting theoretical applications [7] and imple-
+mentations [6]. Coverability is provably a simpler problem that is better suited for applications;
+reachability tools are mostly applied to coverability benchmarks [14]. Yet coverability has ap-
+plications in the verifcation of safety conditions in reactive systems [17, 21]. Such systems
+may require additional data structures to be accurately represented, like counters for example.
+Safety conditions often boil down to whether a particular state can be reached as opposed to a
+particular confguration [8].
+∗University of Warsaw, Warsaw, Poland, f.mazowiecki@mimuw.edu.pl. Supported by the ERC grant INFSYS,
+agreement no. 950398.
+†Centre for Discrete Mathematics and its Applications & Department of Computer Science, University of War-
+wick, Coventry, UK, h.sinclair-banks@warwick.ac.uk. Supported by EPSRC Standard Research Studentship (DTP),
+grant EP/T5179X/1.
+‡Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany, wegrzycki@cs.uni-
+saarland.de. Supported by the ERC grant TIPEA, agreement no. 850979.
+1
+arXiv:2301.13543v1  [cs.FL]  31 Jan 2023
+
+erc
+European Research Council
+Established by the European CommissionNumber of unary counters
+P = NP
+0
+1
+≥ 2
+Number of
+binary counters
+0
+NL-complete [3]
+NL-complete [38]
+NL-complete [35]
+1
+in NC2 ⊆ P [2]
+in NP [this paper]
+≥ 2
+PSPACE-complete [5]
+PSPACE-complete
+PSPACE-complete [35]
+Figure 1: The complexities of coverability in VASS with a fxed number of unary and binary en-
+coded counters. All NL lower bounds arise from the zero counters case, here coverability is di-
+rected graph reachability and that is well known to be NL-complete [3]. In the case of one binary
+counter at least two unary counters, we are not aware of any non-trivial upper bounds below
+PSPACE. When there are at least two binary counters and any number of unary counters, cov-
+erability is PSPACE-complete. The lower bound holds for 2-VASS with two binary counters [5]
+and the upper bound is given by Rackof for any fxed dimension [35]. Recall that coverability
+in general VASS, where the number of counters is not fxed, is EXPSPACE-complete [35].
+Coverability and reachability have been studied for decades. The equivalent model of Petri
+nets was introduced already in the sixties [34]. For general VASS, Lipton proved in 1976 an
+EXPSPACE lower bound that applies to both coverability and reachability [31]. Two years later,
+Rackof proved a matching EXPSPACE upper bound for coverability [35]. Later in 1981, Mayr
+proved that reachability is decidable [32] without providing an upper bound for the algorithm.
+The construction was simplifed by Kosaraju [24] and Lambert [25], and a recent series of papers
+by Leroux and Schmitz ended in 2019 by proving an Ackermann upper bound [27]. A matching
+Ackermann lower bound was published in 2021 by two independent groups [12, 26].
+Plenty of attention has been given to VASS with fxed dimension, that is when the number
+of counters k is invariable, denoted k-VASS. For fxed dimension VASS it matters much whether
+the counter updates are encoded in unary or binary. Already, Rackof gives NL and PSPACE
+upper bounds for coverability in unary encoded and binary encoded k-VASS, respectively [35].
+The coverability problem where there are no counters is just directed graph reachability that is
+NL-complete [3]. Thus, coverability in unary encoded k-VASS is NL-complete, for every fxed k.
+Coverability in binary encoded 1-VASS is in NC2 [2], it can therefore be decided in deterministic
+polynomial time. If there are two or more binary counters, coverability is PSPACE-hard [5] via
+a reduction from reachability in bounded one-counter automata that is PSPACE-complete [18].
+Therefore, coverability in binary encoded k-VASS is PSPACE-complete for every k ≥ 2. See
+Figure 1 for the complexities of coverability in VASS with a fxed number of unary and binary
+encoded counters. This is all in striking contrast to the reachability problem in fxed dimension
+VASS, since reachability in 8-VASS is already known to be nonelementary [13].
+There is a prominent line of work on 2-VASS with various encodings. The seminal paper in
+1979 of Hopcroft and Pansiot [23] shows reachability in 2-VASS is decidable, proving that the
+reachability set is efectively semi-linear. Moreover, in the same paper the authors show, by an
+example, that the 3-VASS reachability set need not be semi-linear. Later, this was improved as
+it was shown that for 2-VASS the reachability relation is efectively semi-linear [28]. This proof
+shows that every 2-VASS can be characterised by a fat model, i.e. where the underlying fnite
+automaton does not contain nested cycles. A more careful analysis of that paper, resulted in
+a PSPACE upper bound result for reachability in binary encoded 2-VASS [5]. Since coverabil-
+ity in binary encoded 2-VASS is PSPACE-hard [5], the authors were able to conclude that both
+coverability and reachability are PSPACE-complete. Just as coverability demonstrated the dif-
+ference encoding makes to complexity, so does reachability; later it was proved that reachability
+in unary encoded 2-VASS is NL-complete [16].
+2
+
+q
+λ
+(100, −1)
+ρ
+(−99, 1)
+Figure 2: Example 2-VASS with one unary counter V . Consider the instance of coverability
+consisting of V , the initial confguration q(0, 1), and the target confguration q(0, 10). Consider
+the path π = λρ λρ · · · λρ ρ · · · ρ which induces a run in V from the initial confguration q(0, 1).
+There are 990 repetitions of the pair of cycles λρ to witness the confguration q(990, 1). The
+cycles alternate so both counters remain non-negative throughout the run. This is followed by
+10 iterations of the cycle ρ so the confguration q(0, 11) is witnessed, achieving coverability of
+the target confguration q(0, 10).
+Our Results and Techniques.
+We consider the coverability problem for 2-VASS with one
+unary counter. Here, updates of one counter are encoded in binary and the updates of the other
+are encoded in unary, see Figure 2 for an example. Notice that the unary counter need not be
+limited to polynomially bounded values. Otherwise, the value of the unary counter could be en-
+coded into the states for an instance of coverability in binary encoded 1-VASS. Furthermore, we
+do not impose any restrictions on the initial and the target confgurations, i.e. both coordinates
+of these vectors are encoded in binary. Our main result is that coverability in 2-VASS with one
+unary counter is in NP.
+Coverability in binary encoded k-VASS is PSPACE-complete, for k ≥ 2. The lower bound
+limits the practicality of applications. Therefore, it is sensible to consider restricted variations
+and quantify their complexity. We remark that coverability in fxed dimension VASS had widely-
+open complexity if there was exactly one binary counter and at least one unary counter. See
+Figure 1 for a summary of the known results.
+The natural starting point is the characterisation of runs via linear path schemes [4]. In-
+tuitively, the authors prove that if coverability or reachability holds then there is a witnessing
+path of a specifc shape. Namely, all paths can be characterised by a bounded language defned
+by a regular expression of the form τ0γ∗
+1τ1 . . . τk−1γ∗
+kτk. Here τ0, . . . , τk are paths that connect
+disjoint cycles γ1, . . . , γk. Since the language is bounded, checking if there is a path for a given
+expression essentially amounts to an instance of integer linear programming. In particular, the
+authors argue that both k and |τ0|+|γ1|+|τ1|+. . .+|τk−1|+|γk|+|τk| are pseudo-polynomially
+bounded [4]. However, a polynomial bound would immediately yield an NP upper bound as such
+a regular expression can be guessed. Given that coverability in 2-VASS with two binary coun-
+ters is PSPACE-hard [5], we cannot simply directly apply the known results when dealing with
+2-VASS with one binary and one unary counter. In Section 3, we provide a detailed discussion
+and a difcult yet motivating example in Figure 3.
+To overcome this problem, we show that coverability can be witnessed by paths in com-
+pressed linear form. We relax the condition of the bounded language, by allowing to nest linear
+forms, provided that the exponents are fxed. Intuitively, an expression of the form (τγ∗τ ′)∗
+is still forbidden, but we allow for (τγeτ ′)∗, where e is fxed but can be exponentially large
+(encoded using polynomially many bits). Such a form easily provides an NP upper bound.
+We rely on two crucial observations to prove that we can focus on paths in compressed linear
+form. First, notice that the ∗ operation in a linear path scheme corresponds to iterating some
+cycle in the VASS. Since γ1, . . . , γk need to be short, one naturally focuses on short cycles. The
+issue is that there are exponentially many cycles of polynomial size. In Section 4 we prove that
+for coverability there are only polynomially many ‘optimal’ cycles. In Section 5 we deal with the
+problem when some cycle γ occurs many times in a linear path scheme witnessing coverability,
+resulting in a polynomial bound on k, the width of the linear path scheme. Then we prove that,
+either we can merge some γi and γj thus reducing the width, or that there is a cycle that has
+positive efect on one counter and non-negative efect on the other counter. Intuitively, in the
+3
+
+latter case, we can reduce the problem to coverability in 1-VASS by pumping such a cycle that
+forces one counter to take an arbitrarily large value. Moreover, such a cycle is witnessed by a
+linear path scheme. Since we need to pump this cycle, we require compressed linear forms to
+describe the repetitions of the cycle.
+We highlight that both our crucial observations rely on that we work with coverability,
+not reachability. We further highlight that we address these crucial observations through our
+technical contributions that often depend on the fact there is one unary counter.
+Further Related Work.
+Asymmetric treatment of the counters has been already considered
+for VASS. Recall that Minsky machines can be seen as VASS with the additional ability of zero-
+testing. For this model coverability is undecidable [33], even with two counters. This raised
+natural questions of what happens where only one of the counters is able to be reset or tested
+for zero. This, and more generally, reachability in VASS with hierarchical zero-tests are known to
+be decidable [36]. There is a further investigation into VASS with one zero-test [20]. Recently,
+work has appeared containing detailed analysis about 2-VASS where counters have diferent
+powers [19, 29]. Finally, one of the most famous open problems in the community is whether
+reachability is decidable for 1-VASS with a pushdown stack. For these systems, coverability is
+known to be decidable [30]. The best known lower bound is that coverability, thus reachability
+also, is PSPACE-hard [15]. Our model, 2-VASS with one unary counter, can be seen as 1-VASS
+with a singleton alphabet pushdown stack.
+The complexity of reachability in binary encoded 3-VASS remains an intriguing open prob-
+lem. It is PSPACE-hard, like in dimension two, and the only known upper bound is primi-
+tive recursive, but not even elementary [27]. Recent works on reachability in fxed dimension
+VASS [11, 9, 13] provide new examples and a better understanding of the VASS model. Inter-
+estingly, many techniques applied to fxed dimension VASS are very closely related to recent
+progress on the nonelementary and Ackermann lower bounds for general VASS [10, 12, 26]. We
+fnally and additionally motivate coverability in VASS with one binary counter and (at least) one
+unary counter as an avenue for fnding new techniques to approach VASS problems with.
+2
+Preliminaries
+Given an integer z ∈ Z we denote bitsize(z) = log2(|z| + 1) + 1. For a vector v := (v1, v2)
+we use (v)1 := v1 and (v)2 := v2 to be the projections to the frst and second coordinates,
+respectively. We use |v|max := max{|v1|, |v2|} + 1 to denote the size of vector v. We write
+v ≤ w if the inequalities hold on each coordinate. We write v < w if at least one of the
+inequalities is strict.
+A 2-VASS with one unary counter V = (Q, T) consists of a fnite set of control states Q
+and a set of transitions T ⊆ Q × Z × {−1, 0, 1} × Q. We shall refer to the frst counter
+as the binary counter and the second counter as the unary counter. The size of V is |V | =
+|Q| + �
+(p,b,u,q)∈T bitsize(b). With |V |max := |Q| + |T| · |T|max we denote the total ‘pseudo-
+polynomial size’ of the automaton, where |T|max denotes the maximum absolute value that
+occurs in the transitions. Note that in a standard 2-VASS both counters are in binary, i.e. the
+domain of updates for the second counter is also Z.
+A path π in V is a, possibly empty, sequence of transitions π = (ti)m
+i=1 such that ti =
+(qi−1, bi, ui, qi) ∈ T. A path is simple if q0, . . . , qm are distinct. A path is a cycle if q0 = qm and
+m > 0 (thus empty cycles are forbidden). We call it a q0-cycle to emphasise the frst and last
+state of the cycle. A cycle is simple if q1, . . . , qm are distinct. A cycle is short if m ≤ |Q|. The
+length of a path is the number of transitions in the path, denoted len(π) = m. We write π[i..j]
+to denote the path that is the subsequence of transitions (ti, . . . , tj) in π.
+A confguration (p, u) ∈ Q × N2, denoted p(u), is a state paired with the current bi-
+nary and unary counter values. A run is a sequence of confgurations (qi(vi))m
+i=0 such that
+4
+
+(qi−1, (vi)1 − (vi−1)1, (vi)2 − (vi−1)2, qi) ∈ T. A run can equivalently be defned by the se-
+quence of confgurations induced by following a path π starting from an initial confguration
+q0(v0). We denote this run q0(v0) π−→ qm(vm). We also write q0(v0) ∗−→ qm(vm) to indicate the
+existence of a run between two confgurations.
+In this paper we study the coverability problem for VASS.
+VASS Coverability
+INPUT:
+A VASS V = (Q, T) and two confgurations p(u) and q(v).
+QUESTION: Does p(u) ∗−→ q(v′) hold, for some v′ ≥ v?
+Do note that the initial confguration p(u) and the target confguration q(v) have both the
+binary and unary components encoded as binary integers. The reachability problem for VASS—
+which we will not study in this paper—requires v′ = v.
+Consider a path π = (ti)m
+i=1, where ti = (qi−1, bi, ui, qi). The efect of π is the sum of the
+counter updates, i.e. the vector eff(π) := �m
+i=1(bi, ui). We often focus on the two projections:
+the binary efect effb(π) := �m
+i=1 bi, and the unary efect effu(π) := �m
+i=1 ui.
+We say that a cycle γ is monotone if eff(γ) ≥ 0 or eff(γ) ≤ 0. Otherwise, we say that
+γ is non-monotone. Note the two variants of a non-monotone cycle: a positive-negative cycle
+effb(γ) > 0 and effu(γ) < 0, and a negative-positive cycle effb(γ) < 0 and effu(γ) > 0.
+Let γ be a cycle. Given e ∈ N we write γe for the path obtained by e repetitions of γ. We refer
+to e as the exponent. A linear path scheme is a regular expression of the form τ0γ∗
+1τ1 · · · τk−1γ∗
+kτk,
+where the paths τ0, τ1, . . . , τk connect disjoint cycles γ1, . . . , γk. Note that a collection of cycles
+is disjoint if no two cycles have a common state. Given ℓ = (τ0, γ1, τ1, . . . , τk−1, γk, τk), we
+say the a path π is in linear form ℓ if π = πℓ = τ0γe1
+1 τ1 · · · τk−1γek
+k τk for some exponents
+e1, . . . , ek. Note that in this defnition every path has a linear form, e.g. τ0 = π is valid. To
+leverage the defnition, we will ask whether paths are in a linear form of certain size. The size of
+a linear form ℓ is �k
+i=0 len(τi) + �k
+i=1 len(γi). The size of πℓ is �k
+i=0 len(τi) + �k
+i=1 len(γi) +
+�k
+i=1 bitsize(ei), i.e. includes the exponents. We refer to k as the width of the linear form.
+3
+Coverability in 2-VASS with One Unary Counter
+In this section we briefy discuss why the state-of-the-art techniques are not enough to prove
+that coverability in 2-VASS with one unary counter is in NP. Blondin et al. [4] show that for a
+given 2-VASS V there exists a set of linear path schemes S such that if p(u) ∗−→ q(v) in V , then
+there exists a path π in a linear path scheme ρ ∈ S such that p(u)
+π−→ q(v). For every linear
+path scheme ρ ∈ S the width of ρ, and therefore the width of every path, is bounded above by
+poly(|Q|, |T|max) [4, Theorem 3.1]. Such a path π is not necessarily a polynomial size witness,
+as the width depends on |T|max polynomially. We provide an example of a 2-VASS with one
+unary counter where the width of every linear form ℓ for a path is exponential in the input size.
+This demonstrates that the combinatorial structure of linear path schemes is not self-sufcient
+to show that there always exists a polynomial size witness of coverability.
+Paths in Compressed Linear Form.
+Nevertheless, there is a natural way to succinctly de-
+scribe the path presented in Figure 3. Let σ = λρ, and note that
+σN =
+�
+tqa αN2 tab βN2 tbqtqptpc γN2 tcd δN2 tdq
+�N
+.
+All paths and cycles are ‘small’, and the bitsize of N and N2 are polynomial in n, so σ itself
+is a path in linear form. We introduce the following generalisation of linear form paths that
+encapsulates the idea behind paths of this kind of arrangement.
+5
+
+q
+p
+a
+b
+c
+d
+λ
+ρ
+(N4, −1)
+(0, 0)
+(−N6 + N, 0)
+(−N, 0)
+(N4, 0)
+(0, 0)
+(−N6 + 1, 1)
+(−N2, 1)
+α
+(N4, −1)
+β
+(−N2, 1)
+γ
+(N4, −1)
+δ
+Figure 3: Example 2-VASS with one unary counter V , where N = 2n, where n is an input
+parameter (thus making N exponentially large). Consider the coverability instance with the
+initial confguration q(0, 1), and the target confguration q(N, 1). Let λ = tqaαN2tabβN2tbq
+and ρ = tqptpcγN2tcdδN2tdq, where txy is the transition from state x to state y. Observe that
+eff(λ) = (N, −1) and eff(ρ) = (−N + 1, 1), thus eff(λρ) = (1, 0). It is easy to then see that
+q(0, 1)
+(λρ)N
+−−−−→ q(N, 1). Intuitively the cycles λ and ρ alternate so both counters remain non-
+negative throughout the run. In Appendix A, we prove that there does not exist a linear form of
+polynomial size for a path that induces a coverability run (Claim A.1).
+Defnition 3.1 (Compressed linear form path). A path π is in compressed linear form if π =
+ρ0σf1
+1 ρ1 · · · ρk−1σfk
+k ρk for some connected paths in linear form ρ0, ρ1, . . . , ρk; cycles in linear form
+σ1, . . . , σk; and exponents f1, . . . , fk. The size of a compressed linear form path is the sum of the
+sizes of all ρi and σi (including the bitsize of their exponents) plus the bitsize of the exponents fi.
+σf1
+1
+· · ·
+σfk
+k
+ρ0
+ρ1
+ρk−1
+ρk
+Figure 4: A compressed linear form path.
+The following theorem is our main contribution.
+Theorem 3.1. Let V be a 2-VASS with one unary counter and fx two confgurations p(u) and q(v).
+If p(u) ∗−→ q(v), then there exists a path in compressed linear form π such that p(u) π−→ q(v′) and
+v′ ≥ v. The size of the compressed linear form path is polynomial in |V |+bitsize(u)+bitsize(v).
+Corollary 3.1. Coverability in 2-VASS with one unary counter is in NP.
+Proof. By Theorem 3.1 it sufces to consider paths in compressed linear form of polynomial
+size, that can be guessed in NP. It sufces to observe that a coverability instance on a given
+compressed linear form amounts to an instance of integer linear programming. Intuitively, this
+is because the nested cycles are fxed. Thus to check whether a run drops below zero it sufces
+to check before applying a cycle and after applying it for the last time (see e.g. [5, Section V,
+Lemma 14]).
+6
+
+We highlight that it is rather unexpected that only one extra ‘level’ of linear form paths is
+enough to obtain polynomial size witnesses of coverability in a 2-VASS with one unary counter,
+since the problem is PSPACE-complete for general 2-VASS. Roughly speaking, the example given
+in Figure 3 observes the most complex behaviour possible and this instance of coverability is
+witnessed by a compressed linear form path. More specifcally, compressed linear form paths
+containing only one linear form cycle sufce as witnesses for coverability in 2-VASS with one
+unary counter. Therefore, all witnesses can be represented by a compressed linear form path
+ρσNτ where ρ and τ are linear form paths to and from the single linear form cycle σ which is
+iterated N times.
+The rest of the paper is dedicated to proving Theorem 3.1. We heavily exploit both distin-
+guishing features of the problem: the fact that one counter receives unary encoded updates (as
+opposed to both counters in binary) and the fact that we aim to assert coverability (as opposed
+to reachability). Our approach is as presented in the introduction. In 4 we observe that we can
+polynomially bound the total number of distinct short cycles. We formalise this and show that
+there are only polynomially many ‘irreplaceable’ short cycles. In 5 we provide a ‘reshufing
+procedure’. If some short cycle γ repeats exponentially many times we aim to modify the path π
+by moving the cycles γ close to each other. Then either every short cycle γ will appear only in
+polynomially many ‘bundles’ γe, or we fnd a cycle σ such that eff(σ) > 0. In the latter case, by
+pumping σ we are essentially left with one counter. Finally, in Section 6 we conclude the proof
+of Theorem 3.1.
+4
+Replacing Short Cycles
+In this section, we show that there are only polynomially many short cycles that need occur in
+a run witnessing coverability. Fix a path π = (qi−1, bi, ui, qi)k
+i=1. Let 0 ≤ ib, iu ≤ k be the frst
+indices such that gb = �ib
+i=1 bi and gu = �iu
+i=1 ui are at their lowest, respectively. Note that
+gb, gu ≤ 0 since by convention if we consider ib, iu = 0 then the sum evaluates to 0. We call and
+denote these two numbers the binary guard grdb(π) = gb and the unary guard grdu(π) = gu.
+The following claim, that is proved in Appendix B, immediately follows from these defnitions.
+Claim 4.1. Both grdb(π[ib + 1..k]) = 0 and grdu(π[iu + 1..k]) = 0.
+Much like the nadir of a cycle in a one-counter net, defned in [1], we defne the binary-
+nadir state as qib, i.e. the frst state in which the binary counter frst attains the lowest value
+when executing π. We call the binary-nadir decomposition π = πb
+1πb
+2, for πb
+1 = π[1..ib] and πb
+2 =
+π[ib + 1..k], as intimated in Claim 4.1. Notice that this decomposition necessitates the binary
+guard of the path π is equal to the binary efect of the prefx πb
+1, grdb(π) = effb(πb
+1) = grdb(πb
+1).
+Furthermore, the sufx of the binary-nadir decomposition has zero binary guard grdb(πb
+2) = 0.
+We primarily utilise binary-nadir states and binary-nadir decompositions, hence the omission
+of matching unary-nadir states and unary-nadir-decompositions.
+Defnition 4.1 (Replaceable cycles). Let γ be a q-cycle and let p be the binary-nadir state of γ.
+We say that γ is replaceable if there exists a q-cycle γ′ with the same binary-nadir state p, such that
+(a) effb(γ′) ≥ effb(γ) and effu(γ′) ≥ effu(γ),
+(b) grdb(γ′) ≥ grdb(γ) and grdu(γ′) ≥ grdu(γ), and
+(c) len(γ′) ≤ len(γ).
+Additionally, at least one inequality is strict and we write γ ≺ γ′.
+We say a cycle is irreplaceable if it is not replaceable. We also say that an irreplaceable q-
+cycle γ with the binary-nadir state p is characterised by the fve values: effb(γ), effu(γ), grdb(γ),
+grdu(γ), and len(γ). The following lemma is proved in Appendix B.
+7
+
+Lemma 4.1 (Replacing cycles). Let π = π1γπ2, where γ is a q-cycle. Suppose p(u) π−→ q(v) then
+the following hold.
+• If γ is replaceable, then there exists an irreplaceable q-cycle γ ≺ γ′ such that p(u)
+π1γ′π2
+−−−−→
+q(v′).
+• If γ is irreplaceable, then for every irreplaceable q-cycle γ′ that has the same characterisation
+as γ, p(u)
+π1γ′π2
+−−−−→ q(v′).
+In both cases v′ ≥ v and len(π) ≥ len(π1γ′π2).
+For convenience, we defne the polynomial R(|Q|) := |Q|4(|Q| + 1)(2|Q| + 1)2.
+Lemma 4.2. There exists at most R(|Q|) many irreplaceable short cycles with diferent character-
+isations.
+Proof. We fx two states q and p and consider only q-cycles γ with the binary-nadir state p. Thus
+in the fnal argument one must multiply everything by |Q|2. Since we consider short cycles, the
+unary efect and the unary guard are small, i.e. −|Q| ≤ effu(γ) ≤ |Q| and −|Q| ≤ grdu(γ) ≤ 0.
+Towards a contradiction, suppose there exists more than |Q|2(|Q|+1)(2|Q|+1)2 many such
+irreplaceable q-cycles with diferent characterisations. By the pigeonhole principle there must
+exist two cycles, denoted in binary-nadir decomposition γ = γ1γ2 and γ′ = γ′
+1γ′
+2, that have the
+same values effu(γ1) = effu(γ′
+1), effu(γ2) = effu(γ′
+2), grdu(γ) = grdu(γ′), len(γ1) = len(γ′
+1),
+and len(γ2) = len(γ′
+2).
+We know that the irreplaceable q-cycles γ and γ′ have diferent characterisations, so it must
+be the case that their binary efects difer effb(γ) ̸= effb(γ′). Otherwise, the cycle with the lesser
+binary guard is replaceable, because the unary efect, unary guard, and length do not difer.
+Without loss of generality, suppose effb(γ) > effb(γ′), then grdb(γ) < grdb(γ′). Otherwise, γ′
+would be replaceable as γ ≺ γ′.
+Now consider the q-cycle σ = γ′
+1γ2, also with the binary-nadir state p. We will show that
+γ ≺ σ contradicting the fact that γ is an irreplaceable q-cycle. First, observe that σ has greater
+binary efect than γ as
+effb(σ) = effb(γ′
+1) + effb(γ2) > effb(γ1) + effb(γ2) = effb(γ),
+where the inequality holds because grdb(γ) < grdb(γ′). Second, σ and γ have equal unary efect
+because effu(γ′
+1) = effu(γ1). Third, we show that σ has a greater binary guard than γ. Since
+γ2 is the sufx of the binary-nadir decomposition of γ, it must be true that grdb(γ2) = 0. By
+Claim 4.1 grdb(σ) = grdb(γ′
+1). Combining these facts, grdb(σ) = grdb(γ′) > grdb(γ). Fourth,
+σ has at least the unary guard of γ because, in particular, the unary guard of the prefx of a path
+is at most the unary guard of the entire path.
+grdu(σ) = min{grdu(γ′
+1), effu(γ′
+1) + grdu(γ2)}
+≥ min{grdu(γ′), effu(γ′
+1) + grdu(γ2)}
+= min{grdu(γ), effu(γ1) + grdu(γ2)} = grdu(γ).
+Fifth and fnally, σ and γ have equal length because len(γ′
+1) = len(γ1). We have at least one
+strict inequality. Thus, we have reached the desired contradiction.
+5
+Reshufling Linear Form Paths
+5.1
+Reshufling Procedure
+There can be many linear forms for a path π. We will try to fnd an ‘optimal’ one, so we
+introduce a cost function to quantify linear forms. Recall that a linear form ℓ is a sequence
+8
+
+of paths τ0, τ1, . . . , τk and a sequence of cycles γ1, . . . , γk. If π is in the linear form ℓ =
+(τ0, γ1, τ1, . . . , τk−1, γk, τk) then we write πℓ = τ0γe1
+1 τ1 · · · τk−1γek
+k τk, where π = πℓ (the in-
+dex is here to stress the exact linear form). For this section, we will consider linear forms only
+containing short cycles γ, they will play a key role in the following arguments.
+We defne a cost function that assigns, to a linear form ℓ, the following pair of naturals
+C(ℓ) :=
+��k
+i=0 len(τi), k
+�
+. For convenience, we defne the polynomial P(|Q|) := 2(|Q|2 +
+1)(|Q|2 + 2) · R(|Q|), where R is the polynomial defned for Lemma 4.2. We say that a linear
+form ℓ is narrow if C(ℓ) ≤ (|Q|(P(|Q|) + 1), P(|Q|)), otherwise we say that ℓ is wide. We say
+that the triple (π′, σ, π′′) is a monotone cycle decomposition of a path π if σ is a monotone cycle,
+π = π′σπ′′, and len(σ) < len(π).
+Lemma 5.1 (Reshufing). Let π be a path such that p(u) π−→ q(v). Then there exists a path ρ such
+that p(u)
+ρ−→ q(w) where w ≥ v, len(ρ) ≤ len(π), and either
+(i) there exists a narrow linear form for ρ, or
+(ii) there exists a monotone cycle decomposition of ρ.
+Proof. We start with a series of preparations. In the early part of this proof, we provide simple
+observations to ascertain some auspicious properties of our path. In the later part of this proof,
+we present the ‘reshufing procedure’ and conclude with one of the cases in the statement of
+this lemma. In this proof we will compare linear forms using the lexicographic order ≺lex, that
+is known to be a linear-order and a well-order. Formally,
+C(ℓ′) ≺lex C(ℓ) ⇐⇒ (C(ℓ′))1 < (C(ℓ))1 or,
+(C(ℓ′))1 = (C(ℓ))1 and (C(ℓ′))2 < (C(ℓ))2.
+We start with a path π′ such that p(u) π′
+−→ q(v′) where v′ ≥ v, len(π′) ≤ len(π), and π′ has
+a linear form ℓ′ that has the least cost among all linear forms for all like-paths. That means there
+does not exist another path π′′ such that p(u) π′′
+−→ q(v′′) where v′′ ≥ v, len(π′′) ≤ len(π), and
+π′′ has a linear form ℓ′′ such that C(ℓ′′) ≺lex C(ℓ′).
+For the frst observation, suppose there exists 0 ≤ i ≤ k such that len(τi) > |Q|. Then the
+path τi can be written as τi = τ ′γτ ′′, where γ is a short cycle. We can defne the linear form ℓ′′
+by modifying ℓ′ where τi is swapped for τ ′γτ ′′. Although this increments the number of cycles
+k, we decrease the total length of the paths as len(τ ′) + len(τ ′′) < len(τi) (recall that empty
+cycles are forbidden). Thus C(ℓ′′) ≺lex C(ℓ′) contradicting the assumption that ℓ has minimum
+cost. Therefore, we assume that len(τi) ≤ |Q| for all 0 ≤ i ≤ k.
+For the second observation, we defne U := {0 ≤ i ≤ m : (vi)2 < |Q|} to be the set of
+indices of confgurations in the run that have unary counter value less than |Q|. Observe that
+if |U| > |Q|2 + 1 then there are two indices 0 < i < j ≤ m such that the two corresponding
+confgurations in the run have matching states qi = qj and equal unary counter values (vi)2 =
+(vj)2. Then, regardless of sign of its binary efect, π′[i..j] is a monotone cycle. Here, case (ii)
+immediately holds by decomposing π′ itself using the monotone cycle π′[i..j], given that i > 0
+and j ≤ m implies len(π′[i..j]) = j − i < m = len(π′). Therefore, we assume |U| ≤ |Q|2 + 1.
+We continue with the aim of satisfying the conditions of case (ii) by fnding a monotone cycle
+decomposition.
+Let d = |{γ1, . . . , γk}| be the number of distinct cycles in the linear form ℓ′. By Lemma 4.1
+and Lemma 4.2, we can assume that d ≤ R(|Q|). Otherwise, we can exchange replaceable
+q-cycles for irreplaceable q-cycles using the frst point in Lemma 4.1. It is possible that for a
+particular characterisation, we can observe more than one irreplaceable q-cycle. Then using the
+second point in Lemma 4.1, we can arbitrarily select one of these irreplaceable q-cycles with
+equal characterisations to exchange all others with. By applying these cycle replacements to π′,
+9
+
+we obtain a diferent path ρ. Defnition 4.1 ensures that we do so without decreasing the efect
+(a), without allowing the counters to take a negative value (b), and without increasing the length
+of the path (c). Therefore p(u)
+ρ−→ q(w) and w ≥ v′ ≥ v, and len(ρ) ≤ len(π′) ≤ len(π). We
+remark since cycles have been exchanged one-for-one, then ρ takes a linear form ℓ with the
+same path segments as ℓ′. Therefore, it is clear that neither the number of cycles k, nor the sum
+of the lengths of the paths between cycles, have changed. We also know that ℓ is a linear form
+for ρ with minimum cost C(ℓ) = C(ℓ′), as per the initialisation in this proof.
+Suppose ρ = ρℓ = τ0γe1
+1 τ1 · · · τk−1γek
+k τk. Let (qj(vj))m
+j=0 be the run obtained by following
+the path ρℓ from the initial confguration q0(v0) = p(u) to the fnal confguration qm(vm) =
+q(w). We may assume that ℓ is wide. Otherwise, case (i) is immediately satisfed. We also
+know that len(ρℓ) ≥ max{(C(ℓ))1, (C(ℓ))2} > P(|Q|). We may also assume that each cycle
+γ1, . . . , γk is non-monotone, i.e. it is positive-negative or negative-positive. Otherwise, case (ii)
+immediately holds by decomposing ρ itself using some monotone cycle γi, given that len(γi) ≤
+|Q| < P(|Q|) < len(ρℓ). Notice this is valid since each ei > 0 by the minimality of C(ℓ),
+otherwise you can write · · · τi−1γ0
+i τi · · · with one less cycle, decreasing (C(ℓ))2.
+From the frst observation, we get �k
+i=0 len(τi) ≤ (k + 1)|Q|. Given that ℓ is wide, either
+|Q|(P(|Q|)+1) < (C(ℓ′))1 = �k
+i=0 len(τi) ≤ (k+1)|Q| that implies P(|Q|) < k, or P(|Q|) <
+(C(ℓ′))2 = k. Regardless, P(|Q|) < k holds. Recall that |U| ≤ |Q|2 + 1 from the second
+observation. Since there are relatively ‘few’ confgurations indexed by U, there must exist a
+relatively ‘distant’ pair of consecutive confgurations indexed by U. More formally, there are i
+and j such that 0 ≤ i < j ≤ k and j − i ≥ 2(|Q|2 + 2)R(|Q|) and all confgurations that occur
+in the run over the path segment τiγei+1
+i+1 · · · γej
+j τj have unary counter value at least |Q|. Notice
+that j − i is the number of cycles in this path segment. Since j − i ≥ 2(|Q|2 + 2)R(|Q|) and by
+pigeonhole principle on the number of irreplaceable cycles, there is a common irreplaceable cycle
+γ repeated at least x = 2(|Q|2 + 2) many times. We will focus on the frst x such occurrences
+of this cycle. Let s1, . . . , sx be the indices of this cycle γ, i.e. γ = γs1 = . . . = γsx. To highlight
+these cycles, we decompose this path segment into
+τiγei+1
+i+1 · · · γej
+j τj = Λ0γf1Λ1 · · · Λx−1γfxΛx,
+where fj := esj and Λj are the concatenated paths (and cycles) in between iterations of γ,
+see Figure 5. To reiterate, we know that all confgurations that occur in the run over this path
+segment have at least |Q| unary counter value and γ is a short cycle.
+Figure 5: The decomposition of the path segment into Λ0γf1Λ1 · · · Λx−1γfxΛx, as above. Notice
+that the unary counter is always at least |Q| as no confgurations indexed by U are present.
+Reshufling Procedure.
+In the rest of the proof we will modify the path segment (above)
+of the path ρℓ with a procedure that we call reshufing. At the end of this procedure we will
+fnd a monotone cycle and satisfy case (ii) of this lemma. We either fnd this cycle directly, or
+we obtain a linear form ℓ′′ such that C(ℓ′′) ≺lex C(ℓ) contradicting the assumption that ℓ has
+minimal cost.
+10
+
+Note that x = 2(|Q|2 + 2) is even, and for every pair of consecutive cycles γ2j−1 and γ2j
+(for 1 < 2j ≤ x), consider the subsegment γf2j−1Λ2j−1γf2j. There are two scenarios depend-
+ing on the variant of the non-monotone cycle γ. In the scenario where γ is positive-negative,
+we move an iteration of γ from right to left obtaining γf2j−1+1Λ2j−1γf2j−1. In the scenario
+where γ is negative-positive, we move an iteration of γ in the opposite direction obtaining
+γf2j−1−1Λ2j−1γf2j+1.
+We repeat this procedure until one of two conditions are met. The frst is when there are no
+iterations of γ on one side, so either f2j−1 or f2j becomes 0. The second is when there appears
+a confguration, in the run over the path subsegment after reshufing, with unary counter value
+less than |Q|. See Figure 6 for a pictorial presentation of reshufing in the scenario where γ is
+positive-negative.
+Figure 6: Reshufing around a path Λ (blue) where γ (red) is positive-negative. Before reshuf-
+fing, the path subsegment · · · γΛγ · · · all confgurations have unary counter value at least |Q|
+in the run (left). After reshufing, the path subsegment · · · γγΛ · · · , there is a confguration
+with unary counter value less than |Q| in the run (right).
+We claim that after each reshufing step, the corresponding run remains executable, so we
+must check that both counters remain non-negative. Notice that by only moving a cycle, the
+total efect of the path subsegment remains the same. Therefore, if the run was executable
+before reshufing, we can safely assume that the prefx before the path subsegment and the
+sufx after the path subsegment are still executable. For that reason, consider the counter values
+of confgurations occurring in the run over the reshufed path subsegment. We focus on a single
+step of the reshufing procedure that concerns the subsegment γf2j−1Λ2j−1γf2j.
+Suppose γ is a positive-negative cycle. Then the reshufing procedure moves γ from right to
+left. We claim that since f2j−1 > 0 and Λ0γf1Λ1 · · · Λ2j−1γf2j−1 is executable, the subsegment
+Λ0γf1Λ1 · · · Λ2j−1γf2j−1+1 is executable from the initial confguration. This is because one
+prerequisite of the reshufing procedure is that all confgurations occurring in the run over the
+path subsegment have at least |Q| unary counter value. Moreover, the cycle γ has length at
+most |Q| so grdu(γ) ≥ −|Q| means the unary counter value remains non-negative. As for the
+binary counter value, since a single execution of γ increases the binary counter and an iteration
+of γ was already executed before reshufing, Λ0γf1Λ1 · · · Λ2j−1γf2j−1+1 is executable. In the
+same way, from the initial confguration, Λ0γf1Λ1 · · · Λ2j−1γf2j−1+1Λ2jγf2j−1
+2j
+is executable.
+This is because effu(γ) ≥ −|Q|, and again, all confgurations occurring in the run over the path
+subsegment have at least |Q| unary counter value, and also because of the monotonicity on the
+binary counter.
+The argument when γ is a negative-positive cycle is analogous. This concludes the correct-
+ness analysis of the reshufing procedure.
+11
+
+Finishing Reshufling.
+We analyse what happens when reshufing is fnished. Suppose that
+there exists a pair 2j − 1 and 2j such that the reshufing fnishes under the frst condition
+where all iterations of γ have been moved to one side of Λ2j−1. In this case we obtain a new
+linear form ℓ′′ for ρ, where one collection of the cycle γ has been removed (decrementing k).
+So (C(ℓ′′))2 = k − 1 < (C(ℓ))2 and the two adjacent path segments can be combined without
+changing the summed length of paths so (C(ℓ′′))1 = (C(ℓ))1. Therefore, C(ℓ′′) ≺lex C(ℓ)
+contradicting the assumption ℓ has the minimal cost.
+Otherwise, for every 1 ≤ j ≤ x/2 the reshufing of pair 2j − 1 and 2j fnishes under
+condition the second condition. So there is a confguration with unary counter value less than
+|Q| in the run induced from the path ρ for each pair 2j − 1 and 2j (see Figure 7). Recall that
+x
+2 = |Q|2 + 2, that is the number of pairs. Akin to the frst observation (in the beginning of
+this proof), we use the pigeonhole principle on the number of such confgurations to obtain
+two confgurations with matching states and equal unary counter values. The path segment
+inducing the part of the run between these two confgurations is a monotone cycle, regardless
+of the binary efect. Again, it must be true that the length of this cycle is less than the length of
+the whole path, so we obtain a monotone cycle decomposition of ρ. Thus case (ii) of the lemma
+holds.
+Figure 7: After reshufing is fnished under condition the second condition, we can fnd a zero
+unary efect cycle using the (sufciently many) confgurations with unary counter less |Q|.
+5.2
+Applying Reshufling
+Lemma 5.1 does not necessarily return a narrow linear form for a path π witnessing coverability.
+Instead it may return a monotone cycle decomposition (ρ, σ, τ) of π. Our next goal is to show
+that there exists polynomial size certifcates for ρ and σ (Lemma 5.2), and then to show that
+there exists a polynomial size certifcate for τ (Lemma 5.4). Like linear forms, there can be many
+monotone cycle decompositions for a path. Following, we will use the cost function assigning
+monotone cycle decompositions to pairs of natural numbers D((ρ, σ, τ)) := (len(ρσ), len(σ)).
+Note that we can compare two decompositions using their cost, even if they are for two diferent
+paths.
+Lemma 5.2. Suppose p(u) ∗−→ q(v) yet there is no narrow linear form ℓ for any path π such that
+p(u) π−→ q(w) and w ≥ v, then there exists a path π′ such that
+(a) p(u) π′
+−→ q(w′) where w′ ≥ v,
+(b) there is a monotone cycle decomposition (ρ, σ, τ) of π′ where eff(σ) > 0, and
+(c) there are narrow linear forms for both ρ and σ.
+Proof. We will again use the lexicographical order ≺lex to compare the cost of monotone cycle
+decompositions. Let π be a path of minimum length such that p(u)
+π−→ q(w) where w ≥ v.
+12
+
+Let c = (ρ, σ, τ) be the monotone cycle decomposition of π that minimizes the cost D(c) under
+the ≺lex order. Such a decomposition must exist, otherwise applying Lemma 5.1 would return
+a narrow linear form ℓ′ for ρ such that p(u)
+ρ−→ q(w′) and w′ ≥ w ≥ v, contradicting an
+assumption of this lemma. Observe that eff(σ) > 0, otherwise one can remove σ and consider
+the shorter path ρτ, contradicting the minimal length of π. Next, we argue that ρ and σ do not
+have monotone cycle decompositions, we then leverage Lemma 5.1 to obtain the narrow linear
+forms required.
+Path ρ cannot be decomposed further.
+Towards a contradiction, assume that there is a
+monotone cycle decomposition c′ = (ρ′, σ′, τ ′) of ρ. Observe that the following monotone
+cycle decomposition c′ = (ρ′, σ′, τ ′στ) of π has lower cost D(c′) ≺lex D(c) as (D(c′))1 =
+len(ρ′) + len(σ′) < len(ρ) + len(σ) = (D(c))1. This contradicts the assumption that (ρ, σ, τ)
+has minimum cost.
+Suppose p(u)
+ρ−→ p′(x). Since there is no monotone cycle decomposition, applying Lemma 5.1
+to ρ returns a path ρ′ with a narrow linear form such that p(u)
+ρ′
+−→ p′(x′) where x′ ≥ x and
+len(ρ′) ≤ len(ρ).
+Cycle σ cannot be decomposed further.
+Towards a contradiction, assume that there is a
+monotone cycle decomposition (ρ′, σ′, τ ′) of σ. Observe that the following monotone cycle
+decomposition c′ = (ρρ′, σ′, τ ′τ) of π has lower cost D(c′) ≺lex D(c) as (D(c′))1 = len(ρ) +
+len(ρ′) + len(σ′) ≤ len(ρ) + len(σ) = (D(c))1 and (D(c′))2 = len(σ′) < len(σ) = (D(c))2.
+This contradicts the assumption that (ρ, σ, τ) has minimum cost.
+Suppose p′(x) σ−→ p′(y). Since there is no monotone cycle decomposition, applying Lemma 5.1
+to σ returns a path σ′ with a narrow linear form such that p′(x) σ′
+−→ p′(y′) where y′ ≥ y and
+len(σ′) ≤ len(σ). In particular, it is also true that eff(σ′) ≥ eff(σ) > 0.
+Replacing ρ′ for ρ and σ′ for σ in π yields a path π′. Clearly if p(u) π−→ q(w) where w ≥ v,
+then p(u) π′
+−→ q(w′) where w′ ≥ w ≥ v. Finally, (ρ′, σ′, τ) is monotone cycle decomposition
+of π′ such that eff(σ′) > 0 and ρ′ and σ′ have narrow linear forms, as required.
+We now aim to obtain a narrow linear form for τ. Note that Lemma 5.2 gives us a monotone
+cycle σ with positive efect on at least one counter, i.e. eff(σ) > 0. By pumping σ we can force
+one of the counters to take an arbitrarily large value (following, the vector x refects this large
+value for Lemma 5.4). Then, loosely speaking, the problem reduces to coverability in 1-VASS.
+However, proving the existence of a polynomial size compressed linear form path in Theorem 3.1
+requires more care. Note that Lemma 5.4 is stated for 2-VASS (not necessarily with one unary
+counter). First we need to recall the following bound on counter values observed throughout
+runs. Recall that |V |max := |Q| + |T| · |T|max is the pseudo-polynomial size of the input.
+Theorem 5.3 (Corollary from Theorem 3.2 in [4]). Consider a 2-VASS (with both counters in bi-
+nary) V = (Q, T) and let p(u) ∗−→ q(v), then there exists a run p(u) = q0(v0), q1(v1), . . . , qm(vm) =
+q(v) such that |v0|max, |v1|max, . . . , |vm|max ≤ (|V |max + |u|max + |v|max)O(1).
+In the following lemma, that is proved in Appendix C, given a 2-VASS V , the initial confgu-
+ration p(u), and target confguration q(v), we write B in place of (|V |max+|u|max+|v|max)O(1)
+from Theorem 5.3 and we fx x = (4B|Q|2|V |2
+max, 0).
+Lemma 5.4. Consider a 2-VASS (with both counters in binary) V = (Q, T) and let p(u) ∗−→ q(v),
+then there exists a narrow linear form path π′ such that p(u + x) π′
+−→ q(v′) for some v′ ≥ v.
+13
+
+6
+Proof of Theorem 3.1
+Before proving Theorem 3.1, we employ the fact that for a general 2-VASS, not necessarily with
+one unary counter, the exponents of cycles in linear forms can be pseudo-polynomially bounded.
+Lemma 6.1 (Corollary from Lemma 18 in [5]). Let π be path in a 2-VASS with a linear form
+π = τ0γf1
+1 τ1 . . . γfk
+k τk such that p(u) π−→ q(v). Then there exist a path π′ = τ0γe1
+1 τ1 · · · τk−1γek
+k τk
+such that p(u)
+π′
+−→ q(v′) where v′ ≥ v and bitsize(e1), . . . , bitsize(ek) are all bounded by a
+polynomial in |V | + bitsize(u) + bitsize(v).
+Proof of Theorem 3.1. Let p(u) π−→ q(v) for some path π. If there is a narrow linear form ℓ for π
+then by Lemma 6.1 we obtain π′ = τ0γe1
+1 τ1 · · · τk−1γek
+k τk such that p(u) π′
+−→ q(v′) where v′ ≥ v
+and bitsize(e1), . . . , bitsize(ek) are all bounded above by a polynomial in |V | + bitsize(u) +
+bitsize(v). Since ℓ is a narrow linear form, we know that k ≤ P(|Q|) so �k
+i=1 len(γi) ≤ k|Q| ≤
+|Q|P(|Q|) and we also know that �k
+i=0 len(τi) ≤ |Q|(P(|Q|) + 1). Together, this implies the
+linear form path π′ is of polynomial size.
+It remains to consider the case when there is no narrow linear form ℓ for π. By Lemma 5.2
+(via Lemma 5.1) there exists a path π′ such that p(u) π′
+−→ q(v′) and v′ ≥ v. Moreover, there is a
+monotone cycle decomposition (ρ, σ, τ) of π′ such that eff(σ) > 0 and there are narrow linear
+forms for both ρ and σ.
+Assume that (eff(σ))1 > 0. This is without loss of generality because if (eff(σ))1 = 0 then
+one can fip the coordinates in V , u and v (for the remainder of the proof it will not matter
+that one counter is unary). Let p′(m) be the confguration such that p(u)
+ρ
+−→ p′(m) στ
+−→ q(v′).
+Observe that since eff(σ) > 0 for every i ∈ N the path ρσi induces the run p(u)
+ρσi
+−−→ p′(m + i ·
+eff(σ)). Consider x = (x)1 = 4B|Q|2|V |2
+max (for Lemma 5.4), clearly x is large enough so that
+p(u)
+ρσx
+−−→ p′(m′) and m′ ≥ m + x. By Lemma 5.4 there exists a narrow linear form for a path
+τ ′ such that p′(m′) τ ′
+−→ q(v′′) and v′′ ≥ v′.
+We conclude by considering the compressed linear form path ρσxτ ′ such that p(u)
+ρσxτ ′
+−−−→
+q(v′′) and v′′ ≥ v′ ≥ v. Since ρ, σ, and τ ′ have narrow linear forms, we can also bound the
+exponents using Lemma 6.1 as in the beginning of this proof. Finally, bitsize(x) is polynomial in
+|V |+bitsize(u)+bitsize(v) much like the exponents of the cycles in the linear forms. Therefore,
+the size of the compressed linear form ρσxτ ′ is polynomial in |V |+bitsize(u)+bitsize(v).
+7
+Conclusion and Future Work
+In this paper we proved that coverability in 2-VASS with one unary counter is in NP, a drop in
+complexity from PSPACE for general 2-VASS. We achieve this by using our new techniques. Most
+notably, we polynomially bounded the number of short cycles that need to be used (Section 4).
+Then, we attempt to fnd a polynomial linear form path by replacing short cycles and reshufing
+the path (Section 5).
+A natural extension is to consider whether coverability in 3-VASS with one binary counter
+and two unary counters is also in NP. The technique for polynomially bounding the number
+of short cycles that need be used can easily be generalised to higher dimension VASS with one
+binary counter and multiple unary counters. However, it is not clear how to modify and use
+our reshufing technique. Another open problem is whether reachability in 2-VASS with one
+unary counter is also in NP. Note that completeness would immediately follow from the fact
+that reachability in binary encoded 1-VASS is NP-hard [22].
+14
+
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+Decidability of Reachability in Vector Addition Systems (Preliminary
+Version). In Harry R. Lewis, Barbara B. Simons, Walter A. Burkhard, and Lawrence H.
+Landweber, editors, Proceedings of the 14th Annual ACM Symposium on Theory of Com-
+puting, May 5-7, 1982, San Francisco, California, USA, pages 267–281. ACM, 1982. doi:
+10.1145/800070.802201.
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+99(1):79–104, 1992. doi:10.1016/0304-3975(92)90173-D.
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+62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO,
+USA, February 7-10, 2022, pages 1241–1252. IEEE, 2021. doi:10.1109/FOCS52979.
+2021.00121.
+[27] Jérôme Leroux and Sylvain Schmitz. Reachability in Vector Addition Systems is Primitive-
+Recursive in Fixed Dimension. In 34th Annual ACM/IEEE Symposium on Logic in Computer
+Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1–13. IEEE, 2019. doi:
+10.1109/LICS.2019.8785796.
+[28] Jérôme Leroux and Grégoire Sutre. On Flatness for 2-Dimensional Vector Addition Sys-
+tems with States. In Philippa Gardner and Nobuko Yoshida, editors, CONCUR 2004 - Con-
+currency Theory, 15th International Conference, London, UK, August 31 - September 3, 2004,
+Proceedings, volume 3170 of Lecture Notes in Computer Science, pages 402–416. Springer,
+2004. doi:10.1007/978-3-540-28644-8\_26.
+[29] Jérôme Leroux and Grégoire Sutre. Reachability in Two-Dimensional Vector Addition Sys-
+tems with States: One Test Is for Free. In Igor Konnov and Laura Kovács, editors, 31st
+International Conference on Concurrency Theory, CONCUR 2020, September 1-4, 2020, Vi-
+enna, Austria (Virtual Conference), volume 171 of LIPIcs, pages 37:1–37:17. Schloss Dagstuhl
+17
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+- Leibniz-Zentrum für Informatik, 2020. doi:10.4230/LIPIcs.CONCUR.2020.
+37.
+[30] Jérôme Leroux, Grégoire Sutre, and Patrick Totzke. On the Coverability Problem for Push-
+down Vector Addition Systems in One Dimension.
+In Magnús M. Halldórsson, Kazuo
+Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Pro-
+gramming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Pro-
+ceedings, Part II, volume 9135 of Lecture Notes in Computer Science, pages 324–336. Springer,
+2015. doi:10.1007/978-3-662-47666-6\_26.
+[31] Richard Lipton. The Reachability Problem Requires Exponential Space. Department of
+Computer Science. Yale University, 62, 1976.
+[32] Ernst W. Mayr. An Algorithm for the General Petri Net Reachability Problem. SIAM J.
+Comput., 13(3):441–460, 1984. doi:10.1137/0213029.
+[33] Marvin L. Minsky. Computation: Finite and Infnite Machines. Prentice-Hall, Inc., 1967.
+[34] C. Petri. Kommunikation mit Automaten, Ph. D. dissertation. University of Bonn, 1962.
+[35] Charles Rackof. The Covering and Boundedness Problems for Vector Addition Systems.
+Theor. Comput. Sci., 6:223–231, 1978. doi:10.1016/0304-3975(78)90036-1.
+[36] Klaus Reinhardt. Reachability in Petri Nets with Inhibitor Arcs. Electron. Notes Theor.
+Comput. Sci., 223:239–264, 2008. doi:10.1016/j.entcs.2008.12.042.
+[37] Sylvain Schmitz.
+The Complexity of Reachability in Vector Addition Systems.
+ACM
+SIGLOG News, 3(1):4–21, 2016.
+URL: https://dl.acm.org/citation.cfm?
+id=2893585.
+[38] Leslie G. Valiant and Mike Paterson. Deterministic One-Counter Automata. J. Comput.
+Syst. Sci., 10(3):340–350, 1975. doi:10.1016/S0022-0000(75)80005-5.
+A
+Proofs of Section 3
+Claim A.1. Given the instance of coverability consisting of V from Figure 3, and the two confg-
+urations q(0, 1) and q(N, 1), there does not exist a path π in linear form of polynomial size such
+that q(0, 1) π−→ q(v′) where v′ ≥ (N, 1).
+Proof. First, let us consider the scenario in a run from the confguration q(0, 1), this is the initial
+confguration. The only option is to take the frst transition tqa of λ, so let us consider a single
+iteration of this cycle. So that the binary counter does not take a negative value, just before
+tbq the last transition in λ, its value must be at least N6 − N. This only be satisfed when
+both α and β are exhausted, so to speak. In other words, α is iterated (exactly N2 many times)
+until the value of the binary counter is less than N2. Then β is iterated (also exactly N2 many
+times) until the value on the unary counter is equal to 0. Therefore, an iteration of the cycle
+λ = tqaαN2tabβN2tbq is taken. This has efect (N, −1), so the confguration q(N, 0) is observed.
+Second, let us consider the scenario in a run from the confguration q(N, 0). This occurs
+after following an iteration of λ, as detailed above. The only option is to take the frst transition
+tqp of ρ, so let us consider a single iteration of this cycle. Like before, so that the binary counter
+does not take a negative value, just before tdq the last transition in ρ, its value must be at least
+N6. This is only satisfed when both γ and δ are exhausted, much like for λ above. In other
+words, γ is iterated (exactly N2 many times) until the value of the binary counter is less than
+N2. Then δ is iterated (also exactly N2 many times) until the value on the unary counter is
+18
+
+equal to 0. Therefore, an iteration of the cycle ρ = tqptpcγN2tcdδN2tdq is taken. This has efect
+(−N + 1, 1), so the confguration q(1, 1) is observed.
+We again are in the same scenario where the frst transition of λ can be iterated. We repeat
+the above arguments, as a result alternating the cycles λ and ρ with the net efect (1, 0). After
+N iterations of the pair of cycles λρ, the confguration q(N, 1) is fnally witnessed.
+The number of iterations of each cycle α and β within λ is equal to N2. The same is true
+for the number of iterations of each cycle γ and δ within λ. Additionally, in order to witness a
+confguration guaranteeing coverability of the target confguration q(N, 1), at least N iterations
+of the pair λρ are required. Given that there are 4N such cycles in this run, it is not possible to
+construct a path in linear form of polynomial size.
+B
+Proofs of Section 4
+Proof of Claim 4.1. Suppose grdb(π[ib + 1..k]) < 0. Then there exists ib + 1 ≤ j ≤ k such that
+grdb(π[ib + 1..k]) = �j
+i=ib+1 bi < 0. Now consider the binary guard of the entire path π.
+grdb(π) =
+bi
+�
+i=1
+bi >
+bi
+�
+i=1
+bi + grdb(π[ib + 1..k]) =
+j
+�
+i=1
+bi
+The prefx path π[1..j] has smaller efect than π[1..ib]. This contradicts the defnition of the
+binary guard of π, so grdb(π[ib+1..k]) ≥ 0. We know that grdb(π[ib+1..k]) ≤ 0, so we conclude
+grdb(π[ib + 1..k]) = 0. The same argument can be used to show grdu(π[iu + 1..k]) = 0.
+Proof of Lemma 4.1. In either case, we refer to the Defnition 4.1. By (a), we have effb(γ′) ≥
+effb(γ) and effu(γ′) ≥ effu(γ) that imply v′ ≥ v. By (b), we have both grdb(γ′) ≥ grdb(γ) and
+grdu(γ′) ≥ grdu(γ) that imply the binary counter and the unary counter remain non-negative.
+By (c), len(γ′) ≤ len(γ), so we can conclude len(π) ≥ len(π1γ′π2).
+C
+Proofs of Section 5
+Proof of Lemma 5.4. Since we work with a 2-VASS we write grd1(π) and grd2(π) (as opposed to
+grdb(π) and grdu(π)) to denote the guards on the frst and second coordinate. Let p(u) π−→ q(v)
+such that π is a bounded run as in Theorem 5.3.
+Somewhat similar to how it is defned in [11], we say a cycle σ is semi-positive if (eff(σ))2 >
+0. We say that (ρ, σ, τ) is a semi-positive cycle decomposition of π if π = ρστ and σ is a
+semi-positive cycle.
+Consider the case when there is no semi-positive cycle decomposition of π. Then from right
+to left we remove maximal length cycles from π, resulting in all cycles being removed from π, to
+obtain a simple path π′. Since we removed only cycles that are not semi-positive, observe that
+eff(π) ≤ eff(π′) + x and grd2(π) ≤ grd2(π′). This holds because len(π′) < |Q| (otherwise
+we could remove another cycle) and grd1(π′) > −|Q| · |V |max. We can write the narrow linear
+form π′ as a single path satisfying the conditions of the lemma.
+We now focus on the case when π contains a semi-positive cycle σ. Notice that we can
+assume that σ is a simple cycle, otherwise σ contains a shorter cycle σ′. If σ′ is a semi-positive
+cycle then either it is simple, or again we can look for a shorter cycle in σ′. If at some point
+we fnd a cycle that is not semi-positive, we delete it shortening previous semi-positive cycles.
+We continue this procedure exhaustively. In the end, since every time we reduce the length of a
+cycle, we obtain a simple semi-positive cycle.
+Let ργ be the shortest prefx of π such that γ is a simple semi-positive cycle, and suppose
+p(u)
+ρ−→ p′(m) (making γ a p′-cycle). Such a ρ and γ must exist by the observation in the previous
+paragraph. Now, let ρ′ be the simple path constructed by consecutively removing simple cycles
+19
+
+from ρ. By minimality of ρ, only cycles that are not semi-positive are removed. Recall that by
+Theorem 5.3, |m|max ≤ B. Thus p(u + (B(1 + |V |max), 0))
+ρ′
+−→ p′(m′) for some m′ ≥ m.
+Let τ be a simple path from p′ to q. Such a path exists as π is a path from p, through q′, to q.
+We defne the path π′ := ρ′γeτ, where e := B · |Q| · |V |max. This provides a narrow linear form
+for π′ as ρ′ is a simple path, γ is a simple cycle, and τ is a simple path. Since γ is simple cycle,
+it is short, thus both grd1(γ) > −|Q| · |V |max and (eff(γe))1 > −e|Q| · |V |max are bounded
+below by −B|Q|2|V |2
+max. Thus p(u + x)
+ρ′γe
+−−→ p′(m′′), where m′′ ≥ v + x
+2 . Finally, since τ is
+simple grd1(τ), (eff(τ))1 ≥ −|Q| · |V |max. Thus p(u + x) π′
+−→ q(v′) for v′ ≥ v.
+20
+
diff --git a/jdFRT4oBgHgl3EQfWTfY/content/tmp_files/load_file.txt b/jdFRT4oBgHgl3EQfWTfY/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..067da60ce75834daa7a4d0ae6275240f555c053f
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+++ b/jdFRT4oBgHgl3EQfWTfY/content/tmp_files/load_file.txt
@@ -0,0 +1,1019 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf,len=1018
+page_content='Coverability in 2-VASS with One Unary Counter is in NP  Filip Mazowiecki ∗  Henry Sinclair-Banks †  Karol Węgrzycki ‡  Abstract  Coverability in Petri nets fnds applications in verifcation of safety properties of reactive  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We study coverability in the equivalent model: Vector Addition Systems with  States (VASS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  A k-VASS can be seen as k counters and a fnite automaton whose transitions are labelled  with k integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Counter values are updated by adding the respective transition labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  A confguration in this system consists of a state and k counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Importantly, the  counters are never allowed to take negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The coverability problem asks whether  one can traverse the k-VASS from the initial confguration to a confguration with at least  the counter values of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  In a well-established line of work on k-VASS, coverability in 2-VASS is already PSPACE-  hard when the integer updates are encoded in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This lower bound limits the practical-  ity of applications, so it is natural to focus on restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In this paper we initiate the study  of 2-VASS with one unary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Here, one counter receives binary encoded updates and  the other receives unary encoded updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Our main result is that coverability in 2-VASS  with one unary counter is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This improves upon the inherited state-of-the-art PSPACE  upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Our main technical contribution is that one only needs to consider runs in a  certain compressed linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  1  Introduction  Vector Addition Systems with States (VASS) are a well-studied class of infnite-state systems  (see the survey [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' These are fnite automata with counters that can be updated, but are never  allowed to take negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus, a confguration consists of a state and a vector over the  natural numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The central decision problems are the reachability and coverability problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The reachability problem asks whether from a given start confguration one can reach the tar-  get confguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The coverability problem is the same except that the target confguration  need not be reached exactly, counter values are allowed to be greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Both problems are not  only mathematically elegant, but they have interesting theoretical applications [7] and imple-  mentations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Coverability is provably a simpler problem that is better suited for applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  reachability tools are mostly applied to coverability benchmarks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Yet coverability has ap-  plications in the verifcation of safety conditions in reactive systems [17, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Such systems  may require additional data structures to be accurately represented, like counters for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Safety conditions often boil down to whether a particular state can be reached as opposed to a  particular confguration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  ∗University of Warsaw, Warsaw, Poland, f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='mazowiecki@mimuw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Supported by the ERC grant INFSYS,  agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' 950398.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  †Centre for Discrete Mathematics and its Applications & Department of Computer Science, University of War-  wick, Coventry, UK, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='sinclair-banks@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Supported by EPSRC Standard Research Studentship (DTP),  grant EP/T5179X/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  ‡Saarland University and Max Planck Institute for Informatics, Saarbrücken, Germany, wegrzycki@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='uni-  saarland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Supported by the ERC grant TIPEA, agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' 850979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='13543v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='FL]  31 Jan 2023  erc  European Research Council  Established by the European CommissionNumber of unary counters  P = NP  0  1  ≥ 2  Number of  binary counters  0  NL-complete [3]  NL-complete [38]  NL-complete [35]  1  in NC2 ⊆ P [2]  in NP [this paper]  ≥ 2  PSPACE-complete [5]  PSPACE-complete  PSPACE-complete [35]  Figure 1: The complexities of coverability in VASS with a fxed number of unary and binary en-  coded counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' All NL lower bounds arise from the zero counters case, here coverability is di-  rected graph reachability and that is well known to be NL-complete [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the case of one binary  counter at least two unary counters, we are not aware of any non-trivial upper bounds below  PSPACE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' When there are at least two binary counters and any number of unary counters, cov-  erability is PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The lower bound holds for 2-VASS with two binary counters [5]  and the upper bound is given by Rackof for any fxed dimension [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recall that coverability  in general VASS, where the number of counters is not fxed, is EXPSPACE-complete [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Coverability and reachability have been studied for decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The equivalent model of Petri  nets was introduced already in the sixties [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For general VASS, Lipton proved in 1976 an  EXPSPACE lower bound that applies to both coverability and reachability [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Two years later,  Rackof proved a matching EXPSPACE upper bound for coverability [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Later in 1981, Mayr  proved that reachability is decidable [32] without providing an upper bound for the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The construction was simplifed by Kosaraju [24] and Lambert [25], and a recent series of papers  by Leroux and Schmitz ended in 2019 by proving an Ackermann upper bound [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A matching  Ackermann lower bound was published in 2021 by two independent groups [12, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Plenty of attention has been given to VASS with fxed dimension, that is when the number  of counters k is invariable, denoted k-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For fxed dimension VASS it matters much whether  the counter updates are encoded in unary or binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Already, Rackof gives NL and PSPACE  upper bounds for coverability in unary encoded and binary encoded k-VASS, respectively [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The coverability problem where there are no counters is just directed graph reachability that is  NL-complete [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus, coverability in unary encoded k-VASS is NL-complete, for every fxed k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Coverability in binary encoded 1-VASS is in NC2 [2], it can therefore be decided in deterministic  polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' If there are two or more binary counters, coverability is PSPACE-hard [5] via  a reduction from reachability in bounded one-counter automata that is PSPACE-complete [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Therefore, coverability in binary encoded k-VASS is PSPACE-complete for every k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' See  Figure 1 for the complexities of coverability in VASS with a fxed number of unary and binary  encoded counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This is all in striking contrast to the reachability problem in fxed dimension  VASS, since reachability in 8-VASS is already known to be nonelementary [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  There is a prominent line of work on 2-VASS with various encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The seminal paper in  1979 of Hopcroft and Pansiot [23] shows reachability in 2-VASS is decidable, proving that the  reachability set is efectively semi-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Moreover, in the same paper the authors show, by an  example, that the 3-VASS reachability set need not be semi-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Later, this was improved as  it was shown that for 2-VASS the reachability relation is efectively semi-linear [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This proof  shows that every 2-VASS can be characterised by a fat model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' where the underlying fnite  automaton does not contain nested cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A more careful analysis of that paper, resulted in  a PSPACE upper bound result for reachability in binary encoded 2-VASS [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since coverabil-  ity in binary encoded 2-VASS is PSPACE-hard [5], the authors were able to conclude that both  coverability and reachability are PSPACE-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Just as coverability demonstrated the dif-  ference encoding makes to complexity, so does reachability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' later it was proved that reachability  in unary encoded 2-VASS is NL-complete [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  2  q  λ  (100, −1)  ρ  (−99, 1)  Figure 2: Example 2-VASS with one unary counter V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Consider the instance of coverability  consisting of V , the initial confguration q(0, 1), and the target confguration q(0, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Consider  the path π = λρ λρ · · · λρ ρ · · · ρ which induces a run in V from the initial confguration q(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  There are 990 repetitions of the pair of cycles λρ to witness the confguration q(990, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The  cycles alternate so both counters remain non-negative throughout the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This is followed by  10 iterations of the cycle ρ so the confguration q(0, 11) is witnessed, achieving coverability of  the target confguration q(0, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Our Results and Techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We consider the coverability problem for 2-VASS with one  unary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Here, updates of one counter are encoded in binary and the updates of the other  are encoded in unary, see Figure 2 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notice that the unary counter need not be  limited to polynomially bounded values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Otherwise, the value of the unary counter could be en-  coded into the states for an instance of coverability in binary encoded 1-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Furthermore, we  do not impose any restrictions on the initial and the target confgurations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' both coordinates  of these vectors are encoded in binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Our main result is that coverability in 2-VASS with one  unary counter is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Coverability in binary encoded k-VASS is PSPACE-complete, for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The lower bound  limits the practicality of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, it is sensible to consider restricted variations  and quantify their complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We remark that coverability in fxed dimension VASS had widely-  open complexity if there was exactly one binary counter and at least one unary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' See  Figure 1 for a summary of the known results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The natural starting point is the characterisation of runs via linear path schemes [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In-  tuitively, the authors prove that if coverability or reachability holds then there is a witnessing  path of a specifc shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Namely, all paths can be characterised by a bounded language defned  by a regular expression of the form τ0γ∗  1τ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' τk−1γ∗  kτk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Here τ0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , τk are paths that connect  disjoint cycles γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since the language is bounded, checking if there is a path for a given  expression essentially amounts to an instance of integer linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In particular, the  authors argue that both k and |τ0|+|γ1|+|τ1|+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='+|τk−1|+|γk|+|τk| are pseudo-polynomially  bounded [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' However, a polynomial bound would immediately yield an NP upper bound as such  a regular expression can be guessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Given that coverability in 2-VASS with two binary coun-  ters is PSPACE-hard [5], we cannot simply directly apply the known results when dealing with  2-VASS with one binary and one unary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Section 3, we provide a detailed discussion  and a difcult yet motivating example in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  To overcome this problem, we show that coverability can be witnessed by paths in com-  pressed linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We relax the condition of the bounded language, by allowing to nest linear  forms, provided that the exponents are fxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Intuitively, an expression of the form (τγ∗τ ′)∗  is still forbidden, but we allow for (τγeτ ′)∗, where e is fxed but can be exponentially large  (encoded using polynomially many bits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Such a form easily provides an NP upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We rely on two crucial observations to prove that we can focus on paths in compressed linear  form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' First, notice that the ∗ operation in a linear path scheme corresponds to iterating some  cycle in the VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , γk need to be short, one naturally focuses on short cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The  issue is that there are exponentially many cycles of polynomial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Section 4 we prove that  for coverability there are only polynomially many ‘optimal’ cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Section 5 we deal with the  problem when some cycle γ occurs many times in a linear path scheme witnessing coverability,  resulting in a polynomial bound on k, the width of the linear path scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then we prove that,  either we can merge some γi and γj thus reducing the width, or that there is a cycle that has  positive efect on one counter and non-negative efect on the other counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Intuitively, in the  3  latter case, we can reduce the problem to coverability in 1-VASS by pumping such a cycle that  forces one counter to take an arbitrarily large value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Moreover, such a cycle is witnessed by a  linear path scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since we need to pump this cycle, we require compressed linear forms to  describe the repetitions of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We highlight that both our crucial observations rely on that we work with coverability,  not reachability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We further highlight that we address these crucial observations through our  technical contributions that often depend on the fact there is one unary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Further Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Asymmetric treatment of the counters has been already considered  for VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recall that Minsky machines can be seen as VASS with the additional ability of zero-  testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For this model coverability is undecidable [33], even with two counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This raised  natural questions of what happens where only one of the counters is able to be reset or tested  for zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This, and more generally, reachability in VASS with hierarchical zero-tests are known to  be decidable [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' There is a further investigation into VASS with one zero-test [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recently,  work has appeared containing detailed analysis about 2-VASS where counters have diferent  powers [19, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Finally, one of the most famous open problems in the community is whether  reachability is decidable for 1-VASS with a pushdown stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For these systems, coverability is  known to be decidable [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The best known lower bound is that coverability, thus reachability  also, is PSPACE-hard [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Our model, 2-VASS with one unary counter, can be seen as 1-VASS  with a singleton alphabet pushdown stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The complexity of reachability in binary encoded 3-VASS remains an intriguing open prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' It is PSPACE-hard, like in dimension two, and the only known upper bound is primi-  tive recursive, but not even elementary [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recent works on reachability in fxed dimension  VASS [11, 9, 13] provide new examples and a better understanding of the VASS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Inter-  estingly, many techniques applied to fxed dimension VASS are very closely related to recent  progress on the nonelementary and Ackermann lower bounds for general VASS [10, 12, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We  fnally and additionally motivate coverability in VASS with one binary counter and (at least) one  unary counter as an avenue for fnding new techniques to approach VASS problems with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  2  Preliminaries  Given an integer z ∈ Z we denote bitsize(z) = log2(|z| + 1) + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For a vector v := (v1, v2)  we use (v)1 := v1 and (v)2 := v2 to be the projections to the frst and second coordinates,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We use |v|max := max{|v1|, |v2|} + 1 to denote the size of vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We write  v ≤ w if the inequalities hold on each coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We write v < w if at least one of the  inequalities is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  A 2-VASS with one unary counter V = (Q, T) consists of a fnite set of control states Q  and a set of transitions T ⊆ Q × Z × {−1, 0, 1} × Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We shall refer to the frst counter  as the binary counter and the second counter as the unary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The size of V is |V | =  |Q| + �  (p,b,u,q)∈T bitsize(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' With |V |max := |Q| + |T| · |T|max we denote the total ‘pseudo-  polynomial size’ of the automaton, where |T|max denotes the maximum absolute value that  occurs in the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note that in a standard 2-VASS both counters are in binary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' the  domain of updates for the second counter is also Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  A path π in V is a, possibly empty, sequence of transitions π = (ti)m  i=1 such that ti =  (qi−1, bi, ui, qi) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A path is simple if q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , qm are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A path is a cycle if q0 = qm and  m > 0 (thus empty cycles are forbidden).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We call it a q0-cycle to emphasise the frst and last  state of the cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A cycle is simple if q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , qm are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A cycle is short if m ≤ |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The  length of a path is the number of transitions in the path, denoted len(π) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We write π[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.j]  to denote the path that is the subsequence of transitions (ti, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , tj) in π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  A confguration (p, u) ∈ Q × N2, denoted p(u), is a state paired with the current bi-  nary and unary counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A run is a sequence of confgurations (qi(vi))m  i=0 such that  4  (qi−1, (vi)1 − (vi−1)1, (vi)2 − (vi−1)2, qi) ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A run can equivalently be defned by the se-  quence of confgurations induced by following a path π starting from an initial confguration  q0(v0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We denote this run q0(v0) π−→ qm(vm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We also write q0(v0) ∗−→ qm(vm) to indicate the  existence of a run between two confgurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  In this paper we study the coverability problem for VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  VASS Coverability  INPUT:  A VASS V = (Q, T) and two confgurations p(u) and q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  QUESTION: Does p(u) ∗−→ q(v′) hold, for some v′ ≥ v?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Do note that the initial confguration p(u) and the target confguration q(v) have both the  binary and unary components encoded as binary integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The reachability problem for VASS—  which we will not study in this paper—requires v′ = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Consider a path π = (ti)m  i=1, where ti = (qi−1, bi, ui, qi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The efect of π is the sum of the  counter updates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' the vector eff(π) := �m  i=1(bi, ui).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We often focus on the two projections:  the binary efect effb(π) := �m  i=1 bi, and the unary efect effu(π) := �m  i=1 ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We say that a cycle γ is monotone if eff(γ) ≥ 0 or eff(γ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Otherwise, we say that  γ is non-monotone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note the two variants of a non-monotone cycle: a positive-negative cycle  effb(γ) > 0 and effu(γ) < 0, and a negative-positive cycle effb(γ) < 0 and effu(γ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Let γ be a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Given e ∈ N we write γe for the path obtained by e repetitions of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We refer  to e as the exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A linear path scheme is a regular expression of the form τ0γ∗  1τ1 · · · τk−1γ∗  kτk,  where the paths τ0, τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , τk connect disjoint cycles γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note that a collection of cycles  is disjoint if no two cycles have a common state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Given ℓ = (τ0, γ1, τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , τk−1, γk, τk), we  say the a path π is in linear form ℓ if π = πℓ = τ0γe1  1 τ1 · · · τk−1γek  k τk for some exponents  e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note that in this defnition every path has a linear form, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' τ0 = π is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' To  leverage the defnition, we will ask whether paths are in a linear form of certain size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The size of  a linear form ℓ is �k  i=0 len(τi) + �k  i=1 len(γi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The size of πℓ is �k  i=0 len(τi) + �k  i=1 len(γi) +  �k  i=1 bitsize(ei), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' includes the exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We refer to k as the width of the linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  3  Coverability in 2-VASS with One Unary Counter  In this section we briefy discuss why the state-of-the-art techniques are not enough to prove  that coverability in 2-VASS with one unary counter is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Blondin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' [4] show that for a  given 2-VASS V there exists a set of linear path schemes S such that if p(u) ∗−→ q(v) in V , then  there exists a path π in a linear path scheme ρ ∈ S such that p(u)  π−→ q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For every linear  path scheme ρ ∈ S the width of ρ, and therefore the width of every path, is bounded above by  poly(|Q|, |T|max) [4, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Such a path π is not necessarily a polynomial size witness,  as the width depends on |T|max polynomially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We provide an example of a 2-VASS with one  unary counter where the width of every linear form ℓ for a path is exponential in the input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  This demonstrates that the combinatorial structure of linear path schemes is not self-sufcient  to show that there always exists a polynomial size witness of coverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Paths in Compressed Linear Form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Nevertheless, there is a natural way to succinctly de-  scribe the path presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let σ = λρ, and note that  σN =  �  tqa αN2 tab βN2 tbqtqptpc γN2 tcd δN2 tdq  �N  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  All paths and cycles are ‘small’, and the bitsize of N and N2 are polynomial in n, so σ itself  is a path in linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We introduce the following generalisation of linear form paths that  encapsulates the idea behind paths of this kind of arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  5  q  p  a  b  c  d  λ  ρ  (N4, −1)  (0, 0)  (−N6 + N, 0)  (−N, 0)  (N4, 0)  (0, 0)  (−N6 + 1, 1)  (−N2, 1)  α  (N4, −1)  β  (−N2, 1)  γ  (N4, −1)  δ  Figure 3: Example 2-VASS with one unary counter V , where N = 2n, where n is an input  parameter (thus making N exponentially large).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Consider the coverability instance with the  initial confguration q(0, 1), and the target confguration q(N, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let λ = tqaαN2tabβN2tbq  and ρ = tqptpcγN2tcdδN2tdq, where txy is the transition from state x to state y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Observe that  eff(λ) = (N, −1) and eff(ρ) = (−N + 1, 1), thus eff(λρ) = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' It is easy to then see that  q(0, 1)  (λρ)N  −−−−→ q(N, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Intuitively the cycles λ and ρ alternate so both counters remain non-  negative throughout the run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Appendix A, we prove that there does not exist a linear form of  polynomial size for a path that induces a coverability run (Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Defnition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 (Compressed linear form path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' A path π is in compressed linear form if π =  ρ0σf1  1 ρ1 · · · ρk−1σfk  k ρk for some connected paths in linear form ρ0, ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , ρk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' cycles in linear form  σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , σk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' and exponents f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The size of a compressed linear form path is the sum of the  sizes of all ρi and σi (including the bitsize of their exponents) plus the bitsize of the exponents fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  σf1  1  · ·  σfk  k  ρ0  ρ1  ρk−1  ρk  Figure 4: A compressed linear form path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The following theorem is our main contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let V be a 2-VASS with one unary counter and fx two confgurations p(u) and q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  If p(u) ∗−→ q(v), then there exists a path in compressed linear form π such that p(u) π−→ q(v′) and  v′ ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The size of the compressed linear form path is polynomial in |V |+bitsize(u)+bitsize(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Coverability in 2-VASS with one unary counter is in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 it sufces to consider paths in compressed linear form of polynomial  size, that can be guessed in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' It sufces to observe that a coverability instance on a given  compressed linear form amounts to an instance of integer linear programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Intuitively, this  is because the nested cycles are fxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus to check whether a run drops below zero it sufces  to check before applying a cycle and after applying it for the last time (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' [5, Section V,  Lemma 14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  6  We highlight that it is rather unexpected that only one extra ‘level’ of linear form paths is  enough to obtain polynomial size witnesses of coverability in a 2-VASS with one unary counter,  since the problem is PSPACE-complete for general 2-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Roughly speaking, the example given  in Figure 3 observes the most complex behaviour possible and this instance of coverability is  witnessed by a compressed linear form path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' More specifcally, compressed linear form paths  containing only one linear form cycle sufce as witnesses for coverability in 2-VASS with one  unary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, all witnesses can be represented by a compressed linear form path  ρσNτ where ρ and τ are linear form paths to and from the single linear form cycle σ which is  iterated N times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The rest of the paper is dedicated to proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We heavily exploit both distin-  guishing features of the problem: the fact that one counter receives unary encoded updates (as  opposed to both counters in binary) and the fact that we aim to assert coverability (as opposed  to reachability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Our approach is as presented in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In 4 we observe that we can  polynomially bound the total number of distinct short cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We formalise this and show that  there are only polynomially many ‘irreplaceable’ short cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In 5 we provide a ‘reshufing  procedure’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' If some short cycle γ repeats exponentially many times we aim to modify the path π  by moving the cycles γ close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then either every short cycle γ will appear only in  polynomially many ‘bundles’ γe, or we fnd a cycle σ such that eff(σ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the latter case, by  pumping σ we are essentially left with one counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Finally, in Section 6 we conclude the proof  of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  4  Replacing Short Cycles  In this section, we show that there are only polynomially many short cycles that need occur in  a run witnessing coverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Fix a path π = (qi−1, bi, ui, qi)k  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let 0 ≤ ib, iu ≤ k be the frst  indices such that gb = �ib  i=1 bi and gu = �iu  i=1 ui are at their lowest, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note that  gb, gu ≤ 0 since by convention if we consider ib, iu = 0 then the sum evaluates to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We call and  denote these two numbers the binary guard grdb(π) = gb and the unary guard grdu(π) = gu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The following claim, that is proved in Appendix B, immediately follows from these defnitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Both grdb(π[ib + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) = 0 and grdu(π[iu + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Much like the nadir of a cycle in a one-counter net, defned in [1], we defne the binary-  nadir state as qib, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' the frst state in which the binary counter frst attains the lowest value  when executing π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We call the binary-nadir decomposition π = πb  1πb  2, for πb  1 = π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.ib] and πb  2 =  π[ib + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k], as intimated in Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notice that this decomposition necessitates the binary  guard of the path π is equal to the binary efect of the prefx πb  1, grdb(π) = effb(πb  1) = grdb(πb  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Furthermore, the sufx of the binary-nadir decomposition has zero binary guard grdb(πb  2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We primarily utilise binary-nadir states and binary-nadir decompositions, hence the omission  of matching unary-nadir states and unary-nadir-decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Defnition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 (Replaceable cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let γ be a q-cycle and let p be the binary-nadir state of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We say that γ is replaceable if there exists a q-cycle γ′ with the same binary-nadir state p, such that  (a) effb(γ′) ≥ effb(γ) and effu(γ′) ≥ effu(γ),  (b) grdb(γ′) ≥ grdb(γ) and grdu(γ′) ≥ grdu(γ), and  (c) len(γ′) ≤ len(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Additionally, at least one inequality is strict and we write γ ≺ γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We say a cycle is irreplaceable if it is not replaceable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We also say that an irreplaceable q-  cycle γ with the binary-nadir state p is characterised by the fve values: effb(γ), effu(γ), grdb(γ),  grdu(γ), and len(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The following lemma is proved in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  7  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 (Replacing cycles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let π = π1γπ2, where γ is a q-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Suppose p(u) π−→ q(v) then  the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  If γ is replaceable, then there exists an irreplaceable q-cycle γ ≺ γ′ such that p(u)  π1γ′π2  −−−−→  q(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  If γ is irreplaceable, then for every irreplaceable q-cycle γ′ that has the same characterisation  as γ, p(u)  π1γ′π2  −−−−→ q(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  In both cases v′ ≥ v and len(π) ≥ len(π1γ′π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  For convenience, we defne the polynomial R(|Q|) := |Q|4(|Q| + 1)(2|Q| + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' There exists at most R(|Q|) many irreplaceable short cycles with diferent character-  isations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We fx two states q and p and consider only q-cycles γ with the binary-nadir state p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus  in the fnal argument one must multiply everything by |Q|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since we consider short cycles, the  unary efect and the unary guard are small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' −|Q| ≤ effu(γ) ≤ |Q| and −|Q| ≤ grdu(γ) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Towards a contradiction, suppose there exists more than |Q|2(|Q|+1)(2|Q|+1)2 many such  irreplaceable q-cycles with diferent characterisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By the pigeonhole principle there must  exist two cycles, denoted in binary-nadir decomposition γ = γ1γ2 and γ′ = γ′  1γ′  2, that have the  same values effu(γ1) = effu(γ′  1), effu(γ2) = effu(γ′  2), grdu(γ) = grdu(γ′), len(γ1) = len(γ′  1),  and len(γ2) = len(γ′  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We know that the irreplaceable q-cycles γ and γ′ have diferent characterisations, so it must  be the case that their binary efects difer effb(γ) ̸= effb(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Otherwise, the cycle with the lesser  binary guard is replaceable, because the unary efect, unary guard, and length do not difer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Without loss of generality, suppose effb(γ) > effb(γ′), then grdb(γ) < grdb(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Otherwise, γ′  would be replaceable as γ ≺ γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Now consider the q-cycle σ = γ′  1γ2, also with the binary-nadir state p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We will show that  γ ≺ σ contradicting the fact that γ is an irreplaceable q-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' First, observe that σ has greater  binary efect than γ as  effb(σ) = effb(γ′  1) + effb(γ2) > effb(γ1) + effb(γ2) = effb(γ),  where the inequality holds because grdb(γ) < grdb(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Second, σ and γ have equal unary efect  because effu(γ′  1) = effu(γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Third, we show that σ has a greater binary guard than γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since  γ2 is the sufx of the binary-nadir decomposition of γ, it must be true that grdb(γ2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 grdb(σ) = grdb(γ′  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Combining these facts, grdb(σ) = grdb(γ′) > grdb(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Fourth,  σ has at least the unary guard of γ because, in particular, the unary guard of the prefx of a path  is at most the unary guard of the entire path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  grdu(σ) = min{grdu(γ′  1), effu(γ′  1) + grdu(γ2)}  ≥ min{grdu(γ′), effu(γ′  1) + grdu(γ2)}  = min{grdu(γ), effu(γ1) + grdu(γ2)} = grdu(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Fifth and fnally, σ and γ have equal length because len(γ′  1) = len(γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We have at least one  strict inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus, we have reached the desired contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  5  Reshufling Linear Form Paths  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1  Reshufling Procedure  There can be many linear forms for a path π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We will try to fnd an ‘optimal’ one, so we  introduce a cost function to quantify linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recall that a linear form ℓ is a sequence  8  of paths τ0, τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , τk and a sequence of cycles γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , γk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' If π is in the linear form ℓ =  (τ0, γ1, τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , τk−1, γk, τk) then we write πℓ = τ0γe1  1 τ1 · · · τk−1γek  k τk, where π = πℓ (the in-  dex is here to stress the exact linear form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For this section, we will consider linear forms only  containing short cycles γ, they will play a key role in the following arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We defne a cost function that assigns, to a linear form ℓ, the following pair of naturals  C(ℓ) :=  ��k  i=0 len(τi), k  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For convenience, we defne the polynomial P(|Q|) := 2(|Q|2 +  1)(|Q|2 + 2) · R(|Q|), where R is the polynomial defned for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We say that a linear  form ℓ is narrow if C(ℓ) ≤ (|Q|(P(|Q|) + 1), P(|Q|)), otherwise we say that ℓ is wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We say  that the triple (π′, σ, π′′) is a monotone cycle decomposition of a path π if σ is a monotone cycle,  π = π′σπ′′, and len(σ) < len(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 (Reshufing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let π be a path such that p(u) π−→ q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then there exists a path ρ such  that p(u)  ρ−→ q(w) where w ≥ v, len(ρ) ≤ len(π), and either  (i) there exists a narrow linear form for ρ, or  (ii) there exists a monotone cycle decomposition of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We start with a series of preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the early part of this proof, we provide simple  observations to ascertain some auspicious properties of our path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the later part of this proof,  we present the ‘reshufing procedure’ and conclude with one of the cases in the statement of  this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In this proof we will compare linear forms using the lexicographic order ≺lex, that  is known to be a linear-order and a well-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Formally,  C(ℓ′) ≺lex C(ℓ) ⇐⇒ (C(ℓ′))1 < (C(ℓ))1 or,  (C(ℓ′))1 = (C(ℓ))1 and (C(ℓ′))2 < (C(ℓ))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We start with a path π′ such that p(u) π′  −→ q(v′) where v′ ≥ v, len(π′) ≤ len(π), and π′ has  a linear form ℓ′ that has the least cost among all linear forms for all like-paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' That means there  does not exist another path π′′ such that p(u) π′′  −→ q(v′′) where v′′ ≥ v, len(π′′) ≤ len(π), and  π′′ has a linear form ℓ′′ such that C(ℓ′′) ≺lex C(ℓ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  For the frst observation, suppose there exists 0 ≤ i ≤ k such that len(τi) > |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then the  path τi can be written as τi = τ ′γτ ′′, where γ is a short cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We can defne the linear form ℓ′′  by modifying ℓ′ where τi is swapped for τ ′γτ ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Although this increments the number of cycles  k, we decrease the total length of the paths as len(τ ′) + len(τ ′′) < len(τi) (recall that empty  cycles are forbidden).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus C(ℓ′′) ≺lex C(ℓ′) contradicting the assumption that ℓ has minimum  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, we assume that len(τi) ≤ |Q| for all 0 ≤ i ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  For the second observation, we defne U := {0 ≤ i ≤ m : (vi)2 < |Q|} to be the set of  indices of confgurations in the run that have unary counter value less than |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Observe that  if |U| > |Q|2 + 1 then there are two indices 0 < i < j ≤ m such that the two corresponding  confgurations in the run have matching states qi = qj and equal unary counter values (vi)2 =  (vj)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then, regardless of sign of its binary efect, π′[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.j] is a monotone cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Here, case (ii)  immediately holds by decomposing π′ itself using the monotone cycle π′[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.j], given that i > 0  and j ≤ m implies len(π′[i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.j]) = j − i < m = len(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, we assume |U| ≤ |Q|2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We continue with the aim of satisfying the conditions of case (ii) by fnding a monotone cycle  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Let d = |{γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , γk}| be the number of distinct cycles in the linear form ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1  and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2, we can assume that d ≤ R(|Q|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Otherwise, we can exchange replaceable  q-cycles for irreplaceable q-cycles using the frst point in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' It is possible that for a  particular characterisation, we can observe more than one irreplaceable q-cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then using the  second point in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1, we can arbitrarily select one of these irreplaceable q-cycles with  equal characterisations to exchange all others with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By applying these cycle replacements to π′,  9  we obtain a diferent path ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Defnition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 ensures that we do so without decreasing the efect  (a), without allowing the counters to take a negative value (b), and without increasing the length  of the path (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore p(u)  ρ−→ q(w) and w ≥ v′ ≥ v, and len(ρ) ≤ len(π′) ≤ len(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We  remark since cycles have been exchanged one-for-one, then ρ takes a linear form ℓ with the  same path segments as ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, it is clear that neither the number of cycles k, nor the sum  of the lengths of the paths between cycles, have changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We also know that ℓ is a linear form  for ρ with minimum cost C(ℓ) = C(ℓ′), as per the initialisation in this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Suppose ρ = ρℓ = τ0γe1  1 τ1 · · · τk−1γek  k τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let (qj(vj))m  j=0 be the run obtained by following  the path ρℓ from the initial confguration q0(v0) = p(u) to the fnal confguration qm(vm) =  q(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We may assume that ℓ is wide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Otherwise, case (i) is immediately satisfed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We also  know that len(ρℓ) ≥ max{(C(ℓ))1, (C(ℓ))2} > P(|Q|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We may also assume that each cycle  γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , γk is non-monotone, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' it is positive-negative or negative-positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Otherwise, case (ii)  immediately holds by decomposing ρ itself using some monotone cycle γi, given that len(γi) ≤  |Q| < P(|Q|) < len(ρℓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notice this is valid since each ei > 0 by the minimality of C(ℓ),  otherwise you can write · · · τi−1γ0  i τi · · · with one less cycle, decreasing (C(ℓ))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  From the frst observation, we get �k  i=0 len(τi) ≤ (k + 1)|Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Given that ℓ is wide, either  |Q|(P(|Q|)+1) < (C(ℓ′))1 = �k  i=0 len(τi) ≤ (k+1)|Q| that implies P(|Q|) < k, or P(|Q|) <  (C(ℓ′))2 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Regardless, P(|Q|) < k holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recall that |U| ≤ |Q|2 + 1 from the second  observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since there are relatively ‘few’ confgurations indexed by U, there must exist a  relatively ‘distant’ pair of consecutive confgurations indexed by U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' More formally, there are i  and j such that 0 ≤ i < j ≤ k and j − i ≥ 2(|Q|2 + 2)R(|Q|) and all confgurations that occur  in the run over the path segment τiγei+1  i+1 · · · γej  j τj have unary counter value at least |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notice  that j − i is the number of cycles in this path segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since j − i ≥ 2(|Q|2 + 2)R(|Q|) and by  pigeonhole principle on the number of irreplaceable cycles, there is a common irreplaceable cycle  γ repeated at least x = 2(|Q|2 + 2) many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We will focus on the frst x such occurrences  of this cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , sx be the indices of this cycle γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' γ = γs1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' = γsx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' To highlight  these cycles, we decompose this path segment into  τiγei+1  i+1 · · · γej  j τj = Λ0γf1Λ1 · · · Λx−1γfxΛx,  where fj := esj and Λj are the concatenated paths (and cycles) in between iterations of γ,  see Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' To reiterate, we know that all confgurations that occur in the run over this path  segment have at least |Q| unary counter value and γ is a short cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Figure 5: The decomposition of the path segment into Λ0γf1Λ1 · · · Λx−1γfxΛx, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notice  that the unary counter is always at least |Q| as no confgurations indexed by U are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Reshufling Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  In the rest of the proof we will modify the path segment (above)  of the path ρℓ with a procedure that we call reshufing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' At the end of this procedure we will  fnd a monotone cycle and satisfy case (ii) of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We either fnd this cycle directly, or  we obtain a linear form ℓ′′ such that C(ℓ′′) ≺lex C(ℓ) contradicting the assumption that ℓ has  minimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  10  Note that x = 2(|Q|2 + 2) is even, and for every pair of consecutive cycles γ2j−1 and γ2j  (for 1 < 2j ≤ x), consider the subsegment γf2j−1Λ2j−1γf2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' There are two scenarios depend-  ing on the variant of the non-monotone cycle γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the scenario where γ is positive-negative,  we move an iteration of γ from right to left obtaining γf2j−1+1Λ2j−1γf2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the scenario  where γ is negative-positive, we move an iteration of γ in the opposite direction obtaining  γf2j−1−1Λ2j−1γf2j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We repeat this procedure until one of two conditions are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The frst is when there are no  iterations of γ on one side, so either f2j−1 or f2j becomes 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The second is when there appears  a confguration, in the run over the path subsegment after reshufing, with unary counter value  less than |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' See Figure 6 for a pictorial presentation of reshufing in the scenario where γ is  positive-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Figure 6: Reshufing around a path Λ (blue) where γ (red) is positive-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Before reshuf-  fing, the path subsegment · · · γΛγ · · · all confgurations have unary counter value at least |Q|  in the run (left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' After reshufing, the path subsegment · · · γγΛ · · · , there is a confguration  with unary counter value less than |Q| in the run (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We claim that after each reshufing step, the corresponding run remains executable, so we  must check that both counters remain non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notice that by only moving a cycle, the  total efect of the path subsegment remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, if the run was executable  before reshufing, we can safely assume that the prefx before the path subsegment and the  sufx after the path subsegment are still executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' For that reason, consider the counter values  of confgurations occurring in the run over the reshufed path subsegment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We focus on a single  step of the reshufing procedure that concerns the subsegment γf2j−1Λ2j−1γf2j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Suppose γ is a positive-negative cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then the reshufing procedure moves γ from right to  left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We claim that since f2j−1 > 0 and Λ0γf1Λ1 · · · Λ2j−1γf2j−1 is executable, the subsegment  Λ0γf1Λ1 · · · Λ2j−1γf2j−1+1 is executable from the initial confguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This is because one  prerequisite of the reshufing procedure is that all confgurations occurring in the run over the  path subsegment have at least |Q| unary counter value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Moreover, the cycle γ has length at  most |Q| so grdu(γ) ≥ −|Q| means the unary counter value remains non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' As for the  binary counter value, since a single execution of γ increases the binary counter and an iteration  of γ was already executed before reshufing, Λ0γf1Λ1 · · · Λ2j−1γf2j−1+1 is executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the  same way, from the initial confguration, Λ0γf1Λ1 · · · Λ2j−1γf2j−1+1Λ2jγf2j−1  2j  is executable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  This is because effu(γ) ≥ −|Q|, and again, all confgurations occurring in the run over the path  subsegment have at least |Q| unary counter value, and also because of the monotonicity on the  binary counter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The argument when γ is a negative-positive cycle is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This concludes the correct-  ness analysis of the reshufing procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  11  Finishing Reshufling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We analyse what happens when reshufing is fnished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Suppose that  there exists a pair 2j − 1 and 2j such that the reshufing fnishes under the frst condition  where all iterations of γ have been moved to one side of Λ2j−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In this case we obtain a new  linear form ℓ′′ for ρ, where one collection of the cycle γ has been removed (decrementing k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  So (C(ℓ′′))2 = k − 1 < (C(ℓ))2 and the two adjacent path segments can be combined without  changing the summed length of paths so (C(ℓ′′))1 = (C(ℓ))1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, C(ℓ′′) ≺lex C(ℓ)  contradicting the assumption ℓ has the minimal cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Otherwise, for every 1 ≤ j ≤ x/2 the reshufing of pair 2j − 1 and 2j fnishes under  condition the second condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' So there is a confguration with unary counter value less than  |Q| in the run induced from the path ρ for each pair 2j − 1 and 2j (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recall that  x  2 = |Q|2 + 2, that is the number of pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Akin to the frst observation (in the beginning of  this proof), we use the pigeonhole principle on the number of such confgurations to obtain  two confgurations with matching states and equal unary counter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The path segment  inducing the part of the run between these two confgurations is a monotone cycle, regardless  of the binary efect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Again, it must be true that the length of this cycle is less than the length of  the whole path, so we obtain a monotone cycle decomposition of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus case (ii) of the lemma  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Figure 7: After reshufing is fnished under condition the second condition, we can fnd a zero  unary efect cycle using the (sufciently many) confgurations with unary counter less |Q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2  Applying Reshufling  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 does not necessarily return a narrow linear form for a path π witnessing coverability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Instead it may return a monotone cycle decomposition (ρ, σ, τ) of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Our next goal is to show  that there exists polynomial size certifcates for ρ and σ (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2), and then to show that  there exists a polynomial size certifcate for τ (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Like linear forms, there can be many  monotone cycle decompositions for a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Following, we will use the cost function assigning  monotone cycle decompositions to pairs of natural numbers D((ρ, σ, τ)) := (len(ρσ), len(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Note that we can compare two decompositions using their cost, even if they are for two diferent  paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Suppose p(u) ∗−→ q(v) yet there is no narrow linear form ℓ for any path π such that  p(u) π−→ q(w) and w ≥ v, then there exists a path π′ such that  (a) p(u) π′  −→ q(w′) where w′ ≥ v,  (b) there is a monotone cycle decomposition (ρ, σ, τ) of π′ where eff(σ) > 0, and  (c) there are narrow linear forms for both ρ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We will again use the lexicographical order ≺lex to compare the cost of monotone cycle  decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let π be a path of minimum length such that p(u)  π−→ q(w) where w ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  12  Let c = (ρ, σ, τ) be the monotone cycle decomposition of π that minimizes the cost D(c) under  the ≺lex order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Such a decomposition must exist, otherwise applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 would return  a narrow linear form ℓ′ for ρ such that p(u)  ρ−→ q(w′) and w′ ≥ w ≥ v, contradicting an  assumption of this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Observe that eff(σ) > 0, otherwise one can remove σ and consider  the shorter path ρτ, contradicting the minimal length of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Next, we argue that ρ and σ do not  have monotone cycle decompositions, we then leverage Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 to obtain the narrow linear  forms required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Path ρ cannot be decomposed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Towards a contradiction, assume that there is a  monotone cycle decomposition c′ = (ρ′, σ′, τ ′) of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Observe that the following monotone  cycle decomposition c′ = (ρ′, σ′, τ ′στ) of π has lower cost D(c′) ≺lex D(c) as (D(c′))1 =  len(ρ′) + len(σ′) < len(ρ) + len(σ) = (D(c))1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This contradicts the assumption that (ρ, σ, τ)  has minimum cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Suppose p(u)  ρ−→ p′(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since there is no monotone cycle decomposition, applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1  to ρ returns a path ρ′ with a narrow linear form such that p(u)  ρ′  −→ p′(x′) where x′ ≥ x and  len(ρ′) ≤ len(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Cycle σ cannot be decomposed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Towards a contradiction, assume that there is a  monotone cycle decomposition (ρ′, σ′, τ ′) of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Observe that the following monotone cycle  decomposition c′ = (ρρ′, σ′, τ ′τ) of π has lower cost D(c′) ≺lex D(c) as (D(c′))1 = len(ρ) +  len(ρ′) + len(σ′) ≤ len(ρ) + len(σ) = (D(c))1 and (D(c′))2 = len(σ′) < len(σ) = (D(c))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  This contradicts the assumption that (ρ, σ, τ) has minimum cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Suppose p′(x) σ−→ p′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since there is no monotone cycle decomposition, applying Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1  to σ returns a path σ′ with a narrow linear form such that p′(x) σ′  −→ p′(y′) where y′ ≥ y and  len(σ′) ≤ len(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In particular, it is also true that eff(σ′) ≥ eff(σ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Replacing ρ′ for ρ and σ′ for σ in π yields a path π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Clearly if p(u) π−→ q(w) where w ≥ v,  then p(u) π′  −→ q(w′) where w′ ≥ w ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Finally, (ρ′, σ′, τ) is monotone cycle decomposition  of π′ such that eff(σ′) > 0 and ρ′ and σ′ have narrow linear forms, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We now aim to obtain a narrow linear form for τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note that Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2 gives us a monotone  cycle σ with positive efect on at least one counter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' eff(σ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By pumping σ we can force  one of the counters to take an arbitrarily large value (following, the vector x refects this large  value for Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then, loosely speaking, the problem reduces to coverability in 1-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  However, proving the existence of a polynomial size compressed linear form path in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1  requires more care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note that Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4 is stated for 2-VASS (not necessarily with one unary  counter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' First we need to recall the following bound on counter values observed throughout  runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recall that |V |max := |Q| + |T| · |T|max is the pseudo-polynomial size of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='3 (Corollary from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Consider a 2-VASS (with both counters in bi-  nary) V = (Q, T) and let p(u) ∗−→ q(v), then there exists a run p(u) = q0(v0), q1(v1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , qm(vm) =  q(v) such that |v0|max, |v1|max, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , |vm|max ≤ (|V |max + |u|max + |v|max)O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  In the following lemma, that is proved in Appendix C, given a 2-VASS V , the initial confgu-  ration p(u), and target confguration q(v), we write B in place of (|V |max+|u|max+|v|max)O(1)  from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='3 and we fx x = (4B|Q|2|V |2  max, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Consider a 2-VASS (with both counters in binary) V = (Q, T) and let p(u) ∗−→ q(v),  then there exists a narrow linear form path π′ such that p(u + x) π′  −→ q(v′) for some v′ ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  13  6  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1  Before proving Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1, we employ the fact that for a general 2-VASS, not necessarily with  one unary counter, the exponents of cycles in linear forms can be pseudo-polynomially bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 (Corollary from Lemma 18 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let π be path in a 2-VASS with a linear form  π = τ0γf1  1 τ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' γfk  k τk such that p(u) π−→ q(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then there exist a path π′ = τ0γe1  1 τ1 · · · τk−1γek  k τk  such that p(u)  π′  −→ q(v′) where v′ ≥ v and bitsize(e1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , bitsize(ek) are all bounded by a  polynomial in |V | + bitsize(u) + bitsize(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let p(u) π−→ q(v) for some path π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' If there is a narrow linear form ℓ for π  then by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 we obtain π′ = τ0γe1  1 τ1 · · · τk−1γek  k τk such that p(u) π′  −→ q(v′) where v′ ≥ v  and bitsize(e1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' , bitsize(ek) are all bounded above by a polynomial in |V | + bitsize(u) +  bitsize(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since ℓ is a narrow linear form, we know that k ≤ P(|Q|) so �k  i=1 len(γi) ≤ k|Q| ≤  |Q|P(|Q|) and we also know that �k  i=0 len(τi) ≤ |Q|(P(|Q|) + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Together, this implies the  linear form path π′ is of polynomial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  It remains to consider the case when there is no narrow linear form ℓ for π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2  (via Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1) there exists a path π′ such that p(u) π′  −→ q(v′) and v′ ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Moreover, there is a  monotone cycle decomposition (ρ, σ, τ) of π′ such that eff(σ) > 0 and there are narrow linear  forms for both ρ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Assume that (eff(σ))1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This is without loss of generality because if (eff(σ))1 = 0 then  one can fip the coordinates in V , u and v (for the remainder of the proof it will not matter  that one counter is unary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let p′(m) be the confguration such that p(u)  ρ  −→ p′(m) στ  −→ q(v′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Observe that since eff(σ) > 0 for every i ∈ N the path ρσi induces the run p(u)  ρσi  −−→ p′(m + i ·  eff(σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Consider x = (x)1 = 4B|Q|2|V |2  max (for Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4), clearly x is large enough so that  p(u)  ρσx  −−→ p′(m′) and m′ ≥ m + x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4 there exists a narrow linear form for a path  τ ′ such that p′(m′) τ ′  −→ q(v′′) and v′′ ≥ v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We conclude by considering the compressed linear form path ρσxτ ′ such that p(u)  ρσxτ ′  −−−→  q(v′′) and v′′ ≥ v′ ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since ρ, σ, and τ ′ have narrow linear forms, we can also bound the  exponents using Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1 as in the beginning of this proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Finally, bitsize(x) is polynomial in  |V |+bitsize(u)+bitsize(v) much like the exponents of the cycles in the linear forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore,  the size of the compressed linear form ρσxτ ′ is polynomial in |V |+bitsize(u)+bitsize(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  7  Conclusion and Future Work  In this paper we proved that coverability in 2-VASS with one unary counter is in NP, a drop in  complexity from PSPACE for general 2-VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We achieve this by using our new techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Most  notably, we polynomially bounded the number of short cycles that need to be used (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Then, we attempt to fnd a polynomial linear form path by replacing short cycles and reshufing  the path (Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  A natural extension is to consider whether coverability in 3-VASS with one binary counter  and two unary counters is also in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The technique for polynomially bounding the number  of short cycles that need be used can easily be generalised to higher dimension VASS with one  binary counter and multiple unary counters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' However, it is not clear how to modify and use  our reshufing technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Another open problem is whether reachability in 2-VASS with one  unary counter is also in NP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Note that completeness would immediately follow from the fact  that reachability in binary encoded 1-VASS is NP-hard [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  14  References  [1] Shaull Almagor, Udi Boker, Piotr Hofman, and Patrick Totzke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Parametrized Universality  Problems for One-Counter Nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Igor Konnov and Laura Kovács, editors, 31st Interna-  tional Conference on Concurrency Theory, CONCUR 2020, September 1-4, 2020, Vienna, Aus-  tria (Virtual Conference), volume 171 of LIPIcs, pages 47:1–47:16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Schloss Dagstuhl - Leibniz-  Zentrum für Informatik, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='CONCUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [2] Shaull Almagor, Nathann Cohen, Guillermo A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Pérez, Mahsa Shirmohammadi, and James  Worrell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Coverability in 1-VASS with Disequality Tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Igor Konnov and Laura Kovács,  editors, 31st International Conference on Concurrency Theory, CONCUR 2020, September 1-4,  2020, Vienna, Austria (Virtual Conference), volume 171 of LIPIcs, pages 38:1–38:20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [3] Sanjeev Arora and Boaz Barak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Computational complexity: a modern approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Cambridge  University Press, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [4] Michael Blondin, Matthias Englert, Alain Finkel, Stefan Göller, Christoph Haase, Ranko  Lazić, Pierre McKenzie, and Patrick Totzke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The Reachability Problem for Two-  Dimensional Vector Addition Systems with States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' ACM, 68(5):34:1–34:43, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='1145/3464794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [5] Michael Blondin, Alain Finkel, Stefan Göller, Christoph Haase, and Pierre McKen-  zie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Reachability in Two-Dimensional Vector Addition Systems with States Is PSPACE-  Complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  In 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS  2015, Kyoto, Japan, July 6-10, 2015, pages 32–43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' IEEE Computer Society, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content=' Directed reachability for infnite-  state systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Jan Friso Groote and Kim Guldstrand Larsen, editors, Tools and Algorithms  for the Construction and Analysis of Systems - 27th International Conference, TACAS 2021,  Held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS  2021, Luxembourg City, Luxembourg, March 27 - April 1, 2021, Proceedings, Part II, volume  12652 of Lecture Notes in Computer Science, pages 3–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1007/  978-3-030-72013-1\\_1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content=' Hu and Moshe Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Vardi, editors, Computer Aided Verifcation, 10th  International Conference, CAV ’98, Vancouver, BC, Canada, June 28 - July 2, 1998, Proceedings,  volume 1427 of Lecture Notes in Computer Science, pages 268–279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Springer, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='  [9] Wojciech Czerwiński, Slawomir Lasota, Ranko Lazic, Jérôme Leroux, and Filip Mazowiecki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Reachability in Fixed Dimension Vector Addition Systems with States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Igor Konnov  and Laura Kovács, editors, 31st International Conference on Concurrency Theory, CONCUR  2020, September 1-4, 2020, Vienna, Austria (Virtual Conference), volume 171 of LIPIcs, pages  48:1–48:21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='CONCUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  15  [10] Wojciech Czerwiński, Sławomir Lasota, Ranko Lazic, Jérôme Leroux, and Filip Mazowiecki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The Reachability Problem for Petri Nets Is Not Elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' ACM, 68(1):7:1–7:28, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='  [11] Wojciech Czerwiński, Sławomir Lasota, Christof Löding, and Radoslaw Piórkowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' New  Pumping Technique for 2-Dimensional VASS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Peter Rossmanith, Pinar Heggernes, and  Joost-Pieter Katoen, editors, 44th International Symposium on Mathematical Foundations of  Computer Science, MFCS 2019, August 26-30, 2019, Aachen, Germany, volume 138 of LIPIcs,  pages 62:1–62:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='MFCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [12] Wojciech Czerwiński and Łukasz Orlikowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Reachability in Vector Addition Systems is  Ackermann-complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In 62nd IEEE Annual Symposium on Foundations of Computer Science,  FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 1229–1240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1109/FOCS52979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='  [13] Wojciech Czerwiński and Łukasz Orlikowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Lower Bounds for the Reachability Problem  in Fixed Dimensional VASSes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Christel Baier and Dana Fisman, editors, LICS ’22: 37th  Annual ACM/IEEE Symposium on Logic in Computer Science, Haifa, Israel, August 2 - 5, 2022,  pages 40:1–40:12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' ACM, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1145/3531130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='3533357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [14] Alex Dixon and Ranko Lazic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' KReach: A Tool for Reachability in Petri Nets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Armin  Biere and David Parker, editors, Tools and Algorithms for the Construction and Analysis of  Systems - 26th International Conference, TACAS 2020, Held as Part of the European Joint  Conferences on Theory and Practice of Software, ETAPS 2020, Dublin, Ireland, April 25-30,  2020, Proceedings, Part I, volume 12078 of Lecture Notes in Computer Science, pages 405–412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Springer, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content=' Pandya, editors, 38th  IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer  16  Science, FSTTCS 2018, December 11-13, 2018, Ahmedabad, India, volume 122 of LIPIcs, pages  31:1–31:14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content=' In  62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO,  USA, February 7-10, 2022, pages 1241–1252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content=' In 34th Annual ACM/IEEE Symposium on Logic in Computer  Science, LICS 2019, Vancouver, BC, Canada, June 24-27, 2019, pages 1–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+page_content='1109/LICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='8785796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [28] Jérôme Leroux and Grégoire Sutre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' On Flatness for 2-Dimensional Vector Addition Sys-  tems with States.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Philippa Gardner and Nobuko Yoshida, editors, CONCUR 2004 - Con-  currency Theory, 15th International Conference, London, UK, August 31 - September 3, 2004,  Proceedings, volume 3170 of Lecture Notes in Computer Science, pages 402–416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Springer,  2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1007/978-3-540-28644-8\\_26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [29] Jérôme Leroux and Grégoire Sutre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Reachability in Two-Dimensional Vector Addition Sys-  tems with States: One Test Is for Free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In Igor Konnov and Laura Kovács, editors, 31st  International Conference on Concurrency Theory, CONCUR 2020, September 1-4, 2020, Vi-  enna, Austria (Virtual Conference), volume 171 of LIPIcs, pages 37:1–37:17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Schloss Dagstuhl  17  Leibniz-Zentrum für Informatik, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4230/LIPIcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='CONCUR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [30] Jérôme Leroux, Grégoire Sutre, and Patrick Totzke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' On the Coverability Problem for Push-  down Vector Addition Systems in One Dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  In Magnús M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Halldórsson, Kazuo  Iwama, Naoki Kobayashi, and Bettina Speckmann, editors, Automata, Languages, and Pro-  gramming - 42nd International Colloquium, ICALP 2015, Kyoto, Japan, July 6-10, 2015, Pro-  ceedings, Part II, volume 9135 of Lecture Notes in Computer Science, pages 324–336.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Springer,  2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1007/978-3-662-47666-6\\_26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [31] Richard Lipton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The Reachability Problem Requires Exponential Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Department of  Computer Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Yale University, 62, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [32] Ernst W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Mayr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' An Algorithm for the General Petri Net Reachability Problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=', 13(3):441–460, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1137/0213029.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [33] Marvin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Minsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Computation: Finite and Infnite Machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Prentice-Hall, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=', 1967.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Petri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Kommunikation mit Automaten, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' dissertation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' University of Bonn, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [35] Charles Rackof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The Covering and Boundedness Problems for Vector Addition Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=', 6:223–231, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1016/0304-3975(78)90036-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [36] Klaus Reinhardt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Reachability in Petri Nets with Inhibitor Arcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notes Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=', 223:239–264, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='entcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='042.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [37] Sylvain Schmitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The Complexity of Reachability in Vector Addition Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  ACM  SIGLOG News, 3(1):4–21, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  URL: https://dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='acm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='org/citation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  id=2893585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  [38] Leslie G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Valiant and Mike Paterson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Deterministic One-Counter Automata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=', 10(3):340–350, 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1016/S0022-0000(75)80005-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  A  Proofs of Section 3  Claim A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Given the instance of coverability consisting of V from Figure 3, and the two confg-  urations q(0, 1) and q(N, 1), there does not exist a path π in linear form of polynomial size such  that q(0, 1) π−→ q(v′) where v′ ≥ (N, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' First, let us consider the scenario in a run from the confguration q(0, 1), this is the initial  confguration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The only option is to take the frst transition tqa of λ, so let us consider a single  iteration of this cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' So that the binary counter does not take a negative value, just before  tbq the last transition in λ, its value must be at least N6 − N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This only be satisfed when  both α and β are exhausted, so to speak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In other words, α is iterated (exactly N2 many times)  until the value of the binary counter is less than N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then β is iterated (also exactly N2 many  times) until the value on the unary counter is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, an iteration of the cycle  λ = tqaαN2tabβN2tbq is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This has efect (N, −1), so the confguration q(N, 0) is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Second, let us consider the scenario in a run from the confguration q(N, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This occurs  after following an iteration of λ, as detailed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The only option is to take the frst transition  tqp of ρ, so let us consider a single iteration of this cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Like before, so that the binary counter  does not take a negative value, just before tdq the last transition in ρ, its value must be at least  N6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This is only satisfed when both γ and δ are exhausted, much like for λ above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In other  words, γ is iterated (exactly N2 many times) until the value of the binary counter is less than  N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then δ is iterated (also exactly N2 many times) until the value on the unary counter is  18  equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Therefore, an iteration of the cycle ρ = tqptpcγN2tcdδN2tdq is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This has efect  (−N + 1, 1), so the confguration q(1, 1) is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We again are in the same scenario where the frst transition of λ can be iterated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We repeat  the above arguments, as a result alternating the cycles λ and ρ with the net efect (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' After  N iterations of the pair of cycles λρ, the confguration q(N, 1) is fnally witnessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  The number of iterations of each cycle α and β within λ is equal to N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The same is true  for the number of iterations of each cycle γ and δ within λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Additionally, in order to witness a  confguration guaranteeing coverability of the target confguration q(N, 1), at least N iterations  of the pair λρ are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Given that there are 4N such cycles in this run, it is not possible to  construct a path in linear form of polynomial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  B  Proofs of Section 4  Proof of Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Suppose grdb(π[ib + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then there exists ib + 1 ≤ j ≤ k such that  grdb(π[ib + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) = �j  i=ib+1 bi < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Now consider the binary guard of the entire path π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  grdb(π) =  bi  �  i=1  bi >  bi  �  i=1  bi + grdb(π[ib + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) =  j  �  i=1  bi  The prefx path π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.j] has smaller efect than π[1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.ib].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This contradicts the defnition of the  binary guard of π, so grdb(π[ib+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We know that grdb(π[ib+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) ≤ 0, so we conclude  grdb(π[ib + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' The same argument can be used to show grdu(π[iu + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='.k]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In either case, we refer to the Defnition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By (a), we have effb(γ′) ≥  effb(γ) and effu(γ′) ≥ effu(γ) that imply v′ ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By (b), we have both grdb(γ′) ≥ grdb(γ) and  grdu(γ′) ≥ grdu(γ) that imply the binary counter and the unary counter remain non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  By (c), len(γ′) ≤ len(γ), so we can conclude len(π) ≥ len(π1γ′π2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  C  Proofs of Section 5  Proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since we work with a 2-VASS we write grd1(π) and grd2(π) (as opposed to  grdb(π) and grdu(π)) to denote the guards on the frst and second coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Let p(u) π−→ q(v)  such that π is a bounded run as in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Somewhat similar to how it is defned in [11], we say a cycle σ is semi-positive if (eff(σ))2 >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We say that (ρ, σ, τ) is a semi-positive cycle decomposition of π if π = ρστ and σ is a  semi-positive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Consider the case when there is no semi-positive cycle decomposition of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Then from right  to left we remove maximal length cycles from π, resulting in all cycles being removed from π, to  obtain a simple path π′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since we removed only cycles that are not semi-positive, observe that  eff(π) ≤ eff(π′) + x and grd2(π) ≤ grd2(π′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This holds because len(π′) < |Q| (otherwise  we could remove another cycle) and grd1(π′) > −|Q| · |V |max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' We can write the narrow linear  form π′ as a single path satisfying the conditions of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We now focus on the case when π contains a semi-positive cycle σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Notice that we can  assume that σ is a simple cycle, otherwise σ contains a shorter cycle σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' If σ′ is a semi-positive  cycle then either it is simple, or again we can look for a shorter cycle in σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' If at some point  we fnd a cycle that is not semi-positive, we delete it shortening previous semi-positive cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We continue this procedure exhaustively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' In the end, since every time we reduce the length of a  cycle, we obtain a simple semi-positive cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Let ργ be the shortest prefx of π such that γ is a simple semi-positive cycle, and suppose  p(u)  ρ−→ p′(m) (making γ a p′-cycle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Such a ρ and γ must exist by the observation in the previous  paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Now, let ρ′ be the simple path constructed by consecutively removing simple cycles  19  from ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' By minimality of ρ, only cycles that are not semi-positive are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Recall that by  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='3, |m|max ≤ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus p(u + (B(1 + |V |max), 0))  ρ′  −→ p′(m′) for some m′ ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  Let τ be a simple path from p′ to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Such a path exists as π is a path from p, through q′, to q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  We defne the path π′ := ρ′γeτ, where e := B · |Q| · |V |max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' This provides a narrow linear form  for π′ as ρ′ is a simple path, γ is a simple cycle, and τ is a simple path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Since γ is simple cycle,  it is short, thus both grd1(γ) > −|Q| · |V |max and (eff(γe))1 > −e|Q| · |V |max are bounded  below by −B|Q|2|V |2  max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus p(u + x)  ρ′γe  −−→ p′(m′′), where m′′ ≥ v + x  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Finally, since τ is  simple grd1(τ), (eff(τ))1 ≥ −|Q| · |V |max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content=' Thus p(u + x) π′  −→ q(v′) for v′ ≥ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
+page_content='  20' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jdFRT4oBgHgl3EQfWTfY/content/2301.13543v1.pdf'}
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+Optimal Decision Tree Policies for Markov Decision Processes
+Dani¨el Vos , Sicco Verwer
+Delft University of Technology
+{d.a.vos, s.e.verwer}@tudelft.nl
+Abstract
+Interpretability of reinforcement learning policies
+is essential for many real-world tasks but learn-
+ing such interpretable policies is a hard problem.
+Particularly rule-based policies such as decision
+trees and rules lists are difficult to optimize due
+to their non-differentiability. While existing tech-
+niques can learn verifiable decision tree policies
+there is no guarantee that the learners generate a
+decision that performs optimally.
+In this work,
+we study the optimization of size-limited decision
+trees for Markov Decision Processes (MPDs) and
+propose OMDTs: Optimal MDP Decision Trees.
+Given a user-defined size limit and MDP formula-
+tion OMDT directly maximizes the expected dis-
+counted return for the decision tree using Mixed-
+Integer Linear Programming. By training optimal
+decision tree policies for different MDPs we em-
+pirically study the optimality gap for existing imita-
+tion learning techniques and find that they perform
+sub-optimally. We show that this is due to an inher-
+ent shortcoming of imitation learning, namely that
+complex policies cannot be represented using size-
+limited trees. In such cases, it is better to directly
+optimize the tree for expected return. While there
+is generally a trade-off between the performance
+and interpretability of machine learning models, we
+find that OMDTs limited to a depth of 3 often per-
+form close to the optimal limit.
+1
+Introduction
+Advances in reinforcement learning using function approxi-
+mation have allowed us to train powerful agents for complex
+problems such as the games of Go and Atari [Schrittwieser et
+al., 2020]. Policies learned using function approximation of-
+ten use neural networks, making them impossible for humans
+to understand. Therefore reinforcement learning is severely
+limited for applications with high-stakes decisions where the
+user has to trust the learned policy.
+Recent work has focused on explaining opaque models
+such as neural networks by attributing prediction importance
+to the input features [Ribeiro et al., 2016; Lundberg and Lee,
+2017]. However, these explanation methods cannot capture
+the full complexity of their models, which can mislead users
+when attempting to understand the predictions [Rudin, 2019].
+Concurrently, there has been much work on interpretable ma-
+chine learning in which the model learned is limited in com-
+plexity to the extent that humans can understand the complete
+model. Particularly decision trees have received much atten-
+tion as they are simple models that are capable of modeling
+non-linear behavior [Lipton, 2018].
+Decision trees are difficult to optimize as they are non-
+differentiable and discontinuous. Previous works have used
+different strategies to overcome the hardness of optimiz-
+ing trees:
+using assumptions or relaxations to make the
+trees differentiable [Gupta et al., 2015; Silva et al., 2020;
+Likmeta et al., 2020], reformulating the MDP into a meta-
+MDP that exclusively models decision tree policies [Topin et
+al., 2021] or extracting trees from a complex teacher [Bastani
+et al., 2018]. While these methods are increasingly success-
+ful in training performant trees they do not offer guarantees
+on this performance.
+Our work takes a first step at bridging the gap between
+the fields of optimal decision trees and reinforcement learn-
+ing. Existing formulations for optimal decision trees assume
+a fixed training set with independent samples. This cannot
+be used in a dynamic setting where actions taken in one state
+influence the best actions in others. Instead, we formulate the
+problem of solving a Markov Decision Process (MDP) us-
+ing a policy represented by a size-limited decision tree (see
+Figure 1) in a single MILP. We link the predictions of the de-
+cision tree policy to the state-action frequencies in the dual
+linear program for solving MDPs. The dual allows us to rea-
+son over policies explicitly, which results in a more efficient
+formulation.
+Our formulation for Optimal MDP Decision
+Trees, OMDTs, optimizes a decision tree policy for a given
+MDP and a tree size limit.
+OMDT produces increasingly
+performant tree policies as runtime progresses and eventually
+proves the optimality of its policy.
+Existing methods for training size-limited decision trees in
+reinforcement learning such as VIPER [Bastani et al., 2018]
+make use of imitation learning where a student tries to learn
+from a powerful teacher policy. We compare the perfomance
+of OMDT and VIPER on a variety of MDPs. Interestingly,
+we show that when training interpretable size-limited trees
+imitation learning performs significantly worse as capacity
+of the learned decision tree is wasted on parts of the state
+
+space that are never reached by the policy. Moreover, VIPER
+cannot prove optimality even if it identifies the optimal solu-
+tion. Regarding the performance-interpretability trade-off we
+show that decision trees of 7 decision nodes are enough to
+perform close to optimally in 8 out of 13 environments. Such
+trees are orders of magnitude smaller than the trees created by
+methods that exactly replicate the optimal policy using size-
+unrestricted trees such as dtcontrol [Ashok et al., 2020].
+2
+Background
+2.1
+Decision Trees
+Decision trees [Breiman et al., 1984; Quinlan, 1986] are sim-
+ple models that execute a series of comparisons between a
+feature value and a threshold in the nodes to arrive at a leaf
+node that contains the prediction value. Due to their sim-
+ple descriptions, size-limited decision trees are easy to under-
+stand for humans [Molnar, 2020]. Particularly, size-limited
+decision trees admit simulatability [Lipton, 2018]: humans
+can use the model to make predictions by hand in reasonable
+time and decomposability: humans can understand each in-
+dividual aspect of the model. The method proposed in this
+paper, OMDT, also offers algorithmic transparency: we can
+trust the learning algorithm to produce models that fulfill cer-
+tain qualities such as global optimality. Therefore decision
+trees are an attractive model class when interpretable policies
+are required.
+2.2
+Markov Decision Processes
+Markov Decision Processes (MDPs) [Bellman, 1957] are the
+processes underlying reinforcement learning problems. An
+MDP models a decision-making process in a stochastic en-
+vironment where the agent has some control over the state
+transitions. An MDP can be described by a tuple ⟨S, A, P, R⟩
+where S contains all states, A the set of actions an agent can
+take, Ps,s′,a the probabilities of transitioning from state s to
+state s′ when taking action a, and Rs,s′,a the reward the agent
+receives when going from state s to state s′ under action a.
+When solving an MDP we want to find a policy π : S → A
+such that when executing its actions the expected sum of re-
+wards (the return) is maximized.
+In this work we define
+policies (w.l.o.g.) as a mapping from states and actions to
+an indicator for whether or not action a is taken in state s:
+π : S × A → {0, 1}.
+if X ≤ 0
+if Y ≤ 1
+if Y ≤ 2
+Left
+Up
+Down
+Right
+yes
+no
+start
+goal
+0
+0
+1
+2
+3
+3
+1
+2
+Figure 1: Depth 2 OMDT on the stochastic Frozenlake 4x4 envi-
+ronment. OMDT proves that no better depth 2 decision tree policy
+exists (discounted return 0.37 with γ = 0.99).
+Value Iteration
+When solving MDPs we generally discount future rewards in
+each step by a user-defined value of 0 < γ < 1 to ensure
+that the optimal policy will generate a finite return. The most
+common approach for optimally solving MDPs is by using
+one of many dynamic programming variants. In this work,
+we focus on value iteration. Value iteration finds a value Vs
+for each state s that holds the expected discounted return for
+taking optimal greedy actions starting from that state. These
+values can be found by iteratively updating Vs until the Bell-
+man equation [Bellman, 1957] is approximately satisfied. We
+will refer to this optimal solution found with value iteration
+as the unrestricted optimal solution as the computed policy
+can be arbitrarily complex.
+3
+Related Work
+3.1
+Learning Decision Tree Policies
+Decision trees have appeared at various parts of the reinforce-
+ment learning pipeline [Glanois et al., 2021] for example in
+modeling the Q-value function or in modeling the policy. In
+this work, we are interested in modeling the policy with a de-
+cision tree as this gives us an interpretable model that we can
+directly use at test time.
+Decision trees make predictions using hard comparisons
+between a single feature value and a threshold the learned
+models can be discontinuous and non-differentiable which
+makes optimization with gradients challenging. One line of
+research focuses on overcoming the non-differentiability of
+decision trees to allow for the use of gradient-based optimiza-
+tion methods. Gupta et al. [Gupta et al., 2015] train decision
+trees that contain linear models in their leaves that can be op-
+timized with gradients but this results in models that are hard
+to interpret. Silva et al. [Silva et al., 2020] first relax the
+constraints that decision nodes select one feature, that leaves
+predict one value, and that thresholds are hard step functions.
+Such relaxed trees are differentiable and can be trained with
+policy gradient methods. By discretizing the relaxed tree they
+end up with an interpretable model that approximates the re-
+laxed model. However, the relaxed tree can get stuck in local
+minima and the discretized tree offers no performance guar-
+antees. Likmeta et al. [Likmeta et al., 2020] consider de-
+cision tree policies for autonomous driving tasks. To make
+the decision tree parameters easily optimizable they fix the
+structure of the tree along with the features used in the de-
+cision nodes. By learning a differentiable hyper-policy over
+decision tree policies they are then able to approximately op-
+timize the models with gradient descent.
+With Iterative Bounding MDPs [Topin et al., 2021] the au-
+thors reformulate the underlying MDP into one where the
+agent implicitly learns a decision tree policy. The method
+can be thought of as a tree agent learning to take actions
+that gather information and a leaf agent learning to take ac-
+tions that work well given the gathered information of the tree
+agent. By reformulating the MDP its decision tree policy can
+be optimized using differentiable function approximators and
+gradient-based optimizers.
+In a separate line of work the goal is to represent a specific,
+usually optimal, policy as a decision tree that is unbounded
+
+in size. These techniques have been developed for policies
+with a single goal state [Br´azdil et al., 2015] and as a tool for
+general controllers [Ashok et al., 2020]: dtcontrol.
+Imitation Learning (VIPER)
+Instead of directly optimizing a decision tree one can also
+try to extract a decision tree policy from a more com-
+plex teacher policy using imitation learning.
+These imi-
+tation learning algorithms turn reinforcement learning into
+a supervised learning problem for which we have success-
+ful decision tree learning algorithms [Breiman et al., 1984;
+Quinlan, 1986]. DAGGER [Ross et al., 2011] (dataset aggre-
+gation) is an algorithm that iteratively collects traces from the
+environment using its current policy and trains a supervised
+model on the union of the current and previous traces. Since
+DAGGER only uses information on the predicted action of
+the teacher policy it ignores extra information on Q-values
+that modern Q-learning algorithms provide. VIPER [Bastani
+et al., 2018] focuses on learning decision trees and improves
+on DAGGER by including Q-value information into the su-
+pervised learning objective. While VIPER generates signif-
+icantly smaller decision trees than DAGGER we will show
+that these trees are not yet optimal with respect to the trade-
+off in size and performance.
+3.2
+Optimal Decision Trees
+The standard algorithms for training decision trees in su-
+pervised learning are greedy heuristics and can learn trees
+that perform arbitrarily poorly [Kearns, 1996].
+Therefore
+in recent years there has been increasing interest in the de-
+sign of algorithms that train decision trees to perform op-
+timally. Early works formulated training decision trees for
+classification and regression and used methods such as dy-
+namic programming to find optimal decision trees [Nijssen
+and Fromont, 2007].
+Mixed-Integer Linear Programming
+based formulations [Bertsimas and Dunn, 2017; Verwer and
+Zhang, 2017] have since become popular. These methods are
+flexible and have been extended to optimize performance un-
+der fairness constraints [Aghaei et al., 2019] or directly op-
+timize adversarial robustness [Vos and Verwer, 2022]. Gen-
+erally, the size of the tree is limited to provide regulariza-
+tion and aid interpretability, then the solver is tasked with
+finding a decision tree that maximizes training performance
+given the size limits.
+The field has since worked on in-
+creasingly efficient optimization techniques using a variety of
+methods such as MILP [Verwer and Zhang, 2019], dynamic
+programming [Demirovi´c et al., 2020; Lin et al., 2020], con-
+straint programming [Verhaeghe et al., 2020], branch-and-
+bound search [Aglin et al., 2020; Aglin et al., 2021] and
+boolean (maximum) satisfiability [Narodytska et al., 2018;
+Hu et al., 2020; Schidler and Szeider, 2021].
+4
+OMDT: Optimal MDP Decision Trees
+In an attempt to bridge the gap between optimal decision
+trees for supervised and reinforcement learning we introduce
+OMDTs: Optimal MDP Decision Trees. OMDT is a Mixed-
+Integer Linear Programming formulation that encodes the
+problem of identifying a decision tree policy that achieves
+maximum discounted return given a user-defined MDP and
++1
+-10
+Markov Decision Process
+under the constraints that:
+objective: maximize expected return (Eq. 1)
+U state-action frequencies obey 
+the MDPs intial and transition 
+probabilities (Eq. 2)
+U the policy selects one action in 
+every state (Eq. 7)
+U state-action frequencies are 0 
+when the policy does not indicate 
+the action is taken (Eq. 8)
+U every decision node selects one 
+split threshold (Eq. 3)
+U obervations go left / right at each 
+node (Eq. 4)
+U every leaf selects an action (Eq. 5)
+U if an observation reaches a leaf 
+that action is taken (Eq. 6)
+Decision Tree
+Formulation based on: 
+Optimal Classification Trees
+Formulation based on:
+standard dual LP
+S = 0
+S = 1
+-10
++0
++0
++1
+if S ≤ 0
+Left
+Right
+yes
+no
+Figure 2: Overview of OMDT’s formulation. We maximize the dis-
+counted return in an MDP under the constraint that the policy is
+represented by a size-limited decision tree.
+size limit. Our formulation can be solved using one of many
+available solvers, in this work we use the state-of-the-art
+solver Gurobi1.
+Intuitively the OMDT formulation consists of two parts:
+a (dual) linear programming formulation for solving MDPs
+and a set of constraints that limits the set of feasible policies
+to decision trees. Figure 2 summarizes OMDT’s formulation
+in natural language. All the notation used in OMDT is sum-
+marized in Table 1.
+4.1
+Constraints
+It is well known that MDPs can be solved using linear pro-
+gramming, the standard linear program is [Puterman, 2014]:
+min.
+�
+s
+p0(s)Vs
+s.t. Vs −
+�
+s′
+γPs,s′,aVs′ ≥
+�
+s′
+Rs,s′,a, ∀s, a
+It is not easy to efficiently add constraints to this formula-
+tion to enforce the policies to be size-limited decision trees
+because it reasons abstractly over policies, i.e. by reasoning
+over the policy’s state values. To create a formulation for de-
+cision tree policies we resort to the standard dual program:
+max.
+�
+s
+�
+a
+xs,a
+�
+s′
+Ps,s′,aRs,s′,a
+(1)
+s.t.
+�
+a
+xs,a−
+�
+s′
+�
+a
+γPs′,s,axs′,a = p0(s), ∀s
+(2)
+1https://www.gurobi.com/
+
+This program uses a measure xs,a of how often the agent
+takes action a in state s, this allows us to add efficient con-
+straints that control the policy of the agent. Intuitively the
+program maximizes the rewards �
+s′ Rs,s′,a weighted by this
+xs,a. The constraints enforce that the frequency by which a
+state is exited is equal to the frequency that the agent is ini-
+tialized in the state or returns to it following the discounted
+transition probabilities γPs,s′,a.
+To enforce the policy to be a size-limited decision tree we
+will later constrain the xs,a values to only be non-zero when
+the agent is supposed to take action a in state s according to
+a tree policy. We will first introduce the variables and con-
+straints required to model the decision tree constraints.
+Modeling Decision Nodes
+Our decision tree formulation is roughly based on OCT [Bert-
+simas and Dunn, 2017] and ROCT [Vos and Verwer, 2022],
+MILP formulations for optimal (OCT) and robust (ROCT)
+classification trees. In these formulations, the shape of the de-
+cision tree is fixed. Like ROCT, we describe a decision node
+m by binary threshold variables bm,j,k, indicating whether
+the kth threshold of feature j is chosen.2 Unlike ROCT, we
+only allow one of these variables to be true over all features
+and possible thresholds:
+�
+j
+�
+k
+bm,j,k = 1,
+∀m
+(3)
+We follow paths through the tree to map observations to
+leaves. In each node m we decide the direction ds,m that the
+observation of state s takes (left=0 or right=1 of the threshold
+k). ROCT uses two variables per state-node pair to model di-
+rections ds,m to account for perturbations in the observations.
+Since we are optimizing for an MDP without uncertainty in
+the observations we only require one variable ds,m per state-
+node pair.
+We further improve over ROCT by determining ds,m us-
+ing only one constraint per state-node pair instead of a sep-
+arate constraint per state-node-feature triple.
+For this, we
+pre-compute a function side(s, j, k) which indicates for each
+feature-threshold pair (j, k) and every observation s the side
+of k that s is on for feature j (left=0 or right=1), i.e. whether
+Xsj > k holds. This formulation is not limited to the predi-
+cates ‘≤’ or ‘>’ however and can be easily extended to other
+predicates in the pre-computation of side(s, j, k). The follow-
+ing then forces the direction ds,m to be equal to the direction
+of the indicated threshold:
+ds,m =
+�
+j
+�
+k
+side(s, j, k) bm,j,k,
+∀s, m
+(4)
+The variables ds,m represent the direction of an observa-
+tion’s path at a decision node. Together, the d variables allow
+us to follow an observation’s path through the tree which we
+use to identify the leaf that it reaches. Important in this for-
+mulation, compared to existing binary encodings, is that it
+requires no big-M constraints to describe these paths. This
+2In practice, the possible values for threshold k depends on the
+chosen feature j. We do not model this dependence for convenience
+of notation.
+makes the relaxation stronger and therefore the solver give
+much better bounds than using the big-M style formulations
+from ROCT.
+Modeling Policy Actions
+Decision leaves only have one set of binary decision vari-
+ables: ct,a encoding whether or not leaf t predicts action a.
+We want each leaf to select exactly one action:
+�
+a
+ct,a = 1,
+∀t
+(5)
+As mentioned before we can follow an observation’s path
+through the tree by using their ds,m path variables. One can
+linearize an implication of a conjunction of binary variables
+as follows:
+x1 ∧ x2 ∧ ... ∧ xn =⇒ y
+≡ x1 + x2 + ... + xn − n + 1 ≤ y
+If an observation reaches leaf t and the leaf predicts action a
+then we want to force the policy πs,a to take that action in the
+associated state s. Using the aforementioned equivalence we
+add the constraint:
+�
+m∈Al(t)
+(1−ds,m) +
+�
+m∈Ar(t)
+ds,m
++ ct,a − |A(t)| ≤ πs,a,
+∀s, a, t
+(6)
+This constraint forces the agent to take the action indicated by
+the leaf. To prevent the agent from taking other actions that
+were not indicated we force it to only take a single action in
+each state (giving a deterministic policy):
+�
+a
+πs,a = 1,
+∀s
+(7)
+Now we have indicators πs,a that mark what action is taken
+by the agent. To link this back to the MDP linear program-
+ming formulation that we use to optimize the policy, we set
+the xs,a variables. We need to set xs,a = 0 if πs,a = 0, else
+xs,a should be unbounded. We encode this using a big-M
+formulation:
+xs,a ≤ Mπs,a,
+∀s, a
+(8)
+M should be chosen as small as possible, but larger or equal
+to the largest value that xs,a can take. We use the fact that we
+are optimizing the MDP using discount factor γ to compute
+an upper bound on xs,a and set M =
+1
+1−γ , proof is given in
+the appendix.
+4.2
+Complete Formulation
+The runtime of MILP solvers grows worst-case exponentially
+with respect to formulation size therefore it is important to
+limit the scale of the formulation. The number of variables in
+our formulation grows with O(|S||J||TD|+|A||TL|+|S||A|)
+which follows from their indices in Table 1. The number of
+constraints grows with the order O(|S||TD| + |S||A||TL|) as
+it is dominated by the constraints that determine ds,m at each
+node (Equation 4) and constraints that force πs,a according to
+the tree (Equation 6). OMDT’s complete MILP formulation
+is summarized below:
+max
+�
+s
+�
+a
+xs,a
+�
+s′
+Ps,s′,aRs,s′,a
+(1)
+
+Name
+Kind Description
+bm,j,k
+bin.
+Tree uses feat. j and threshold k in node m
+ct,a
+bin.
+Tree selects action a in leaf t
+ds,m
+bin.
+Observation of s goes left / right in node m
+πs,a
+bin.
+Policy takes action a in state s
+xs,a
+cont. Frequency of action a taken in state s
+Ps,s′,a
+const. Probability of transition s→s′ with action a
+Rs,s′,a
+const. Reward for transition s→s′ with action a
+p0(s)
+const. Probability of starting in state s
+γ
+const. Discount factor
+Xij
+const. Feature j’s value of observation i
+side(s,j,k) const. Side state s is on for thresh. k and feat. j
+a ∈ A
+set
+Set of actions in MDP
+s ∈ S
+set
+Set of states in MDP
+i=1..|S| set
+Observation and state indices
+j ∈ J
+set
+Set of feature indices
+k = 1..K set
+Indices of all possible feature thresholds
+m ∈ TD
+set
+Set of decision nodes in the tree
+t ∈ TL
+set
+Set of leaves in the tree
+A(t)
+set
+Set of ancestors of leaf t
+Al(t)
+set
+... that have t in their left path
+Ar(t)
+set
+... that have t in their right path
+Table 1: Summary of notation used in OMDT.
+s.t.
+�
+a
+xs,a −
+�
+s′
+�
+a
+γPs′,s,axs′,a = p0(s),
+∀s
+(2)
+�
+j
+�
+k
+bm,j,k = 1,
+∀m
+(3)
+ds,m =
+�
+j
+�
+k
+side(s, j, k) bm,j,k,
+∀s, m
+(4)
+�
+a
+ct,a = 1,
+∀t
+(5)
+�
+m∈Al(t)
+(1−ds,m) +
+�
+m∈Ar(t)
+ds,m + ct,a−|A(t)| ≤ πs,a,
+∀s, a, t
+(6)
+�
+a
+πs,a = 1,
+∀s
+(7)
+xs,a ≤ Mπs,a,
+∀s, a
+(8)
+5
+Results
+We present experiments comparing the performance of
+OMDTs with VIPER and dtcontrol.
+Viper uses imitation
+learning to extract a size-limited decision tree from a teacher
+policy and dtcontrol learns an unrestricted tree that exactly
+copies the teacher’s behavior. To provide a fair comparison
+we have trained VIPER and dtcontrol with an optimal teacher
+by first solving the MDP with value iteration and then ex-
+tracting all Q values, both methods ran with default param-
+eters. We also implemented and ran our experiments on in-
+terpretable Differentiable Decision Trees [Silva et al., 2020]
+but excluded these models from our analysis as they did not
+outperform a random policy. The full code for OMDT and
+our experiments can be found on GitHub3. All of our experi-
+ments ran on a Linux machine with 16 Intel Xeon CPU cores
+and 72 GB of RAM total and used Gurobi 10.0.0 with default
+parameters. Each method ran on a single CPU core.
+5.1
+Environments
+For comparison we implemented 13 environments based on
+well known MDPs from the literature, the sizes of these
+MDPs are given in Table 2. All MDPs were pre-processed
+such that states that are unreachable from the initial states are
+removed. We briefly describe the environments below but re-
+fer to the appendix for complete descriptions.
+In 3d navigation the agent controls a robots in a 5x5x5
+world and attempts to reach from one corner to the other end
+with each voxel having a chance to make the robot disappear.
+blackjack is a simplified version of the famous casino game
+where we assume an infinite sized-deck and only the actions
+‘skip’ or ‘hit’. frozenlake is a grid world where the agent
+attempts to go from the start state to the end state without
+falling into holes, actions are stochastic so the agent will not
+always move in the intended direction (e.g. the action ‘up’
+will only surely not send the agent ‘down’). inventory man-
+agement models a company with limited inventory size that
+has decide how many items to order to meet its customer’s
+demand while minimizing cost. system administrator refers
+to a computer network where computers randomly crash and
+an administrator has to decide which computer to spend time
+on rebooting, a crashed computer has an increased probabil-
+ity of crashing a neighboring computer. tictactoe vs random
+is the well known game of tic-tac-toe when played against
+an opponent that makes random moves. In tiger vs antelope
+the tiger attempts to catch the antelope that randomly jumps
+away from the tiger in a grid world. traffic intersection de-
+scribes a perpendicular intersection where traffic flows in at
+different rates and the operator decides when to switch the
+traffic lights. xor is an MDP constructed such that the states
+randomly lie on a plain, the agent gets 1 reward for taking the
+action according to an XOR function and -1 for a mistake.
+The XOR problem is notoriously difficult for greedy decision
+tree learning algorithms.
+5.2
+Performance-Interpretability Trade-off
+It is often assumed that there is a trade-off in the performance
+and interpretability of machine learning models [Gunning and
+Aha, 2019] since interpretable models necessarily lack com-
+plexity but this assumption is not always true [Rudin, 2019].
+We aim to answer whether the performance-interpretability
+trade-off occurs in a variety of MDPs by training size-limited
+decision trees and comparing their performance to the opti-
+mal solutions that were not restricted in complexity. We vi-
+sualize the normalized return of depth 3 OMDTs and unre-
+stricted dtcontrol trees in Figure 3. Returns were normalized
+such that 0 corresponds to the return of a random policy and
+1 to an optimal one. Since small deterministic decision tree
+policies are limited in the number of distinct actions that they
+take an optimal tree can perform worse than a random policy.
+Experiments were repeated 3 times and runs were limited to 2
+3https://github.com/tudelft-cda-lab/OMDT
+
+normalized return
+MILP
+runtime (s)
+MDP
+|S|
+|A|
+VIPER
+OMDT
+5 mins.
+OMDT
+2 hrs.
+vars.
+constrs.
+trees
+VIPER
+OMDT
+optimal
+3d navigation
+125
+6
+1.00 ±.00
+.81 ±.10
+1.00 ±.00
+2,528
+7,890
+1014
+2,090 ±55
+315 ±89
+blackjack
+533
+2
+1.00 ±.00
+1.00 ±.00
+1.00 ±.00
+6,187
+14,406
+1014
+2,248 ±27
+408 ±85
+frozenlake 4x4
+16
+4
+.67 ±.00
+.96 ±.00
+.96 ±.00
+328
+735
+1010
+74 ±3
+2 ±0
+frozenlake 8x8
+64
+4
+.83 ±.06
+.95 ±.00
+.95 ±.00
+1,104
+2,895
+1013
+178 ±5
+98 ±30
+frozenlake 12x12
+144
+4
+.19 ±.09
+.63 ±.03
+.68 ±.04
+2,360
+6,495
+1014
+196 ±38
+timeout
+inv. management
+101
+100
+1.00 ±.00
+.37 ±.37
+1.00 ±.00
+22,414
+91,824
+1030
+2,254 ±86
+2,533 ±540
+sysadmin 1
+256
+9
+.88 ±.01
+.85 ±.01
+.92 ±.00
+6,584
+23,055
+1014
+2,265 ±37
+timeout
+sysadmin 2
+256
+9
+.59 ±.00
+.23 ±.06
+.58 ±.01
+6,584
+23,055
+1014
+2,257 ±7
+timeout
+sysadmin tree
+128
+8
+.57 ±.04
+.48 ±.07
+.70 ±.09
+3,106
+10,383
+1013
+2,136 ±72
+timeout
+tictactoe vs random
+2,424
+9
+.80 ±.01
+-.06 ±.00
+.43 ±.18
+61,239
+218,175
+1020
+21 ±3
+timeout
+tiger vs antelope
+626
+5
+-.10 ±.02
+-.17 ±.19
+.52 ±.03
+10,850
+33,819
+1015
+490 ±243
+timeout
+traffic intersection
+361
+2
+.98 ±.00
+.99 ±.00
+1.00 ±.00
+4,127
+9,762
+1011
+2,188 ±121
+1,219 ±177
+xor
+200
+2
+.34 ±.06
+1.00 ±.00
+1.00 ±.00
+5,016
+5,415
+1021
+1,999 ±123
+50 ±0
+Table 2: Comparison of depth 3 trees trained with VIPER and OMDT on 13 MDPs, experiments were repeated 3 times and runs were limited
+to 2 hours of runtime. OMDT solves some MDPs in 5 minutes but significantly improves when given 2 hours of runtime. While 2 hours
+are enough for OMDT to achieve greater or equal scores to VIPER in most MDPs, OMDT needs more runtime to outperform VIPER on the
+large tictactoe instance. OMDT was able to identify the optimal size-limited tree and prove its optimality in 7 instances.
+hours. We consider an OMDT optimal when the relative gap
+between its objective and bound is proven to be smaller than
+0.01%.
+While it is debatable what the precise size limits are for
+decision trees to be interpretable [Lipton, 2018] we use trees
+of depth 3 which implies that a tree has at most 8 leaves. We
+find that in all environments, OMDTs of depth 3 improve on
+the performance of random policies, and in 8 out of 13 en-
+vironments the policy gets close to optimal. Decision trees
+trained with dtcontrol always achieve the optimal normalized
+return of 1 since they exactly mimic the optimal policy. How-
+ever, dtcontrol produces large trees that are not interpretable
+to humans. When run on 3d navigation for example, dtcon-
+trol produces a tree of 68 decision nodes which is very com-
+plex for humans to understand. OMDT produces a tree of 7
+decision nodes which performs equally well.
+Overall, our results demonstrate that for many environ-
+ments there is no performance-interpretability trade-off: sim-
+ple policies represented by size-limited trees perform approx-
+imately as well as the unrestricted optimal policy.
+5.3
+Direct Optimization versus Imitation Learning
+The above conclusion holds when the policy is learned to op-
+timality under the constraint that it has to be a small tree, e.g.,
+using OMDT. Frequently used techniques such as VIPER do
+not directly enforce this constraint but aim to imitate the unre-
+stricted optimal policy. We now show that this comes at a cost
+when the unrestricted policy is too complex to be represented
+using a small tree.
+VIPER trains its decision trees by imitating high Q val-
+ues of the optimal policies while OMDT directly maximizes
+expected return. In Table 2 we list the normalized return (0
+for random policies, 1 for optimal policies) for VIPER and
+OMDT with respectively 5 minutes and 2 hours of runtime.
+After 5 minutes OMDT improves performance over random
+0.00
+0.25
+0.50
+0.75
+1.00
+normalized return
+3d_navigation
+xor
+blackjack
+inv. management
+traffic_intersection
+frozenlake_4x4
+frozenlake_8x8
+sysadmin_1
+sysadmin_tree
+frozenlake_12x12
+sysadmin_2
+tiger_vs_antelope
+tictactoe_vs_random
+2
+3
+2
+5
+2
+7
+2
+9
+tree size (nodes)
+omdt
+dtcontrol
+Figure 3: (top) Normalized return and bounds for OMDT trees of
+depth 3, optimal policies score 1 while uniform random policies
+score 0. (bottom) Log of tree sizes for OMDT (maximum depth
+3) and dtcontrol. Dtcontrol always produces an optimal policy but
+the trees are orders of magnitude larger than OMDT.
+policies but often needs more time to improve over VIPER.
+After 2 hours OMDT’s policies win on 11 out 13 environ-
+ments. For instances with large state space such as tictac-
+toe vs random OMDT needs more than 2 hours to improve
+over VIPER.
+Shortcomings of Imitation Learning
+Overall, given sufficient run-time, OMDT produces better
+policies than VIPER. This cannot be easily solved by giving
+VIPER more run-time but is an inherent problem of imitation
+learning. To illustrate this, we investigate the results on the
+
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+G
+(a) Optimal: 92% success
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+G
+(b) OMDT (depth 3): 66% success
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10 11
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+G
+(c) VIPER (depth 3): 11% success
+Figure 4: Paths taken on 10,000 Frozenlake 12x12 runs. The agent starts at (0, 0) and attempts to reach the goal tile ‘G’ while avoiding holes.
+Actions are indicated by arrows and are somewhat stochastic, i.e. an action of ‘up’ will send the agent ‘left’, ‘up’ or ‘right’ (but never down)
+with equal probability. VIPER fails to produce a good policy because it spends capacity of its tree mimicking parts of the complex teacher
+policy that its simple student policy will never reach. OMDT achieves a greater success rate by directly optimizing a simple policy.
+frozenlake environments. Table 2 demonstrates that imita-
+tion learning (VIPER) can perform far from optimal on these
+environments, while direct optimization (OMDT) performs
+closer to the unrestricted optimal solution. It is insightful to
+understand why this happens.
+In theory, imitation learning performs optimally in the
+limit [Ross et al., 2011] but this result requires the student
+policy to be as expressive as the teacher policy. This is not
+the case for size-limited decision trees. When VIPER learns
+a policy for frozenlake 12x12 it tries to imitate a complex pol-
+icy using a small tree that cannot represent all teacher policy
+actions. This results in VIPER spending capacity of its deci-
+sion tree on parts of the state space that will never be reached
+under its student policy. In Figure 4 we visualize the paths
+that the agents took on 10,000 runs and indicate the policies
+with arrows. VIPER creates leaves that control action in the
+right section of the grid world (indicated in red). The optimal
+teacher policy often visits this section but the simple student
+does not. By directly optimizing a decision tree policy using
+OMDT, the policy spends its capacity on parts of the state
+space that it actually reaches. As a result, VIPER cannot pre-
+vent actions that send its agent into holes on the left part of the
+grid world (indicated in red). OMDT actively avoids these.
+5.4
+Runtime
+Runtime for solving Mixed-Integer Linear Programming for-
+mulations scales worst-case exponentially which makes it im-
+portant to understand how solvers operate on complex formu-
+lations such as OMDT. We solved OMDTs for a depth of 3
+for a maximum of 2 hours and display the results in Table
+2. The table compares the runtimes of VIPER and solving
+OMDT to optimality. If the solver does not prove optimal-
+ity within 2 hours we denote it as ‘timeout’. We also denote
+the number of possible decision tree policies computed as:
+|TB|possible splits × |TL||A| . It provides an estimate of how many
+decision tree policies are possible and shows that enumerat-
+ing trees with brute force is intractable.
+OMDT solves a simple environment such as Frozenlake
+4x4 (16 states, 4 actions) to optimality within 2 seconds
+but runtime grows for larger environments such as inventory
+management (101 states, 100 actions) which took on aver-
+age 2533 seconds. VIPER needs roughly 2250 seconds of
+runtime for every MDP and runs significantly faster on some
+MDPs. This is because VIPER spends much time evaluating
+policies on the environment and some environments quickly
+reach terminal states which results in short episodes. While
+OMDT was able to prove optimality on only 7 out of 13 en-
+vironments within 2 hours, OMDT finds good policies before
+this time on 12 out of 13 environments.
+6
+Conclusion
+We propose OMDT, a Mixed-Integer Linear Programming
+formulation for training optimal size-limited decision trees
+for Markov Decision Processes. Our results show that for
+simple environments such as blackjack we do not have to
+trade off interpretability for performance: OMDTs of depth
+3 achieve near-optimal performance. On Frozenlake 12x12,
+OMDT outperforms VIPER by more than 100%.
+OMDT sets a foundation for extending supervised opti-
+mal decision tree learning techniques for to the reinforcement
+learning setting. Still, OMDT requires a full specification of
+the Markov Decision Process. Imitation learning techniques
+such as VIPER can instead also learn from a simulation envi-
+ronment. Therefore, future work should focus on closing the
+gap between the theoretical bound supplied by OMDT and
+the practical performance achieved by algorithms that require
+only simulation access to optimize interpretable decision tree
+policies in reinforcement learning. Additionally, future work
+can incorporate factored MDPs into OMDT’s formulation to
+scale up to larger state spaces.
+
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+
+A
+Environment Descriptions
+We describe the environments in additional detail below. All
+environments in the paper were solved with a discount rate
+of γ = 0.99. Such a value ensures that the solvers generate
+policies that have good performance even for environments
+that takes hundreds of steps to solve. All MDPs were pre-
+processed such that states that are unreachable from the initial
+states are removed.
+A.1
+3d navigation
+While policies in 2d scenarios are often easy to visualize and
+interpret policies in higher dimensions are harder to under-
+stand. Therefore we extend the 2d navigation problem from
+ICAP 2011 IPPC competition4 to 3 dimensions. The agent
+controls a robots in a 5x5x5 world and attempts to reach from
+one corner to the other end with each voxel having a chance
+to make the robot disappear. The disappearing chances were
+independently sampled from a uniform distribution on [0, 1].
+A.2
+blackjack
+The version of blackjack that we used5 is a slightly simplified
+version of the one that often appears in casinos. Specifically,
+we make the following assumptions:
+• There is no doubling down and splitting action.
+• The player does not have knowledge about other players
+at the board.
+• Aces always account for a value of 11.
+• Cards are drawn from an infinitely sized deck, i.e. drawn
+with replacement.
+The player always starts in the state where no cards are
+dealt, the dealer receives a random card, followed by the
+player receiving a random card. The player can ’Draw’ cards
+until they cross the total of 21 or until they ’Skip’. When that
+happens the dealer receives cards until their total is over 17.
+In the end, the reward is computed as follows:
+• -1 if the player has a lower total than the dealer or if the
+player went over 21.
+• 0 if the player and dealer end up with the same total.
+• +1 if the player has a higher total than the dealer or if the
+dealer went over 21.
+• +1.5 if the player gets to exactly 21
+A.3
+frozenlake
+Frozenlake is a maze-like game played on a 2D grid where
+the player attempts to move from the start state (S) to the goal
+state (G) without falling into holes (H). The player walks over
+frozen tiles (F) that make the actions stochastic:
+• The action ’Up’ sends a player in the directions left, up,
+or right with probability 1/3 each.
+• The action ’Right’ sends a player in the directions up,
+right, or down with probability 1/3 each.
+4http://users.cecs.anu.edu.au/∼ssanner/IPPC 2011/
+5https://gist.github.com/iiLaurens/
+ba9c479e71ee4ceef816ad50b87d9ebd
+• The action ’Down’ sends a player in the directions right,
+down, or left with probability 1/3 each.
+• The action ’Left’ sends a player in the directions down,
+left, or up with probability 1/3 each.
+We used the default 4x4 and 8x8 maps as defined by the
+OpenAI gym6 and defined a new map for 12x12. The maps
+are given below (in the same format that the environment
+uses):
+4x4
+"SFFF",
+"FHFH",
+"FFFH",
+"HFFG",
+8x8
+"SFFFFFFF",
+"FFFFFFFF",
+"FFFHFFFF",
+"FFFFFHFF",
+"FFFHFFFF",
+"FHHFFFHF",
+"FHFFHFHF",
+"FFFHFFFG",
+12x12
+"SFFFFFFFFFFF",
+"FFFFFFFFFFFF",
+"FFFHFFFFFFFH",
+"FFFFFHFFFFFF",
+"FFFHFFFFFFFF",
+"FHHFFFHFFHFF",
+"FHFFHFHFFFFF",
+"FFFHFFFFFFFF",
+"FFFFFFFFHFFF",
+"HFFFFHFFFFHH",
+"FFFFFFGFFFFF",
+"FFFFFFFFFFFF",
+A.4
+inventory management
+In inventory management, the player is the owner of a shop
+that stocks items and sells them to customers. Our environ-
+ment is adapted from the implementation by Paul Hendricks7.
+Every day the player can choose ‘Don’t buy’ or ‘Buy x’ items.
+We used the following parameters for the problem:
+• The maximum inventory size is 100 (this creates 101
+states since the inventory can have 0 up to and includ-
+ing 100 items)
+• Ordering any items comes with a fixed cost of -10
+• Ordering items costs -2 per item
+• Holding an item in storage costs -1 per item
+• Selling an item grants the player +4
+• The number of customers follows a Poisson distribution
+with expectation λ = 15 (customers per day)
+A.5
+system administrator
+The system administrator task refers to a computer network
+where computers randomly crash and an administrator has to
+decide which computer to spend time on rebooting. This en-
+vironment is based on one of the MDPs of the ICAPS 2011
+IPPC competition4. When a computer crashes it has an in-
+creased probability in following time steps to crash a neigh-
+boring computer. The topology of the network can be varied
+to create MDPs with various properties. The 3 topologies
+considered in this work are visualized in Figure 5.
+The system administrator is then allowed to reboot one
+machine at every time step or wait. Rebooting a computer
+is penalized with a reward of -0.45 but will always fix the
+crashed computer. At each time step every computer that has
+6https://github.com/openai/gym/blob/master/gym/envs/toy text/
+frozen lake.py
+7https://github.com/paulhendricks/gym-inventory/tree/master
+
+(a) tree
+(b) random 1
+(c) random 2
+Figure 5: System administrator topologies.
+not crashed will reward the system administrator with +1. We
+refer to a computer being on (1) or crashed (0) as the ‘status’,
+the probability of being on in the next step is then given by:
+ratio on neighbors × (0.05 + 0.9 × status)
+A.6
+tictactoe vs random
+The well known tic-tac-toe game is played on a 3x3 grid
+where players take turns putting a cross / circle in an empty
+square. The player that manages to put 3 of their symbols in a
+row / column / diagonal wins the game. While 2 player zero-
+sum games are not MDPs, they become MDPs when we fix
+the strategy of one of the players. In this case the opponent
+plays moves uniformly at random. It is worth noting that this
+turns the game that is normally deterministic into a stochastic
+environment. The player gets to start the game, i.e. they play
+with the crosses.
+A.7
+tiger vs antelope
+The tiger vs antelope game is played on a 5x5 grid where the
+agent controls the tiger and tries to catch an antelope. Every
+turn the agent can move ‘up’, ‘down’, ‘left’ or ‘right’ and
+if the antelope is not caught it responds by jumping in any
+direction away from the player uniformly at random.
+A.8
+traffic intersection
+The traffic intersection scenario that we considered describes
+a perpendicular intersection where traffic flows in at differ-
+ent rates and the operator decides when to switch the traffic
+lights. At each side of the intersection 5 cars can fit in a row
+and from the moment the traffic light switches we count up to
+5 time steps in which one side of the intersection is waiting.
+Cars arrive with probability 0.1 per time step on one side of
+the intersection and 0.5 on the other side.
+The operator gets rewarded +1 for each car making it
+through the intersection and gets penalized for flipping the
+color of the traffic signs and for waiting cars. Flipping the
+color of the traffic lights incurs a reward of -2. For each wait-
+ing car the traffic operator gets a reward of −0.1 × 2wait time.
+A.9
+xor
+The XOR problem is notoriously difficult for greedy decision
+tree learning algorithms, we generate an MDP that mimics
+the structure of the supervised learning problem. Specifically
+we draw 200 samples in 2 dimensions uniformly at random.
+The agent gets 1 reward for taking the action a according to
+an XOR function and -1 for a mistake:
+R(x, y, a) =
+�1
+round(x) + round(y)
+mod 2 = a
+−1
+round(x) + round(y)
+mod 2 = a
+Each action sends the agent to a state uniformly at random.
+B
+Linearizing Implications of Conjunctions
+In the main text, we constructed one of OMDT’s constraints
+using the fact that:
+x1 ∧ x2 ∧ ... ∧ xn =⇒ y ≡ x1 + x2 + ... + xn − n + 1 ≤ y
+We give a short proof of this statement. First, we rewrite the
+logical formula into conjunctive normal form. In this case a
+single disjunctive clause.
+x1 ∧ x2 ∧ ... ∧ xn =⇒ y
+≡ ¬(x1 ∧ x2 ∧ ... ∧ xn) ∨ y
+≡ ¬x1 ∨ ¬x2 ∨ ... ∨ ¬xn ∨ y
+A disjunctive constraint a∨b∨c can be trivially expressed as
+the linear constraint a + b + c ≥ 1. Intuitively any variable a,
+b or c needs to be true. Now we rewrite to linear constraints:
+¬x1 ∨ ¬x2 ∨ ... ∨ ¬xn ∨ y
+≡ (1 − x1) + (1 − x2) + ... + (1 − xn) + y ≥ 1
+≡ x1 + x2 + ... + xn − n + 1 ≤ y
+C
+Big-M Value
+For the constraints that force xs,a according to the policy πs,a
+(Equation 8) we used a big-M formulation with M =
+1
+1−γ
+as an upper bound on xs,a. To prove this upper bound we
+construct an MDP in which we maximize xs,a then show that
+this value equals our bound. Consider an MDP with one state
+s∗ and one action a∗ such that the agent always starts in that
+state (p0(s∗) = 1) and action a∗ will always return to s∗
+with probability Ps∗,s∗,a∗ = 1. Clearly, this maximizes the
+frequency xs∗,a∗. Substituting into Equation 2 we find:
+�
+a
+xs,a = p0(s) +
+�
+s′
+�
+a
+γPs′,s,axs′,a
+(2)
+xs∗,a∗ = p0(s∗) + γxs∗,a∗ =
+1
+1 − γ
+D
+Performance at Varying Depths
+In the main text we discussed performance of trees at a depth
+of 3 since these trees are still easily interpretable but more
+powerful than trees of depth 1 or 2. Normalized returns after
+2 hours of runtime for varying depth are presented in Table 3,
+Table 4 shows the runtime needed to prove optimality of the
+solutions for varying depths.
+E
+Size-Limited Policies
+We claim that since our models are interpretable since they
+are limited in size to an extend that a human can easily follow
+the predictions of the models [Lipton, 2018]. The decision
+trees of depth 3 created by OMDT and VIPER are visualized
+for inspection in Figure 6 and Figure 7 respectively.
+
+32
+5
+33
+0
+5
+6
+4depth 1
+depth 2
+depth 3
+depth 4
+MDP
+OMDT
+VIPER
+OMDT
+VIPER
+OMDT
+VIPER
+OMDT
+VIPER
+3d navigation
+-.00 ± .00
+-.00 ± .00
+.06 ± .00
+.01 ± .00
+1.00 ± .00
+1.00 ± .00
+1.00 ± .00
+1.00 ± .00
+blackjack
+.98 ± .00
+.98 ± .00
+1.00 ± .00
+.98 ± .00
+1.00 ± .00
+1.00 ± .00
+1.00 ± .00
+1.00 ± .00
+frozenlake 4x4
+.19 ± .00
+.04 ± .06
+.67 ± .00
+.46 ± .00
+.96 ± .00
+.67 ± .00
+1.00 ± .00
+1.00 ± .00
+frozenlake 8x8
+.74 ± .00
+.72 ± .01
+.93 ± .00
+.90 ± .02
+.95 ± .00
+.83 ± .06
+.98 ± .00
+.95 ± .00
+frozenlake 12x12
+.06 ± .00
+.04 ± .02
+.34 ± .00
+.30 ± .00
+.68 ± .04
+.19 ± .09
+.81 ± .01
+.19 ± .09
+inv. management
+.99 ± .00
+.98 ± .00
+1.00 ± .00
+.99 ± .00
+1.00 ± .00
+1.00 ± .00
+1.00 ± .00
+1.00 ± .00
+sysadmin 1
+.84 ± .00
+.83 ± .00
+.89 ± .00
+.86 ± .03
+.92 ± .00
+.88 ± .01
+.91 ± .00
+.92 ± .01
+sysadmin 2
+.27 ± .00
+.27 ± .00
+.55 ± .00
+.55 ± .00
+.57 ± .02
+.59 ± .00
+.67 ± .04
+.72 ± .00
+sysadmin tree
+-.03 ± .00
+-.03 ± .00
+.37 ± .00
+.26 ± .00
+.71 ± .08
+.57 ± .04
+.86 ± .04
+.86 ± .00
+tictactoe vs random
+-.06 ± .00
+-.06 ± .00
+.61 ± .00
+.61 ± .00
+.43 ± .18
+.80 ± .01
+.68 ± .10
+.83 ± .00
+tiger vs antelope
+-.36 ± .00
+-.64 ± .06
+.19 ± .05
+-.31 ± .17
+.50 ± .05
+-.10 ± .02
+.64 ± .03
+.23 ± .10
+traffic intersection
+.50 ± .00
+.42 ± .00
+.98 ± .00
+.98 ± .00
+1.00 ± .00
+.98 ± .00
+1.00 ± .00
+1.00 ± .00
+xor
+.15 ± .00
+.05 ± .00
+1.00 ± .00
+.13 ± .00
+1.00 ± .00
+.34 ± .06
+1.00 ± .00
+.90 ± .03
+Table 3: Normalized returns of OMDT and VIPER when varying the maximum depth of the decision tree, runtime was limited to 2 hours
+and experiments repeated with 3 random seeds. For depths until 3 OMDT finds optimal or high quality trees within 2 hours while for depth 4
+OMDT will need more runtime to improve on the scores produced by VIPER’s trees on 3 environments.
+MDP
+depth 1
+depth 2
+depth 3
+depth 4
+3d navigation
+9 ±0
+258 ±27
+315 ±89
+223 ±25
+blackjack
+36 ±5
+72 ±7
+408 ±85
+598 ±99
+frozenlake 4x4
+1 ±0
+1 ±0
+2 ±0
+2 ±0
+frozenlake 8x8
+2 ±0
+9 ±2
+98 ±30
+3134 ±350
+frozenlake 12x12
+10 ±2
+349 ±57
+timeout
+timeout
+inv. management
+453 ±42
+6597 ±507
+2533 ±540
+3548 ±732
+sysadmin 1
+129 ±10
+5059 ±542
+timeout
+timeout
+sysadmin 2
+488 ±53
+timeout
+timeout
+timeout
+sysadmin tree
+74 ±10
+4547 ±767
+timeout
+timeout
+tictactoe vs random
+2989 ±61
+timeout
+timeout
+timeout
+tiger vs antelope
+1695 ±210
+timeout
+timeout
+timeout
+traffic intersection
+33 ±1
+104 ±28
+1219 ±177
+5595
+xor
+113 ±10
+43 ±1
+50 ±0
+98 ±27
+Table 4: Runtimes until OMDT proves optimality when varying the depth of the trees, runtime was limited to 2 hours and experiments repeated
+with 3 random seeds. Proving optimality for deeper trees usually takes more runtime but since a deeper tree allows the tree’s objective value
+to be closer to the bound runtime can reduce e.g. in 3d navigation depth 4. OMDT only proved optimality for traffic intersection depth 4 in
+one of the runs.
+
+if z <= 2
+if z <= 0
+yes
+if z <= 3
+no
+if x <= 1
+if x <= 2
+right
+forward
+right
+forward
+if y <= 3
+if x <= 2
+up
+forward
+up
+right
+(a) 3d navigation
+if dealer_total <= 9.000
+if dealer_total <= 6.000
+yes
+if dealer_total <= 10.000
+no
+if player_total <= 11.000
+if player_total <= 14.000
+Draw
+Skip
+Draw
+Skip
+if player_total <= 13.000
+if player_total <= 11.000
+Draw
+Skip
+Draw
+Skip
+(b) blackjack
+if X <= 0.000
+if Y <= 1.000
+yes
+if Y <= 2.000
+no
+Left
+if Y <= 2.000
+Up
+Down
+if X <= 1.000
+if X <= 1.000
+Down
+Left
+Right
+Down
+(c) frozenlake 4x4
+if X <= 6.000
+if X <= 5.000
+yes
+if Y <= 1.000
+no
+if Y <= 0.000
+if Y <= 3.000
+Right
+Up
+Right
+Up
+if Y <= 0.000
+if Y <= 6.000
+Right
+Down
+Right
+Left
+(d) frozenlake 8x8
+if X <= 1.000
+if Y <= 7.000
+yes
+if X <= 6.000
+no
+if Y <= 6.000
+if X <= 0.000
+Left
+Down
+Up
+Right
+if Y <= 10.000
+if Y <= 10.000
+Down
+Right
+Down
+Left
+(e) frozenlake 12x12
+if inventory <= 4
+if inventory <= 2
+yes
+if inventory <= 20
+no
+if inventory <= 1
+if inventory <= 3
+Buy 24
+Buy 9
+Buy 8
+Buy 7
+if inventory <= 16
+if inventory <= 21
+Don't buy
+Buy 7
+Buy 3
+Don't buy
+(f) inventory management
+if computer_2_running <= 0
+if computer_0_running <= 0
+yes
+if computer_5_running <= 0
+no
+if computer_6_running <= 0
+reboot_computer_0
+reboot_computer_0
+reboot_computer_4
+if computer_6_running <= 0
+if computer_1_running <= 0
+reboot_computer_0
+reboot_computer_4
+reboot_computer_1
+reboot_computer_0
+(g) system administrator 1
+if computer_1_running <= 0
+reboot_computer_1
+yes
+if computer_2_running <= 0
+no
+if computer_4_running <= 0
+if computer_4_running <= 0
+reboot_computer_4
+reboot_computer_2
+reboot_computer_4
+reboot_computer_3
+(h) system administrator 2
+if computer_5_running <= 0
+if computer_2_running <= 0
+yes
+if computer_0_running <= 0
+no
+if computer_0_running <= 0
+if computer_4_running <= 0
+reboot_computer_0
+reboot_computer_2
+reboot_computer_4
+reboot_computer_5
+if computer_6_running <= 1
+if computer_6_running <= 0
+reboot_computer_0
+wait
+reboot_computer_6
+reboot_computer_1
+(i) system administrator tree
+if bottom_right_cross <= 0
+if center_circle <= -1
+yes
+if bottom_right_cross <= 0
+no
+if bottom_center_free <= 0
+bottom_right
+top_right
+bottom_right
+if bottom_center_free <= 0
+if bottom_center_free <= 0
+bottom_center
+bottom_right
+bottom_left
+bottom_center
+(j) tictactoe vs random
+if antelope_x <= 3
+if tiger_y <= 2
+yes
+if antelope_y <= 3
+no
+if antelope_y <= 0
+if tiger_x <= 1
+down
+up
+down
+left
+if tiger_y <= 2
+if tiger_x <= 3
+right
+down
+wait
+up
+(k) tiger vs antelope
+if green_side <= 0
+if cars_right <= 0
+yes
+if waiting_time <= 1
+no
+if waiting_time <= 2
+if cars_left <= 1
+wait
+switch_light
+switch_light
+wait
+if cars_left <= 1
+if cars_left <= 0
+wait
+switch_light
+wait
+switch_light
+(l) traffic intersection
+if X <= 0.982
+if X <= 0.497
+yes
+xor
+no
+if Y <= 0.501
+if Y <= 0.490
+not_xor
+xor
+xor
+not_xor
+(m) xor
+Figure 6: Depth 3 trees produced by OMDT within 2 hours of runtime with one fixed seed.
+
+if x <= 2
+if x <= 1
+yes
+if y <= 3
+no
+right
+if z <= 0
+up
+right
+if z <= 2
+if z <= 3
+up
+left
+up
+right
+(a) 3d navigation
+if player_total <= 11.500
+Draw
+yes
+if player_total <= 13.500
+no
+if dealer_total <= 6.500
+if player_total <= 14.500
+Skip
+Draw
+Skip
+Skip
+(b) blackjack
+if Y <= 1.500
+if X <= 0.500
+yes
+if X <= 0.500
+no
+Left
+if Y <= 0.500
+Up
+Left
+Up
+if Y <= 2.500
+Down
+Right
+(c) frozenlake 4x4
+if Y <= 2.500
+if X <= 6.500
+yes
+if X <= 6.500
+no
+if X <= 5.500
+if Y <= 0.500
+Up
+Right
+Right
+Down
+if Y <= 3.500
+Right
+Right
+Up
+(d) frozenlake 8x8
+if X <= 10.500
+if Y <= 6.500
+yes
+if Y <= 2.000
+no
+if X <= 0.500
+if Y <= 8.500
+Left
+Left
+Down
+Right
+if Y <= 0.500
+if Y <= 7.500
+Left
+Up
+Down
+Up
+(e) frozenlake 12x12
+if inventory <= 0
+Buy 10
+yes
+if inventory <= 4
+no
+if inventory <= 3
+if inventory <= 16
+Buy 8
+Buy 6
+Don't buy
+Buy 6
+(f) inventory management
+if computer_6_running <= 0
+if computer_0_running <= 0
+yes
+if computer_5_running <= 0
+no
+if computer_4_running <= 0
+if computer_2_running <= 0
+reboot_computer_0
+reboot_computer_5
+reboot_computer_0
+reboot_computer_0
+if computer_4_running <= 0
+if computer_0_running <= 0
+reboot_computer_0
+reboot_computer_4
+reboot_computer_5
+reboot_computer_2
+(g) system administrator 1
+if computer_1_running <= 0
+if computer_7_running <= 0
+yes
+if computer_4_running <= 0
+no
+if computer_6_running <= 0
+if computer_2_running <= 0
+reboot_computer_1
+reboot_computer_1
+reboot_computer_2
+reboot_computer_1
+if computer_3_running <= 0
+if computer_2_running <= 0
+reboot_computer_4
+reboot_computer_4
+reboot_computer_2
+reboot_computer_3
+(h) system administrator 2
+if computer_1_running <= 0
+if computer_0_running <= 0
+yes
+if computer_3_running <= 0
+no
+reboot_computer_0
+reboot_computer_1
+if computer_2_running <= 0
+if computer_5_running <= 0
+reboot_computer_2
+reboot_computer_3
+reboot_computer_5
+reboot_computer_4
+(i) system administrator tree
+if top_center_circle <= 0
+if top_right_cross <= 0
+yes
+if center_free <= 0
+no
+if bottom_right_free <= 0
+if top_left_cross <= 0
+bottom_left
+top_right
+top_left
+top_center
+if bottom_left_free <= 0
+if bottom_left_free <= 0
+bottom_right
+bottom_left
+bottom_right
+center
+(j) tictactoe vs random
+if antelope_x <= 2
+if tiger_y <= 2
+yes
+if tiger_y <= 1
+no
+if antelope_y <= 2
+if antelope_y <= 2
+up
+up
+down
+left
+right
+if antelope_y <= 3
+up
+wait
+(k) tiger vs antelope
+if cars_left <= 0
+if green_side <= 0
+yes
+if waiting_time <= 1
+no
+if waiting_time <= 1
+wait
+wait
+switch_light
+if cars_left <= 1
+if green_side <= 0
+switch_light
+wait
+switch_light
+switch_light
+(l) traffic intersection
+if Y <= 0.042
+if X <= 0.359
+yes
+if X <= 0.904
+no
+not_xor
+xor
+if Y <= 0.214
+if Y <= 0.527
+not_xor
+xor
+xor
+not_xor
+(m) xor
+Figure 7: Depth 3 trees produced by VIPER with one fixed seed.
+
diff --git a/kdFPT4oBgHgl3EQf2DV-/content/tmp_files/load_file.txt b/kdFPT4oBgHgl3EQf2DV-/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..273e582c6a9017c0106340d8a67feaf9d4fd9794
--- /dev/null
+++ b/kdFPT4oBgHgl3EQf2DV-/content/tmp_files/load_file.txt
@@ -0,0 +1,1191 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf,len=1190
+page_content='Optimal Decision Tree Policies for Markov Decision Processes  Dani¨el Vos , Sicco Verwer  Delft University of Technology  {d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='vos, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='verwer}@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='nl  Abstract  Interpretability of reinforcement learning policies  is essential for many real-world tasks but learn-  ing such interpretable policies is a hard problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Particularly rule-based policies such as decision  trees and rules lists are difficult to optimize due  to their non-differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' While existing tech-  niques can learn verifiable decision tree policies  there is no guarantee that the learners generate a  decision that performs optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  In this work,  we study the optimization of size-limited decision  trees for Markov Decision Processes (MPDs) and  propose OMDTs: Optimal MDP Decision Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Given a user-defined size limit and MDP formula-  tion OMDT directly maximizes the expected dis-  counted return for the decision tree using Mixed-  Integer Linear Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' By training optimal  decision tree policies for different MDPs we em-  pirically study the optimality gap for existing imita-  tion learning techniques and find that they perform  sub-optimally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We show that this is due to an inher-  ent shortcoming of imitation learning, namely that  complex policies cannot be represented using size-  limited trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In such cases, it is better to directly  optimize the tree for expected return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' While there  is generally a trade-off between the performance  and interpretability of machine learning models, we  find that OMDTs limited to a depth of 3 often per-  form close to the optimal limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  1  Introduction  Advances in reinforcement learning using function approxi-  mation have allowed us to train powerful agents for complex  problems such as the games of Go and Atari [Schrittwieser et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Policies learned using function approximation of-  ten use neural networks, making them impossible for humans  to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Therefore reinforcement learning is severely  limited for applications with high-stakes decisions where the  user has to trust the learned policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Recent work has focused on explaining opaque models  such as neural networks by attributing prediction importance  to the input features [Ribeiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Lundberg and Lee,  2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' However, these explanation methods cannot capture  the full complexity of their models, which can mislead users  when attempting to understand the predictions [Rudin, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Concurrently, there has been much work on interpretable ma-  chine learning in which the model learned is limited in com-  plexity to the extent that humans can understand the complete  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Particularly decision trees have received much atten-  tion as they are simple models that are capable of modeling  non-linear behavior [Lipton, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Decision trees are difficult to optimize as they are non-  differentiable and discontinuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Previous works have used  different strategies to overcome the hardness of optimiz-  ing trees:  using assumptions or relaxations to make the  trees differentiable [Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Likmeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020], reformulating the MDP into a meta-  MDP that exclusively models decision tree policies [Topin et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2021] or extracting trees from a complex teacher [Bastani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' While these methods are increasingly success-  ful in training performant trees they do not offer guarantees  on this performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Our work takes a first step at bridging the gap between  the fields of optimal decision trees and reinforcement learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Existing formulations for optimal decision trees assume  a fixed training set with independent samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This cannot  be used in a dynamic setting where actions taken in one state  influence the best actions in others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Instead, we formulate the  problem of solving a Markov Decision Process (MDP) us-  ing a policy represented by a size-limited decision tree (see  Figure 1) in a single MILP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We link the predictions of the de-  cision tree policy to the state-action frequencies in the dual  linear program for solving MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The dual allows us to rea-  son over policies explicitly, which results in a more efficient  formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Our formulation for Optimal MDP Decision  Trees, OMDTs, optimizes a decision tree policy for a given  MDP and a tree size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  OMDT produces increasingly  performant tree policies as runtime progresses and eventually  proves the optimality of its policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Existing methods for training size-limited decision trees in  reinforcement learning such as VIPER [Bastani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2018]  make use of imitation learning where a student tries to learn  from a powerful teacher policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We compare the perfomance  of OMDT and VIPER on a variety of MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Interestingly,  we show that when training interpretable size-limited trees  imitation learning performs significantly worse as capacity  of the learned decision tree is wasted on parts of the state  space that are never reached by the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Moreover, VIPER  cannot prove optimality even if it identifies the optimal solu-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Regarding the performance-interpretability trade-off we  show that decision trees of 7 decision nodes are enough to  perform close to optimally in 8 out of 13 environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Such  trees are orders of magnitude smaller than the trees created by  methods that exactly replicate the optimal policy using size-  unrestricted trees such as dtcontrol [Ashok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  2  Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='1  Decision Trees  Decision trees [Breiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Quinlan, 1986] are sim-  ple models that execute a series of comparisons between a  feature value and a threshold in the nodes to arrive at a leaf  node that contains the prediction value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Due to their sim-  ple descriptions, size-limited decision trees are easy to under-  stand for humans [Molnar, 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Particularly, size-limited  decision trees admit simulatability [Lipton, 2018]: humans  can use the model to make predictions by hand in reasonable  time and decomposability: humans can understand each in-  dividual aspect of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The method proposed in this  paper, OMDT, also offers algorithmic transparency: we can  trust the learning algorithm to produce models that fulfill cer-  tain qualities such as global optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Therefore decision  trees are an attractive model class when interpretable policies  are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='2  Markov Decision Processes  Markov Decision Processes (MDPs) [Bellman, 1957] are the  processes underlying reinforcement learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' An  MDP models a decision-making process in a stochastic en-  vironment where the agent has some control over the state  transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' An MDP can be described by a tuple ⟨S, A, P, R⟩  where S contains all states, A the set of actions an agent can  take, Ps,s′,a the probabilities of transitioning from state s to  state s′ when taking action a, and Rs,s′,a the reward the agent  receives when going from state s to state s′ under action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  When solving an MDP we want to find a policy π : S → A  such that when executing its actions the expected sum of re-  wards (the return) is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  In this work we define  policies (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=') as a mapping from states and actions to  an indicator for whether or not action a is taken in state s:  π : S × A → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  if X ≤ 0  if Y ≤ 1  if Y ≤ 2  Left  Up  Down  Right  yes  no  start  goal  0  0  1  2  3  3  1  2  Figure 1: Depth 2 OMDT on the stochastic Frozenlake 4x4 envi-  ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT proves that no better depth 2 decision tree policy  exists (discounted return 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='37 with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='99).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Value Iteration  When solving MDPs we generally discount future rewards in  each step by a user-defined value of 0 < γ < 1 to ensure  that the optimal policy will generate a finite return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The most  common approach for optimally solving MDPs is by using  one of many dynamic programming variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In this work,  we focus on value iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Value iteration finds a value Vs  for each state s that holds the expected discounted return for  taking optimal greedy actions starting from that state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' These  values can be found by iteratively updating Vs until the Bell-  man equation [Bellman, 1957] is approximately satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We  will refer to this optimal solution found with value iteration  as the unrestricted optimal solution as the computed policy  can be arbitrarily complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  3  Related Work  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='1  Learning Decision Tree Policies  Decision trees have appeared at various parts of the reinforce-  ment learning pipeline [Glanois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2021] for example in  modeling the Q-value function or in modeling the policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In  this work, we are interested in modeling the policy with a de-  cision tree as this gives us an interpretable model that we can  directly use at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Decision trees make predictions using hard comparisons  between a single feature value and a threshold the learned  models can be discontinuous and non-differentiable which  makes optimization with gradients challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' One line of  research focuses on overcoming the non-differentiability of  decision trees to allow for the use of gradient-based optimiza-  tion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' [Gupta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2015] train decision  trees that contain linear models in their leaves that can be op-  timized with gradients but this results in models that are hard  to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' [Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020] first relax the  constraints that decision nodes select one feature, that leaves  predict one value, and that thresholds are hard step functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Such relaxed trees are differentiable and can be trained with  policy gradient methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' By discretizing the relaxed tree they  end up with an interpretable model that approximates the re-  laxed model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' However, the relaxed tree can get stuck in local  minima and the discretized tree offers no performance guar-  antees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Likmeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' [Likmeta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020] consider de-  cision tree policies for autonomous driving tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' To make  the decision tree parameters easily optimizable they fix the  structure of the tree along with the features used in the de-  cision nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' By learning a differentiable hyper-policy over  decision tree policies they are then able to approximately op-  timize the models with gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  With Iterative Bounding MDPs [Topin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2021] the au-  thors reformulate the underlying MDP into one where the  agent implicitly learns a decision tree policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The method  can be thought of as a tree agent learning to take actions  that gather information and a leaf agent learning to take ac-  tions that work well given the gathered information of the tree  agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' By reformulating the MDP its decision tree policy can  be optimized using differentiable function approximators and  gradient-based optimizers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  In a separate line of work the goal is to represent a specific,  usually optimal, policy as a decision tree that is unbounded  in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' These techniques have been developed for policies  with a single goal state [Br´azdil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2015] and as a tool for  general controllers [Ashok et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020]: dtcontrol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Imitation Learning (VIPER)  Instead of directly optimizing a decision tree one can also  try to extract a decision tree policy from a more com-  plex teacher policy using imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  These imi-  tation learning algorithms turn reinforcement learning into  a supervised learning problem for which we have success-  ful decision tree learning algorithms [Breiman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Quinlan, 1986].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' DAGGER [Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2011] (dataset aggre-  gation) is an algorithm that iteratively collects traces from the  environment using its current policy and trains a supervised  model on the union of the current and previous traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Since  DAGGER only uses information on the predicted action of  the teacher policy it ignores extra information on Q-values  that modern Q-learning algorithms provide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' VIPER [Bastani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2018] focuses on learning decision trees and improves  on DAGGER by including Q-value information into the su-  pervised learning objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' While VIPER generates signif-  icantly smaller decision trees than DAGGER we will show  that these trees are not yet optimal with respect to the trade-  off in size and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='2  Optimal Decision Trees  The standard algorithms for training decision trees in su-  pervised learning are greedy heuristics and can learn trees  that perform arbitrarily poorly [Kearns, 1996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Therefore  in recent years there has been increasing interest in the de-  sign of algorithms that train decision trees to perform op-  timally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Early works formulated training decision trees for  classification and regression and used methods such as dy-  namic programming to find optimal decision trees [Nijssen  and Fromont, 2007].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Mixed-Integer Linear Programming  based formulations [Bertsimas and Dunn, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Verwer and  Zhang, 2017] have since become popular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' These methods are  flexible and have been extended to optimize performance un-  der fairness constraints [Aghaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2019] or directly op-  timize adversarial robustness [Vos and Verwer, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Gen-  erally, the size of the tree is limited to provide regulariza-  tion and aid interpretability, then the solver is tasked with  finding a decision tree that maximizes training performance  given the size limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The field has since worked on in-  creasingly efficient optimization techniques using a variety of  methods such as MILP [Verwer and Zhang, 2019], dynamic  programming [Demirovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020], con-  straint programming [Verhaeghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020], branch-and-  bound search [Aglin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Aglin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2021] and  boolean (maximum) satisfiability [Narodytska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Schidler and Szeider, 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  4  OMDT: Optimal MDP Decision Trees  In an attempt to bridge the gap between optimal decision  trees for supervised and reinforcement learning we introduce  OMDTs: Optimal MDP Decision Trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT is a Mixed-  Integer Linear Programming formulation that encodes the  problem of identifying a decision tree policy that achieves  maximum discounted return given a user-defined MDP and  +1  10  Markov Decision Process  under the constraints that:  objective: maximize expected return (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 1)  U state-action frequencies obey  the MDPs intial and transition  probabilities (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 2)  U the policy selects one action in  every state (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 7)  U state-action frequencies are 0  when the policy does not indicate  the action is taken (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 8)  U every decision node selects one  split threshold (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 3)  U obervations go left / right at each  node (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 4)  U every leaf selects an action (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 5)  U if an observation reaches a leaf  that action is taken (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' 6)  Decision Tree  Formulation based on:  Optimal Classification Trees  Formulation based on:  standard dual LP  S = 0  S = 1  10  +0  +0  +1  if S ≤ 0  Left  Right  yes  no  Figure 2: Overview of OMDT’s formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We maximize the dis-  counted return in an MDP under the constraint that the policy is  represented by a size-limited decision tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  size limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Our formulation can be solved using one of many  available solvers, in this work we use the state-of-the-art  solver Gurobi1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Intuitively the OMDT formulation consists of two parts:  a (dual) linear programming formulation for solving MDPs  and a set of constraints that limits the set of feasible policies  to decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Figure 2 summarizes OMDT’s formulation  in natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' All the notation used in OMDT is sum-  marized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='1  Constraints  It is well known that MDPs can be solved using linear pro-  gramming, the standard linear program is [Puterman, 2014]:  min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  �  s  p0(s)Vs  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Vs −  �  s′  γPs,s′,aVs′ ≥  �  s′  Rs,s′,a, ∀s, a  It is not easy to efficiently add constraints to this formula-  tion to enforce the policies to be size-limited decision trees  because it reasons abstractly over policies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' by reasoning  over the policy’s state values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' To create a formulation for de-  cision tree policies we resort to the standard dual program:  max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  �  s  �  a  xs,a  �  s′  Ps,s′,aRs,s′,a  (1)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  �  a  xs,a−  �  s′  �  a  γPs′,s,axs′,a = p0(s), ∀s  (2)  1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='gurobi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='com/  This program uses a measure xs,a of how often the agent  takes action a in state s, this allows us to add efficient con-  straints that control the policy of the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Intuitively the  program maximizes the rewards �  s′ Rs,s′,a weighted by this  xs,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The constraints enforce that the frequency by which a  state is exited is equal to the frequency that the agent is ini-  tialized in the state or returns to it following the discounted  transition probabilities γPs,s′,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  To enforce the policy to be a size-limited decision tree we  will later constrain the xs,a values to only be non-zero when  the agent is supposed to take action a in state s according to  a tree policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We will first introduce the variables and con-  straints required to model the decision tree constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Modeling Decision Nodes  Our decision tree formulation is roughly based on OCT [Bert-  simas and Dunn, 2017] and ROCT [Vos and Verwer, 2022],  MILP formulations for optimal (OCT) and robust (ROCT)  classification trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In these formulations, the shape of the de-  cision tree is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Like ROCT, we describe a decision node  m by binary threshold variables bm,j,k, indicating whether  the kth threshold of feature j is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='2 Unlike ROCT, we  only allow one of these variables to be true over all features  and possible thresholds:  �  j  �  k  bm,j,k = 1,  ∀m  (3)  We follow paths through the tree to map observations to  leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In each node m we decide the direction ds,m that the  observation of state s takes (left=0 or right=1 of the threshold  k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' ROCT uses two variables per state-node pair to model di-  rections ds,m to account for perturbations in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Since we are optimizing for an MDP without uncertainty in  the observations we only require one variable ds,m per state-  node pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  We further improve over ROCT by determining ds,m us-  ing only one constraint per state-node pair instead of a sep-  arate constraint per state-node-feature triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  For this, we  pre-compute a function side(s, j, k) which indicates for each  feature-threshold pair (j, k) and every observation s the side  of k that s is on for feature j (left=0 or right=1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' whether  Xsj > k holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This formulation is not limited to the predi-  cates ‘≤’ or ‘>’ however and can be easily extended to other  predicates in the pre-computation of side(s, j, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The follow-  ing then forces the direction ds,m to be equal to the direction  of the indicated threshold:  ds,m =  �  j  �  k  side(s, j, k) bm,j,k,  ∀s, m  (4)  The variables ds,m represent the direction of an observa-  tion’s path at a decision node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Together, the d variables allow  us to follow an observation’s path through the tree which we  use to identify the leaf that it reaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Important in this for-  mulation, compared to existing binary encodings, is that it  requires no big-M constraints to describe these paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This  2In practice, the possible values for threshold k depends on the  chosen feature j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We do not model this dependence for convenience  of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  makes the relaxation stronger and therefore the solver give  much better bounds than using the big-M style formulations  from ROCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Modeling Policy Actions  Decision leaves only have one set of binary decision vari-  ables: ct,a encoding whether or not leaf t predicts action a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  We want each leaf to select exactly one action:  �  a  ct,a = 1,  ∀t  (5)  As mentioned before we can follow an observation’s path  through the tree by using their ds,m path variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' One can  linearize an implication of a conjunction of binary variables  as follows:  x1 ∧ x2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' ∧ xn =⇒ y  ≡ x1 + x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' + xn − n + 1 ≤ y  If an observation reaches leaf t and the leaf predicts action a  then we want to force the policy πs,a to take that action in the  associated state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Using the aforementioned equivalence we  add the constraint:  �  m∈Al(t)  (1−ds,m) +  �  m∈Ar(t)  ds,m  + ct,a − |A(t)| ≤ πs,a,  ∀s, a, t  (6)  This constraint forces the agent to take the action indicated by  the leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' To prevent the agent from taking other actions that  were not indicated we force it to only take a single action in  each state (giving a deterministic policy):  �  a  πs,a = 1,  ∀s  (7)  Now we have indicators πs,a that mark what action is taken  by the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' To link this back to the MDP linear program-  ming formulation that we use to optimize the policy, we set  the xs,a variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We need to set xs,a = 0 if πs,a = 0, else  xs,a should be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We encode this using a big-M  formulation:  xs,a ≤ Mπs,a,  ∀s, a  (8)  M should be chosen as small as possible, but larger or equal  to the largest value that xs,a can take.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We use the fact that we  are optimizing the MDP using discount factor γ to compute  an upper bound on xs,a and set M =  1  1−γ , proof is given in  the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='2  Complete Formulation  The runtime of MILP solvers grows worst-case exponentially  with respect to formulation size therefore it is important to  limit the scale of the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The number of variables in  our formulation grows with O(|S||J||TD|+|A||TL|+|S||A|)  which follows from their indices in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The number of  constraints grows with the order O(|S||TD| + |S||A||TL|) as  it is dominated by the constraints that determine ds,m at each  node (Equation 4) and constraints that force πs,a according to  the tree (Equation 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT’s complete MILP formulation  is summarized below:  max  �  s  �  a  xs,a  �  s′  Ps,s′,aRs,s′,a  (1)  Name  Kind Description  bm,j,k  bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Tree uses feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' j and threshold k in node m  ct,a  bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Tree selects action a in leaf t  ds,m  bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Observation of s goes left / right in node m  πs,a  bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Policy takes action a in state s  xs,a  cont.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Frequency of action a taken in state s  Ps,s′,a  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Probability of transition s→s′ with action a  Rs,s′,a  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Reward for transition s→s′ with action a  p0(s)  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Probability of starting in state s  γ  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Discount factor  Xij  const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Feature j’s value of observation i  side(s,j,k) const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Side state s is on for thresh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' k and feat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' j  a ∈ A  set  Set of actions in MDP  s ∈ S  set  Set of states in MDP  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='.|S| set  Observation and state indices  j ∈ J  set  Set of feature indices  k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='.K set  Indices of all possible feature thresholds  m ∈ TD  set  Set of decision nodes in the tree  t ∈ TL  set  Set of leaves in the tree  A(t)  set  Set of ancestors of leaf t  Al(t)  set  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' that have t in their left path  Ar(t)  set  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' that have t in their right path  Table 1: Summary of notation used in OMDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  �  a  xs,a −  �  s′  �  a  γPs′,s,axs′,a = p0(s),  ∀s  (2)  �  j  �  k  bm,j,k = 1,  ∀m  (3)  ds,m =  �  j  �  k  side(s, j, k) bm,j,k,  ∀s, m  (4)  �  a  ct,a = 1,  ∀t  (5)  �  m∈Al(t)  (1−ds,m) +  �  m∈Ar(t)  ds,m + ct,a−|A(t)| ≤ πs,a,  ∀s, a, t  (6)  �  a  πs,a = 1,  ∀s  (7)  xs,a ≤ Mπs,a,  ∀s, a  (8)  5  Results  We present experiments comparing the performance of  OMDTs with VIPER and dtcontrol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Viper uses imitation  learning to extract a size-limited decision tree from a teacher  policy and dtcontrol learns an unrestricted tree that exactly  copies the teacher’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' To provide a fair comparison  we have trained VIPER and dtcontrol with an optimal teacher  by first solving the MDP with value iteration and then ex-  tracting all Q values, both methods ran with default param-  eters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We also implemented and ran our experiments on in-  terpretable Differentiable Decision Trees [Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020]  but excluded these models from our analysis as they did not  outperform a random policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The full code for OMDT and  our experiments can be found on GitHub3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' All of our experi-  ments ran on a Linux machine with 16 Intel Xeon CPU cores  and 72 GB of RAM total and used Gurobi 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='0 with default  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Each method ran on a single CPU core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='1  Environments  For comparison we implemented 13 environments based on  well known MDPs from the literature, the sizes of these  MDPs are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' All MDPs were pre-processed  such that states that are unreachable from the initial states are  removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We briefly describe the environments below but re-  fer to the appendix for complete descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  In 3d navigation the agent controls a robots in a 5x5x5  world and attempts to reach from one corner to the other end  with each voxel having a chance to make the robot disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  blackjack is a simplified version of the famous casino game  where we assume an infinite sized-deck and only the actions  ‘skip’ or ‘hit’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' frozenlake is a grid world where the agent  attempts to go from the start state to the end state without  falling into holes, actions are stochastic so the agent will not  always move in the intended direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' the action ‘up’  will only surely not send the agent ‘down’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' inventory man-  agement models a company with limited inventory size that  has decide how many items to order to meet its customer’s  demand while minimizing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' system administrator refers  to a computer network where computers randomly crash and  an administrator has to decide which computer to spend time  on rebooting, a crashed computer has an increased probabil-  ity of crashing a neighboring computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' tictactoe vs random  is the well known game of tic-tac-toe when played against  an opponent that makes random moves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In tiger vs antelope  the tiger attempts to catch the antelope that randomly jumps  away from the tiger in a grid world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' traffic intersection de-  scribes a perpendicular intersection where traffic flows in at  different rates and the operator decides when to switch the  traffic lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' xor is an MDP constructed such that the states  randomly lie on a plain, the agent gets 1 reward for taking the  action according to an XOR function and -1 for a mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The XOR problem is notoriously difficult for greedy decision  tree learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='2  Performance-Interpretability Trade-off  It is often assumed that there is a trade-off in the performance  and interpretability of machine learning models [Gunning and  Aha, 2019] since interpretable models necessarily lack com-  plexity but this assumption is not always true [Rudin, 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  We aim to answer whether the performance-interpretability  trade-off occurs in a variety of MDPs by training size-limited  decision trees and comparing their performance to the opti-  mal solutions that were not restricted in complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We vi-  sualize the normalized return of depth 3 OMDTs and unre-  stricted dtcontrol trees in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Returns were normalized  such that 0 corresponds to the return of a random policy and  1 to an optimal one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Since small deterministic decision tree  policies are limited in the number of distinct actions that they  take an optimal tree can perform worse than a random policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Experiments were repeated 3 times and runs were limited to 2  3https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='com/tudelft-cda-lab/OMDT  normalized return  MILP  runtime (s)  MDP  |S|  |A|  VIPER  OMDT  5 mins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  OMDT  2 hrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  vars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  constrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  trees  VIPER  OMDT  optimal  3d navigation  125  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='00  2,528  7,890  1014  2,090 ±55  315 ±89  blackjack  533  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='52 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='03  10,850  33,819  1015  490 ±243  timeout  traffic intersection  361  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='98 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='99 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  4,127  9,762  1011  2,188 ±121  1,219 ±177  xor  200  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='34 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  5,016  5,415  1021  1,999 ±123  50 ±0  Table 2: Comparison of depth 3 trees trained with VIPER and OMDT on 13 MDPs, experiments were repeated 3 times and runs were limited  to 2 hours of runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT solves some MDPs in 5 minutes but significantly improves when given 2 hours of runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' While 2 hours  are enough for OMDT to achieve greater or equal scores to VIPER in most MDPs, OMDT needs more runtime to outperform VIPER on the  large tictactoe instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT was able to identify the optimal size-limited tree and prove its optimality in 7 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We consider an OMDT optimal when the relative gap  between its objective and bound is proven to be smaller than  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='01%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  While it is debatable what the precise size limits are for  decision trees to be interpretable [Lipton, 2018] we use trees  of depth 3 which implies that a tree has at most 8 leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We  find that in all environments, OMDTs of depth 3 improve on  the performance of random policies, and in 8 out of 13 en-  vironments the policy gets close to optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Decision trees  trained with dtcontrol always achieve the optimal normalized  return of 1 since they exactly mimic the optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' How-  ever, dtcontrol produces large trees that are not interpretable  to humans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' When run on 3d navigation for example, dtcon-  trol produces a tree of 68 decision nodes which is very com-  plex for humans to understand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT produces a tree of 7  decision nodes which performs equally well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Overall, our results demonstrate that for many environ-  ments there is no performance-interpretability trade-off: sim-  ple policies represented by size-limited trees perform approx-  imately as well as the unrestricted optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='3  Direct Optimization versus Imitation Learning  The above conclusion holds when the policy is learned to op-  timality under the constraint that it has to be a small tree, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=',  using OMDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Frequently used techniques such as VIPER do  not directly enforce this constraint but aim to imitate the unre-  stricted optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We now show that this comes at a cost  when the unrestricted policy is too complex to be represented  using a small tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  VIPER trains its decision trees by imitating high Q val-  ues of the optimal policies while OMDT directly maximizes  expected return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In Table 2 we list the normalized return (0  for random policies, 1 for optimal policies) for VIPER and  OMDT with respectively 5 minutes and 2 hours of runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  After 5 minutes OMDT improves performance over random  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  normalized return  3d_navigation  xor  blackjack  inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' management  traffic_intersection  frozenlake_4x4  frozenlake_8x8  sysadmin_1  sysadmin_tree  frozenlake_12x12  sysadmin_2  tiger_vs_antelope  tictactoe_vs_random  2  3  2  5  2  7  2  9  tree size (nodes)  omdt  dtcontrol  Figure 3: (top) Normalized return and bounds for OMDT trees of  depth 3, optimal policies score 1 while uniform random policies  score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' (bottom) Log of tree sizes for OMDT (maximum depth  3) and dtcontrol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Dtcontrol always produces an optimal policy but  the trees are orders of magnitude larger than OMDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  policies but often needs more time to improve over VIPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  After 2 hours OMDT’s policies win on 11 out 13 environ-  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' For instances with large state space such as tictac-  toe vs random OMDT needs more than 2 hours to improve  over VIPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Shortcomings of Imitation Learning  Overall, given sufficient run-time, OMDT produces better  policies than VIPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This cannot be easily solved by giving  VIPER more run-time but is an inherent problem of imitation  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' To illustrate this, we investigate the results on the  0  1  2  3  4  5  6  7  8  9  10 11  0  1  2  3  4  5  6  7  8  9  10  11  G  (a) Optimal: 92% success  0  1  2  3  4  5  6  7  8  9  10 11  0  1  2  3  4  5  6  7  8  9  10  11  G  (b) OMDT (depth 3): 66% success  0  1  2  3  4  5  6  7  8  9  10 11  0  1  2  3  4  5  6  7  8  9  10  11  G  (c) VIPER (depth 3): 11% success  Figure 4: Paths taken on 10,000 Frozenlake 12x12 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The agent starts at (0, 0) and attempts to reach the goal tile ‘G’ while avoiding holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Actions are indicated by arrows and are somewhat stochastic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' an action of ‘up’ will send the agent ‘left’, ‘up’ or ‘right’ (but never down)  with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' VIPER fails to produce a good policy because it spends capacity of its tree mimicking parts of the complex teacher  policy that its simple student policy will never reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT achieves a greater success rate by directly optimizing a simple policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  frozenlake environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Table 2 demonstrates that imita-  tion learning (VIPER) can perform far from optimal on these  environments, while direct optimization (OMDT) performs  closer to the unrestricted optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' It is insightful to  understand why this happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  In theory, imitation learning performs optimally in the  limit [Ross et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2011] but this result requires the student  policy to be as expressive as the teacher policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This is not  the case for size-limited decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' When VIPER learns  a policy for frozenlake 12x12 it tries to imitate a complex pol-  icy using a small tree that cannot represent all teacher policy  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This results in VIPER spending capacity of its deci-  sion tree on parts of the state space that will never be reached  under its student policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In Figure 4 we visualize the paths  that the agents took on 10,000 runs and indicate the policies  with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' VIPER creates leaves that control action in the  right section of the grid world (indicated in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The optimal  teacher policy often visits this section but the simple student  does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' By directly optimizing a decision tree policy using  OMDT, the policy spends its capacity on parts of the state  space that it actually reaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' As a result, VIPER cannot pre-  vent actions that send its agent into holes on the left part of the  grid world (indicated in red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT actively avoids these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='4  Runtime  Runtime for solving Mixed-Integer Linear Programming for-  mulations scales worst-case exponentially which makes it im-  portant to understand how solvers operate on complex formu-  lations such as OMDT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We solved OMDTs for a depth of 3  for a maximum of 2 hours and display the results in Table  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The table compares the runtimes of VIPER and solving  OMDT to optimality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' If the solver does not prove optimal-  ity within 2 hours we denote it as ‘timeout’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We also denote  the number of possible decision tree policies computed as:  |TB|possible splits × |TL||A| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' It provides an estimate of how many  decision tree policies are possible and shows that enumerat-  ing trees with brute force is intractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  OMDT solves a simple environment such as Frozenlake  4x4 (16 states, 4 actions) to optimality within 2 seconds  but runtime grows for larger environments such as inventory  management (101 states, 100 actions) which took on aver-  age 2533 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' VIPER needs roughly 2250 seconds of  runtime for every MDP and runs significantly faster on some  MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This is because VIPER spends much time evaluating  policies on the environment and some environments quickly  reach terminal states which results in short episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' While  OMDT was able to prove optimality on only 7 out of 13 en-  vironments within 2 hours, OMDT finds good policies before  this time on 12 out of 13 environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  6  Conclusion  We propose OMDT, a Mixed-Integer Linear Programming  formulation for training optimal size-limited decision trees  for Markov Decision Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Our results show that for  simple environments such as blackjack we do not have to  trade off interpretability for performance: OMDTs of depth  3 achieve near-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' On Frozenlake 12x12,  OMDT outperforms VIPER by more than 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  OMDT sets a foundation for extending supervised opti-  mal decision tree learning techniques for to the reinforcement  learning setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Still, OMDT requires a full specification of  the Markov Decision Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Imitation learning techniques  such as VIPER can instead also learn from a simulation envi-  ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Therefore, future work should focus on closing the  gap between the theoretical bound supplied by OMDT and  the practical performance achieved by algorithms that require  only simulation access to optimize interpretable decision tree  policies in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Additionally, future work  can incorporate factored MDPs into OMDT’s formulation to  scale up to larger state spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  References  [Aghaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2019] Sina Aghaei, Mohammad Javad Azizi,  and Phebe Vayanos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Learning optimal and fair decision  trees for non-discriminative decision-making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In Proceed-  ings of the AAAI Conference on Artificial Intelligence, vol-  ume 33, pages 1418–1426, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Aglin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020] Ga¨el Aglin, Siegfried Nijssen, and Pierre  Schaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Learning optimal decision trees using caching  branch-and-bound search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='  Mastering atari,  go, chess and shogi by planning with a learned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Nature, 588(7839):604–609, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Silva et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020] Andrew Silva, Matthew Gombolay, Tay-  lor Killian, Ivan Jimenez, and Sung-Hyun Son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Optimiza-  tion methods for interpretable differentiable decision trees  applied to reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In International confer-  ence on artificial intelligence and statistics, pages 1855–  1865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' PMLR, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Tange, 2021] Ole Tange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Gnu parallel 20210622 (’protase-  vich’), June 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' GNU Parallel is a general parallelizer  to run multiple serial command line programs in parallel  without changing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Topin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2021] Nicholay Topin, Stephanie Milani, Fei  Fang, and Manuela Veloso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Iterative bounding mdps:  Learning interpretable policies via non-interpretable meth-  ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In Proceedings of the AAAI Conference on Artificial  Intelligence, volume 35, pages 9923–9931, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Verhaeghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=', 2020] H´elene Verhaeghe, Siegfried Ni-  jssen, Gilles Pesant, Claude-Guy Quimper, and Pierre  Schaus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Learning optimal decision trees using constraint  programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Constraints, 25(3):226–250, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Verwer and Zhang, 2017] Sicco  Verwer  and  Yingqian  Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Learning decision trees with flexible constraints  and objectives using integer optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In International  Conference on AI and OR Techniques in Constraint  Programming for Combinatorial Optimization Problems,  pages 94–103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Verwer and Zhang, 2019] Sicco  Verwer  and  Yingqian  Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Learning optimal classification trees using a  binary linear program formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In Proceedings of the  AAAI conference on artificial intelligence, volume 33,  pages 1625–1632, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  [Vos and Verwer, 2022] Dani¨el Vos and Sicco Verwer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Ro-  bust optimal classification trees against adversarial exam-  ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In Proceedings of the AAAI Conference on Artificial  Intelligence, volume 36, pages 8520–8528, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  A  Environment Descriptions  We describe the environments in additional detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' All  environments in the paper were solved with a discount rate  of γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Such a value ensures that the solvers generate  policies that have good performance even for environments  that takes hundreds of steps to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' All MDPs were pre-  processed such that states that are unreachable from the initial  states are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='1  3d navigation  While policies in 2d scenarios are often easy to visualize and  interpret policies in higher dimensions are harder to under-  stand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Therefore we extend the 2d navigation problem from  ICAP 2011 IPPC competition4 to 3 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The agent  controls a robots in a 5x5x5 world and attempts to reach from  one corner to the other end with each voxel having a chance  to make the robot disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The disappearing chances were  independently sampled from a uniform distribution on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='2  blackjack  The version of blackjack that we used5 is a slightly simplified  version of the one that often appears in casinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Specifically,  we make the following assumptions:  There is no doubling down and splitting action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The player does not have knowledge about other players  at the board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Aces always account for a value of 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Cards are drawn from an infinitely sized deck, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' drawn  with replacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The player always starts in the state where no cards are  dealt, the dealer receives a random card, followed by the  player receiving a random card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The player can ’Draw’ cards  until they cross the total of 21 or until they ’Skip’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' When that  happens the dealer receives cards until their total is over 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  In the end, the reward is computed as follows:  -1 if the player has a lower total than the dealer or if the  player went over 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  0 if the player and dealer end up with the same total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  +1 if the player has a higher total than the dealer or if the  dealer went over 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='5 if the player gets to exactly 21  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='3  frozenlake  Frozenlake is a maze-like game played on a 2D grid where  the player attempts to move from the start state (S) to the goal  state (G) without falling into holes (H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The player walks over  frozen tiles (F) that make the actions stochastic:  The action ’Up’ sends a player in the directions left, up,  or right with probability 1/3 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The action ’Right’ sends a player in the directions up,  right, or down with probability 1/3 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  4http://users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='cecs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='anu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='au/∼ssanner/IPPC 2011/  5https://gist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='com/iiLaurens/  ba9c479e71ee4ceef816ad50b87d9ebd  The action ’Down’ sends a player in the directions right,  down, or left with probability 1/3 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The action ’Left’ sends a player in the directions down,  left, or up with probability 1/3 each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  We used the default 4x4 and 8x8 maps as defined by the  OpenAI gym6 and defined a new map for 12x12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The maps  are given below (in the same format that the environment  uses):  4x4  "SFFF",  "FHFH",  "FFFH",  "HFFG",  8x8  "SFFFFFFF",  "FFFFFFFF",  "FFFHFFFF",  "FFFFFHFF",  "FFFHFFFF",  "FHHFFFHF",  "FHFFHFHF",  "FFFHFFFG",  12x12  "SFFFFFFFFFFF",  "FFFFFFFFFFFF",  "FFFHFFFFFFFH",  "FFFFFHFFFFFF",  "FFFHFFFFFFFF",  "FHHFFFHFFHFF",  "FHFFHFHFFFFF",  "FFFHFFFFFFFF",  "FFFFFFFFHFFF",  "HFFFFHFFFFHH",  "FFFFFFGFFFFF",  "FFFFFFFFFFFF",  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='4  inventory management  In inventory management, the player is the owner of a shop  that stocks items and sells them to customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Our environ-  ment is adapted from the implementation by Paul Hendricks7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Every day the player can choose ‘Don’t buy’ or ‘Buy x’ items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  We used the following parameters for the problem:  The maximum inventory size is 100 (this creates 101  states since the inventory can have 0 up to and includ-  ing 100 items)  Ordering any items comes with a fixed cost of -10  Ordering items costs -2 per item  Holding an item in storage costs -1 per item  Selling an item grants the player +4  The number of customers follows a Poisson distribution  with expectation λ = 15 (customers per day)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='5  system administrator  The system administrator task refers to a computer network  where computers randomly crash and an administrator has to  decide which computer to spend time on rebooting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' This en-  vironment is based on one of the MDPs of the ICAPS 2011  IPPC competition4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' When a computer crashes it has an in-  creased probability in following time steps to crash a neigh-  boring computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The topology of the network can be varied  to create MDPs with various properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The 3 topologies  considered in this work are visualized in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The system administrator is then allowed to reboot one  machine at every time step or wait.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Rebooting a computer  is penalized with a reward of -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='45 but will always fix the  crashed computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' At each time step every computer that has  6https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='com/openai/gym/blob/master/gym/envs/toy text/  frozen lake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='py  7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='com/paulhendricks/gym-inventory/tree/master  (a) tree  (b) random 1  (c) random 2  Figure 5: System administrator topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  not crashed will reward the system administrator with +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' We  refer to a computer being on (1) or crashed (0) as the ‘status’,  the probability of being on in the next step is then given by:  ratio on neighbors × (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='05 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='9 × status)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='6  tictactoe vs random  The well known tic-tac-toe game is played on a 3x3 grid  where players take turns putting a cross / circle in an empty  square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The player that manages to put 3 of their symbols in a  row / column / diagonal wins the game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' While 2 player zero-  sum games are not MDPs, they become MDPs when we fix  the strategy of one of the players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In this case the opponent  plays moves uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' It is worth noting that this  turns the game that is normally deterministic into a stochastic  environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The player gets to start the game, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' they play  with the crosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='7  tiger vs antelope  The tiger vs antelope game is played on a 5x5 grid where the  agent controls the tiger and tries to catch an antelope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Every  turn the agent can move ‘up’, ‘down’, ‘left’ or ‘right’ and  if the antelope is not caught it responds by jumping in any  direction away from the player uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='8  traffic intersection  The traffic intersection scenario that we considered describes  a perpendicular intersection where traffic flows in at differ-  ent rates and the operator decides when to switch the traffic  lights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' At each side of the intersection 5 cars can fit in a row  and from the moment the traffic light switches we count up to  5 time steps in which one side of the intersection is waiting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  Cars arrive with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='1 per time step on one side of  the intersection and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='5 on the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The operator gets rewarded +1 for each car making it  through the intersection and gets penalized for flipping the  color of the traffic signs and for waiting cars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Flipping the  color of the traffic lights incurs a reward of -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' For each wait-  ing car the traffic operator gets a reward of −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='1 × 2wait time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='9  xor  The XOR problem is notoriously difficult for greedy decision  tree learning algorithms, we generate an MDP that mimics  the structure of the supervised learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Specifically  we draw 200 samples in 2 dimensions uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  The agent gets 1 reward for taking the action a according to  an XOR function and -1 for a mistake:  R(x, y, a) =  �1  round(x) + round(y)  mod 2 = a  −1  round(x) + round(y)  mod 2 = a  Each action sends the agent to a state uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  B  Linearizing Implications of Conjunctions  In the main text, we constructed one of OMDT’s constraints  using the fact that:  x1 ∧ x2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' ∧ xn =⇒ y ≡ x1 + x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' + xn − n + 1 ≤ y  We give a short proof of this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' First, we rewrite the  logical formula into conjunctive normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' In this case a  single disjunctive clause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  x1 ∧ x2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' ∧ xn =⇒ y  ≡ ¬(x1 ∧ x2 ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' ∧ xn) ∨ y  ≡ ¬x1 ∨ ¬x2 ∨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' ∨ ¬xn ∨ y  A disjunctive constraint a∨b∨c can be trivially expressed as  the linear constraint a + b + c ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Intuitively any variable a,  b or c needs to be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Now we rewrite to linear constraints:  ¬x1 ∨ ¬x2 ∨ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' ∨ ¬xn ∨ y  ≡ (1 − x1) + (1 − x2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' + (1 − xn) + y ≥ 1  ≡ x1 + x2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' + xn − n + 1 ≤ y  C  Big-M Value  For the constraints that force xs,a according to the policy πs,a  (Equation 8) we used a big-M formulation with M =  1  1−γ  as an upper bound on xs,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' To prove this upper bound we  construct an MDP in which we maximize xs,a then show that  this value equals our bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Consider an MDP with one state  s∗ and one action a∗ such that the agent always starts in that  state (p0(s∗) = 1) and action a∗ will always return to s∗  with probability Ps∗,s∗,a∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Clearly, this maximizes the  frequency xs∗,a∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Substituting into Equation 2 we find:  �  a  xs,a = p0(s) +  �  s′  �  a  γPs′,s,axs′,a  (2)  xs∗,a∗ = p0(s∗) + γxs∗,a∗ =  1  1 − γ  D  Performance at Varying Depths  In the main text we discussed performance of trees at a depth  of 3 since these trees are still easily interpretable but more  powerful than trees of depth 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Normalized returns after  2 hours of runtime for varying depth are presented in Table 3,  Table 4 shows the runtime needed to prove optimality of the  solutions for varying depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  E  Size-Limited Policies  We claim that since our models are interpretable since they  are limited in size to an extend that a human can easily follow  the predictions of the models [Lipton, 2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' The decision  trees of depth 3 created by OMDT and VIPER are visualized  for inspection in Figure 6 and Figure 7 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  32  5  33  0  5  6  4depth 1  depth 2  depth 3  depth 4  MDP  OMDT  VIPER  OMDT  VIPER  OMDT  VIPER  OMDT  VIPER  3d navigation  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='06 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='01 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  blackjack  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='98 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='98 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  frozenlake 4x4  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='19 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='96 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  frozenlake 8x8  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='74 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='95 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  frozenlake 12x12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='06 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='04 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='34 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='30 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='68 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='04  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='19 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='09  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='81 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='19 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='09  inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' management  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='99 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='98 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='99 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  sysadmin 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='84 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='83 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='89 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='86 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='03  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='92 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='88 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='91 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='92 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='01  sysadmin 2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='27 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='27 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='55 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='55 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='57 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='59 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='67 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='04  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='72 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  sysadmin tree  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='03 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='03 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='37 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='26 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='71 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='08  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='57 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='04  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='86 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='04  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='86 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  tictactoe vs random  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='06 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='06 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='61 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='61 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='43 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='18  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='80 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='01  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='68 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='10  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='83 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  tiger vs antelope  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='36 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='64 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='06  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='19 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='31 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='17  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='50 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='05  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='10 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='02  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='64 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='03  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='23 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='10  traffic intersection  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='50 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='42 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='98 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='98 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='98 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  xor  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='15 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='05 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='13 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='34 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='06  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='00  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='90 ± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='03  Table 3: Normalized returns of OMDT and VIPER when varying the maximum depth of the decision tree, runtime was limited to 2 hours  and experiments repeated with 3 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' For depths until 3 OMDT finds optimal or high quality trees within 2 hours while for depth 4  OMDT will need more runtime to improve on the scores produced by VIPER’s trees on 3 environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  MDP  depth 1  depth 2  depth 3  depth 4  3d navigation  9 ±0  258 ±27  315 ±89  223 ±25  blackjack  36 ±5  72 ±7  408 ±85  598 ±99  frozenlake 4x4  1 ±0  1 ±0  2 ±0  2 ±0  frozenlake 8x8  2 ±0  9 ±2  98 ±30  3134 ±350  frozenlake 12x12  10 ±2  349 ±57  timeout  timeout  inv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' management  453 ±42  6597 ±507  2533 ±540  3548 ±732  sysadmin 1  129 ±10  5059 ±542  timeout  timeout  sysadmin 2  488 ±53  timeout  timeout  timeout  sysadmin tree  74 ±10  4547 ±767  timeout  timeout  tictactoe vs random  2989 ±61  timeout  timeout  timeout  tiger vs antelope  1695 ±210  timeout  timeout  timeout  traffic intersection  33 ±1  104 ±28  1219 ±177  5595  xor  113 ±10  43 ±1  50 ±0  98 ±27  Table 4: Runtimes until OMDT proves optimality when varying the depth of the trees,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' runtime was limited to 2 hours and experiments repeated  with 3 random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' Proving optimality for deeper trees usually takes more runtime but since a deeper tree allows the tree’s objective value  to be closer to the bound runtime can reduce e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' in 3d navigation depth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content=' OMDT only proved optimality for traffic intersection depth 4 in  one of the runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  if z <= 2  if z <= 0  yes  if z <= 3  no  if x <= 1  if x <= 2  right  forward  right  forward  if y <= 3  if x <= 2  up  forward  up  right  (a) 3d navigation  if dealer_total <= 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if dealer_total <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  yes  if dealer_total <= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  no  if player_total <= 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if player_total <= 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  Draw  Skip  Draw  Skip  if player_total <= 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if player_total <= 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  Draw  Skip  Draw  Skip  (b) blackjack  if X <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if Y <= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  yes  if Y <= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  no  Left  if Y <= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  Up  Down  if X <= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if X <= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  Down  Left  Right  Down  (c) frozenlake 4x4  if X <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if X <= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  yes  if Y <= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  no  if Y <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if Y <= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  Right  Up  Right  Up  if Y <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if Y <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  Right  Down  Right  Left  (d) frozenlake 8x8  if X <= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if Y <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  yes  if X <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  no  if Y <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if X <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  Left  Down  Up  Right  if Y <= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  if Y <= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Down  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Right  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Down  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Left  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(e) frozenlake 12x12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='no  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 24  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 9  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 21  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content="Don't buy  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 7  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content="Don't buy  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(f) inventory management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_2_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='no  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='reboot_computer_0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='reboot_computer_4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='if computer_1_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='reboot_computer_2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_6_running <= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_6_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='wait  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(i) system administrator tree  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if bottom_right_cross <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if center_circle <= -1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if bottom_right_cross <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='no  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if bottom_center_free <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='bottom_right  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+page_content='bottom_right  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if bottom_center_free <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if bottom_center_free <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='bottom_center  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='bottom_right  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='bottom_left  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='bottom_center  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(j) tictactoe vs random  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if antelope_x <= 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if tiger_y <= 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if antelope_y <= 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='no  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if antelope_y <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if tiger_x <= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='down  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='up  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='down  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='left  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if tiger_y <= 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if tiger_x <= 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='right  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='down  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='wait  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='up  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(k) tiger vs antelope  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if green_side <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if cars_right <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if waiting_time <= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='no  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if waiting_time <= 2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if cars_left <= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='wait  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='switch_light  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='switch_light  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='wait  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if cars_left <= 1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if cars_left <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='wait  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='switch_light  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='wait  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='switch_light  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(l) traffic intersection  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if X <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='982  if X <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='497  yes  xor  no  if Y <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='501  if Y <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='490  not_xor  xor  xor  not_xor  (m) xor  Figure 6: Depth 3 trees produced by OMDT within 2 hours of runtime with one fixed seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='  if x <= 2  if x <= 1  yes  if y <= 3  no  right  if z <= 0  up  right  if z <= 2  if z <= 3  up  left  up  right  (a) 3d navigation  if player_total <= 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  Draw  yes  if player_total <= 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  no  if dealer_total <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  if player_total <= 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  Skip  Draw  Skip  Skip  (b) blackjack  if Y <= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  if X <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  yes  if X <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  no  Left  if Y <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  Up  Left  Up  if Y <= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  Down  Right  (c) frozenlake 4x4  if Y <= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  if X <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  yes  if X <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  no  if X <= 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  if Y <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  Up  Right  Right  Down  if Y <= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  Right  Right  Up  (d) frozenlake 8x8  if X <= 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  if Y <= 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  yes  if Y <= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='000  no  if X <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  if Y <= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  Left  Left  Down  Right  if Y <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  if Y <= 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='500  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Left  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Up  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Down  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Up  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(e) frozenlake 12x12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='no  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if inventory <= 16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content="Don't buy  " metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='Buy 6  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='(f) inventory management  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_6_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_0_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='yes  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_5_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='no  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_4_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='if computer_2_running <= 0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
+page_content='reboot_computer_5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kdFPT4oBgHgl3EQf2DV-/content/2301.13185v1.pdf'}
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+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+1
+DepthP+P: Metric Accurate Monocular
+Depth Estimation using Planar and Parallax
+Sadra Safadoust
+ssafadoust20@ku.edu.tr
+Fatma Güney
+fguney@ku.edu.tr
+KUIS AI Center
+Koç University
+Istanbul, Turkey
+Abstract
+Current self-supervised monocular depth estimation methods are mostly based on
+estimating a rigid-body motion representing camera motion. These methods suffer from
+the well-known scale ambiguity problem in their predictions. We propose DepthP+P, a
+method that learns to estimate outputs in metric scale by following the traditional planar
+parallax paradigm. We first align the two frames using a common ground plane which
+removes the effect of the rotation component in the camera motion. With two neural
+networks, we predict the depth and the camera translation, which is easier to predict
+alone compared to predicting it together with rotation. By assuming a known camera
+height, we can then calculate the induced 2D image motion of a 3D point and use it for
+reconstructing the target image in a self-supervised monocular approach. We perform
+experiments on the KITTI driving dataset and show that the planar parallax approach,
+which only needs to predict camera translation, can be a metrically accurate alternative
+to the current methods that rely on estimating 6DoF camera motion.
+1
+Introduction
+Understanding the 3D structure of a scene is fairly easy for human beings. We can easily
+reason about our surroundings and decompose them into different objects. Having this abil-
+ity is crucial for autonomous vehicles to be able to drive in different environments. Training
+deep networks for estimating depth has proven successful in computer vision research. How-
+ever, many such methods are supervised and require ground truth depth which is costly to
+achieve. Another line of work uses a stereo setup that must be carefully calibrated. Both
+of these approaches cannot use the vast amount of unlabeled videos that are easily available
+for training. On the other hand, self-supervised monocular depth estimation methods that do
+not rely on stereo supervision do not suffer from these limitations and, in practice, have been
+closing the gap with their supervised or stereo counterparts.
+Current self-supervised monocular depth estimation approaches all follow the same basic
+idea proposed in [37]. They use a pose network to estimate the ego-motion between a source
+frame and the target frame and a depth network to estimate the depth of the target image.
+These estimations can then be used to sample pixels from the source image to synthesize
+the target frame. The difference between the target frame and the synthesized can be used
+as the source of supervision for training the networks. In this paper, we propose another
+approach to synthesize the target image. Our approach, DepthP+P, illustrated in Fig. 1,
+uses the traditional planar parallax formulation [15, 25], which decomposes the motion into
+a planar homography and a residual parallax. Consider a plane in the scene and its motion
+arXiv:2301.02092v1  [cs.CV]  5 Jan 2023
+
+2
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+[
+h11    h12    h13
+h21    h22    h23
+h31    h32    h33]
+Depth
+Pose
+H=
+Is
+Iw
+D
+t
+It
+Figure 1: Overview of our Approach. Using the source image Is and the target image It,
+we first calculate the homography H that aligns the road plane across these two images. We
+then warp Is according to H and obtain the aligned image Iw. The aligned image Iw and the
+target image It are input to the pose network which estimates the camera translation t only.
+The depth network takes the It and produces a metric accurate depth map ˆD.
+represented by a homography from the source to the target image. By first warping the
+source image according to this homography, the motion of the plane is canceled. Then the
+residual image motion depends on two factors: (1) the deviations of the scene structure from
+the plane, i.e. the depth of points and their perpendicular distance to the plane and (2) only
+the translational motion of the camera. Autonomous driving is a perfect use case for this
+approach because there is typically a planar surface in front of the vehicle, i.e. the road.
+However, it is important to note that the plane in the planar parallax formulation does not
+necessarily have to be a real plane and can also be a virtual plane, but choosing the road as
+the planar surface makes it easier to implement in practice. Moreover, our approach does not
+rely on the availability of a plane to predict depth during inference.
+In this approach, we first align the road plane between the source and target images.
+This is achieved by calculating the homography between the road regions in two frames and
+then warping the source frame according to the homography to obtain the aligned image.
+By doing so, the road regions in the aligned image and target image match. The residual
+motion between the aligned image and the target image can be explained as follows: We
+first estimate the depth of each pixel with a monocular depth network and back-project them
+into 3D. Then using a known camera height, we can calculate the perpendicular distance of
+each point to the road. In addition, we estimate the translation between the camera origins.
+Note that this is different from the typical monocular depth approach [10, 37] which needs
+to estimate both the rotation and translation components. Finally, the target image can be
+synthesized from the aligned image using the calculated residual parallax as shown in Fig. 2.
+The planar parallax approach for self-supervised monocular estimations has a number
+of advantages over the previous paradigm. Firstly, it is much easier to optimize because it
+removes the ambiguities associated with predicting rotational camera motion [14]. Secondly,
+it can produce metric accurate outputs. Previous monocular depth methods can estimate
+depth and motion up to a scale. Typically, during inference, ground truth depth data is used
+to scale the predicted depth values such that the median of the predicted depth is equal to
+that of ground truth depth [37]. Our approach is able to predict metric accurate depth without
+needing ground truth depth data by only assuming a known camera height.
+
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+3
+2
+Related Work
+2.1
+Self-Supervised Monocular Depth
+View Synthesis:
+Garg et al. [6] were the first to propose a method that uses view syn-
+thesis as an objective for depth estimation from single images. Monodepth [9] uses Spatial
+Transformer Networks (STNs) [16] to synthesize the images in a fully-differentiable way.
+SfmLearner [37] generalizes view synthesis to temporally consecutive images by using an-
+other network to predict the relative pose between them. Zhan et al. [36] use stereo sequences
+to perform view synthesis using temporally consecutive pairs as well as the left-right pairs,
+enabling them to benefit from both monocular and stereo supervision. In addition to image
+reconstruction, they also use feature reconstruction as supervision. Similarly, by going be-
+yond pixel-wise reconstruction error, Mahjourian et al. [19] propose to use a 3D point cloud
+alignment loss to enforce the estimated point clouds and the camera pose to be consistent
+temporally. Wang et al. [29] use direct visual odometry in a differentiable manner to solve
+for ego-motion using the estimated depth.
+In addition to depth and camera pose, several methods estimate optical flow for residual
+motion. After predicting the camera motion, GeoNet [35] estimates the remaining object
+motion using optical flow. In order to prevent the errors of camera pose or depth predictions
+from propagating to flow estimations, DF-Net [39] enforces consistency between optical flow
+and the flow induced by the depth and pose predictions. GLNet [4] uses epipolar constraint
+for optical flow, along with other geometric constraints, further improving the performance.
+EPC++ [18] proposes a holistic 3D motion parser that uses predicted depth, pose, and op-
+tical flow to estimate segmentation masks for dynamic objects and their motion as well as
+background motion. Ranjan et al. [20] jointly train networks for depth, pose, optical flow,
+and motion segmentation so that they can use geometric constraints on the static regions and
+generic optical flow on moving objects. MonoDepthSeg [24] proposes to jointly estimate
+depth, independently moving regions, and their motion with an efficient architecture.
+Some approaches keep the original framework with a depth and a pose network but im-
+prove the performance with better loss functions, improved network architectures, and inno-
+vative design choices. When estimating depth at multiple scales, Monodepth2 [10] proposes
+to first upsample the estimated low-scale depths to the input image size and then calculate
+the photometric loss at that scale. Monodepth2 also proposes to calculate the minimum of
+reprojection errors per pixel instead of averaging them when synthesizing the target image
+from multiple views to prevent blurry depth estimations. PackNet [11] changes the architec-
+ture of the depth network and uses 3D convolutions to learn to preserve spatial information
+using symmetrical 3D packing and unpacking blocks for predicting depth.
+Scale Ambiguity:
+Self-supervised monocular depth estimation models suffer from the
+scale ambiguity problem, and the depth and pose outputs of such models are in an unknown
+scale. The median scaling technique used by many previous methods does not actually
+solve this problem because it relies on ground truth depth data during inference which is not
+always easily available. Bian et al. [2] introduce a loss to minimize normalized differences of
+depth maps across the entire sequence. This makes the estimations globally scale-consistent.
+However, although this means that the predictions are at the same scale, that specific scale is
+still unknown, and the median scaling is still required during evaluation.
+There are a number of monocular methods that can output depth estimations in absolute
+scale. Roussel et al. [22] use a network that was pre-trained with stereo pairs on a dataset
+
+4
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+and finetunes it on another dataset while maintaining the metric scale. Guizilini et al. [11]
+propose a version of their PackNet that uses ground truth camera velocity and the timestamps
+of images to enforce the estimations to be metrically accurate. Bartoccioni et al. [1] supervise
+their depth predictions with a sparse LiDAR. However, all of these approaches rely on ground
+truth data from extra sensors during training.
+There are a few other methods that do not require additional supervision and only use the
+camera height to achieve depth estimations in metric units similar to the proposed method.
+DNet [32] estimates the ground plane during inference and, using the real height of the
+camera, recovers the scale of the predictions. However, it needs a ground plane to be visible
+during the test time. In other words, they do not train their depth outputs to be in absolute
+scale. Rather, they recover the scale of the estimations with another module during test time.
+Wagstaff and Kelly [28] train a network that learns the metric scale during training using
+camera height. They introduce a plane segmentation network and propose a three-staged
+training procedure for training the depth estimation model in metric scale. First, they train
+an unscaled depth network and then use it to train the plane segmentation network. Finally,
+they train a new metrically accurate depth network using the pre-trained plane segmentation
+network. Similar to [28], we also learn the metric scale during training, but we do not need
+a multi-stage process, nor do we rely on the existence of a ground plane during inference,
+differently from previous work [32].
+2.2
+Planar Parallax
+The Planar Parallax paradigm, also called Plane + Parallax (P+P), has been used to under-
+stand the 3D structure of a scene from multiple images by decomposing the motion into a
+planar homography and a residual parallax. Sawhney [25] proposes a formulation for the
+residual parallax that uses depth and distance to the plane. Irani et al. [15] use this for-
+mulation to derive a rigidity constraint between pairs of points over multiple images. Irani
+et al. [13] derive trifocal constraints and use them to propose a simple method for new view
+synthesis. In a follow-up work [14], they extend the planar parallax method to more than
+two uncalibrated frames.
+More recently, MR-Flow [31] uses P+P to refine the optical flow estimations with rigidity
+constraints. Chaney et al. [3] use P+P to estimate the height of points in the scene with event-
+based cameras. We propose a method to use the P+P formulation within the view synthesis
+framework for self-supervised monocular depth estimation.
+3
+Methodology
+Despite the success of current self-supervised monocular depth estimation approaches, they
+suffer from scale ambiguity. i.e. the estimated depth values are in an unknown scale. There-
+fore, in order to evaluate and compare these methods, they are usually normalized using the
+median scaling approach [37]. Here, we propose an approach that predicts depth maps in
+metric scale without using any ground truth depth supervision.
+3.1
+DepthP+P
+Our approach is based on the Planar Parallax decomposition which has been studied in detail
+before [15, 25]. We first introduce it here to establish our notation and then build our method
+
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+5
+x
+p
+p′
+pw
+PW
+H
+Π
+It
+Is
+C
+C′
+C
+x
+h
+NTx
+dc
+x
+z
+y
+Π
+Z
+N
+Figure 2: Visualization of the Planar Parallax. Left: The 3D point x is projected to points
+p and p′ on the target image It and the source image Is respectively. Using the homography
+H induced by the plane Π, the point p′ will be transformed to point pw on the target image.
+Right: Calculating h, distance of x to the Π using the camera height dc and the normal vector
+N of the plane. C and C′ are the camera centers of It and Is and Z is the depth of the point.
+to predict depth following that notation.
+Notation:
+Let Π be a 3D plane and H be the homography aligning Π between the target
+image It and the source image Is. Let p and p′ be the images of the 3D point x = [X,Y,Z]T on
+the It and Is respectively. As shown on the left in Fig. 2, we can warp p′ by the homography
+H and obtain the image point pw:
+pw ∼ Hp′
+(1)
+where we omit the conversion to the homogenous coordinates. Note that by warping the
+source image Is, we obtain the aligned image Iw such that the plane Π matches between
+them. The displacement between pw and p can be computed as follows:
+pw −p =
+γ
+dc −γtz
+(tzp−Kt)
+(2)
+where K is the camera intrinsic, t = [tx,ty,tz]T is the translation vector between the It and Is,
+and dc is the distance between the camera for the source view to the plane Π. The structure
+is represented by γ = h
+Z where h is the distance of x to Π. Note that when x lies on the plane
+Π, i.e. h = 0, we will have pw = p.
+DepthP+P: Following the typical self-supervised monocular depth approach, our frame-
+work has two networks, one for estimating depth and another for estimating the translation
+between frames. Note that, unlike other methods, we do not need to estimate the rotation
+between the two views. Precisely, our pose network takes the source and images Is,Iw and
+outputs the translation vector t. The depth network takes the target image It and outputs the
+depth map ˆD for It.
+For every pixel p = [x,y], let ˆD(p) denote its estimated depth. We backproject p using
+the camera intrinsics and the estimated depth to obtain the corresponding 3D point ˆx in the
+camera coordinate system as follows:
+ˆx = ˆD(p) K−1 [x,y,1]T.
+(3)
+
+6
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+Therefore, as demonstrated on the right in Fig. 2, we have the following:
+ˆh = dc −NT ˆx,
+ˆγ =
+ˆh
+ˆD(p)
+(4)
+where N is the normal vector of the plane Π, ˆh is the estimated distance of the point ˆx to
+the plane Π and ˆγ is our estimate of the structure variable γ. As a result, we obtain all the
+parameters required to use Eq. (2) to reconstruct the target image It by warping the aligned
+image Iw resulting in ˆIw. In other words, for each pixel p on the It, we calculate the pw using
+(2) according to the depth and translation predicted by our two networks and then inverse
+warp Iw and obtain ˆIw to reconstruct It:
+It(p) ≈ ˆIw(p) = Iw(pw)
+(5)
+We minimize the difference between It and ˆIw for supervision as explained in the Section 3.2.
+In order to obtain the aligned images, we perform a pre-processing step on the dataset.
+We calculate a homography for every consecutive frame by using the road as the plane Π and
+warp the frames according to the calculated homographies. In other words, we calculate a
+homography H for every target image It and source image Is pair, and then warp Is according
+to H to obtain the warped source image Iw. We explain the details of this pre-processing step
+in Section 4.1.
+3.2
+Self-Supervised Training Loss
+In our approach, we define our photometric loss function as the linear combination of the
+L1 distance and the structural similarity (SSIM) [30] to minimize the difference between the
+target image It and the reconstructed image ˆIw. Our photometric loss is therefore defined as
+follows:
+Lphoto(p) = (1−α)|It(p)− ˆIw(p)|+ α
+2
+�
+1−SSIM(It,ˆIw)(p)
+�
+(6)
+where we set α = 0.85. Note that for every target image we consider two aligned images.
+One from warping the previous frame, and one from warping the next frame. We use the
+per-pixel minimum reprojection error introduced in [10] and calculate the minimum of the
+Lphoto for each pixel across the previous and next aligned images. We also define Lsmooth
+as an edge-aware smoothness loss over the mean-normalized inverse depth estimates [29] to
+encourage the depth predictions to be locally smooth. Our total loss function is a combination
+of Lsmooth and Lphoto averaged over all N pixels:
+L = 1
+N ∑
+p
+λ Lsmooth(p)+min
+w (Lphoto(p))
+(7)
+where min
+w
+calculates the minimum over the previous and next aligned frames and λ is a
+hyperparameter controlling the effect of the loss terms.
+3.3
+Network Architecture
+Our depth network is based on the U-Net architecture [21]. We use a ResNet [12] pre-trained
+on the ImageNet [23] as the encoder for our depth network, and for the decoder we use the
+architecture similar to the one used by [10]. The difference is that we directly estimate depth
+
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+7
+by multiplying the output of the last sigmoid layer by 250, which is the maximum depth
+value that can be predicted, instead of estimating disparity as in [10]. The depth network
+takes as input a single target image It and outputs the per-pixel depth estimates. Note that
+the output of our depth decoder is in metric scale.
+In DepthP+P, our second network takes Iw and It and outputs only the translation vector
+between the views. The network is similar to the pose network proposed in [10], except that
+the output is a 3-element vector representing the translation and is metric scale in our case.
+4
+Experiments
+4.1
+Dataset
+KITTI: We use the Eigen split [5] of the KITTI dataset [7, 8] to train and evaluate our model.
+We use all of the images in the split for which we could accurately estimate the homography
+for aligning the road between consecutive images as explained in the next paragraph. This
+results in 45000 training and 1769 validation samples. We evaluate our model on the 697 test
+images in the split using the original ground truth provided by LiDAR. We also report results
+using the improved ground truth for 652 test images provided by Uhrig et al [27]. They use
+a stereo-reconstruction method to remove the outliers in LiDAR points and increase the
+ground truth density by accumulating laser scans which result in high-quality ground truth
+data. The camera height in this dataset is dc = 1.65 and we assume that the road is completely
+horizontal, i.e. N = [0,1,0]T.
+Pre-processing the dataset for DepthP+P: In order to use our P+P approach, we need to
+calculate the homography between the consecutive frames and warp the source frame accord-
+ing to the estimated homographies. Since we work on the driving scenarios on KITTI, we
+choose the “road” as our plane Π which is visible in most of the frames. For calculating the
+homography, we need to find a set of (at least 4) corresponding pairs of road pixels between
+a source view Is and the target view It, i.e. two consecutive images. For this purpose, we use
+the optical flow between Is and It using [26] to find the corresponding pixels. We then use
+[38] to select only the pixels that belong to the semantic class “road”. Using the correspond-
+ing pairs of road pixels, we estimate the homography H using OpenCV’s RANSAC-based
+robust method. We do this to find the homography H for all of the consecutive pairs of
+frames on KITTI. Note that for any consecutive pair of frames I1,I2 and the homography H
+between them, we use H to warp I1 towards I2 and also use H−1 to warp I2 towards I1.
+4.2
+Depth Estimation Results
+In Table 1, we report the depth estimation results of our method on the KITTI Eigen split
+using both the original and the improved ground truth. To the best of our knowledge, this
+is the first time that a deep learning model has been trained with view synthesis through the
+planar parallax paradigm (Eq. (2)). All of the previous methods are trained based on estimat-
+ing the pose whereas our method introduces a novel approach. We can see that our method
+achieves significantly better results than the initial models by predicting the pose and depth.
+After the initial proposal of SfMLearner by Zhou et al. [37], several improvements have been
+proposed to improve its performance. Therefore, we believe that similar improvements can
+
+8
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+Method
+Scale
+Lower Better
+Higher Better
+Abs Rel
+Sq Rel
+RMSE
+RMSElog
+δ < 1.25
+δ < 1.252
+δ < 1.253
+Original Ground Truth
+Zhou et al. [37]
+
+0.183
+1.595
+6.709
+0.270
+0.734
+0.902
+0.959
+Yang et al. [34]
+
+0.182
+1.481
+6.501
+0.267
+0.725
+0.906
+0.963
+Mahjourian et al. [19]
+
+0.163
+1.240
+6.220
+0.250
+0.762
+0.916
+0.968
+Yin et al. [35]
+
+0.149
+1.060
+5.567
+2.226
+0.796
+0.935
+0.975
+Wang et al. [29]
+
+0.151
+1.257
+5.583
+0.228
+0.810
+0.936
+0.974
+Zou et al. [39]
+
+0.150
+1.124
+5.507
+0.223
+0.806
+0.933
+0.973
+Yang et al. [33]
+
+0.162
+1.352
+6.276
+0.252
+-
+-
+-
+Ranjan et al. [20]
+
+0.148
+1.149
+5.464
+0.226
+0.815
+0.935
+0.973
+Luo et al. [18]
+
+0.141
+1.029
+5.350
+0.216
+0.816
+0.941
+0.976
+Chen et al. [4]
+
+0.135
+1.070
+5.230
+0.210
+0.841
+0.948
+0.980
+Godard et al. [10]
+
+0.110
+0.831
+4.642
+0.187
+0.883
+0.962
+0.982
+Guizilini et al. [11]
+
+0.111
+0.785
+4.601
+0.189
+0.878
+0.960
+0.982
+Safadoust et al. [24]
+
+0.110
+0.792
+4.700
+0.189
+0.881
+0.960
+0.982
+Xue et al. [32]
+
+0.118
+0.925
+4.918
+0.199
+0.862
+0.953
+0.979
+Wagstaff and Kelly [28]
+
+0.123
+0.996
+5.253
+0.213
+0.840
+0.947
+0.978
+DepthP+P (Ours)
+
+0.152
+1.322
+6.185
+0.239
+0.781
+0.920
+0.970
+Improved GT [27]
+Zhou et al. [37]
+
+0.176
+1.532
+6.129
+0.244
+0.758
+0.921
+0.971
+Mahjourian et al. [19]
+
+0.134
+0.983
+5.501
+0.203
+0.827
+0.944
+0.981
+Yin et al. [35]
+
+0.132
+0.994
+5.240
+0.193
+0.833
+0.953
+0.985
+Wang et al. [29]
+
+0.126
+0.866
+4.932
+0.185
+0.851
+0.958
+0.986
+Ranjan et al. [20]
+
+0.123
+0.881
+4.834
+0.181
+0.860
+0.959
+0.985
+Luo et al. [18]
+
+0.120
+0.789
+4.755
+0.177
+0.856
+0.961
+0.987
+Godard et al. [10]
+
+0.085
+0.468
+3.672
+0.128
+0.921
+0.985
+0.995
+Safadoust et al. [24]
+
+0.085
+0.458
+3.779
+0.131
+0.919
+0.985
+0.996
+Guizilini et al. [11]
+
+0.078
+0.420
+3.485
+0.121
+0.931
+0.986
+0.996
+DepthP+P (Ours)
+
+0.134
+1.042
+5.566
+0.199
+0.820
+0.946
+0.983
+Table 1: Quantitative Results for Monocular Training on KITTI. This table compares
+our proposed approach, DepthP+P, to previous approaches on the KITTI dataset that were
+trained only with monocular supervision. The scale column specifies whether the method
+can estimate depth in metric scale. We provide results with the original and improved ground
+truth. We show the results for the input resolution 640 × 192. The best method in each
+column is shown in bold and the second best is underlined.
+follow our model as future work to make it perform better than our initial proposal as well
+as the other state-of-the-art models that are trained to estimate the full pose.
+Note that [32] is not trained to estimate metrically accurate depth. Instead, its depth
+network outputs depth in an unknown scale, and then during inference, it needs a ground
+plane to be visible on the image to recover the scale of the network. When the ground
+plane is not visible on the image, [32] fails completely as shown in Fig. 3. As can be seen
+in this figure, this image from the KITTI dataset does not have a ground plane, and [32]
+cannot recover the scale and produces completely wrong estimates. While our method needs
+a ground plane during training, it does not rely on the availability of the ground plane during
+inference, therefore it can still perform well. For reference, the absolute relative (Abs Rel)
+error of [32] on Fig. 3 is 1.178, while our model achieves a 0.252 error. [28] achieves better
+results by using a pre-trained plane segmentation network in addition to the depth network,
+while our approach can achieve comparable results without a separate segmentation network.
+DepthP+P can also be trained with additional stereo supervision. In the proposed ap-
+proach, we obtain monocular supervision from the P+P paradigm. In addition, using the
+known camera baseline and the estimated depth, we can warp the other image in the stereo
+setup to the input image for additional supervision signal. In Table 2, we report the per-
+
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+9
+Method
+Lower Better
+Higher Better
+Abs Rel
+Sq Rel
+RMSE
+RMSElog
+δ < 1.25
+δ < 1.252
+δ < 1.253
+Original GT
+Li et al. [17]
+0.183
+1.730
+6.570
+0.268
+-
+-
+-
+Zhan et al. [36]
+0.135
+1.132
+5.585
+0.229
+0.820
+0.933
+0.971
+Luo et al. [18]
+0.128
+0.935
+5.011
+0.209
+0.831
+0.945
+0.979
+Godard et al. [10]
+0.106
+0.818
+4.750
+0.196
+0.874
+0.957
+0.979
+DepthP+P (ResNet18)
+0.110
+0.907
+4.888
+0.199
+0.867
+0.954
+0.979
+DepthP+P (ResNet50)
+0.106
+0.900
+4.828
+0.198
+0.871
+0.954
+0.979
+Improved GT
+Zhan et al. [36]
+0.130
+1.520
+5.184
+0.205
+0.859
+0.955
+0.981
+Luo et al. [18]
+0.123
+0.754
+4.453
+0.172
+0.863
+0.964
+0.989
+Godard et al. [10]
+0.080
+0.466
+3.681
+0.127
+0.926
+0.985
+0.995
+DepthP+P (ResNet18)
+0.088
+0.572
+3.905
+0.138
+0.911
+0.981
+0.994
+DepthP+P (ResNet50)
+0.084
+0.543
+3.784
+0.134
+0.916
+0.982
+0.995
+Table 2: Quantitative Results on KITTI with Additional Stereo Supervision. We com-
+pare DepthP+P to previous approaches that use additional stereo supervision on KITTI.
+Stereo supervision significantly improves the results of DepthP+P model.
+By using
+ResNet50, our model performs on par with Monodepth2 [10].
+Figure 3: Qualitative Comparison. We compare the absolute relative error of our depth
+estimation method (left) with DNet [32] (right) on an image from the KITTI dataset without
+a ground plane. The colorbar on the right shows the values of the absolute relative error
+metric. We cap the max error at the value of 1.0 for visualization. We can see that [32]
+completely fails to estimate the metric depth due to the wrong scale recovery because there
+is no ground plane in the image, while our model does not have this issue and can still
+perform well. The absolute relative error for [32] is 1.178 while it is 0.252 for our method.
+formances of the methods that also use stereo supervision for training. Using stereo super-
+vision significantly improves the performance of our DepthP+P model, outperforming all
+methods except for Monodepth2 [10]. We show that by using a ResNet50 backbone instead
+of ResNet18, DepthP+P can obtain comparable results to Monodepth2 [10].
+5
+Conclusion and Future Work
+In this paper, we presented a new approach to self-supervised monocular depth estimation
+following the traditional planar parallax paradigm. We showed that our approach is able to
+produce metrically accurate depth estimates by using a known camera height. Unlike pre-
+vious methods that rely on estimating the full rigid-body motion of the camera, our method
+only needs to estimate the camera translation. We discussed the advantage of our method
+compared to the other scale-aware depth prediction methods. We see our approach as a first
+step to unlocking the potential of the plane and parallax for efficient and metric-accurate
+depth estimation. An exciting future direction can focus on detecting moving foreground
+objects by checking the violations in the plane and parallax constraints [15].
+
+1.0
+0.8
+0.6
+0.4
+0.2
+0.010
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+References
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+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+13
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+Computer Vision (ECCV), pages 36–53, 2018.
+
+14
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+Supplementary Material for
+DepthP+P: Metric Accurate Monocular
+Depth Estimation using Planar and Parallax
+Sadra Safadoust
+ssafadoust20@ku.edu.tr
+Fatma Güney
+fguney@ku.edu.tr
+KUIS AI Center
+Koç University
+Istanbul, Turkey
+In Section A of this supplementary document, we provide the derivation of the residual
+parallax (Equation 2 in the main paper) [15]. In Section B, we investigate the effect of using
+a more accurate estimation of the normal vector of the road. Finally, in Section C, we provide
+additional qualitative results of our model.
+A
+Derivation
+Let C and C′ be the camera centers of the target view It and the source view Is, respectively.
+Also, let x = [X,Y,Z]T and x′ = [X′,Y′,Z′]T be the coordinates of a 3D point with respect
+to C and C′ respectively. We can relate x and x′ as follows:
+x = Rx′ +t
+(8)
+where R is the rotation and t = [tx,ty,tz]T is the translation between C and C′. According to
+the Fig 2 (right), we can calculate h, the perpendicular distance of the 3D point to the plane
+Π as:
+h = dc −NTx′
+(9)
+Note that dc is the height of C′, the camera of the source view Is. However, h is invariant
+with respect to the cameras. We can rewrite the above equation as:
+1 = h+NTx′
+dc
+(10)
+Therefore, by substituting this into Eq. (8), we have:
+x = Rx′ +t
+�h+NTx′
+dc
+�
+=
+�
+R+ tNT
+dc
+�
+x′ + h
+dc
+t
+(11)
+Let K and K′ denote the camera intrinsic for the target view and the source view. Then
+p = [x,y,1]T = 1
+ZKx and p′ = [x′,y′,1] = 1
+Z′ K′x′ represent the pixel coordinates of our 3D
+point in the target and source view, respectively. Therefore Eq. (11) can be written as:
+ZK−1p =
+�
+R+ tNT
+dc
+�
+Z′K′−1p′ + h
+dc
+t
+(12)
+
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+15
+By multiplying both sides by 1
+Z′ K we obtain:
+Z
+Z′ p = K
+�
+R+ tNT
+dc
+�
+K′−1
+�
+��
+�
+H
+p′ +
+h
+Z′dc
+Kt
+(13)
+Z
+Z′ p = Hp′ +
+h
+Z′dc
+Kt
+(14)
+where H = K
+�
+R+ tNT
+dc
+�
+K′−1 is the 3 × 3 homography matrix associated with the plane Π
+between the source and the target view. Note that in Eq. (14), the third component of the
+vector in two sides of the equation should be equal. Therefore:
+Z
+Z′ = H3p′ +
+h
+Z′dc
+K3t
+(15)
+where H3 and K3 are the third row of H and K. Since K3t = tz, we therefore have:
+Z
+Z′ = H3p′ +
+h
+Z′dc
+tz
+(16)
+By dividing each side of Eq. (14) by each side of Eq. (16) we obtain:
+p =
+Hp′ +
+h
+Z′dc Kt
+H3p′ +
+h
+Z′dc tz
+(17)
+Adding and subtracting Hp′
+H3p′ to the right-hand side yields:
+p = Hp′
+H3p′ − Hp′
+H3p′ +
+Hp′ +
+h
+Z′dc Kt
+H3p′ +
+h
+Z′dc tz
+(18)
+= Hp′
+H3p′ −
+Hp′ �
+H3p′ +
+h
+Z′dc tz
+�
+H3p′
+�
+H3p′ +
+h
+Z′dc tz
+� +
+H3p′ �
+Hp′ +
+h
+Z′dc Kt
+�
+H3p′
+�
+H3p′ +
+h
+Z′dc tz
+�
+(19)
+= Hp′
+H3p′ −
+Hp′ �
+h
+Z′dc tz
+�
+H3p′
+�
+H3p′ +
+h
+Z′dc tz
+� +
+H3p′ �
+h
+Z′dc Kt
+�
+H3p′
+�
+H3p′ +
+h
+Z′dc tz
+�
+(20)
+Substituting Eq. (16) into the denominators of Eq. (20) results in:
+p = Hp′
+H3p′ −
+Hp′ �
+h
+Z′dc tz
+�
+H3p′ � Z
+Z′
+�
++
+H3p′ �
+h
+Z′dc Kt
+�
+H3p′ � Z
+Z′
+�
+(21)
+= Hp′
+H3p′ − htz
+Zdc
+Hp′
+H3p′ + h
+Zdc
+Kt
+(22)
+The point pw = [xw,yw,1]T = Hp′
+H3p′ is the point p′ transformed by the homography matrix H.
+Thus, Eq. (22) can be simplified to:
+p = pw − htz
+Zdc
+pw + h
+Zdc
+Kt
+(23)
+
+16
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+Subtracting htz
+Zdc p from both sides, we obtain:
+p− htz
+Zdc
+p = pw − htz
+Zdc
+pw + h
+Zdc
+Kt− htz
+Zdc
+p
+(24)
+p
+�
+1− htz
+Zdc
+�
+= pw
+�
+1− htz
+Zdc
+�
++ h
+Zdc
+Kt− htz
+Zdc
+p
+(25)
+By dividing both sides by 1− htz
+Zdc we get:
+p = pw +
+h
+Zdc Kt− htz
+Zdc p
+1− htz
+Zdc
+(26)
+By rearranging the terms and defining γ = h
+Z we obtain:
+pw −p =
+htz
+Zdc p−
+h
+Zdc Kt
+1− htz
+Zdc
+(27)
+=
+htz
+Z p− h
+ZKt
+dc − htz
+Z
+(28)
+= γtzp−γKt
+dc −γtz
+(29)
+=
+γ
+dc −γtz
+(tzp−Kt)
+(30)
+B
+Ablation Study on Normal Vector
+In our experiments in the paper, we have assumed that the road plane is horizontal with
+respect to the camera. In other words, we have assumed that N = [0,1,0]T. However, the road
+planes are not always completely flat and can be titled. For example, one side can be higher
+than the other. Or consider an uphill where the road is sloping upwards. To analyze this, we
+perform an experiment where we calculate the normal vector of the road using ground truth
+depth. Concretely, during training, we back-project the road pixels to 3D using their ground
+truth depth and fit a plane to the obtained 3D points. We then use the normal vector of the
+fitted plane as our vector N in Equation 4 in the main paper. We report the results in Table 3.
+Method
+Abs Rel
+Sq Rel
+RMSE
+RMSElog
+δ < 1.25
+δ < 1.252
+δ < 1.253
+DepthP+P (Fixed Normal)
+0.152
+1.322
+6.185
+0.239
+0.781
+0.920
+0.970
+DepthP+P (GT Normal)
+0.142
+1.127
+5.922
+0.238
+0.780
+0.921
+0.971
+Table 3: Effect of Calculating Normal Vector of Road. This table analyzes the effect
+of accurately predicting the normal vector N of the road on the performance. Fixed Normal
+assumes that the ground is perfectly horizontal. GT Normal uses the ground truth depth data,
+generates the point clouds for the road region, fits a plane and calculates the normal vector.
+GT Normal performs better than Fixed Normal, highlighting the fact that a more accurate
+normal vector calculation will improve the performance.
+
+SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P
+17
+It can be seen that by estimating N using this approach, we can achieve much better results.
+In this experiment, the ground truth values were only used to calculate the normal vector N.
+Therefore, we conclude that our method can benefit from a more accurate estimation of the
+normal vector. A future study can focus on predicting N more accurately during training, as
+was done in [32] in testing phase, in Planar Parallax framework.
+C
+Qualitative Results
+In Fig. 4, we provide qualitative results of our models on the KITTI dataset in comparison
+to Monodepth2 [10] and DNet [32]. Our stereo model produces sharp outputs and captures
+the boundaries of the objects very well, and neither of our models suffer from artefacts such
+as the wrong estimation for the road lane line in the last row.
+Input Image
+Monodepth2 [10]
+DNet [32]
+Ours-Mono
+Ours-Stereo
+Figure 4: Qualitative Results. In each row, for an input image, we show the results of
+Monodepth2 [10] and DNet [32] and compare them to our models in the last two columns.
+Ours-Mono refers to our ResNet18 model trained only on monocular sequences, and Ours-
+Stereo refers to our ResNet50 model that was trained using additional stereo supervision.
+
+中Lr7P33
\ No newline at end of file
diff --git a/mtA0T4oBgHgl3EQfJf_Z/content/tmp_files/load_file.txt b/mtA0T4oBgHgl3EQfJf_Z/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,955 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf,len=954
+page_content='SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  1  DepthP+P: Metric Accurate Monocular  Depth Estimation using Planar and Parallax  Sadra Safadoust  ssafadoust20@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='tr  Fatma Güney  fguney@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='tr  KUIS AI Center  Koç University  Istanbul, Turkey  Abstract  Current self-supervised monocular depth estimation methods are mostly based on  estimating a rigid-body motion representing camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' These methods suffer from  the well-known scale ambiguity problem in their predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We propose DepthP+P, a  method that learns to estimate outputs in metric scale by following the traditional planar  parallax paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We first align the two frames using a common ground plane which  removes the effect of the rotation component in the camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' With two neural  networks, we predict the depth and the camera translation, which is easier to predict  alone compared to predicting it together with rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' By assuming a known camera  height, we can then calculate the induced 2D image motion of a 3D point and use it for  reconstructing the target image in a self-supervised monocular approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We perform  experiments on the KITTI driving dataset and show that the planar parallax approach,  which only needs to predict camera translation, can be a metrically accurate alternative  to the current methods that rely on estimating 6DoF camera motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  1  Introduction  Understanding the 3D structure of a scene is fairly easy for human beings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We can easily  reason about our surroundings and decompose them into different objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Having this abil-  ity is crucial for autonomous vehicles to be able to drive in different environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Training  deep networks for estimating depth has proven successful in computer vision research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' How-  ever, many such methods are supervised and require ground truth depth which is costly to  achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Another line of work uses a stereo setup that must be carefully calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Both  of these approaches cannot use the vast amount of unlabeled videos that are easily available  for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' On the other hand, self-supervised monocular depth estimation methods that do  not rely on stereo supervision do not suffer from these limitations and, in practice, have been  closing the gap with their supervised or stereo counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Current self-supervised monocular depth estimation approaches all follow the same basic  idea proposed in [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' They use a pose network to estimate the ego-motion between a source  frame and the target frame and a depth network to estimate the depth of the target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  These estimations can then be used to sample pixels from the source image to synthesize  the target frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The difference between the target frame and the synthesized can be used  as the source of supervision for training the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In this paper, we propose another  approach to synthesize the target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Our approach, DepthP+P, illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' 1,  uses the traditional planar parallax formulation [15, 25], which decomposes the motion into  a planar homography and a residual parallax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Consider a plane in the scene and its motion  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='02092v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='CV]  5 Jan 2023  2  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  [  h11    h12    h13  h21    h22    h23  h31    h32    h33]  Depth  Pose  H=  Is  Iw  D  t  It  Figure 1: Overview of our Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Using the source image Is and the target image It,  we first calculate the homography H that aligns the road plane across these two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We  then warp Is according to H and obtain the aligned image Iw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The aligned image Iw and the  target image It are input to the pose network which estimates the camera translation t only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  The depth network takes the It and produces a metric accurate depth map ˆD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  represented by a homography from the source to the target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' By first warping the  source image according to this homography, the motion of the plane is canceled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Then the  residual image motion depends on two factors: (1) the deviations of the scene structure from  the plane, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' the depth of points and their perpendicular distance to the plane and (2) only  the translational motion of the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Autonomous driving is a perfect use case for this  approach because there is typically a planar surface in front of the vehicle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  However, it is important to note that the plane in the planar parallax formulation does not  necessarily have to be a real plane and can also be a virtual plane, but choosing the road as  the planar surface makes it easier to implement in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Moreover, our approach does not  rely on the availability of a plane to predict depth during inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  In this approach, we first align the road plane between the source and target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  This is achieved by calculating the homography between the road regions in two frames and  then warping the source frame according to the homography to obtain the aligned image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  By doing so, the road regions in the aligned image and target image match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The residual  motion between the aligned image and the target image can be explained as follows: We  first estimate the depth of each pixel with a monocular depth network and back-project them  into 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Then using a known camera height, we can calculate the perpendicular distance of  each point to the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In addition, we estimate the translation between the camera origins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Note that this is different from the typical monocular depth approach [10, 37] which needs  to estimate both the rotation and translation components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Finally, the target image can be  synthesized from the aligned image using the calculated residual parallax as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  The planar parallax approach for self-supervised monocular estimations has a number  of advantages over the previous paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Firstly, it is much easier to optimize because it  removes the ambiguities associated with predicting rotational camera motion [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Secondly,  it can produce metric accurate outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Previous monocular depth methods can estimate  depth and motion up to a scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Typically, during inference, ground truth depth data is used  to scale the predicted depth values such that the median of the predicted depth is equal to  that of ground truth depth [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Our approach is able to predict metric accurate depth without  needing ground truth depth data by only assuming a known camera height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  3  2  Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1  Self-Supervised Monocular Depth  View Synthesis:  Garg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [6] were the first to propose a method that uses view syn-  thesis as an objective for depth estimation from single images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Monodepth [9] uses Spatial  Transformer Networks (STNs) [16] to synthesize the images in a fully-differentiable way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  SfmLearner [37] generalizes view synthesis to temporally consecutive images by using an-  other network to predict the relative pose between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Zhan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [36] use stereo sequences  to perform view synthesis using temporally consecutive pairs as well as the left-right pairs,  enabling them to benefit from both monocular and stereo supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In addition to image  reconstruction, they also use feature reconstruction as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Similarly, by going be-  yond pixel-wise reconstruction error, Mahjourian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [19] propose to use a 3D point cloud  alignment loss to enforce the estimated point clouds and the camera pose to be consistent  temporally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [29] use direct visual odometry in a differentiable manner to solve  for ego-motion using the estimated depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  In addition to depth and camera pose, several methods estimate optical flow for residual  motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' After predicting the camera motion, GeoNet [35] estimates the remaining object  motion using optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In order to prevent the errors of camera pose or depth predictions  from propagating to flow estimations, DF-Net [39] enforces consistency between optical flow  and the flow induced by the depth and pose predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' GLNet [4] uses epipolar constraint  for optical flow, along with other geometric constraints, further improving the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  EPC++ [18] proposes a holistic 3D motion parser that uses predicted depth, pose, and op-  tical flow to estimate segmentation masks for dynamic objects and their motion as well as  background motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Ranjan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [20] jointly train networks for depth, pose, optical flow,  and motion segmentation so that they can use geometric constraints on the static regions and  generic optical flow on moving objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' MonoDepthSeg [24] proposes to jointly estimate  depth, independently moving regions, and their motion with an efficient architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Some approaches keep the original framework with a depth and a pose network but im-  prove the performance with better loss functions, improved network architectures, and inno-  vative design choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' When estimating depth at multiple scales, Monodepth2 [10] proposes  to first upsample the estimated low-scale depths to the input image size and then calculate  the photometric loss at that scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Monodepth2 also proposes to calculate the minimum of  reprojection errors per pixel instead of averaging them when synthesizing the target image  from multiple views to prevent blurry depth estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' PackNet [11] changes the architec-  ture of the depth network and uses 3D convolutions to learn to preserve spatial information  using symmetrical 3D packing and unpacking blocks for predicting depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Scale Ambiguity:  Self-supervised monocular depth estimation models suffer from the  scale ambiguity problem, and the depth and pose outputs of such models are in an unknown  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The median scaling technique used by many previous methods does not actually  solve this problem because it relies on ground truth depth data during inference which is not  always easily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Bian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [2] introduce a loss to minimize normalized differences of  depth maps across the entire sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' This makes the estimations globally scale-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  However, although this means that the predictions are at the same scale, that specific scale is  still unknown, and the median scaling is still required during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  There are a number of monocular methods that can output depth estimations in absolute  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Roussel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [22] use a network that was pre-trained with stereo pairs on a dataset  4  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  and finetunes it on another dataset while maintaining the metric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Guizilini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [11]  propose a version of their PackNet that uses ground truth camera velocity and the timestamps  of images to enforce the estimations to be metrically accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Bartoccioni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [1] supervise  their depth predictions with a sparse LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' However, all of these approaches rely on ground  truth data from extra sensors during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  There are a few other methods that do not require additional supervision and only use the  camera height to achieve depth estimations in metric units similar to the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  DNet [32] estimates the ground plane during inference and, using the real height of the  camera, recovers the scale of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' However, it needs a ground plane to be visible  during the test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In other words, they do not train their depth outputs to be in absolute  scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Rather, they recover the scale of the estimations with another module during test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Wagstaff and Kelly [28] train a network that learns the metric scale during training using  camera height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' They introduce a plane segmentation network and propose a three-staged  training procedure for training the depth estimation model in metric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' First, they train  an unscaled depth network and then use it to train the plane segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Finally,  they train a new metrically accurate depth network using the pre-trained plane segmentation  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Similar to [28], we also learn the metric scale during training, but we do not need  a multi-stage process, nor do we rely on the existence of a ground plane during inference,  differently from previous work [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='2  Planar Parallax  The Planar Parallax paradigm, also called Plane + Parallax (P+P), has been used to under-  stand the 3D structure of a scene from multiple images by decomposing the motion into a  planar homography and a residual parallax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Sawhney [25] proposes a formulation for the  residual parallax that uses depth and distance to the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Irani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [15] use this for-  mulation to derive a rigidity constraint between pairs of points over multiple images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Irani  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [13] derive trifocal constraints and use them to propose a simple method for new view  synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In a follow-up work [14], they extend the planar parallax method to more than  two uncalibrated frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  More recently, MR-Flow [31] uses P+P to refine the optical flow estimations with rigidity  constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Chaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [3] use P+P to estimate the height of points in the scene with event-  based cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We propose a method to use the P+P formulation within the view synthesis  framework for self-supervised monocular depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  3  Methodology  Despite the success of current self-supervised monocular depth estimation approaches, they  suffer from scale ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' the estimated depth values are in an unknown scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' There-  fore, in order to evaluate and compare these methods, they are usually normalized using the  median scaling approach [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Here, we propose an approach that predicts depth maps in  metric scale without using any ground truth depth supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1  DepthP+P  Our approach is based on the Planar Parallax decomposition which has been studied in detail  before [15, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We first introduce it here to establish our notation and then build our method  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  5  x  p  p′  pw  PW  H  Π  It  Is  C  C′  C  x  h  NTx  dc  x  z  y  Π  Z  N  Figure 2: Visualization of the Planar Parallax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Left: The 3D point x is projected to points  p and p′ on the target image It and the source image Is respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Using the homography  H induced by the plane Π, the point p′ will be transformed to point pw on the target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Right: Calculating h, distance of x to the Π using the camera height dc and the normal vector  N of the plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' C and C′ are the camera centers of It and Is and Z is the depth of the point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  to predict depth following that notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Notation:  Let Π be a 3D plane and H be the homography aligning Π between the target  image It and the source image Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Let p and p′ be the images of the 3D point x = [X,Y,Z]T on  the It and Is respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' As shown on the left in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' 2, we can warp p′ by the homography  H and obtain the image point pw:  pw ∼ Hp′  (1)  where we omit the conversion to the homogenous coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Note that by warping the  source image Is, we obtain the aligned image Iw such that the plane Π matches between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The displacement between pw and p can be computed as follows:  pw −p =  γ  dc −γtz  (tzp−Kt)  (2)  where K is the camera intrinsic, t = [tx,ty,tz]T is the translation vector between the It and Is,  and dc is the distance between the camera for the source view to the plane Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The structure  is represented by γ = h  Z where h is the distance of x to Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Note that when x lies on the plane  Π, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' h = 0, we will have pw = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  DepthP+P: Following the typical self-supervised monocular depth approach, our frame-  work has two networks, one for estimating depth and another for estimating the translation  between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Note that, unlike other methods, we do not need to estimate the rotation  between the two views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Precisely, our pose network takes the source and images Is,Iw and  outputs the translation vector t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The depth network takes the target image It and outputs the  depth map ˆD for It.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  For every pixel p = [x,y], let ˆD(p) denote its estimated depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We backproject p using  the camera intrinsics and the estimated depth to obtain the corresponding 3D point ˆx in the  camera coordinate system as follows:  ˆx = ˆD(p) K−1 [x,y,1]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  (3)  6  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  Therefore, as demonstrated on the right in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' 2, we have the following:  ˆh = dc −NT ˆx,  ˆγ =  ˆh  ˆD(p)  (4)  where N is the normal vector of the plane Π, ˆh is the estimated distance of the point ˆx to  the plane Π and ˆγ is our estimate of the structure variable γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' As a result, we obtain all the  parameters required to use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (2) to reconstruct the target image It by warping the aligned  image Iw resulting in ˆIw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In other words, for each pixel p on the It, we calculate the pw using  (2) according to the depth and translation predicted by our two networks and then inverse  warp Iw and obtain ˆIw to reconstruct It:  It(p) ≈ ˆIw(p) = Iw(pw)  (5)  We minimize the difference between It and ˆIw for supervision as explained in the Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  In order to obtain the aligned images, we perform a pre-processing step on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  We calculate a homography for every consecutive frame by using the road as the plane Π and  warp the frames according to the calculated homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In other words, we calculate a  homography H for every target image It and source image Is pair, and then warp Is according  to H to obtain the warped source image Iw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We explain the details of this pre-processing step  in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='2  Self-Supervised Training Loss  In our approach, we define our photometric loss function as the linear combination of the  L1 distance and the structural similarity (SSIM) [30] to minimize the difference between the  target image It and the reconstructed image ˆIw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Our photometric loss is therefore defined as  follows:  Lphoto(p) = (1−α)|It(p)− ˆIw(p)|+ α  2  �  1−SSIM(It,ˆIw)(p)  �  (6)  where we set α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Note that for every target image we consider two aligned images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  One from warping the previous frame, and one from warping the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We use the  per-pixel minimum reprojection error introduced in [10] and calculate the minimum of the  Lphoto for each pixel across the previous and next aligned images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We also define Lsmooth  as an edge-aware smoothness loss over the mean-normalized inverse depth estimates [29] to  encourage the depth predictions to be locally smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Our total loss function is a combination  of Lsmooth and Lphoto averaged over all N pixels:  L = 1  N ∑  p  λ Lsmooth(p)+min  w (Lphoto(p))  (7)  where min  w  calculates the minimum over the previous and next aligned frames and λ is a  hyperparameter controlling the effect of the loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='3  Network Architecture  Our depth network is based on the U-Net architecture [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We use a ResNet [12] pre-trained  on the ImageNet [23] as the encoder for our depth network, and for the decoder we use the  architecture similar to the one used by [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The difference is that we directly estimate depth  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  7  by multiplying the output of the last sigmoid layer by 250, which is the maximum depth  value that can be predicted, instead of estimating disparity as in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The depth network  takes as input a single target image It and outputs the per-pixel depth estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Note that  the output of our depth decoder is in metric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  In DepthP+P, our second network takes Iw and It and outputs only the translation vector  between the views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The network is similar to the pose network proposed in [10], except that  the output is a 3-element vector representing the translation and is metric scale in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  4  Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1  Dataset  KITTI: We use the Eigen split [5] of the KITTI dataset [7, 8] to train and evaluate our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  We use all of the images in the split for which we could accurately estimate the homography  for aligning the road between consecutive images as explained in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' This  results in 45000 training and 1769 validation samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We evaluate our model on the 697 test  images in the split using the original ground truth provided by LiDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We also report results  using the improved ground truth for 652 test images provided by Uhrig et al [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' They use  a stereo-reconstruction method to remove the outliers in LiDAR points and increase the  ground truth density by accumulating laser scans which result in high-quality ground truth  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The camera height in this dataset is dc = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='65 and we assume that the road is completely  horizontal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' N = [0,1,0]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Pre-processing the dataset for DepthP+P: In order to use our P+P approach, we need to  calculate the homography between the consecutive frames and warp the source frame accord-  ing to the estimated homographies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Since we work on the driving scenarios on KITTI, we  choose the “road” as our plane Π which is visible in most of the frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' For calculating the  homography, we need to find a set of (at least 4) corresponding pairs of road pixels between  a source view Is and the target view It, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' two consecutive images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' For this purpose, we use  the optical flow between Is and It using [26] to find the corresponding pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We then use  [38] to select only the pixels that belong to the semantic class “road”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Using the correspond-  ing pairs of road pixels, we estimate the homography H using OpenCV’s RANSAC-based  robust method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We do this to find the homography H for all of the consecutive pairs of  frames on KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Note that for any consecutive pair of frames I1,I2 and the homography H  between them, we use H to warp I1 towards I2 and also use H−1 to warp I2 towards I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' This table compares  our proposed approach, DepthP+P, to previous approaches on the KITTI dataset that were  trained only with monocular supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The scale column specifies whether the method  can estimate depth in metric scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We provide results with the original and improved ground  truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We show the results for the input resolution 640 × 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The best method in each  column is shown in bold and the second best is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  follow our model as future work to make it perform better than our initial proposal as well  as the other state-of-the-art models that are trained to estimate the full pose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Note that [32] is not trained to estimate metrically accurate depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Instead, its depth  network outputs depth in an unknown scale, and then during inference, it needs a ground  plane to be visible on the image to recover the scale of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' When the ground  plane is not visible on the image, [32] fails completely as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' As can be seen  in this figure, this image from the KITTI dataset does not have a ground plane, and [32]  cannot recover the scale and produces completely wrong estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' While our method needs  a ground plane during training, it does not rely on the availability of the ground plane during  inference, therefore it can still perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' For reference, the absolute relative (Abs Rel)  error of [32] on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' 3 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='178, while our model achieves a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='252 error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' [28] achieves better  results by using a pre-trained plane segmentation network in addition to the depth network,  while our approach can achieve comparable results without a separate segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  DepthP+P can also be trained with additional stereo supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content='995  Table 2: Quantitative Results on KITTI with Additional Stereo Supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We com-  pare DepthP+P to previous approaches that use additional stereo supervision on KITTI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Stereo supervision significantly improves the results of DepthP+P model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  By using  ResNet50, our model performs on par with Monodepth2 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Figure 3: Qualitative Comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We compare the absolute relative error of our depth  estimation method (left) with DNet [32] (right) on an image from the KITTI dataset without  a ground plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The colorbar on the right shows the values of the absolute relative error  metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We cap the max error at the value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='0 for visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We can see that [32]  completely fails to estimate the metric depth due to the wrong scale recovery because there  is no ground plane in the image, while our model does not have this issue and can still  perform well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' The absolute relative error for [32] is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='178 while it is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='252 for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  formances of the methods that also use stereo supervision for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Using stereo super-  vision significantly improves the performance of our DepthP+P model, outperforming all  methods except for Monodepth2 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We show that by using a ResNet50 backbone instead  of ResNet18, DepthP+P can obtain comparable results to Monodepth2 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  5  Conclusion and Future Work  In this paper, we presented a new approach to self-supervised monocular depth estimation  following the traditional planar parallax paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We showed that our approach is able to  produce metrically accurate depth estimates by using a known camera height.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Unlike pre-  vious methods that rely on estimating the full rigid-body motion of the camera, our method  only needs to estimate the camera translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We discussed the advantage of our method  compared to the other scale-aware depth prediction methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We see our approach as a first  step to unlocking the potential of the plane and parallax for efficient and metric-accurate  depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' An exciting future direction can focus on detecting moving foreground  objects by checking the violations in the plane and parallax constraints [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content='010  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  References  [1] Florent Bartoccioni, Éloi Zablocki, Patrick Pérez, Matthieu Cord, and Karteek Ala-  hari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' In Advances in Neural Information Processing Systems  (NeurIPS), pages 35–45, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' on Image Pro-  cessing (TIP), 13(4):600–612, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' IEEE International Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' on Intelligent Robots  and Systems (IROS), pages 2330–2337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' on Computer  Vision and Pattern Recognition (CVPR), pages 225–234, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' In  Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
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+page_content=' Unsupervised learning of monocular depth estimation and visual odom-  etry with deep feature reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' on Computer Vision and  Pattern Recognition (CVPR), pages 340–349, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  13  [37] Tinghui Zhou, Matthew Brown, Noah Snavely, and David G Lowe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Unsupervised  learning of depth and ego-motion from video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' on Computer Vision  and Pattern Recognition (CVPR), pages 1851–1858, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  [38] Yi Zhu, Karan Sapra, Fitsum A Reda, Kevin J Shih, Shawn Newsam, Andrew Tao, and  Bryan Catanzaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Improving semantic segmentation via video propagation and label  relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' IEEE Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' on Computer Vision and Pattern Recognition (CVPR),  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  [39] Yuliang Zou, Zelun Luo, and Jia-Bin Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' DF-Net: Unsupervised joint learning  of depth and flow using cross-task consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' of the European Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' on  Computer Vision (ECCV), pages 36–53, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  14  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  Supplementary Material for  DepthP+P: Metric Accurate Monocular  Depth Estimation using Planar and Parallax  Sadra Safadoust  ssafadoust20@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='tr  Fatma Güney  fguney@ku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='tr  KUIS AI Center  Koç University  Istanbul, Turkey  In Section A of this supplementary document, we provide the derivation of the residual  parallax (Equation 2 in the main paper) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In Section B, we investigate the effect of using  a more accurate estimation of the normal vector of the road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Finally, in Section C, we provide  additional qualitative results of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  A  Derivation  Let C and C′ be the camera centers of the target view It and the source view Is, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Also, let x = [X,Y,Z]T and x′ = [X′,Y′,Z′]T be the coordinates of a 3D point with respect  to C and C′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We can relate x and x′ as follows:  x = Rx′ +t  (8)  where R is the rotation and t = [tx,ty,tz]T is the translation between C and C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' According to  the Fig 2 (right), we can calculate h, the perpendicular distance of the 3D point to the plane  Π as:  h = dc −NTx′  (9)  Note that dc is the height of C′, the camera of the source view Is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' However, h is invariant  with respect to the cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We can rewrite the above equation as:  1 = h+NTx′  dc  (10)  Therefore, by substituting this into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (8), we have:  x = Rx′ +t  �h+NTx′  dc  �  =  �  R+ tNT  dc  �  x′ + h  dc  t  (11)  Let K and K′ denote the camera intrinsic for the target view and the source view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Then  p = [x,y,1]T = 1  ZKx and p′ = [x′,y′,1] = 1  Z′ K′x′ represent the pixel coordinates of our 3D  point in the target and source view, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (11) can be written as:  ZK−1p =  �  R+ tNT  dc  �  Z′K′−1p′ + h  dc  t  (12)  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  15  By multiplying both sides by 1  Z′ K we obtain:  Z  Z′ p = K  �  R+ tNT  dc  �  K′−1  �  ��  �  H  p′ +  h  Z′dc  Kt  (13)  Z  Z′ p = Hp′ +  h  Z′dc  Kt  (14)  where H = K  �  R+ tNT  dc  �  K′−1 is the 3 × 3 homography matrix associated with the plane Π  between the source and the target view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Note that in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (14), the third component of the  vector in two sides of the equation should be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Therefore:  Z  Z′ = H3p′ +  h  Z′dc  K3t  (15)  where H3 and K3 are the third row of H and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Since K3t = tz, we therefore have:  Z  Z′ = H3p′ +  h  Z′dc  tz  (16)  By dividing each side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (14) by each side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (16) we obtain:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Hp′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc Kt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(17)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Adding and subtracting Hp′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ to the right-hand side yields:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p = Hp′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ − Hp′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Hp′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc Kt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(18)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='= Hp′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Hp′ �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Hp′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc Kt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(19)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='= Hp′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Hp′ �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ �  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc Kt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='H3p′ +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z′dc tz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(20)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (16) into the denominators of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (20) results in:  p = Hp′  H3p′ −  Hp′ �  h  Z′dc tz  �  H3p′ � Z  Z′  �  +  H3p′ �  h  Z′dc Kt  �  H3p′ � Z  Z′  �  (21)  = Hp′  H3p′ − htz  Zdc  Hp′  H3p′ + h  Zdc  Kt  (22)  The point pw = [xw,yw,1]T = Hp′  H3p′ is the point p′ transformed by the homography matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' (22) can be simplified to:  p = pw − htz  Zdc  pw + h  Zdc  Kt  (23)  16  SADRA SAFADOUST,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' FATMA GÜNEY: DEPTHP+P  Subtracting htz  Zdc p from both sides,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' we obtain:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p = pw − htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='pw + h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Kt− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(24)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='= pw  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='+ h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Kt− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(25)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='By dividing both sides by 1− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc we get:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='p = pw +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc Kt− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc p  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(26)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='By rearranging the terms and defining γ = h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z we obtain:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='pw −p =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc p−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc Kt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='1− htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Zdc  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(27)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z p− h  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='ZKt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='dc − htz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Z  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(28)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='= γtzp−γKt  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='dc −γtz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(29)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='=  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='γ  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='dc −γtz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(tzp−Kt)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='(30)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='Ablation Study on Normal Vector  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='In our experiments in the paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' we have assumed that the road plane is horizontal with  respect to the camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In other words, we have assumed that N = [0,1,0]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' However, the road  planes are not always completely flat and can be titled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' For example, one side can be higher  than the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Or consider an uphill where the road is sloping upwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' To analyze this, we  perform an experiment where we calculate the normal vector of the road using ground truth  depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Concretely, during training, we back-project the road pixels to 3D using their ground  truth depth and fit a plane to the obtained 3D points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We then use the normal vector of the  fitted plane as our vector N in Equation 4 in the main paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' We report the results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Method  Abs Rel  Sq Rel  RMSE  RMSElog  δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='25  δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='252  δ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='253  DepthP+P (Fixed Normal)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='152  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='322  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='185  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='239  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='781  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='920  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='970  DepthP+P (GT Normal)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='142  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='127  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='922  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='238  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='780  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='921  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='971  Table 3: Effect of Calculating Normal Vector of Road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' This table analyzes the effect  of accurately predicting the normal vector N of the road on the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Fixed Normal  assumes that the ground is perfectly horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' GT Normal uses the ground truth depth data,  generates the point clouds for the road region, fits a plane and calculates the normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  GT Normal performs better than Fixed Normal, highlighting the fact that a more accurate  normal vector calculation will improve the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  SADRA SAFADOUST, FATMA GÜNEY: DEPTHP+P  17  It can be seen that by estimating N using this approach, we can achieve much better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  In this experiment, the ground truth values were only used to calculate the normal vector N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Therefore, we conclude that our method can benefit from a more accurate estimation of the  normal vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' A future study can focus on predicting N more accurately during training, as  was done in [32] in testing phase, in Planar Parallax framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  C  Qualitative Results  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' 4, we provide qualitative results of our models on the KITTI dataset in comparison  to Monodepth2 [10] and DNet [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' Our stereo model produces sharp outputs and captures  the boundaries of the objects very well, and neither of our models suffer from artefacts such  as the wrong estimation for the road lane line in the last row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Input Image  Monodepth2 [10]  DNet [32]  Ours-Mono  Ours-Stereo  Figure 4: Qualitative Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content=' In each row, for an input image, we show the results of  Monodepth2 [10] and DNet [32] and compare them to our models in the last two columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  Ours-Mono refers to our ResNet18 model trained only on monocular sequences, and Ours-  Stereo refers to our ResNet50 model that was trained using additional stereo supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
+page_content='  中Lr7P33' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtA0T4oBgHgl3EQfJf_Z/content/2301.02092v1.pdf'}
diff --git a/mtE4T4oBgHgl3EQfuA2V/content/tmp_files/2301.05229v1.pdf.txt b/mtE4T4oBgHgl3EQfuA2V/content/tmp_files/2301.05229v1.pdf.txt
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+Topological Floquet engineering using two frequencies in two dimensions
+Yixiao Wang,1 Anne-Sophie Walter,1 Gregor Jotzu,2 and Konrad Viebahn1, ∗
+1Institute for Quantum Electronics, ETH Zurich, 8093 Zurich, Switzerland
+2Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany
+(Dated: January 13, 2023)
+Using two-frequency driving in two dimensions opens up new possibilites for Floquet engineering,
+which range from controlling specific symmetries to tuning the properties of resonant gaps.
+In
+this work, we study two-band lattice models subject to two-tone Floquet driving and analyse the
+resulting effective Floquet bandstructures both numerically and analytically. On the one hand, we
+extend the methodology of Sandholzer et al. [10.1103/PhysRevResearch.4.013056] from one to two
+dimensions and find competing topological phases in a simple Bravais lattice when the two resonant
+drives at 1ω and 2ω interfere. On the other hand, we explore driving-induced symmetry breaking
+in the hexagonal lattice, in which the breaking of either inversion or time-reversal symmetry can
+be tuned independently via the Floquet modulation. Possible applications of our work include a
+simpler generation of topological bands for ultracold atoms, and the realisation of non-linear Hall
+effects as well as Haldane’s parity anomaly in inversion-symmetric parent lattices.
+I.
+INTRODUCTION
+Topological phenomena emerge naturally for electrons
+under the effect of strong magnetic fields, exemplified by
+the quantum Hall effect (QHE) [1].
+In these phenom-
+ena, the underlying physical mechanism is the breaking
+of time-reversal symmetry due to the magnetic field. Sev-
+eral years after the discovery of the QHE it was noticed
+that topological insulators can also arise without an am-
+bient magnetic field, such as the anomalous quantum Hall
+state [2] and the quantum spin Hall effect [3]. Today,
+anomalous topological phases can be realised by Floquet
+engineering, that is, periodic driving of a quantum sys-
+tem. So far, Floquet engineered topological phenomena
+have been largely limited to single-frequency protocols,
+applied to optical lattices [4–13], real materials [14–17],
+photonic and plasmonic waveguides [18–20], and other
+synthetic systems [21–24].
+Recently, bichromatic and
+multi-frequency Floquet engineering has emerged as a
+powerful strategy to enhance the driving capabilities.
+These include time-reversal or spatial symmetry break-
+ing [25–35] and interference between different Floquet
+harmonics [36–42]. The combination of both two-path
+interference and time-reversal symmetry breaking has
+led to the proposal [43] and experimental demonstra-
+tion [44] of a topological pump in one-dimensional lattice.
+To our knowledge, all experimental results in the multi-
+frequency regime have been limited to one-dimensional
+driving patterns, motivating the need to find novel strate-
+gies for two-dimensional driving.
+In this work, we use the idea of two-tone Floquet engi-
+neering to generate novel topological bandstructures and
+topological phase transitions from simple parent lattices
+in two dimensions. On the one hand, we consider reso-
+nant driving for the lowest two bands of a triangular lat-
+tice. Although this model only features a single potential
+∗ viebahnk@phys.ethz.ch
+minimum per unit cell, we are able to drive topological
+transitions by tuning the amplitude and relative phase
+between 1ω and 2ω drives. The simplicity of this scheme
+could open up new pathways for realising strongly cor-
+related topological insulators, which have remained out
+of reach in existing methods.
+On the other hand, we
+apply two-frequency driving to an inversion-symmetric
+(graphene-like) hexagonal lattice. This driving scheme
+enables the selective breaking of inversion symmetry
+while maintaining time-reversal symmetry, giving access
+to a new class of Floquet Hamiltoninans. Thus, we find
+that an external breaking of inversion symmetry is not
+necessary to drive the celebrated parity anomaly in the
+Haldane model [2]. The two-frequency approach generi-
+cally applies to Floquet-driven systems in lattices, rang-
+ing from condensed matter to synthetic quantum matter.
+The remainder of this paper is structured as follows.
+The relevant two-tone driving waveforms, and possible
+experimental implementations, are introduced in section
+II. Afterwards, we show how resonant 1ω–2ω driving in
+a triangular lattice leads to topological phase transitions
+(section III). Section IV discusses the effective breaking of
+time-reversal and inversion symmetry under off-resonant
+driving in a hexagonal lattice.
+II.
+1ω–2ω FLOQUET DRIVING IN TWO
+DIMENSIONS
+Two-frequency driving can be applied, in princple, to
+any physical system which has been employed for Floquet
+engineering so far [14]. As a generic starting point, we
+consider a particle confined to a static potential Vlat(r),
+and subject to an oscillating force
+ˆHcm(τ) = ˆp2
+2m + Vlat(r) − F(τ) · ˆr ,
+(1)
+with
+F(τ) ≡ eE(τ) = −m¨rm(τ) = − ˙A(τ) .
+(2)
+arXiv:2301.05229v1  [cond-mat.quant-gas]  12 Jan 2023
+
+2
+FIG. 1.
+Single- and two-frequency Lissajous curves for two-
+dimensional driving.
+The plots show the real-space modu-
+lation figures rm(τ) (Eq. 4). These shapes are topologically
+equivalent to those of the time-dependent force F(τ), since the
+two are related to one another by taking the second derivative
+(Eq. 2). (a-c) Circular strong driving at 1ω (a), circular weak
+driving 2ω (b), or both drives combined (c) can be used to
+resonantly address a higher-band transition. The thin line in
+(c) represents the dominant 1ω waveform, the same as (a).
+(d-f) Superposition of an elliptical driving field at 1ω and a
+linear field at 2ω with relative phases ϕy1 = 0 (d), 0.2π (e),
+and 0.5π (f). (g-i) Superposition of two circular drives with
+relative phase ϕx2 = 0.9π, ϕy1 = 0.5π, ϕy2 = 0.4π, driving
+strength Kx2 = Ky2 = 0.7, and Kx1 = Ky1 = 0.5 (g), 0.65
+(h), and 0.8 (i).
+In solid state systems, the driving force F(τ) can directly
+correspond to the oscillating field E(τ) of a laser coupling
+to free electrons [17, 45]. Equivalently, the application of
+periodic driving can be seen as ‘minimal coupling’ using
+the vector potential as q → q−A(τ)/ℏ. Furthermore, col-
+lective modes such as phonons [46] or magnons [47] can
+also be involved. In waveguide-based synthetic lattices,
+transverse movements of the waveguides provide an ef-
+fective inertial force [19]. For optical lattices, the driving
+force can be provided by an oscillating optical or mag-
+netic gradient [48]. Alternatively, an inertial force can
+result from a periodic displacement of the entire lattice
+potential, an approach known as ‘lattice shaking’ [4, 5].
+In this case, the Hamiltonian in the lab frame is
+ˆHlab(τ) = ˆp2
+2m + Vlat [r − rm(τ)] ,
+(3)
+related to Eq. 2 by two unitary transformations [49]. In
+summary, the derivations in this work directly carry over
+from engineered quantum platforms, such as ultracold
+atoms in optical lattices, to condensed matter systems.
+We define the two-dimensional driving pattern rm(τ)
+as
+xm(τ) =
+Ax1 cos(ωτ)
++Ax2 cos(2ωτ + ϕx2)
+(4)
+ym(τ) =Ay1 cos(ωτ + ϕy1)+Ay2 cos(2ωτ + ϕy2) .
+These two-tone driving waveforms are visualised as Lis-
+sajous curves in Fig. 1 for the relevant choices of driving
+parameters (phases and amplitudes) used in this work.
+The amplitude of the vector potential is given by the di-
+mensionless driving strength Kαβ = mωβAαβa/ℏ, where
+a is the lattice spacing and Aαβ is the real-space am-
+plitude.
+The index α ∈ {x, y} denotes the direction,
+whereas β ∈ {1, 2} indicates the frequency.
+Resonant circular 1ω–2ω driving. The combination of
+a strong circular 1ω drive (Fig. 1a, ϕy1 = π/2) with a
+weak circular 2ω drive (Fig. 1b, ϕx2 = 0, ϕy2 = π/2)
+leads to a modulated circular pattern (Fig. 1c). Due to
+two-path interference when addressing the s–p resonance
+of the triangular lattice, this driving pattern can be used
+to drive topological transitions in two dimensions, similar
+to the topological pump of ref. [44].
+Off-resonant 1ω–2ω driving to access spatial and tem-
+poral symmetries. Alternatively, an elliptical driving field
+at 1ω and a linearly polarised laser at 2ω (Ax2 = 0, ϕy2 =
+0) leads to a competition between the breaking of in-
+version symmetry and time-reversal symmetry. For in-
+stance, the choice ϕy1 = 0 results in a ‘boomerang’
+pattern (Fig. 1d), which breaks inversion symmetry but
+not time-reversal symmetry.
+Conversely, the case of
+ϕy1 = π/2 (Fig. 1f) gives a rounded ‘kite’ shape in which
+the breaking of time-reversal symmetry dominates over
+the breaking of inversion symmetry. Tuning the phase
+ϕy1 thus allows to interpolate between the two symme-
+try breaking situations (Fig. 1e), thereby accessing novel
+topological regimes.
+Instead of combining linear and elliptical drives,
+the competition between inversion-breaking and time-
+reversal-breaking can be achieved by superimposing two
+circular drives of opposite helicity (Fig. 1g-i). The way in
+which inversion symmetry is broken in this case depends
+on the relative orientation of the driving pattern to the
+lattice geometry. This can be tuned by increasing ϕx2
+and ϕy2 by the same amount (i.e. by delaying the two
+frequency drives with respect to each other), which ro-
+tates the driving pattern around the origin (Fig. 1g, see
+also refs. [29, 33, 34]). Driving a topological transition
+via changing only a relative phase can be experimentally
+advantageous in condensed matter systems where chang-
+ing polarisation or amplitude can lead to spurious effects.
+Experimental implementation. In this work we focus
+on a triangular and a hexagonal lattice. A corresponding
+optical lattice potential, and the resulting tight-binding
+parameters, can be found in Appendix A. In waveguide-
+based systems, the lattice geometry can generally be cho-
+
+a)
+c)
+1.0
+a
+1.0
+0.5
+time t (2π/w)
+V0.0
+x
+y
+0.5
+b)
+1.0
+0.0
+0.5
+y (a.u.)
+a.u.
+10.0
+d)
+f)
+e
+.0
+.0
+1.0
+a
+time
+time
+0.5
+0.5
+V0.0
+V0.0
+10.0
+x
+x
+x
+y
+y
+y
+h)
+g)
+i)
+1.0
+1.0
+1.0
+T
+time
+time
+time
+0.5
+0.5
+0.5
+Q
+10.0
+0.0
+10.0
+x
+y
+x
+y
+x
+y3
+FIG. 2.
+Resonant band coupling in the triangular lattice (a),
+with relevant parameters t1 = −0.05, t2 = −0.07, t3 = −0.07,
+ε = −0.66 in units of Erec and dimensionless inter-band cou-
+pling ηsp = 0.19. Resonant one-photon or two-photon pro-
+cesses lead to the coupling of lower and higher bands, corre-
+sponding to the green and black arrows in (b), respectively.
+Floquet driving lifts the degeneracy of the static band struc-
+ture in the rotating frame (c) and opens a band gap (d). The
+ring-shape minima of the band structure leads to a ring-shape
+Berry curvature (e).
+(f) shows a trivial spatially localized
+Berry curvature under two-frequency driving, which features
+a strong Berry curvature dipole.
+sen freely [19]. The time-dependence of the driving term
+is encoded in an additional spatial coordinate, which can
+accommodate multi-frequency driving. Finally, in solid
+state systems, the laser field corresponding to the driving
+term at 2ω can be generated via second-harmonic gener-
+ation, ensuring that it is phase-stable with respect to the
+1ω term. The relative phase between the two can then
+be tuned using a dispersive material of varying thickness.
+Tight-binding Hamiltonians for two-dimensional Flo-
+quet engineering.
+By expanding the original Hamilto-
+nian (Eq. 1) in a Wannier basis and transforming it to
+quasimomentum space, we get the following expression
+�
+n
+�
+εnˆa†
+nˆan −
+Z
+�
+i=1
+�
+tn,i
+�
+ei(θi(τ)+q·bi)ˆa†
+nˆan + h.c.
+��
+− F(τ) · ˆr
+�
+n′
+ηnn′ˆa†
+nˆan′
+�
+.
+(5)
+Here, εn, tn,i, ηnn′ are the band centre energies, nearest-
+neighbour tunnelling matrix elements, and interband
+coupling elements, respectively, for bands n, n′ (Ap-
+pendix B). bi = ja1+ka2 are the nearest-neighbour tun-
+neling vectors and i indexes the in-equivalent neighbours
+up to the coordination number Z; a1, a2 correspond to
+the primitive translation vectors in two dimensions. The
+time-periodic Peierls phases are θi(τ) =
+m
+ℏ bi · ˙rm(τ).
+In second quantisation, ˆan denotes the annihilation op-
+erator in band n.
+The first line of Eq. 5 describes
+static and time-dependent contributions to intra-band
+processes whereas the second line is the inter-band cou-
+pling that arises from periodically forcing the system.
+The coupling elements ηnn′ can be understood as dipole
+matrix elements of Bloch states [41]. Consequently, the
+first line is diagonal in quasimomentum and band index
+n, whereas the second line contributes off-diagonal ele-
+ments to the Hamiltonian.
+We use the Hamiltonian in Eq. 5 as the starting point
+for the Floquet analysis. If the periodic drive resonantly
+addresses the gap between s and p bands, we addition-
+ally employ a unitary transformation to ‘rotate away’ the
+energy difference (sec. III). In order to evaluate the effec-
+tive Floquet Hamiltonian, we employ two complementary
+methods.
+On the one hand, we calculate the effective
+Hamiltonian analytically in inverse powers of the driving
+frequency ω using the well-known high-frequency expan-
+sion (HFE) [50–54].
+Since we work in the weak driv-
+ing regime the HFE remains valid. On the other hand,
+we can numerically obtain the effective Hamiltonian via
+the Trotter decomposition [55]. We use this numerical
+method to validate the results of the analytic calcula-
+tion and to justify the truncation of the high-frequency
+expansion a posteriori. To characterise the band topolo-
+gies, the Berry curvature is evaluated via the eigenstates
+in the discretised two-dimensional Brillouin zone. The re-
+sulting Chern number is the surface integral of the Berry
+curvature over the closed Brillouin torus [56, 57].
+III.
+RESONANTLY DRIVEN TRIANGULAR
+LATTICE
+The first situation we consider is a simple triangular
+lattice in which the lowest two bands (s and p) are reso-
+nantly coupled with two frequencies, inspired by ref. [43]
+and building on previous works using single-frequency
+driving [58–60].
+In the non-shaken case the static Hamiltonian for a
+triangular tight-binding model is
+ˆH =
+�
+ε +
+3
+�
+i=1
+ti cos(q · bi)
+�
+σz .
+(6)
+The nearest-neighbour tunneling vectors are b1
+=
+(−1, 1)b, b2 = (1, 0)b, b3 = (0, 1)b and q = (qx, qy)
+in the rotated coordinate ˆe′
+x =
+ˆex+ˆey
+√
+2 , ˆe′
+y =
+−ˆex+ˆey
+√
+2
+(Fig. 2a). The simplest way to couple s and p bands is
+
+a)
+e
+X
+(元 / b
+b)
+0.5-1
+0
+0.5
+3
+0
+nw
+(kHz)
+2hw
+hw
+3
+3
+0.:
+X
+M
+0.5
+qx(元/b)
+d)
+e
+q(元/b)
+1×(元/b)
+20.5-1
+=0.5-1
+0
+0
+0.5
+0.5
+0.1
+0.5
+(zH)
+(kHz)
+0
+0.0
+3
+3
+-0.5
+-0.1
+.0.5
+0.5
+0
+0.5
+qx(元/b)
+qx(元/b) 0.54
+to apply single-frequency driving, parametrised by Eq. 4
+with Ax2 = Ay2 = 0 and denoting ϕy1 as ϕ.
+Starting from Eq. 5 the time-dependent Hamiltonian
+can be written as
+ˆH(τ) =
+�
+ε +
+3
+�
+i=1
+ti cos [q · bi + θi(τ)]
+�
+σz+
+[ηspℏωKx cos(ωτ) + ηspℏωKy cos(ωτ + ϕ)] σx,
+(7)
+where ε = (εs − εp)/2 = −∆ε/2 with ∆ε being the
+gap size. The other parameters are ti = ts,i − tp,i with
+i = 1, 2, 3, the Peierls phases are θi(τ) = − 1
+ℏ
+� τ
+0 F(τ ′) ·
+bi dτ ′ = −[Kx sin(ωτ)ˆx+Ky sin(ωτ +ϕ)ˆy]·bi/b, and the
+Pauli matrices are σj with j = x, y, z. The dimension-
+less driving strengths are denoted by Kx = mωAx1a/ℏ,
+Ky = mωAy1a/ℏ, respectively.
+The tight-binding pa-
+rameters, including the tunneling amplitudes, inter-band
+coupling and band center energies, are obtained from a
+triangular lattice potential by evaluating the Wannier
+functions (Appendix B, Fig. 2a).
+Single-frequency circular driving resonant with the gap.
+If the frequency ω is near-resonant with the gap, corre-
+sponding to the dominant one-photon process, we apply
+the unitary UR(τ) = exp(−iωτσz/2) to eliminate the en-
+ergy gap between the two bands (Fig. 2d). This gives
+ˆH(τ) =
+�
+ε + ℏω/2 +
+3
+�
+i=1
+ti cos [q · bi + θi(τ)]
+�
+σz
++ [ηspℏωKx cos(ωτ) + ηspℏωKy cos(ωτ + ϕ)]
+× [σx cos(ωτ) − σy sin(ωτ)] /2.
+(8)
+The terms proportional to the identity matrix can be
+omitted without affecting the topology of the bands.
+By analysing Eq. 8 in the high-frequency expan-
+sion (Appendix C), we find a ring-shaped gap opening
+(Fig. 2e, see also refs. [13, 61]). However, the off-diagonal
+terms are dominanted by constants. Physically, this cor-
+responds to on-site couplings between the s and p bands,
+which yield topologically trivial bands (Fig. 3a).
+Single-frequency elliptical driving resonant with half
+the gap. Driving the system at half the gap energy (ℏω =
+12.8t1 = 0.64Erec) leads to neighbouring-site interband
+couplings which can give rise to topological bands. Start-
+ing from the waveforms in Eq. 4 with Kx2 = Ky2 = 0 and
+applying the unitary UR(τ) = exp(−iωτσz) gives
+ˆH(τ) =
+�
+ε + ℏω +
+3
+�
+i=1
+ti cos [q · bi + θi(τ)]
+�
+σz
++ [ηspℏωKx cos(ωτ) + ηspℏωKy cos(ωτ + ϕ)]
+× [σx cos(2ωτ) − σy sin(2ωτ)]/2,
+(9)
+where ε+ℏω ≈ 0. In this case, the only non-zero Fourier
+components of the drive are ±1ω and ±3ω.
+Contrary
+to the direct one-photon resonance, the inter-band cou-
+plings happen via two-photon processes (Fig. 2b). We
+now find quasimomentum-dependent σx and σy terms
+FIG. 3.
+Phase diagrams of the triangular lattice un-
+der resonant driving. Single-frequency driving on resonance
+(ℏω
+= 25.6t1
+= 1.28Erec, Ky
+= 0.5, ε + ℏω/2 ≃ 0)
+leads to trivial bands (a), whereas driving on half-resonance
+(ℏω = 12.8t1 = 0.64Erec, Ky = 0.5, ε + ℏω ≃ 0) in-
+duces topological bands (b).
+The trivial regions in (b) are
+caused by bands being shifted out of resonance due to the
+ac-Stark effect. (c) shows the competition between the two
+resonant processes induced by the two-frequency driving with
+Kx1 = 0.4, Ky1 = 0.5, Kx2 = 0.02, Ky2 = 0.02. (d) resulting
+band gap evaluated for a cut through (c) along the red line.
+in the effective Hamiltonian (Appendix C), leading to
+topologically non-trivial regions in the phase diagram
+(Fig. 3b, see also ref. [60]). Lines at Kx = 0 and ϕ = 0
+have zero extent in the phase diagram of Fig. 3b, pre-
+venting the observation of a transition from C ̸= 0 to
+C = 0 in a realistic experiment.
+While large driving
+amplitudes (beyond Kx = 0.5) render the model topo-
+logically trivial, this effect is not due to an interference
+between different processes, but rather due to the ac-
+Stark effect that shifts the bands out of resonance [4].
+Combining the above two resonant schemes at 1ω and
+2ω introduces genuine topological transitions as function
+of driving phase, as outlined in the following.
+Two-frequency resonant elliptical driving.
+The com-
+petition between resonant 1ω and 2ω driving results in
+an interplay between topological phases with C = 0 and
+C = ±1. We now consider the full two-tone waveforms
+of Eq. 4 for fixed [Kx1, Kx2, Ky1, Ky2], and tuneable ϕy1
+
+b)
+a)
+Single frequency resonant
+Single frequency half resonant
+元
+元
+≤2
+C=-1
+6
+9
+Phase
+Phase 
+C=0
+C=0
+0
+0
+元
+2
+C=1
+2
+一元L
+一元
+0.25
+0.5
+0.75
+0
+0.25
+0
+0.5
+0.75
+Driving strength Kx
+Driving strength Kx
+c)
+d)
+Two-frequency (lw + 2w) driving
+元
+元
+巫2
+巫2
+Phase Pyl
+Phase Py1
+C=0
+0
+0
+2
+一元
+巫2
+0
+0
+0.04
+0.08
+一π元
+元
+Phase Py2
+Band gap size (ti)5
+and ϕy2 (ϕx2 = 0). This Floquet scheme allows to cross
+topological transitions purely by changing the phase be-
+tween 1ω and 2ω drives, as is evident from the distinct re-
+gions of C = 0 and C ̸= 0 in the phase diagram (Fig. 3c).
+The values of the induced band gaps are on the order of
+0.1 to 0.2 tunnelling energies, which can be enlarged by
+choosing stronger driving amplitudes.
+Realising topological band structures in a Bravais lat-
+tice, such as the simple triangular lattice considered here,
+has important implications. For instance, it simplifies the
+generation of topological bands in the context of ultracold
+atoms. Up to now, two-dimensional topological models
+in optical lattices have been realised either in bipartite
+lattices [10, 12, 60], square lattices with moving superlat-
+tices [8, 9, 11], or by employing spin-orbit coupling [62].
+These implementations rely on the relative phase stabil-
+ity between lattice or Raman laser beams, leading to a
+significant technological overhead. An insufficient phase
+stability may have been an obstacle for realising strongly
+correlated phases in two-dimensional topological lattices.
+The triangular lattice, on the contrary, can be built by
+simply superimposing three standing waves in the plane
+with no active stabilisation. Therefore, the 1ω–2ω Flo-
+quet scheme could provide an avenue towards realising
+correlated topological states of matter. In addition to its
+conceptual simplicity, the two-tone driving leads to the
+appearance of Berry curvature along a ring-shaped gap,
+which can be asymmetrically distributed (Fig. 2f), partic-
+ularly in the topologically trivial regions of Fig. 3c. This
+supports a non-zero Berry curvature dipole, enabled by
+periodic driving, although the underlying lattice is trivial
+FIG. 4.
+Off-resonant driving in the hexagonal lattice (a)
+with tunnelings t1 = t2 = t3 = 0.06 in units of Erec. The
+static band structure features two degenerate Dirac points
+(b). Floquet driving lifts the degeneracy and opens a band
+gap (c) by effectively breaking symmetries.
+and inversion-symmetric. This finding is relevant beyond
+the field of ultracold atoms and it could enable the obser-
+vation of nonlinear Hall effects which so far have relied
+on materials with broken inversion symmetry [63].
+IV.
+OFF-RESONANTLY DRIVEN HEXAGONAL
+LATTICE
+The second application of the two-frequency driving
+method considers the hexagonal lattice in brick-wall con-
+figuration, shown in Fig. 4a.
+This potential can be
+realised in the same setup as the triangular lattice of
+sec. III, but the derivation also applies to 120◦ honey-
+comb lattices [12] as well as real graphene [17].
+We start from the tight-binding Hamiltonian
+ˆHq(τ) =
+� 3
+�
+i=1
+ti cos [q · bi + θi(τ)]
+�
+σx+
+� 3
+�
+i=1
+ti sin [q · bi + θi(τ)]
+�
+σy,
+(10)
+The nearest-neighbour tunneling vectors are b1 = (0, 1)b,
+b2 = (−1, 1)b and b3 = (−1, 0)b in the rotated coordi-
+nate system {ˆe′
+x, ˆe′
+y} (Fig. 4a). The waveforms of Eq. 4
+result in the following Peierls phases:
+θ1(τ) = −Ky1 sin(ωτ + ϕy1) − Ky2 sin(2ωτ + ϕy2) = −θ3(τ),
+θ2(τ) = Kx1 sin(ωτ) + Kx2 sin(2ωτ + ϕx2),
+(11)
+with the driving strengths being Kx1 = mωAx1a/ℏ,
+Kx2 = 2mωAx2a/ℏ, Ky1 = mωAy1a/ℏ, and Ky2 =
+2mωAy2a/ℏ, as before. Since we do not consider inter-
+banding couplings here, there is no need to apply uni-
+taries and the high-frequency expansion can be directly
+conducted.
+Single-frequency elliptical driving. The choice Kx2 =
+Ky2 = 0, Kx1 > 0 corresponds to elliptical driving, giv-
+ing rise to topological or trivial bands. For any Ky1 > 0
+and ϕy1 mod π ̸= 0 we recover the well-known topo-
+logical gap opening at both Dirac points due to time-
+reversal symmetry breaking (Fig. 4b-c) [64]. The limits
+of Ky1 = 0 or ϕy1 mod π = 0 correspond to linearly
+polarised driving, leaving time-reversal symmetry intact
+and both Dirac points remain closed. However, the triv-
+ial region with C = 0 is only a ‘line’ of zero extent in the
+phase diagram (Fig. 5a, as before in Fig. 3b). Usually, a
+manifestly broken inversion symmetry of the underlying
+lattice is required to cross a topological phase transition
+and induce Haldane’s ‘parity anomaly’, thereby extend-
+ing the trivial ‘lines’ in the phase diagram to finite re-
+gions [2, 10]. In the following, we demonstrate that in-
+version symmetry can be broken by employing the two-
+tone Floquet drive only. Therefore, a topological phase
+transition between trivial and non-trivial regions can be
+achieved purely via driving.
+
+a)
+t1
+3
+α1
+q(π/b)
+(元/b）
+b)
+-0.5-1
+0.5-1
+0
+0.5
+0.5
+0.5
+0.5
+(zHD)
+(zHD)
+0.0
+0.0
+3
+3
+-0.5
+-0.5
+0.5
+-0.5
+-0.5
+qr(π
+x(元/b)6
+FIG. 5.
+Phase diagrams of the hexagonal lattice under off-resonant driving (ℏω = 20.8t1 = 1.25Erec). Single-frequency
+elliptical driving (Ky1 = 0.9, Kx2 = Ky2 = 0) can break time-reversal symmetry and result in a non-zero Chern number (a).
+The breaking of the inversion and time-reversal symmetries can be independently controlled by adding a linearly polarized field
+(Ky1 = Ky2 = 0.9, Kx2 = 0 and ϕx2 = ϕy2 = 0) (b), as well as using two circular fields with opposite polarization (Kx1 = Ky1,
+Kx2 = Ky2 = 0.7, ϕy1 = π/2 and ϕy2 = ϕx2 − π/2) (c). The transition paths I - II - III marked in (b) and (c) correspond to
+the Lissajous curves in Fig. 1d-f and Fig. 1g-i, respectively.
+Two-frequency driving. Starting from the above case
+of single-frequency elliptical driving, we add linearly po-
+larized ‘light’ by setting Kx2 = 0 while Ky2 ̸= 0. The
+driving field further breaks inversion symmetry and the
+competition between the breaking of inversion and time-
+reversal symmetry can be observed in Fig. 5b-c.
+The
+topological phase transition can simply be induced by
+tuning the phase along the marked path, corresponding
+to the Lissajous curves in Fig. 1d-f.
+Alternatively, constraining Kx1 = Ky1, Kx2 = Ky2,
+ϕy1 = π/2, and ϕy2 = ϕx2 − π/2 in Eq. 4, the 1ω − 2ω
+driving scheme corresponds to two counter-rotating, cir-
+cularly polarized fields. Clearly, when one circular driv-
+ing field is much larger than the other, the resulting
+Chern number will correspond to the helicity of the dom-
+inant field. However, when the amplitude of one circu-
+lar field approaches the other one, the Lissajous curve
+becomes a ‘cloverleaf’ pattern and the breaking of time-
+reversal symmetry is suppressed (Fig. 1g-i). This compe-
+tition allows to interpolate between topologically trivial
+and non-trivial settings (with either positive or negative
+mass terms) purely by changing driving phases. The re-
+sulting gap sizes are similar to the known results in the
+Haldane model [10]. Interestingly, we found that a pure
+Dirac Hamiltonian subject to two-frequency driving does
+not show a corresponding gap opening in the case of bro-
+ken inversion symmetry. While time-reversal symmetry
+breaking via circular driving can be captured within the
+approximate Dirac Hamiltonian [17], the inversion sym-
+metry breaking relies on the presence of a lattice po-
+tential. Mathematically, this can be seen from the fact
+that the driven Dirac Hamiltonian remains linear and
+its Fourier components at ω and 2ω will not mix in the
+high-frequency expansion. Conceptually, a specific driv-
+ing pattern favours a gap opening specific to one sublat-
+tice of the honeycomb, which does not apply to the Dirac
+Hamiltonian.
+We also studied the behaviour of inversion symmetry
+breaking beyond the validity of the high-frequency ex-
+pansion, using numerically calculated band structures.
+Here, the gap opening scales roughly as 1/ω4 which be-
+comes dominant in the low-frequency regime.
+There-
+fore, the 1ω–2ω scheme could be particularly interest-
+ing for phase-controlled topological transitions in laser-
+driven graphene, which could be readily implemented in
+setups such as used in ref. [17].
+V.
+CONCLUSION
+We have shown that two-frequency Floquet driving in
+two dimensions gives rise to rich topological phase dia-
+grams, both for resonant and off-resonant modulation.
+We identify the relative phase between the 1ω and 2ω
+drives as an important control parameter over the sym-
+metry properties and the resulting topologies in the effec-
+tive Floquet Hamiltonians. In many experiments, phase-
+only control is advantageous over other parameters, such
+as frequencies or amplitudes. For periodically driven real
+materials, such as graphene [17], incoherent excitations
+and electron relaxation can strongly affect the measure-
+ments, potentially obscuring any topological response.
+Incoherent effects often depend on the polarisation of
+the incident light, especially close to the contacts of a
+sample. Several of the driving schemes proposed in this
+work, particularly the case of counter-rotating, circular
+drives, depend only on the relative phase between the
+light fields. Therefore, we believe that our approach will
+
+a)
+lw elliptical driving
+b)
+lw elliptical + 2w linear driving
+c)
+lw + 2w circular driving
+CIII
+C=-1
+π-2
+C=-1
+≤2
+C=-1
+I
+Phase Py1
+Phase Py1
+C=0
+C=0
+I
+0.5
+C=1
+0.25
+C=1
+C=1
+一元L
+一元L
+-2
+0.25
+0.5
+0.75
+0.25
+0.5
+一π元
+0
+0
+0
+0.75
+1
+Driving strength Kx1
+Driving strength Kx1
+Phase x27
+lead to novel applications in manipulation and detection
+of topological phenomena.
+By combining the schemes
+considered here with spin-dependent Floquet engineer-
+ing [48], it may become possible to access the entire ‘pe-
+riodic table’ of topological insulators [65].
+ACKNOWLEDGMENTS
+We acknowledge funding by the Swiss National Sci-
+ence Foundation (Grants No. 182650 and NCCR-QSIT)
+and European Research Council advanced grant TransQ
+(Grant No. 742579).
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+
+10
+Appendix A: Optical lattice potential
+Lattices of various geometric configurations can be readily realised in the setup of ref. [66], that is,
+Vlat(x, y) =
+− Vx cos(kLx + θ/2)2 − Vx cos(kLx)2 − Vy cos(kLy)2
+− 2
+�
+VxVy cos(kLx) cos(kLy) cos(φ),
+(A1)
+where kL = 2π/λ. The spacing between neighbouring minima of a one-dimensional standing wave is a = λ/2, the
+recoil energy is Erec = (ℏkL)2/(2m), the wavelength λ is 1064 nm, and the atoms are potassium-40. The following
+choice of parameters: Vx = 0.6, Vx = 0.3, Vy = 2.8 all in units of Erec, θ = π, and φ = 0 yields a triangular lattice.
+The hexagonal lattice can be realized by setting Vx = 12, Vx = 0.8 and Vy = 4.65. Compared to the tight-binding
+lattices, we have the length of the unit cells b =
+√
+2a = λ/
+√
+2. The tight-binding parameters can be evaluated from
+the realistic lattice potential (Appendix B), yielding t1 = −0.05, t2 = −0.07, t3 = −0.07, ε = −0.66 in units of Erec
+(triangular lattice, ηsp = 0.19) and t1 = t2 = t3 = 0.06 in units of Erec (hexagonal lattice).
+Appendix B: Derivation of the Hamiltonian
+We start from Hamiltonian in co-moving frame,
+ˆHcm(τ) = ˆp2
+2m + Vlat(ˆr) − F(τ) · ˆr ,
+(B1)
+where ˆp and ˆr are momentum and position operators, respectively, and F = −m¨rm(τ) = − ˙A(τ). The Hamiltonian has
+two parts. The first two terms include the static lattice Hamiltonian and the last term represents a time-dependent
+dispersion.
+Next, we expand ˆHcm(τ) in Wannier basis [67],
+wn,R(r) ≡ wn(r − R) =
+V
+(2π)2
+�
+BZ
+e−ik·Rφn,k(r) dk ,
+(B2)
+where n is band index, φn,k(r) = eikrun,k(r) is Bloch state, R = n1a1 + n2a2 is arbitrary lattice vector, and a1, a2
+are primitive translation vectors in two dimensions.
+In second quantization, we can rewrite the field operator in Wannier basis as
+ˆψ(r) =
+�
+n,R
+w∗
+n,R(r)ˆan,R ,
+(B3)
+where ˆan,R is the annihilation operator.
+The first part of the co-moving Hamiltonian can be expanded as
+ˆH0(τ) =
+�
+n,m,R,R’
+ˆa†
+n,Rˆam,R’
+�
+wn,R(r)
+� ˆp2
+2m + Vlat(ˆr)
+�
+w∗
+m,R’(r) dr .
+(B4)
+Using the fact that the static Hamiltonian cannot mix different bands and the Wannier functions are exponentially
+localized, we only consider the onsite and nearest-neighbour couplings in individual bands,
+ˆH0(τ) =
+�
+n,R
+�
+ˆa†
+n,Rˆan,Rεn +
+�
+bi
+�
+ˆa†
+n,Rˆan,R+biti + h.c.
+��
+,
+(B5)
+εn =
+�
+wn,R(r)
+� ˆp2
+2m + Vlat(ˆr)
+�
+w∗
+n,R(r) dr ,
+(B6)
+ti =
+�
+wn,R(r)
+� ˆp2
+2m + Vlat(ˆr)
+�
+w∗
+n,R+bi(r) dr .
+(B7)
+
+11
+bi = ja1 + ka2 are the nearest-neighbour tunneling vectors. In the configuration of triangular and hexagonal lattices,
+i ∈ {1, 2, 3}. εn is the band center energy and ti is the nearest-neighbour tunneling amplitude.
+Now we consider the driving term which can couple different bands on-site but has negligible inter-site effects, i.e.
+ˆH1(τ) = −
+�
+n,m,R,R’
+ˆa†
+n,Rˆam,R’
+�
+wn,R(r) [F(τ) · ˆr] w∗
+m,R’(r) dr
+= −
+�
+n,R
+ˆa†
+n,Rˆan,R
+�
+wn,R(r) [F(τ) · ˆr] w∗
+n,R(r) dr −
+�
+n,m̸=n,R
+ˆa†
+n,Rˆam,R
+�
+wn,R(r) [F(τ) · ˆr] w∗
+m,R(r) dr
+= −
+�
+n,R
+�
+�ˆa†
+n,Rˆan,RF(τ) · R +
+�
+m̸=n
+ˆa†
+n,Rˆam,RaF(τ) · ηnm ˆr′
+�
+� .
+(B8)
+Here we define the inter-band coupling as
+ηnm = 1
+b
+����
+�
+wn,R(r)ˆrw∗
+m,R(r) dr
+���� .
+(B9)
+By numerically evaluating ηsp according to the method in [66, 68], we justify that the dipole-like coupling term
+�
+wm,R(r)ˆrw∗
+n,R(r) dr between s and p band is along the direction ˆr′ = ˆr = ˆex+ˆey. In addition, we evaluate tunnelings
+and band center energies through calculating the Wannier functions of atoms in a realistic lattice potential.
+The on-site driving term ˆa†
+n,Rˆan,RF(τ) · R breaks translation symmetry, so we need to rotate it away by applying
+the unitary
+ˆU(τ) = exp
+�
+−i
+�
+Rn
+χR(τ)ˆa†
+n,Rˆan,R
+�
+,
+(B10)
+ˆH′(τ) = ˆU †(τ) ˆH(τ) ˆU(τ) − iℏ ˆU †(τ) ∂
+∂τ
+ˆU(τ),
+(B11)
+where χR(τ) = m
+ℏ R · ˙rm(τ). Then, transforming the Hamiltonian into quasi-momentum space and only considering
+the lowest two bands, we get the final Hamiltonian,
+ˆHq(τ) =
+�
+n∈{s,p}
+�
+εnˆa†
+nˆan +
+Z
+�
+i=1
+�
+tn,i
+�
+ei(θi(τ)+q·bi)ˆa†
+nˆan + h.c.
+��
+− F(τ) · ˆr
+�
+m∈{s,p}̸=n
+ηnmˆa†
+nˆam
+�
+,
+(B12)
+with the time-periodic Peierls phase θi(τ) = χR+bi(τ) − χR(τ) = m
+ℏ bi · ˙rm(τ).
+Appendix C: Analytical Hamiltonian by the high-frequency expansion
+In the following, we write down the effective Floquet Hamiltonian, analytically derived using the high-frequency
+expansion (HFE) in tight-binding approximation. The idea of the HFE is to decompose the original time-dependent
+Hamiltonian into Fourier components and obtain the effective Hamiltonians, then neglect the small high frequency
+terms. The effective Hamiltonian is decomposed into terms of different orders,
+ˆHeff =
+∞
+�
+n=0
+ˆH(n)
+eff ,
+(C1)
+which is in inverse power of ω, i.e.
+ˆH(n)
+eff
+∼ ω−n. If the frequency is much higher than other energy scales in the
+system, the high-order effective Hamiltonian can be truncated. Here we only use the first two orders [52],
+ˆH(0)
+eff = 1
+T
+� T
+0
+ˆH(τ) dτ ≡ ˆH0 ,
+(C2)
+
+12
+ˆH(0)
+eff
+σxηspℏω[Kx + Ky cos(ϕ)]/2
+σyηspℏωKy sin(ϕ)/2
+σz{ε + ℏω/2 + t1 cos[(qx − qy)b]J0(2Kx) + t2 cos(qxb)[J0(Kx)J0(Ky) − 2J1(Kx)J1(Ky) cos(ϕ)+
+2J2(Kx)J2(Ky) cos(2ϕ)] + t3 cos(qyb)[J0(Kx)J0(Ky) + 2J1(Kx)J1(Ky) cos(ϕ) + 2J2(Kx)J2(Ky) cos(2ϕ)]}
+ˆH(1)
+eff,2
+σxηsp{−t1 cos[(qx − qy)b]J2(2Kx)[Kx + Ky cos(ϕ)] − t2 cos(qxb){J1(Kx)J1(Ky)[Ky + Kx cos(ϕ)]+
+J0(Ky)J2(Kx)[Kx + Ky cos(ϕ)] + J0(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]} + t3 cos(qyb){J1(Kx)J1(Ky)
+[Ky + Kx cos(ϕ)] − J0(Ky)J2(Kx)[Kx + Ky cos(ϕ)] − J0(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]}}/2
+σyηsp sin(ϕ){t1 cos[(qx − qy)b]J2(2Kx)Ky − t2 cos(qxb){J1(Kx)J1(Ky)Kx − J0(Ky)J2(Kx)Ky + J0(Kx)J2(Ky)
+[Ky + 2Kx cos(ϕ)]} + t3 cos(qyb){J1(Kx)J1(Ky)Kx + J0(Ky)J2(Kx)Ky − J0(Kx)J2(Ky)[Ky + 2Kx cos(ϕ)]}}/2
+σzη2ℏω[K2
+x + K2
+y + 2KxKy cos(ϕ)]/8
+TABLE I. Effective Hamiltonian for the triangular lattice under single frequency driving resonant with the gap
+ˆH(1)
+eff = 1
+ℏω
+∞
+�
+l=1
+1
+l
+�
+ˆHl, ˆH−l
+�
+=
+∞
+�
+l=1
+ˆH(1)
+eff,l .
+(C3)
+Here, ˆHl are the Fourier components of the time-dependent Hamiltonian, ˆH(τ) = �∞
+l=−∞ ˆHleilωτ.
+Table I is the effective Hamiltonian for the triangular lattice when the driving field is resonant with the gap energy.
+The constant terms are much larger than the quasimomentum dependent terms in off-diagonal, which means the
+on-site inter-band couplings are dominant. We safely neglect the Bessel functions higher than the second order as the
+driving strength K ≤ 1 remains in weak driving regime.
+Table II shows the case of single-frequency driving resonant with the half the gap. Now the off-diagonal terms are
+dominantly quasimomentum dependent. The Hamiltonian for the triangular lattice in two-frequency driving scheme
+are not listed here.
+ˆH(0)
+eff
+σz{ε + ℏω + t1 cos[(qx − qy)b]J0(2Kx) + t2 cos(qxb)[J0(Kx)J0(Ky) − 2J1(Kx)J1(Ky) cos(ϕ)+
+2J2(Kx)J2(Ky) cos(2ϕ)] + t3 cos(qyb)[J0(Kx)J0(Ky) + 2J1(Kx)J1(Ky) cos(ϕ) + 2J2(Kx)J2(Ky) cos(2ϕ)]}
+ˆH(1)
+eff,1
+σxηsp{−t1 sin[(qx − qy)b]J1(2Kx)Ky sin(ϕ) − t2 sin(qxb) sin(ϕ){KyJ0(Ky)J1(Kx) + KxJ1(Ky)J2(Kx)−
+J1(Kx)J2(Ky)[Ky + 2Kx cos(ϕ) + 2Ky cos(2ϕ)] + J0(Kx)J1(Ky)[Kx + 2Ky cos(ϕ)]}
+−t3 sin(qyb) sin(2ϕ){KyJ0(Kx)J1(Ky) + J1(Kx)J2(Ky)[Kx + 2Ky cos(ϕ)]}}
+σyηsp{t1 sin[(qx − qy)b]J1(2Kx)[Kx + Ky cos(ϕ)] + t2 sin(qxb){J0(Ky)J1(Kx)[Kx + Ky cos(ϕ)]−
+J1(Ky)J2(Kx)[Ky + Kx cos(ϕ)] + J1(Ky)J0(Kx)[Kx cos(ϕ) + Ky cos(2ϕ)] − J1(Kx)J2(Ky)
+[Kx cos(2ϕ) + Ky cos(3ϕ)]} + t3 sin(qyb) sin(2ϕ){−J0(Ky)J1(Kx)[Kx + Ky cos(ϕ)] − J1(Ky)J2(Kx)
+[Ky + Kx cos(ϕ)] + J1(Ky)J0(Kx)[Kx cos(ϕ) + Ky cos(2ϕ)] + J1(Kx)J2(Ky)[Kx cos(2ϕ) + Ky cos(3ϕ)]}}
+σzη2
+spℏω[K2
+x + K2
+y + 2KxKy cos(ϕ)]/4
+ˆH(1)
+eff,3
+σxηsp{−t2 sin(qxb){KxJ1(Ky)J2(Kx) + J1(Kx)J2(Ky)[Ky + 2Kx cos(ϕ)]}
++t3 sin(qyb){−KxJ1(Ky)J2(Kx) + J1(Kx)J2(Ky)[Ky + 2Kx cos(ϕ)]}}/3
+σyηsp{t2 sin(qxb){J1(Ky)J2(Kx)[Ky + Kx cos(ϕ)] + J1(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]}
++t3 sin(qyb){J1(Ky)J2(Kx)[Ky + Kx cos(ϕ)] − J1(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]}}/3
+σzη2
+spℏω[K2
+x + K2
+y + 2KxKy cos(ϕ)]/12
+TABLE II. Effective Hamiltonian for the triangular lattice under single frequency driving resonant with the half the gap
+For the hexagonal lattice, Table III gives the general Hamiltonian under single-frequency off-resonant driving, while
+Table IV is the Hamiltonian under two-frequency off-resonant driving. Note that the second order Bessel functions
+are omitted in ˆH(1)
+eff in Table IV but kept in the numerical calculations.
+
+13
+ˆH(0)
+eff
+σx{[t1 cos(qyb) + t3 cos(qxb)]J0(Ky1) + t2 cos[(qx − qy)b]J0(Kx1)}
+σy{[t1 sin(qyb) − t3 sin(qxb)]J0(Ky1) − t2 sin[(qx − qy)b]J0(Kx1)}
+ˆH(1)
+eff
+−σz
+�∞
+n=1
+4
+nℏω t2Jn(Kx1)Jn(Ky1)
+�
+(−1)n+1t1 sin(qxb) + t3 sin(qyb)
+�
+sin(nϕy1)
+TABLE III. Effective Hamiltonian for the hexagonal lattice under single frequency off-resonant driving
+ˆH(0)
+eff
+σx{[t1 cos(qyb) + t3 cos(qxb)]J0(Ky1)J0(Ky2) + t2 cos[(qx − qy)b]J0(Kx1)J0(Kx2)−
+2[t1 sin(qxb) + t3 sin(qyb)]J1(Ky2)J2(Ky1) sin(2ϕy1 − ϕy2) + 2t2 sin[(qx − qy)b]J1(Kx2)J2(Kx1) sin(ϕx2)}
+σy{[t1 sin(qyb) − t3 sin(qxb)]J0(Ky1)J0(Ky2) − t2 sin[(qx − qy)b]J0(Kx1)J0(Kx2)+
+2[t1 cos(qxb) − t3 cos(qyb)]J1(Ky2)J2(Ky1) sin(2ϕy1 − ϕy2) + 2t2 cos[(qx − qy)b]J1(Kx2)J2(Kx1) sin(ϕx2)}
+ˆH(1)
+eff
+σz
+2
+3ℏω {6J0(Ky2)J1(Ky1)[t2J1(Kx1)J1(Kx2)(t1 cos(qxb) − t3 cos(qyb)) cos(ϕx2 − ϕy1) + (t2
+1 − t2
+3)
+J1(Ky1)J1(Ky2) cos(2ϕy1 − ϕy2)] − 6t2J0(Kx2)J1(Kx1)[t2J1(Kx1)J1(Kx2) cos(ϕx2) + J1(Ky1)J1(Ky2)
+(t1 cos(qxb) + t3 cos(qyb)) cos(ϕy1 − ϕy2) − J1(Ky1)J0(Ky2)(t1 sin(qxb) + t3 sin(qyb)) sin(ϕy1)]
+−t2J1(Kx2)J1(Ky2)[3J0(Kx1)J0(Ky1)(t1 sin(qxb) + t3 sin(qyb)) sin(ϕx2 − ϕy2) − 2J1(Kx1)J1(Ky1)
+(t1 sin(qxb) − t3 sin(qyb))(sin(ϕx2 − ϕy1 − ϕy2) + 3 sin(ϕx2 + ϕy1 − ϕy2))]}
+TABLE IV. Effective Hamiltonian for the hexagonal lattice under two frequency off-resonant driving
+
diff --git a/mtE4T4oBgHgl3EQfuA2V/content/tmp_files/load_file.txt b/mtE4T4oBgHgl3EQfuA2V/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..82d7ee325e256d7de6ec7c5e317096356edaf87a
--- /dev/null
+++ b/mtE4T4oBgHgl3EQfuA2V/content/tmp_files/load_file.txt
@@ -0,0 +1,742 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf,len=741
+page_content='Topological Floquet engineering using two frequencies in two dimensions  Yixiao Wang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1 Anne-Sophie Walter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1 Gregor Jotzu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='2 and Konrad Viebahn1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' ∗  1Institute for Quantum Electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' ETH Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 8093 Zurich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Switzerland  2Max Planck Institute for the Structure and Dynamics of Matter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 22761 Hamburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Germany  (Dated: January 13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2023)  Using two-frequency driving in two dimensions opens up new possibilites for Floquet engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  which range from controlling specific symmetries to tuning the properties of resonant gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In  this work, we study two-band lattice models subject to two-tone Floquet driving and analyse the  resulting effective Floquet bandstructures both numerically and analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' On the one hand, we  extend the methodology of Sandholzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' [10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1103/PhysRevResearch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='013056] from one to two  dimensions and find competing topological phases in a simple Bravais lattice when the two resonant  drives at 1ω and 2ω interfere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' On the other hand, we explore driving-induced symmetry breaking  in the hexagonal lattice, in which the breaking of either inversion or time-reversal symmetry can  be tuned independently via the Floquet modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Possible applications of our work include a  simpler generation of topological bands for ultracold atoms, and the realisation of non-linear Hall  effects as well as Haldane’s parity anomaly in inversion-symmetric parent lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  INTRODUCTION  Topological phenomena emerge naturally for electrons  under the effect of strong magnetic fields, exemplified by  the quantum Hall effect (QHE) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In these phenom-  ena, the underlying physical mechanism is the breaking  of time-reversal symmetry due to the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Sev-  eral years after the discovery of the QHE it was noticed  that topological insulators can also arise without an am-  bient magnetic field, such as the anomalous quantum Hall  state [2] and the quantum spin Hall effect [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Today,  anomalous topological phases can be realised by Floquet  engineering, that is, periodic driving of a quantum sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' So far, Floquet engineered topological phenomena  have been largely limited to single-frequency protocols,  applied to optical lattices [4–13], real materials [14–17],  photonic and plasmonic waveguides [18–20], and other  synthetic systems [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Recently, bichromatic and  multi-frequency Floquet engineering has emerged as a  powerful strategy to enhance the driving capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  These include time-reversal or spatial symmetry break-  ing [25–35] and interference between different Floquet  harmonics [36–42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The combination of both two-path  interference and time-reversal symmetry breaking has  led to the proposal [43] and experimental demonstra-  tion [44] of a topological pump in one-dimensional lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  To our knowledge, all experimental results in the multi-  frequency regime have been limited to one-dimensional  driving patterns, motivating the need to find novel strate-  gies for two-dimensional driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In this work, we use the idea of two-tone Floquet engi-  neering to generate novel topological bandstructures and  topological phase transitions from simple parent lattices  in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' On the one hand, we consider reso-  nant driving for the lowest two bands of a triangular lat-  tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Although this model only features a single potential  ∗ viebahnk@phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='ch  minimum per unit cell, we are able to drive topological  transitions by tuning the amplitude and relative phase  between 1ω and 2ω drives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The simplicity of this scheme  could open up new pathways for realising strongly cor-  related topological insulators, which have remained out  of reach in existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  On the other hand, we  apply two-frequency driving to an inversion-symmetric  (graphene-like) hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' This driving scheme  enables the selective breaking of inversion symmetry  while maintaining time-reversal symmetry, giving access  to a new class of Floquet Hamiltoninans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Thus, we find  that an external breaking of inversion symmetry is not  necessary to drive the celebrated parity anomaly in the  Haldane model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The two-frequency approach generi-  cally applies to Floquet-driven systems in lattices, rang-  ing from condensed matter to synthetic quantum matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The relevant two-tone driving waveforms, and possible  experimental implementations, are introduced in section  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Afterwards, we show how resonant 1ω–2ω driving in  a triangular lattice leads to topological phase transitions  (section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Section IV discusses the effective breaking of  time-reversal and inversion symmetry under off-resonant  driving in a hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  1ω–2ω FLOQUET DRIVING IN TWO  DIMENSIONS  Two-frequency driving can be applied, in princple, to  any physical system which has been employed for Floquet  engineering so far [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' As a generic starting point, we  consider a particle confined to a static potential Vlat(r),  and subject to an oscillating force  ˆHcm(τ) = ˆp2  2m + Vlat(r) − F(τ) · ˆr ,  (1)  with  F(τ) ≡ eE(τ) = −m¨rm(τ) = − ˙A(τ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (2)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='05229v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='quant-gas]  12 Jan 2023  2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Single- and two-frequency Lissajous curves for two-  dimensional driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The plots show the real-space modu-  lation figures rm(τ) (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' These shapes are topologically  equivalent to those of the time-dependent force F(τ), since the  two are related to one another by taking the second derivative  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' (a-c) Circular strong driving at 1ω (a), circular weak  driving 2ω (b), or both drives combined (c) can be used to  resonantly address a higher-band transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The thin line in  (c) represents the dominant 1ω waveform, the same as (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (d-f) Superposition of an elliptical driving field at 1ω and a  linear field at 2ω with relative phases ϕy1 = 0 (d), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='2π (e),  and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5π (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' (g-i) Superposition of two circular drives with  relative phase ϕx2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='9π, ϕy1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5π, ϕy2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='4π, driving  strength Kx2 = Ky2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='7, and Kx1 = Ky1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5 (g), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='65  (h), and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='8 (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In solid state systems, the driving force F(τ) can directly  correspond to the oscillating field E(τ) of a laser coupling  to free electrons [17, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Equivalently, the application of  periodic driving can be seen as ‘minimal coupling’ using  the vector potential as q → q−A(τ)/ℏ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Furthermore, col-  lective modes such as phonons [46] or magnons [47] can  also be involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In waveguide-based synthetic lattices,  transverse movements of the waveguides provide an ef-  fective inertial force [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' For optical lattices, the driving  force can be provided by an oscillating optical or mag-  netic gradient [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Alternatively, an inertial force can  result from a periodic displacement of the entire lattice  potential, an approach known as ‘lattice shaking’ [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In this case, the Hamiltonian in the lab frame is  ˆHlab(τ) = ˆp2  2m + Vlat [r − rm(τ)] ,  (3)  related to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2 by two unitary transformations [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In  summary, the derivations in this work directly carry over  from engineered quantum platforms, such as ultracold  atoms in optical lattices, to condensed matter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  We define the two-dimensional driving pattern rm(τ)  as  xm(τ) =  Ax1 cos(ωτ)  +Ax2 cos(2ωτ + ϕx2)  (4)  ym(τ) =Ay1 cos(ωτ + ϕy1)+Ay2 cos(2ωτ + ϕy2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  These two-tone driving waveforms are visualised as Lis-  sajous curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1 for the relevant choices of driving  parameters (phases and amplitudes) used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The amplitude of the vector potential is given by the di-  mensionless driving strength Kαβ = mωβAαβa/ℏ, where  a is the lattice spacing and Aαβ is the real-space am-  plitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The index α ∈ {x, y} denotes the direction,  whereas β ∈ {1, 2} indicates the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Resonant circular 1ω–2ω driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The combination of  a strong circular 1ω drive (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1a, ϕy1 = π/2) with a  weak circular 2ω drive (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1b, ϕx2 = 0, ϕy2 = π/2)  leads to a modulated circular pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Due to  two-path interference when addressing the s–p resonance  of the triangular lattice, this driving pattern can be used  to drive topological transitions in two dimensions, similar  to the topological pump of ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Off-resonant 1ω–2ω driving to access spatial and tem-  poral symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Alternatively, an elliptical driving field  at 1ω and a linearly polarised laser at 2ω (Ax2 = 0, ϕy2 =  0) leads to a competition between the breaking of in-  version symmetry and time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' For in-  stance, the choice ϕy1 = 0 results in a ‘boomerang’  pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1d), which breaks inversion symmetry but  not time-reversal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Conversely, the case of  ϕy1 = π/2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1f) gives a rounded ‘kite’ shape in which  the breaking of time-reversal symmetry dominates over  the breaking of inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Tuning the phase  ϕy1 thus allows to interpolate between the two symme-  try breaking situations (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1e), thereby accessing novel  topological regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Instead of combining linear and elliptical drives,  the competition between inversion-breaking and time-  reversal-breaking can be achieved by superimposing two  circular drives of opposite helicity (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1g-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The way in  which inversion symmetry is broken in this case depends  on the relative orientation of the driving pattern to the  lattice geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' This can be tuned by increasing ϕx2  and ϕy2 by the same amount (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' by delaying the two  frequency drives with respect to each other), which ro-  tates the driving pattern around the origin (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1g, see  also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' [29, 33, 34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Driving a topological transition  via changing only a relative phase can be experimentally  advantageous in condensed matter systems where chang-  ing polarisation or amplitude can lead to spurious effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Experimental implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In this work we focus  on a triangular and a hexagonal lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' A corresponding  optical lattice potential, and the resulting tight-binding  parameters, can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In waveguide-  based systems, the lattice geometry can generally be cho-  a)  c)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  a  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  time t (2π/w)  V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  x  y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  y (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=')  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  d)  f)  e  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  a  time  time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  x  x  x  y  y  y  h)  g)  i)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  T  time  time  time  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  Q  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  x  y  x  y  x  y3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Resonant band coupling in the triangular lattice (a),  with relevant parameters t1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='05, t2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='07, t3 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='07,  ε = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='66 in units of Erec and dimensionless inter-band cou-  pling ηsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Resonant one-photon or two-photon pro-  cesses lead to the coupling of lower and higher bands, corre-  sponding to the green and black arrows in (b), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Floquet driving lifts the degeneracy of the static band struc-  ture in the rotating frame (c) and opens a band gap (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The  ring-shape minima of the band structure leads to a ring-shape  Berry curvature (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (f) shows a trivial spatially localized  Berry curvature under two-frequency driving, which features  a strong Berry curvature dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  sen freely [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The time-dependence of the driving term  is encoded in an additional spatial coordinate, which can  accommodate multi-frequency driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Finally, in solid  state systems, the laser field corresponding to the driving  term at 2ω can be generated via second-harmonic gener-  ation, ensuring that it is phase-stable with respect to the  1ω term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The relative phase between the two can then  be tuned using a dispersive material of varying thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Tight-binding Hamiltonians for two-dimensional Flo-  quet engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  By expanding the original Hamilto-  nian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1) in a Wannier basis and transforming it to  quasimomentum space, we get the following expression  �  n  �  εnˆa†  nˆan −  Z  �  i=1  �  tn,i  �  ei(θi(τ)+q·bi)ˆa†  nˆan + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ��  − F(τ) · ˆr  �  n′  ηnn′ˆa†  nˆan′  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (5)  Here, εn, tn,i, ηnn′ are the band centre energies, nearest-  neighbour tunnelling matrix elements, and interband  coupling elements, respectively, for bands n, n′ (Ap-  pendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' bi = ja1+ka2 are the nearest-neighbour tun-  neling vectors and i indexes the in-equivalent neighbours  up to the coordination number Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' a1, a2 correspond to  the primitive translation vectors in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The  time-periodic Peierls phases are θi(τ) =  m  ℏ bi · ˙rm(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In second quantisation, ˆan denotes the annihilation op-  erator in band n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The first line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 5 describes  static and time-dependent contributions to intra-band  processes whereas the second line is the inter-band cou-  pling that arises from periodically forcing the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The coupling elements ηnn′ can be understood as dipole  matrix elements of Bloch states [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Consequently, the  first line is diagonal in quasimomentum and band index  n, whereas the second line contributes off-diagonal ele-  ments to the Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  We use the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 5 as the starting point  for the Floquet analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' If the periodic drive resonantly  addresses the gap between s and p bands, we addition-  ally employ a unitary transformation to ‘rotate away’ the  energy difference (sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In order to evaluate the effec-  tive Floquet Hamiltonian, we employ two complementary  methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  On the one hand, we calculate the effective  Hamiltonian analytically in inverse powers of the driving  frequency ω using the well-known high-frequency expan-  sion (HFE) [50–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Since we work in the weak driv-  ing regime the HFE remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' On the other hand,  we can numerically obtain the effective Hamiltonian via  the Trotter decomposition [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' We use this numerical  method to validate the results of the analytic calcula-  tion and to justify the truncation of the high-frequency  expansion a posteriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' To characterise the band topolo-  gies, the Berry curvature is evaluated via the eigenstates  in the discretised two-dimensional Brillouin zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The re-  sulting Chern number is the surface integral of the Berry  curvature over the closed Brillouin torus [56, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  RESONANTLY DRIVEN TRIANGULAR  LATTICE  The first situation we consider is a simple triangular  lattice in which the lowest two bands (s and p) are reso-  nantly coupled with two frequencies, inspired by ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' [43]  and building on previous works using single-frequency  driving [58–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In the non-shaken case the static Hamiltonian for a  triangular tight-binding model is  ˆH =  �  ε +  3  �  i=1  ti cos(q · bi)  �  σz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (6)  The nearest-neighbour tunneling vectors are b1  =  (−1, 1)b, b2 = (1, 0)b, b3 = (0, 1)b and q = (qx, qy)  in the rotated coordinate ˆe′  x =  ˆex+ˆey  √  2 , ˆe′  y =  −ˆex+ˆey  √  2  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The simplest way to couple s and p bands is  a)  e  X  (元 / b  b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5-1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  3  0  nw  (kHz)  2hw  hw  3  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=':  X  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  qx(元/b)  d)  e  q(元/b)  1×(元/b)  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5-1  =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5-1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  (zH)  (kHz)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  3  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  qx(元/b)  qx(元/b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='54  to apply single-frequency driving, parametrised by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4  with Ax2 = Ay2 = 0 and denoting ϕy1 as ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Starting from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 5 the time-dependent Hamiltonian  can be written as  ˆH(τ) =  �  ε +  3  �  i=1  ti cos [q · bi + θi(τ)]  �  σz+  [ηspℏωKx cos(ωτ) + ηspℏωKy cos(ωτ + ϕ)] σx,  (7)  where ε = (εs − εp)/2 = −∆ε/2 with ∆ε being the  gap size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The other parameters are ti = ts,i − tp,i with  i = 1, 2, 3, the Peierls phases are θi(τ) = − 1  ℏ  � τ  0 F(τ ′) ·  bi dτ ′ = −[Kx sin(ωτ)ˆx+Ky sin(ωτ +ϕ)ˆy]·bi/b, and the  Pauli matrices are σj with j = x, y, z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The dimension-  less driving strengths are denoted by Kx = mωAx1a/ℏ,  Ky = mωAy1a/ℏ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The tight-binding pa-  rameters, including the tunneling amplitudes, inter-band  coupling and band center energies, are obtained from a  triangular lattice potential by evaluating the Wannier  functions (Appendix B, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Single-frequency circular driving resonant with the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  If the frequency ω is near-resonant with the gap, corre-  sponding to the dominant one-photon process, we apply  the unitary UR(τ) = exp(−iωτσz/2) to eliminate the en-  ergy gap between the two bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' This gives  ˆH(τ) =  �  ε + ℏω/2 +  3  �  i=1  ti cos [q · bi + θi(τ)]  �  σz  + [ηspℏωKx cos(ωτ) + ηspℏωKy cos(ωτ + ϕ)]  × [σx cos(ωτ) − σy sin(ωτ)] /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (8)  The terms proportional to the identity matrix can be  omitted without affecting the topology of the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  By analysing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 8 in the high-frequency expan-  sion (Appendix C), we find a ring-shaped gap opening  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2e, see also refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' [13, 61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' However, the off-diagonal  terms are dominanted by constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Physically, this cor-  responds to on-site couplings between the s and p bands,  which yield topologically trivial bands (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Single-frequency elliptical driving resonant with half  the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Driving the system at half the gap energy (ℏω =  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='8t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='64Erec) leads to neighbouring-site interband  couplings which can give rise to topological bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Start-  ing from the waveforms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4 with Kx2 = Ky2 = 0 and  applying the unitary UR(τ) = exp(−iωτσz) gives  ˆH(τ) =  �  ε + ℏω +  3  �  i=1  ti cos [q · bi + θi(τ)]  �  σz  + [ηspℏωKx cos(ωτ) + ηspℏωKy cos(ωτ + ϕ)]  × [σx cos(2ωτ) − σy sin(2ωτ)]/2,  (9)  where ε+ℏω ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In this case, the only non-zero Fourier  components of the drive are ±1ω and ±3ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Contrary  to the direct one-photon resonance, the inter-band cou-  plings happen via two-photon processes (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' We  now find quasimomentum-dependent σx and σy terms  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Phase diagrams of the triangular lattice un-  der resonant driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Single-frequency driving on resonance  (ℏω  = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='6t1  = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='28Erec, Ky  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5, ε + ℏω/2 ≃ 0)  leads to trivial bands (a), whereas driving on half-resonance  (ℏω = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='8t1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='64Erec, Ky = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5, ε + ℏω ≃ 0) in-  duces topological bands (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The trivial regions in (b) are  caused by bands being shifted out of resonance due to the  ac-Stark effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' (c) shows the competition between the two  resonant processes induced by the two-frequency driving with  Kx1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='4, Ky1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5, Kx2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='02, Ky2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' (d) resulting  band gap evaluated for a cut through (c) along the red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  in the effective Hamiltonian (Appendix C), leading to  topologically non-trivial regions in the phase diagram  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 3b, see also ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Lines at Kx = 0 and ϕ = 0  have zero extent in the phase diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 3b, pre-  venting the observation of a transition from C ̸= 0 to  C = 0 in a realistic experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  While large driving  amplitudes (beyond Kx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5) render the model topo-  logically trivial, this effect is not due to an interference  between different processes, but rather due to the ac-  Stark effect that shifts the bands out of resonance [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Combining the above two resonant schemes at 1ω and  2ω introduces genuine topological transitions as function  of driving phase, as outlined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Two-frequency resonant elliptical driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The com-  petition between resonant 1ω and 2ω driving results in  an interplay between topological phases with C = 0 and  C = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' We now consider the full two-tone waveforms  of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4 for fixed [Kx1, Kx2, Ky1, Ky2], and tuneable ϕy1  b)  a)  Single frequency resonant  Single frequency half resonant  元  元  ≤2  C=-1  6  9  Phase  Phase  C=0  C=0  0  0  元  2  C=1  2  一元L  一元  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='75  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='75  Driving strength Kx  Driving strength Kx  c)  d)  Two-frequency (lw + 2w) driving  元  元  巫2  巫2  Phase Pyl  Phase Py1  C=0  0  0  2  一元  巫2  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='08  一π元  元  Phase Py2  Band gap size (ti)5  and ϕy2 (ϕx2 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' This Floquet scheme allows to cross  topological transitions purely by changing the phase be-  tween 1ω and 2ω drives, as is evident from the distinct re-  gions of C = 0 and C ̸= 0 in the phase diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The values of the induced band gaps are on the order of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='2 tunnelling energies, which can be enlarged by  choosing stronger driving amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Realising topological band structures in a Bravais lat-  tice, such as the simple triangular lattice considered here,  has important implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' For instance, it simplifies the  generation of topological bands in the context of ultracold  atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Up to now, two-dimensional topological models  in optical lattices have been realised either in bipartite  lattices [10, 12, 60], square lattices with moving superlat-  tices [8, 9, 11], or by employing spin-orbit coupling [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  These implementations rely on the relative phase stabil-  ity between lattice or Raman laser beams, leading to a  significant technological overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' An insufficient phase  stability may have been an obstacle for realising strongly  correlated phases in two-dimensional topological lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The triangular lattice, on the contrary, can be built by  simply superimposing three standing waves in the plane  with no active stabilisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Therefore, the 1ω–2ω Flo-  quet scheme could provide an avenue towards realising  correlated topological states of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In addition to its  conceptual simplicity, the two-tone driving leads to the  appearance of Berry curvature along a ring-shaped gap,  which can be asymmetrically distributed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 2f), partic-  ularly in the topologically trivial regions of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' This  supports a non-zero Berry curvature dipole, enabled by  periodic driving, although the underlying lattice is trivial  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Off-resonant driving in the hexagonal lattice (a)  with tunnelings t1 = t2 = t3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='06 in units of Erec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The  static band structure features two degenerate Dirac points  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Floquet driving lifts the degeneracy and opens a band  gap (c) by effectively breaking symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  and inversion-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' This finding is relevant beyond  the field of ultracold atoms and it could enable the obser-  vation of nonlinear Hall effects which so far have relied  on materials with broken inversion symmetry [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  OFF-RESONANTLY DRIVEN HEXAGONAL  LATTICE  The second application of the two-frequency driving  method considers the hexagonal lattice in brick-wall con-  figuration, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  This potential can be  realised in the same setup as the triangular lattice of  sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' III, but the derivation also applies to 120◦ honey-  comb lattices [12] as well as real graphene [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  We start from the tight-binding Hamiltonian  ˆHq(τ) =  � 3  �  i=1  ti cos [q · bi + θi(τ)]  �  σx+  � 3  �  i=1  ti sin [q · bi + θi(τ)]  �  σy,  (10)  The nearest-neighbour tunneling vectors are b1 = (0, 1)b,  b2 = (−1, 1)b and b3 = (−1, 0)b in the rotated coordi-  nate system {ˆe′  x, ˆe′  y} (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The waveforms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4  result in the following Peierls phases:  θ1(τ) = −Ky1 sin(ωτ + ϕy1) − Ky2 sin(2ωτ + ϕy2) = −θ3(τ),  θ2(τ) = Kx1 sin(ωτ) + Kx2 sin(2ωτ + ϕx2),  (11)  with the driving strengths being Kx1 = mωAx1a/ℏ,  Kx2 = 2mωAx2a/ℏ, Ky1 = mωAy1a/ℏ, and Ky2 =  2mωAy2a/ℏ, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Since we do not consider inter-  banding couplings here, there is no need to apply uni-  taries and the high-frequency expansion can be directly  conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Single-frequency elliptical driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The choice Kx2 =  Ky2 = 0, Kx1 > 0 corresponds to elliptical driving, giv-  ing rise to topological or trivial bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' For any Ky1 > 0  and ϕy1 mod π ̸= 0 we recover the well-known topo-  logical gap opening at both Dirac points due to time-  reversal symmetry breaking (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4b-c) [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The limits  of Ky1 = 0 or ϕy1 mod π = 0 correspond to linearly  polarised driving, leaving time-reversal symmetry intact  and both Dirac points remain closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' However, the triv-  ial region with C = 0 is only a ‘line’ of zero extent in the  phase diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 5a, as before in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Usually, a  manifestly broken inversion symmetry of the underlying  lattice is required to cross a topological phase transition  and induce Haldane’s ‘parity anomaly’, thereby extend-  ing the trivial ‘lines’ in the phase diagram to finite re-  gions [2, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In the following, we demonstrate that in-  version symmetry can be broken by employing the two-  tone Floquet drive only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Therefore, a topological phase  transition between trivial and non-trivial regions can be  achieved purely via driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  a)  t1  3  α1  q(π/b)  (元/b）  b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5-1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  (zHD)  (zHD)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='0  3  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  qr(π  x(元/b)6  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Phase diagrams of the hexagonal lattice under off-resonant driving (ℏω = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='8t1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='25Erec).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Single-frequency  elliptical driving (Ky1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='9, Kx2 = Ky2 = 0) can break time-reversal symmetry and result in a non-zero Chern number (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The breaking of the inversion and time-reversal symmetries can be independently controlled by adding a linearly polarized field  (Ky1 = Ky2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='9, Kx2 = 0 and ϕx2 = ϕy2 = 0) (b), as well as using two circular fields with opposite polarization (Kx1 = Ky1,  Kx2 = Ky2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='7, ϕy1 = π/2 and ϕy2 = ϕx2 − π/2) (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The transition paths I - II - III marked in (b) and (c) correspond to  the Lissajous curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1d-f and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1g-i, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Two-frequency driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Starting from the above case  of single-frequency elliptical driving, we add linearly po-  larized ‘light’ by setting Kx2 = 0 while Ky2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The  driving field further breaks inversion symmetry and the  competition between the breaking of inversion and time-  reversal symmetry can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 5b-c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The  topological phase transition can simply be induced by  tuning the phase along the marked path, corresponding  to the Lissajous curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1d-f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Alternatively, constraining Kx1 = Ky1, Kx2 = Ky2,  ϕy1 = π/2, and ϕy2 = ϕx2 − π/2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 4, the 1ω − 2ω  driving scheme corresponds to two counter-rotating, cir-  cularly polarized fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Clearly, when one circular driv-  ing field is much larger than the other, the resulting  Chern number will correspond to the helicity of the dom-  inant field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' However, when the amplitude of one circu-  lar field approaches the other one, the Lissajous curve  becomes a ‘cloverleaf’ pattern and the breaking of time-  reversal symmetry is suppressed (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 1g-i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' This compe-  tition allows to interpolate between topologically trivial  and non-trivial settings (with either positive or negative  mass terms) purely by changing driving phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The re-  sulting gap sizes are similar to the known results in the  Haldane model [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Interestingly, we found that a pure  Dirac Hamiltonian subject to two-frequency driving does  not show a corresponding gap opening in the case of bro-  ken inversion symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' While time-reversal symmetry  breaking via circular driving can be captured within the  approximate Dirac Hamiltonian [17], the inversion sym-  metry breaking relies on the presence of a lattice po-  tential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Mathematically, this can be seen from the fact  that the driven Dirac Hamiltonian remains linear and  its Fourier components at ω and 2ω will not mix in the  high-frequency expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Conceptually, a specific driv-  ing pattern favours a gap opening specific to one sublat-  tice of the honeycomb, which does not apply to the Dirac  Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  We also studied the behaviour of inversion symmetry  breaking beyond the validity of the high-frequency ex-  pansion, using numerically calculated band structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Here, the gap opening scales roughly as 1/ω4 which be-  comes dominant in the low-frequency regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  There-  fore, the 1ω–2ω scheme could be particularly interest-  ing for phase-controlled topological transitions in laser-  driven graphene, which could be readily implemented in  setups such as used in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  CONCLUSION  We have shown that two-frequency Floquet driving in  two dimensions gives rise to rich topological phase dia-  grams, both for resonant and off-resonant modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  We identify the relative phase between the 1ω and 2ω  drives as an important control parameter over the sym-  metry properties and the resulting topologies in the effec-  tive Floquet Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In many experiments, phase-  only control is advantageous over other parameters, such  as frequencies or amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' For periodically driven real  materials, such as graphene [17], incoherent excitations  and electron relaxation can strongly affect the measure-  ments, potentially obscuring any topological response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Incoherent effects often depend on the polarisation of  the incident light, especially close to the contacts of a  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Several of the driving schemes proposed in this  work, particularly the case of counter-rotating, circular  drives, depend only on the relative phase between the  light fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Therefore, we believe that our approach will  a)  lw elliptical driving  b)  lw elliptical + 2w linear driving  c)  lw + 2w circular driving  CIII  C=-1  π-2  C=-1  ≤2  C=-1  I  Phase Py1  Phase Py1  C=0  C=0  I  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  C=1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='25  C=1  C=1  一元L  一元L  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='5  一π元  0  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='75  1  Driving strength Kx1  Driving strength Kx1  Phase x27  lead to novel applications in manipulation and detection  of topological phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  By combining the schemes  considered here with spin-dependent Floquet engineer-  ing [48], it may become possible to access the entire ‘pe-  riodic table’ of topological insulators [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  We acknowledge funding by the Swiss National Sci-  ence Foundation (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' 182650 and NCCR-QSIT)  and European Research Council advanced grant TransQ  (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
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+page_content=' [66], that is,  Vlat(x, y) =  − Vx cos(kLx + θ/2)2 − Vx cos(kLx)2 − Vy cos(kLy)2  − 2  �  VxVy cos(kLx) cos(kLy) cos(φ),  (A1)  where kL = 2π/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
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+page_content=' The following  choice of parameters: Vx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
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+page_content='3, Vy = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
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+page_content='  The hexagonal lattice can be realized by setting Vx = 12, Vx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='8 and Vy = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
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+page_content=' Compared to the tight-binding  lattices, we have the length of the unit cells b =  √  2a = λ/  √  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The tight-binding parameters can be evaluated from  the realistic lattice potential (Appendix B), yielding t1 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
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+page_content='07, ε = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='66 in units of Erec  (triangular lattice, ηsp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='19) and t1 = t2 = t3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='06 in units of Erec (hexagonal lattice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Appendix B: Derivation of the Hamiltonian  We start from Hamiltonian in co-moving frame,  ˆHcm(τ) = ˆp2  2m + Vlat(ˆr) − F(τ) · ˆr ,  (B1)  where ˆp and ˆr are momentum and position operators, respectively, and F = −m¨rm(τ) = − ˙A(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The Hamiltonian has  two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The first two terms include the static lattice Hamiltonian and the last term represents a time-dependent  dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Next, we expand ˆHcm(τ) in Wannier basis [67],  wn,R(r) ≡ wn(r − R) =  V  (2π)2  �  BZ  e−ik·Rφn,k(r) dk ,  (B2)  where n is band index, φn,k(r) = eikrun,k(r) is Bloch state, R = n1a1 + n2a2 is arbitrary lattice vector, and a1, a2  are primitive translation vectors in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  In second quantization, we can rewrite the field operator in Wannier basis as  ˆψ(r) =  �  n,R  w∗  n,R(r)ˆan,R ,  (B3)  where ˆan,R is the annihilation operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The first part of the co-moving Hamiltonian can be expanded as  ˆH0(τ) =  �  n,m,R,R’  ˆa†  n,Rˆam,R’  �  wn,R(r)  � ˆp2  2m + Vlat(ˆr)  �  w∗  m,R’(r) dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (B4)  Using the fact that the static Hamiltonian cannot mix different bands and the Wannier functions are exponentially  localized, we only consider the onsite and nearest-neighbour couplings in individual bands,  ˆH0(τ) =  �  n,R  �  ˆa†  n,Rˆan,Rεn +  �  bi  �  ˆa†  n,Rˆan,R+biti + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ��  ,  (B5)  εn =  �  wn,R(r)  � ˆp2  2m + Vlat(ˆr)  �  w∗  n,R(r) dr ,  (B6)  ti =  �  wn,R(r)  � ˆp2  2m + Vlat(ˆr)  �  w∗  n,R+bi(r) dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (B7)  11  bi = ja1 + ka2 are the nearest-neighbour tunneling vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In the configuration of triangular and hexagonal lattices,  i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' εn is the band center energy and ti is the nearest-neighbour tunneling amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Now we consider the driving term which can couple different bands on-site but has negligible inter-site effects, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ˆH1(τ) = −  �  n,m,R,R’  ˆa†  n,Rˆam,R’  �  wn,R(r) [F(τ) · ˆr] w∗  m,R’(r) dr  = −  �  n,R  ˆa†  n,Rˆan,R  �  wn,R(r) [F(τ) · ˆr] w∗  n,R(r) dr −  �  n,m̸=n,R  ˆa†  n,Rˆam,R  �  wn,R(r) [F(τ) · ˆr] w∗  m,R(r) dr  = −  �  n,R  �  �ˆa†  n,Rˆan,RF(τ) · R +  �  m̸=n  ˆa†  n,Rˆam,RaF(τ) · ηnm ˆr′  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (B8)  Here we define the inter-band coupling as  ηnm = 1  b  ����  �  wn,R(r)ˆrw∗  m,R(r) dr  ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (B9)  By numerically evaluating ηsp according to the method in [66, 68], we justify that the dipole-like coupling term  �  wm,R(r)ˆrw∗  n,R(r) dr between s and p band is along the direction ˆr′ = ˆr = ˆex+ˆey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' In addition, we evaluate tunnelings  and band center energies through calculating the Wannier functions of atoms in a realistic lattice potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The on-site driving term ˆa†  n,Rˆan,RF(τ) · R breaks translation symmetry, so we need to rotate it away by applying  the unitary  ˆU(τ) = exp  �  −i  �  Rn  χR(τ)ˆa†  n,Rˆan,R  �  ,  (B10)  ˆH′(τ) = ˆU †(τ) ˆH(τ) ˆU(τ) − iℏ ˆU †(τ) ∂  ∂τ  ˆU(τ),  (B11)  where χR(τ) = m  ℏ R · ˙rm(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Then, transforming the Hamiltonian into quasi-momentum space and only considering  the lowest two bands, we get the final Hamiltonian,  ˆHq(τ) =  �  n∈{s,p}  �  εnˆa†  nˆan +  Z  �  i=1  �  tn,i  �  ei(θi(τ)+q·bi)ˆa†  nˆan + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ��  − F(τ) · ˆr  �  m∈{s,p}̸=n  ηnmˆa†  nˆam  �  ,  (B12)  with the time-periodic Peierls phase θi(τ) = χR+bi(τ) − χR(τ) = m  ℏ bi · ˙rm(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Appendix C: Analytical Hamiltonian by the high-frequency expansion  In the following, we write down the effective Floquet Hamiltonian, analytically derived using the high-frequency  expansion (HFE) in tight-binding approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The idea of the HFE is to decompose the original time-dependent  Hamiltonian into Fourier components and obtain the effective Hamiltonians, then neglect the small high frequency  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The effective Hamiltonian is decomposed into terms of different orders,  ˆHeff =  ∞  �  n=0  ˆH(n)  eff ,  (C1)  which is in inverse power of ω, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ˆH(n)  eff  ∼ ω−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' If the frequency is much higher than other energy scales in the  system, the high-order effective Hamiltonian can be truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Here we only use the first two orders [52],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ˆH(0)  eff = 1  T  � T  0  ˆH(τ) dτ ≡ ˆH0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (C2)  12  ˆH(0)  eff  σxηspℏω[Kx + Ky cos(ϕ)]/2  σyηspℏωKy sin(ϕ)/2  σz{ε + ℏω/2 + t1 cos[(qx − qy)b]J0(2Kx) + t2 cos(qxb)[J0(Kx)J0(Ky) − 2J1(Kx)J1(Ky) cos(ϕ)+  2J2(Kx)J2(Ky) cos(2ϕ)] + t3 cos(qyb)[J0(Kx)J0(Ky) + 2J1(Kx)J1(Ky) cos(ϕ) + 2J2(Kx)J2(Ky) cos(2ϕ)]}  ˆH(1)  eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σxηsp{−t1 cos[(qx − qy)b]J2(2Kx)[Kx + Ky cos(ϕ)] − t2 cos(qxb){J1(Kx)J1(Ky)[Ky + Kx cos(ϕ)]+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='J0(Ky)J2(Kx)[Kx + Ky cos(ϕ)] + J0(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]} + t3 cos(qyb){J1(Kx)J1(Ky)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='[Ky + Kx cos(ϕ)] − J0(Ky)J2(Kx)[Kx + Ky cos(ϕ)] − J0(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]}}/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σyηsp sin(ϕ){t1 cos[(qx − qy)b]J2(2Kx)Ky − t2 cos(qxb){J1(Kx)J1(Ky)Kx − J0(Ky)J2(Kx)Ky + J0(Kx)J2(Ky)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='[Ky + 2Kx cos(ϕ)]} + t3 cos(qyb){J1(Kx)J1(Ky)Kx + J0(Ky)J2(Kx)Ky − J0(Kx)J2(Ky)[Ky + 2Kx cos(ϕ)]}}/2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σzη2ℏω[K2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='x + K2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='y + 2KxKy cos(ϕ)]/8  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Effective Hamiltonian for the triangular lattice under single frequency driving resonant with the gap  ˆH(1)  eff = 1  ℏω  ∞  �  l=1  1  l  �  ˆHl, ˆH−l  �  =  ∞  �  l=1  ˆH(1)  eff,l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  (C3)  Here, ˆHl are the Fourier components of the time-dependent Hamiltonian, ˆH(τ) = �∞  l=−∞ ˆHleilωτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Table I is the effective Hamiltonian for the triangular lattice when the driving field is resonant with the gap energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  The constant terms are much larger than the quasimomentum dependent terms in off-diagonal, which means the  on-site inter-band couplings are dominant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' We safely neglect the Bessel functions higher than the second order as the  driving strength K ≤ 1 remains in weak driving regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  Table II shows the case of single-frequency driving resonant with the half the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Now the off-diagonal terms are  dominantly quasimomentum dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' The Hamiltonian for the triangular lattice in two-frequency driving scheme  are not listed here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  ˆH(0)  eff  σz{ε + ℏω + t1 cos[(qx − qy)b]J0(2Kx) + t2 cos(qxb)[J0(Kx)J0(Ky) − 2J1(Kx)J1(Ky) cos(ϕ)+  2J2(Kx)J2(Ky) cos(2ϕ)] + t3 cos(qyb)[J0(Kx)J0(Ky) + 2J1(Kx)J1(Ky) cos(ϕ) + 2J2(Kx)J2(Ky) cos(2ϕ)]}  ˆH(1)  eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σxηsp{−t1 sin[(qx − qy)b]J1(2Kx)Ky sin(ϕ) − t2 sin(qxb) sin(ϕ){KyJ0(Ky)J1(Kx) + KxJ1(Ky)J2(Kx)−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='J1(Kx)J2(Ky)[Ky + 2Kx cos(ϕ) + 2Ky cos(2ϕ)] + J0(Kx)J1(Ky)[Kx + 2Ky cos(ϕ)]}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='−t3 sin(qyb) sin(2ϕ){KyJ0(Kx)J1(Ky) + J1(Kx)J2(Ky)[Kx + 2Ky cos(ϕ)]}}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σyηsp{t1 sin[(qx − qy)b]J1(2Kx)[Kx + Ky cos(ϕ)] + t2 sin(qxb){J0(Ky)J1(Kx)[Kx + Ky cos(ϕ)]−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='J1(Ky)J2(Kx)[Ky + Kx cos(ϕ)] + J1(Ky)J0(Kx)[Kx cos(ϕ) + Ky cos(2ϕ)] − J1(Kx)J2(Ky)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='[Kx cos(2ϕ) + Ky cos(3ϕ)]} + t3 sin(qyb) sin(2ϕ){−J0(Ky)J1(Kx)[Kx + Ky cos(ϕ)] − J1(Ky)J2(Kx)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='[Ky + Kx cos(ϕ)] + J1(Ky)J0(Kx)[Kx cos(ϕ) + Ky cos(2ϕ)] + J1(Kx)J2(Ky)[Kx cos(2ϕ) + Ky cos(3ϕ)]}}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σzη2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='spℏω[K2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='x + K2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='y + 2KxKy cos(ϕ)]/4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='ˆH(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='3  σxηsp{−t2 sin(qxb){KxJ1(Ky)J2(Kx) + J1(Kx)J2(Ky)[Ky + 2Kx cos(ϕ)]}  +t3 sin(qyb){−KxJ1(Ky)J2(Kx) + J1(Kx)J2(Ky)[Ky + 2Kx cos(ϕ)]}}/3  σyηsp{t2 sin(qxb){J1(Ky)J2(Kx)[Ky + Kx cos(ϕ)] + J1(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]}  +t3 sin(qyb){J1(Ky)J2(Kx)[Ky + Kx cos(ϕ)] − J1(Kx)J2(Ky)[Ky cos(ϕ) + Kx cos(2ϕ)]}}/3  σzη2  spℏω[K2  x + K2  y + 2KxKy cos(ϕ)]/12  TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Effective Hamiltonian for the triangular lattice under single frequency driving resonant with the half the gap  For the hexagonal lattice, Table III gives the general Hamiltonian under single-frequency off-resonant driving, while  Table IV is the Hamiltonian under two-frequency off-resonant driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Note that the second order Bessel functions  are omitted in ˆH(1)  eff in Table IV but kept in the numerical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='  13  ˆH(0)  eff  σx{[t1 cos(qyb) + t3 cos(qxb)]J0(Ky1) + t2 cos[(qx − qy)b]J0(Kx1)}  σy{[t1 sin(qyb) − t3 sin(qxb)]J0(Ky1) − t2 sin[(qx − qy)b]J0(Kx1)}  ˆH(1)  eff  −σz  �∞  n=1  4  nℏω t2Jn(Kx1)Jn(Ky1)  �  (−1)n+1t1 sin(qxb) + t3 sin(qyb)  �  sin(nϕy1)  TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Effective Hamiltonian for the hexagonal lattice under single frequency off-resonant driving  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='ˆH(0)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='eff  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σx{[t1 cos(qyb) + t3 cos(qxb)]J0(Ky1)J0(Ky2) + t2 cos[(qx − qy)b]J0(Kx1)J0(Kx2)−  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='2[t1 sin(qxb) + t3 sin(qyb)]J1(Ky2)J2(Ky1) sin(2ϕy1 − ϕy2) + 2t2 sin[(qx − qy)b]J1(Kx2)J2(Kx1) sin(ϕx2)}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σy{[t1 sin(qyb) − t3 sin(qxb)]J0(Ky1)J0(Ky2) − t2 sin[(qx − qy)b]J0(Kx1)J0(Kx2)+  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='2[t1 cos(qxb) − t3 cos(qyb)]J1(Ky2)J2(Ky1) sin(2ϕy1 − ϕy2) + 2t2 cos[(qx − qy)b]J1(Kx2)J2(Kx1) sin(ϕx2)}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='ˆH(1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='eff  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='σz  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='3ℏω {6J0(Ky2)J1(Ky1)[t2J1(Kx1)J1(Kx2)(t1 cos(qxb) − t3 cos(qyb)) cos(ϕx2 − ϕy1) + (t2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='1 − t2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='3)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='J1(Ky1)J1(Ky2) cos(2ϕy1 − ϕy2)] − 6t2J0(Kx2)J1(Kx1)[t2J1(Kx1)J1(Kx2) cos(ϕx2) + J1(Ky1)J1(Ky2)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='(t1 cos(qxb) + t3 cos(qyb)) cos(ϕy1 − ϕy2) − J1(Ky1)J0(Ky2)(t1 sin(qxb) + t3 sin(qyb)) sin(ϕy1)]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='−t2J1(Kx2)J1(Ky2)[3J0(Kx1)J0(Ky1)(t1 sin(qxb) + t3 sin(qyb)) sin(ϕx2 − ϕy2) − 2J1(Kx1)J1(Ky1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='(t1 sin(qxb) − t3 sin(qyb))(sin(ϕx2 − ϕy1 − ϕy2) + 3 sin(ϕx2 + ϕy1 − ϕy2))]}  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content='TABLE IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
+page_content=' Effective Hamiltonian for the hexagonal lattice under two frequency off-resonant driving' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE4T4oBgHgl3EQfuA2V/content/2301.05229v1.pdf'}
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+1st Place Solution for ECCV 2022 OOD-CV
+Challenge Image Classification Track
+Yilu Guo1, Xingyue Shi2,⋆, Weijie Chen1,3,†, Shicai Yang1, Di Xie1,
+Shiliang Pu1, and Yueting Zhuang3
+1 Hikvision Research Institute, Hangzhou China
+2 Peking University Shenzhen Graduate School, Shenzhen, China
+3 Zhejiang University, Hangzhou, China
+{guoyilu5, chenweijie5, yangshicai, xiedi, pushiliang.hri}@hikvision.com,
+shixy@stu.pku.edu.cn, {chenweijie, yzhuang}@zju.edu.cn
+Abstract. OOD-CV challenge‡ is an out-of-distribution generalization
+task. In this challenge, our core solution can be summarized as that
+Noisy Label Learning Is A Strong Test-Time Domain Adap-
+tation Optimizer. Briefly speaking, our main pipeline can be divided
+into two stages, a pre-training stage for domain generalization and a
+test-time training stage for domain adaptation. We only exploit labeled
+source data in the pre-training stage and only exploit unlabeled target
+data in the test-time training stage. In the pre-training stage, we pro-
+pose a simple yet effective Mask-Level Copy-Paste data augmentation
+strategy to enhance out-of-distribution generalization ability so as to re-
+sist shape, pose, context, texture, occlusion, and weather domain shifts
+in this challenge. In the test-time training stage, we use the pre-trained
+model to assign noisy label for the unlabeled target data, and propose
+a Label-Periodically-Updated DivideMix method for noisy label
+learning. After integrating Test-Time Augmentation and Model Ensem-
+ble strategies, our solution ranks the first place on the Image Classifi-
+cation Leaderboard of the OOD-CV Challenge. Code will be released in
+https://github.com/hikvision-research/OOD-CV.
+Keywords: Mask-Level Copy-Paste, Out-of-Distribution Generalization,
+Test-Time Domain Adaptation, Noisy Label Learning
+1
+Introduction
+Following the existing works, such as SSNLL [2] for Image Classification and
+SFOD [6] for Object Detection, we put forth the idea that Noisy Label Learn-
+ing Is A Strong Test-Time Domain Adaptation Optimizer. 1) Before
+Test-Time Domain Adaptation, it is an essential prerequisite to pre-train a
+⋆ Internship in Hikvision Research Institute
+† Corresponding author
+‡ https://www.ood-cv.org/
+arXiv:2301.04795v1  [cs.CV]  12 Jan 2023
+
+2
+Y. Guo et al.
+mask-level copy-paste
+Epoch e
+Co-Divide
+Model
+A
+Model
+B
+Epoch (e-1)
+GMM
+GMM
+Model
+A
+Model
+B
+MixMatch
+MixMatch
+Model
+customized
+augmentation
+② Test Time Domain Adaptation via Label-Periodically-Updated DivideMix
+rule-based 
+weather simulation
+autoaug
+cutmix
+Model
+A
+Model
+B
+generate  
+pseudo label
+label updated periodically every n epoch
+Mask-Level Copy-Paste Data Augmentaion
+task-unrelated
+foreground
+task-related
+foreground
+task-unrelated
+image
+task-related 
+image
+context
+augmentation
+occlusion
+augmentation
+affine transformation 
+color transformation
+random resize
+shape, pose, texture
+augmentation
+① Train Strong Baseline Models Using Diverse Data Augmentation
+two different weight models
+Training with different random seeds
+Fig. 1. The pipeline of our proposed framework, which is composed of a model pre-
+training stage by merely exploiting the labeled source data and a test-time domain
+adaptation stage by merely exploiting the unlabeled target data. Note that Mask-Level
+Copy-Paste is developed upon a weakly-supervised semantic segmentation method us-
+ing only image-level label.
+strong baseline model that generalizes well to the out-of-distribution data. An
+instinctual method is to stack diverse strong data augmentation strategies on
+the source data in order to resist diverse domain shifts. To this end, besides
+
+WatthMatthew Rose581ECCV 2022 OOD-CV Challenge Image Classification Track
+3
+the conventional data augmentation, we develop a novel Mask-Level Copy-
+Paste data augmentation method. Specifically, given the image-level label, we
+adopt a state-of-the-art weakly-supervised semantic segmentation method MCT-
+former [9] to segment the foreground objects on the ImageNet-1K and ROBIN
+training datasets. In this way, we can apply affine transformation and color
+jittering to augment the foreground objects (against shape, pose and texture
+domain shifts), and then paste the task-related foreground objects onto task-
+unrelated image (against context domain shift) and paste the task-unrelated
+foreground objects onto task-related image (against occlusion domain shift). 2)
+After model pre-training, pseudo labels of target data inferred by the source
+pre-trained model can be viewed as the noisy labels. Under this consideration,
+existing noisy label learning methods, such as DivideMix [5], can be naturally
+used for test-time domain adaptation. In this challenge, we propose a Label-
+Periodically-Updated DivideMix, which can correct the noisy labels in time
+while avoiding overfitting the noisy labels. After integrating Test-Time Augmen-
+tation and Model Ensemble with diverse hyper-parameters, our solution ranks
+the first place on the Image Classification Leaderboard of the OOD-CV Chal-
+lenge https://codalab.lisn.upsaclay.fr/competitions/6781#results.
+Our main contribution can be summarized as follows:
+– Pretraining Stage: Mask-Level Copy-Paste. We propose a novel Mask-
+Level Copy-Paste data augmentation strategy to resist diverse domain shifts.
+– Adaptation Stage: Label-Periodically-Updated DivideMix. We con-
+sider test-time training as a noisy label learning problem. We modify the
+DivideMix by updating noisy labels periodically and use strong-weak aug-
+mentation to get a significant performance improvement on the OOD data.
+2
+Method
+2.1
+Mask-Level Copy-Paste.
+We propose a novel Mask-Level Copy-Paste data augmentation strategy to resist
+diverse domain shifts. Specifically, we train a weakly-supervised semantic seg-
+mentation (WSSS) model [9] by exploiting the image-level label on ImageNet-1K
+and ROBIN training datasets. We then segment the foreground objects by test-
+ing this WSSS model. According to whether the class label is related to the task
+in this challenge or not, the foreground objects can be divided into task-related
+and task-unrelated parts. Similarly, the images from ImageNet-1K and ROBIN
+training datasets can also be divided into task-related and task-unrelated parts.
+In this way, we can apply randomly affine transformation and color transforma-
+tion to augment the foreground objects (against shape, pose and texture domain
+shifts), and then paste the task-related foreground objects onto task-unrelated
+image (against context domain shift) and paste the task-unrelated foreground
+objects onto task-related image (against occlusion domain shift). By stacking
+with other data augmentation strategies, including AutoAug [3], CutMix [10]
+and rule-based weather simulation methods [4] (against weather domain shift),
+we can get a pre-trained model with a strong domain generalization ability.
+
+4
+Y. Guo et al.
+2.2
+Label-Periodically-Updated DivideMix.
+We consider test time domain adaptation as a noisy label learning problem.
+And we regard the strong pre-trained model as a noisy annotator on the unla-
+beled test set. After obtaining noisy labels, we modify the DivideMix [5] into a
+label-periodically-updated one, which can correct the noisy labels in time while
+avoiding overfitting the noisy labels. Furthermore, different from the original
+DivideMix, we employ the popular strong-weak augmentation in the MixMatch
+component, which uses weak augmentation for pseudo labeling and strong aug-
+mentation for model optimization.
+3
+Technique Details
+3.1
+Dependencies
+– Transformer Backbone: Swinv2 [7]
+– Dataset: ROBIN [11], ImageNet-1K [8]
+– Data Augmentation: AutoAug [3], CutMix [10], Rule-Based Weather Sim-
+ulation [4]
+– Weakly-Supervised Semantic Segmentation: MCTformer [9]
+– Noisy Label Learning Method: DivideMix [5]
+3.2
+Training Description
+Baseline model. Comparing different model architectures, we select Swinv2-
+B [7] pretrained on the ImageNet-1K dataset as the baseline to develop our
+method. Swinv2 is a hierarchical vision Transformer whose representation is
+determined by a shifted windowing scheme, which introduces locality to the
+model and improves computation efficiency.
+Data augmentation. 1) We design a Mask-Level Copy-Paste data augmen-
+tation strategy against different nuisances based on weakly-supervised semantic
+segmentation. An image-level supervised semantic segmentation model is trained
+on the ROBIN training dataset via MCTformer [9]. After getting the segmented
+masks, we can apply randomly affine transformation (object distortion, rotation,
+horizontal and vertical flipping) and color transformation to augment the fore-
+ground objects (against shape, pose and texture domain shifts), and then paste
+the task-related foreground objects onto task-unrelated image (against context
+domain shift). In addition, we train another image-level supervised semantic seg-
+mentation model on the images from task-unrelated classes in the ImageNet-1K
+dataset. Then we randomly paste the task-unrelated foreground objects onto
+task-related image to imitate object occlusion (against occlusion domain shift).
+2) Rule-based weather simulation. We follow ImageNet-C [4] to implement four
+image transformations simulating four common weather conditions, including
+rain, snow, fog and sunshine (against weather domain shift). The detailed im-
+plementation is based on three Python libraries, including cv2, skimage and
+
+ECCV 2022 OOD-CV Challenge Image Classification Track
+5
+scipy.ndimage. 3) We also adopt two common data augmentation strategies
+like AutoAug [3] and CutMix [10]. By stacking multiple data augmentations, we
+can obtain a Strong Baseline model.
+Implementation details. We use the cross-entropy loss with label smoothing
+as the loss function, and the smoothing factor is set to 0.1. Swinv2-B model is
+trained with 8 GPUs and 32 samples per GPU. We use the SGD optimizer with
+momentum 0.9. The learning rate is adjusted according to the cosine decaying
+policy [8] and the initial learning rate is set to 0.01. The warm-up [8] strategy
+is applied over the first 3 epochs, gradually increasing the learning rate linearly
+from 1e-6 to the initial value of the cosine schedule. The weight decay is set to
+2e-5. The model is trained for 100 epochs and is stopped when the accuracy on
+the validation set no longer increases.
+3.3
+Testing Description
+Test-time domain adaptation. The classifier trained on the training set suf-
+fers great performance degradation on the out-of-distribution test set. Regarding
+the trained classifier as a noisy annotator on the unlabeled test set, the test-time
+training procedure can be considered as an instance of noisy-label learning. To
+this end, DivideMix [5] is modified to conduct test-time noisy-label learning as
+shown in Fig.1. Firstly, the classifier trained with only training set assigns noisy
+pseudo-labels to the unlabeled test set, denoted as the test-time training set.
+Then, two models (A and B) are trained by different random seeds or hyper-
+parameters on the training set for co-teaching. Every epoch, a model divides
+the noisy training set into a labeled subset (mostly clean) X and an unlabeled
+subset (mostly noisy) U based on the assumption that clean samples have lower
+losses. DivideMix performs label co-refinement on X and label co-guessing on U
+to enable the two models to teach each other. After obtaining ˆX (and ˆU) which
+contain multiple augmented views of labeled (and unlabeled) samples and their
+refined (and guessed) labels, DivideMix follows MixMatch [1] to mix-up ˆX and
+ˆU, and calculate the cross-entropy loss and the MSE (mean square error) loss on
+them respectively. Specifically, we use weak augmented views for getting refined
+(guessed) labels and use strong augmented views (AutoAug) for training. Dur-
+ing DivideMix, the performance might degenerate after a number of epochs. We
+update noisy pseudo-labels every 3 epochs, which can correct the noisy labels in
+time while avoiding overfitting the noisy labels. Also, we rebuild the source code
+of DivideMix project from single-GPU to multi-GPU to speed up training.
+Implementation details. During test time training, only AutoAug is utilized
+for strong data augmentation. The model are trained with 8 GPUs and 6 samples
+per GPU. We use the SGD optimizer with momentum 0.9. The learning rate is
+set to 0.02 and the weight decay is set to 5e-4. To get the labeled subset X from
+the noisy test-time training set, we set the clean probability threshold (p) to 0.8,
+which is higher than the default value 0.5 in the original DivideMix [5], so as to
+achieve more reliable pseudo-labeled samples.
+
+6
+Y. Guo et al.
+Testing phase. During testing, the TenCrop inference are utilized which crops
+the given image into four corners and the central crop plus the horizontal flipped
+version of these crops.
+4
+Experimental Results
+4.1
+Ablation Study
+Methods
+IID
+OOD
+Shape Pose Context Texture Occlusion Weather Avg.
+Swinv2-B
+90.90 85.97 90.56
+87.67
+95.25
+83.63
+92.09
+89.19
+Swinv2-B + AutoAug + CutMix
+90.94 86.99 90.46
+88.73
+95.77
+86.32
+92.38
+90.11
+Strong Baseline
+90.80 87.09 90.56
+89.26
+95.25
+91.74
+93.34
+91.21
+Strong Baseline + TTT
+87.10 88.99 93.77
+96.30
+96.81
+93.84
+95.18
+94.15
+Strong Baseline + TTT + TenCrop 87.10 89.40 94.23
+97.18
+97.35
+94.96
+95.55
+94.78
+Table 1. Some ablation results on phase 1 of the challenge. “TTT” is short for Test-
+Time Training. “Avg.” is the average accuracy of the six domains.
+Table 1 shows the ablation results on phase 1 of the challenge. The techniques
+can improve the OOD accuracy from 89.19% to 94.78%. It is much higher than
+the performance (92.65%) on the leaderboard of phase 1. Since there are more
+OOD data than IID data in the test set, the IID performance degenerates after
+test time training.
+4.2
+Ensembles And Fusion Strategies
+Settings
+IID
+OOD
+Shape Pose Context Texture Occlusion Weather Avg.
+p=0.85
+85.40 84.09 84.81
+87.39
+76.93
+95.02
+85.74
+85.74
+p=0.85 w/ a different seed 90.22 84.77 82.93
+86.08
+73.96
+97.32
+86.60
+85.28
+p=0.8
+89.23 84.09 85.62
+87.46
+75.21
+97.73
+87.60
+86.29
+p=0.8 w/ a different epoch 90.88 83.27 85.89
+87.46
+77.10
+97.70
+86.40
+86.30
+Model ensemble
+90.68 85.11 85.48
+88.28
+77.50
+97.99
+87.80
+87.03
+Table 2. Final results on phase 2 of the challenge. The 2nd-row model is set as the
+anchor model. The last-row (6th-row) result is the ensemble from the models in 2nd-
+5th rows via an entropy-guided model adaptive ensemble strategy. Note that p is a
+hyper-parameter in DivideMix to split noisy data into a labeled set and an unlabeled
+set for MixMatch training.
+We ensemble 4 models trained by different hyper-parameters via an entropy-
+guided model adaptive ensemble strategy. Specifically, we choose a model as an
+
+ECCV 2022 OOD-CV Challenge Image Classification Track
+7
+anchor model. For each image, we trust the result if the entropy of the prediction
+from the anchor model is small enough (less than 2/3 mean entropy of the test
+set). Otherwise, we use the results from 4 models for ensemble. In this way, the
+inference time can be reduced. The 4 models are trained by different random
+seeds or hyper-parameters, different p and different training epochs in DivideMix.
+After ensemble, the OOD accuracy is improved from 86.30% to 87.03%.
+4.3
+Model Complexity
+complexity
+Parameters
+Train Time
+Single model
+22.01GFLOPs
+86.9M
+12hours * 8GPUs
+TenCrop+Ensemble 880.4GFLOPs
+347.6M
+48hours * 8GPUs
+Table 3. Method complexity analysis. Four models with different training hyper-
+parameters are used for model ensemble. Eight Nvidia Tesla V100 GPUs are used
+for testing the Train Time.
+Please refer to Table 3 for model complexity analysis.
+5
+Conclusion
+The OOD-CV challenge deepens our understanding of model robustness under
+domain shifts. We thank the organizers for their great work in the challenge.
+References
+1. Berthelot, D., Carlini, N., Goodfellow, I., Papernot, N., Oliver, A., Raffel,
+C.A.: Mixmatch: A holistic approach to semi-supervised learning. In: Wal-
+lach, H., Larochelle, H., Beygelzimer, A., d'Alch´e-Buc, F., Fox, E., Garnett,
+R. (eds.) Advances in Neural Information Processing Systems. vol. 32. Curran
+Associates, Inc. (2019), https://proceedings.neurips.cc/paper/2019/file/
+1cd138d0499a68f4bb72bee04bbec2d7-Paper.pdf
+2. Chen, W., Lin, L., Yang, S., Xie, D., Pu, S., Zhuang, Y.: Self-supervised noisy label
+learning for source-free unsupervised domain adaptation. In: IROS (2022)
+3. Cubuk, E.D., Zoph, B., Man´e, D., Vasudevan, V., Le, Q.V.: Autoaugment: Learning
+augmentation policies from data. CoRR abs/1805.09501 (2018), http://arxiv.
+org/abs/1805.09501
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+page_content='1st Place Solution for ECCV 2022 OOD-CV  Challenge Image Classification Track  Yilu Guo1, Xingyue Shi2,⋆, Weijie Chen1,3,†, Shicai Yang1, Di Xie1,  Shiliang Pu1, and Yueting Zhuang3  1 Hikvision Research Institute, Hangzhou China  2 Peking University Shenzhen Graduate School, Shenzhen, China  3 Zhejiang University, Hangzhou, China  {guoyilu5, chenweijie5, yangshicai, xiedi, pushiliang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='hri}@hikvision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='com,  shixy@stu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
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+page_content='cn, {chenweijie, yzhuang}@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
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+page_content='cn  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' OOD-CV challenge‡ is an out-of-distribution generalization  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In this challenge, our core solution can be summarized as that  Noisy Label Learning Is A Strong Test-Time Domain Adap-  tation Optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Briefly speaking, our main pipeline can be divided  into two stages, a pre-training stage for domain generalization and a  test-time training stage for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We only exploit labeled  source data in the pre-training stage and only exploit unlabeled target  data in the test-time training stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In the pre-training stage, we pro-  pose a simple yet effective Mask-Level Copy-Paste data augmentation  strategy to enhance out-of-distribution generalization ability so as to re-  sist shape, pose, context, texture, occlusion, and weather domain shifts  in this challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In the test-time training stage, we use the pre-trained  model to assign noisy label for the unlabeled target data, and propose  a Label-Periodically-Updated DivideMix method for noisy label  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' After integrating Test-Time Augmentation and Model Ensem-  ble strategies, our solution ranks the first place on the Image Classifi-  cation Leaderboard of the OOD-CV Challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Code will be released in  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='com/hikvision-research/OOD-CV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Keywords: Mask-Level Copy-Paste, Out-of-Distribution Generalization,  Test-Time Domain Adaptation, Noisy Label Learning  1  Introduction  Following the existing works, such as SSNLL [2] for Image Classification and  SFOD [6] for Object Detection, we put forth the idea that Noisy Label Learn-  ing Is A Strong Test-Time Domain Adaptation Optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' 1) Before  Test-Time Domain Adaptation, it is an essential prerequisite to pre-train a  ⋆ Internship in Hikvision Research Institute  † Corresponding author  ‡ https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='ood-cv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='org/  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
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+page_content='CV]  12 Jan 2023  2  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='mask-level copy-paste  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Epoch e  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Co-Divide  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Epoch (e-1)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='GMM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='GMM  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='MixMatch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='MixMatch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='customized  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='② Test Time Domain Adaptation via Label-Periodically-Updated DivideMix  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='rule-based  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='weather simulation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='autoaug  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='cutmix  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='A  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Model  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='B  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='generate  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='pseudo label  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='label updated periodically every n epoch  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='Mask-Level Copy-Paste Data Augmentaion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='task-unrelated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='foreground  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='task-related  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='foreground  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='task-unrelated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='task-related  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='context  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='occlusion  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='augmentation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='affine transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='color transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='random resize  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='shape,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' pose,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' texture  augmentation  ① Train Strong Baseline Models Using Diverse Data Augmentation  two different weight models  Training with different random seeds  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The pipeline of our proposed framework, which is composed of a model pre-  training stage by merely exploiting the labeled source data and a test-time domain  adaptation stage by merely exploiting the unlabeled target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Note that Mask-Level  Copy-Paste is developed upon a weakly-supervised semantic segmentation method us-  ing only image-level label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  strong baseline model that generalizes well to the out-of-distribution data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' An  instinctual method is to stack diverse strong data augmentation strategies on  the source data in order to resist diverse domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' To this end, besides  WatthMatthew Rose581ECCV 2022 OOD-CV Challenge Image Classification Track  3  the conventional data augmentation, we develop a novel Mask-Level Copy-  Paste data augmentation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Specifically, given the image-level label, we  adopt a state-of-the-art weakly-supervised semantic segmentation method MCT-  former [9] to segment the foreground objects on the ImageNet-1K and ROBIN  training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In this way, we can apply affine transformation and color  jittering to augment the foreground objects (against shape, pose and texture  domain shifts), and then paste the task-related foreground objects onto task-  unrelated image (against context domain shift) and paste the task-unrelated  foreground objects onto task-related image (against occlusion domain shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' 2)  After model pre-training, pseudo labels of target data inferred by the source  pre-trained model can be viewed as the noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Under this consideration,  existing noisy label learning methods, such as DivideMix [5], can be naturally  used for test-time domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In this challenge, we propose a Label-  Periodically-Updated DivideMix, which can correct the noisy labels in time  while avoiding overfitting the noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' After integrating Test-Time Augmen-  tation and Model Ensemble with diverse hyper-parameters, our solution ranks  the first place on the Image Classification Leaderboard of the OOD-CV Chal-  lenge https://codalab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='lisn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='upsaclay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='fr/competitions/6781#results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Our main contribution can be summarized as follows:  – Pretraining Stage: Mask-Level Copy-Paste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We propose a novel Mask-  Level Copy-Paste data augmentation strategy to resist diverse domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  – Adaptation Stage: Label-Periodically-Updated DivideMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We con-  sider test-time training as a noisy label learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We modify the  DivideMix by updating noisy labels periodically and use strong-weak aug-  mentation to get a significant performance improvement on the OOD data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  2  Method  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='1  Mask-Level Copy-Paste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  We propose a novel Mask-Level Copy-Paste data augmentation strategy to resist  diverse domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Specifically, we train a weakly-supervised semantic seg-  mentation (WSSS) model [9] by exploiting the image-level label on ImageNet-1K  and ROBIN training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We then segment the foreground objects by test-  ing this WSSS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' According to whether the class label is related to the task  in this challenge or not, the foreground objects can be divided into task-related  and task-unrelated parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Similarly, the images from ImageNet-1K and ROBIN  training datasets can also be divided into task-related and task-unrelated parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  In this way, we can apply randomly affine transformation and color transforma-  tion to augment the foreground objects (against shape, pose and texture domain  shifts), and then paste the task-related foreground objects onto task-unrelated  image (against context domain shift) and paste the task-unrelated foreground  objects onto task-related image (against occlusion domain shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' By stacking  with other data augmentation strategies, including AutoAug [3], CutMix [10]  and rule-based weather simulation methods [4] (against weather domain shift),  we can get a pre-trained model with a strong domain generalization ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  4  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='2  Label-Periodically-Updated DivideMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  We consider test time domain adaptation as a noisy label learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  And we regard the strong pre-trained model as a noisy annotator on the unla-  beled test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' After obtaining noisy labels, we modify the DivideMix [5] into a  label-periodically-updated one, which can correct the noisy labels in time while  avoiding overfitting the noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Furthermore, different from the original  DivideMix, we employ the popular strong-weak augmentation in the MixMatch  component, which uses weak augmentation for pseudo labeling and strong aug-  mentation for model optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  3  Technique Details  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='1  Dependencies  – Transformer Backbone: Swinv2 [7]  – Dataset: ROBIN [11], ImageNet-1K [8]  – Data Augmentation: AutoAug [3], CutMix [10], Rule-Based Weather Sim-  ulation [4]  – Weakly-Supervised Semantic Segmentation: MCTformer [9]  – Noisy Label Learning Method: DivideMix [5]  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='2  Training Description  Baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Comparing different model architectures, we select Swinv2-  B [7] pretrained on the ImageNet-1K dataset as the baseline to develop our  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Swinv2 is a hierarchical vision Transformer whose representation is  determined by a shifted windowing scheme, which introduces locality to the  model and improves computation efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' 1) We design a Mask-Level Copy-Paste data augmen-  tation strategy against different nuisances based on weakly-supervised semantic  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' An image-level supervised semantic segmentation model is trained  on the ROBIN training dataset via MCTformer [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' After getting the segmented  masks, we can apply randomly affine transformation (object distortion, rotation,  horizontal and vertical flipping) and color transformation to augment the fore-  ground objects (against shape, pose and texture domain shifts), and then paste  the task-related foreground objects onto task-unrelated image (against context  domain shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In addition, we train another image-level supervised semantic seg-  mentation model on the images from task-unrelated classes in the ImageNet-1K  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Then we randomly paste the task-unrelated foreground objects onto  task-related image to imitate object occlusion (against occlusion domain shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  2) Rule-based weather simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We follow ImageNet-C [4] to implement four  image transformations simulating four common weather conditions, including  rain, snow, fog and sunshine (against weather domain shift).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The detailed im-  plementation is based on three Python libraries, including cv2, skimage and  ECCV 2022 OOD-CV Challenge Image Classification Track  5  scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='ndimage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' 3) We also adopt two common data augmentation strategies  like AutoAug [3] and CutMix [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' By stacking multiple data augmentations, we  can obtain a Strong Baseline model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We use the cross-entropy loss with label smoothing  as the loss function, and the smoothing factor is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Swinv2-B model is  trained with 8 GPUs and 32 samples per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We use the SGD optimizer with  momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The learning rate is adjusted according to the cosine decaying  policy [8] and the initial learning rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The warm-up [8] strategy  is applied over the first 3 epochs, gradually increasing the learning rate linearly  from 1e-6 to the initial value of the cosine schedule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The weight decay is set to  2e-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The model is trained for 100 epochs and is stopped when the accuracy on  the validation set no longer increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='3  Testing Description  Test-time domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The classifier trained on the training set suf-  fers great performance degradation on the out-of-distribution test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Regarding  the trained classifier as a noisy annotator on the unlabeled test set, the test-time  training procedure can be considered as an instance of noisy-label learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' To  this end, DivideMix [5] is modified to conduct test-time noisy-label learning as  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Firstly, the classifier trained with only training set assigns noisy  pseudo-labels to the unlabeled test set, denoted as the test-time training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Then, two models (A and B) are trained by different random seeds or hyper-  parameters on the training set for co-teaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Every epoch, a model divides  the noisy training set into a labeled subset (mostly clean) X and an unlabeled  subset (mostly noisy) U based on the assumption that clean samples have lower  losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' DivideMix performs label co-refinement on X and label co-guessing on U  to enable the two models to teach each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' After obtaining ˆX (and ˆU) which  contain multiple augmented views of labeled (and unlabeled) samples and their  refined (and guessed) labels, DivideMix follows MixMatch [1] to mix-up ˆX and  ˆU, and calculate the cross-entropy loss and the MSE (mean square error) loss on  them respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Specifically, we use weak augmented views for getting refined  (guessed) labels and use strong augmented views (AutoAug) for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Dur-  ing DivideMix, the performance might degenerate after a number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We  update noisy pseudo-labels every 3 epochs, which can correct the noisy labels in  time while avoiding overfitting the noisy labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Also, we rebuild the source code  of DivideMix project from single-GPU to multi-GPU to speed up training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' During test time training, only AutoAug is utilized  for strong data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The model are trained with 8 GPUs and 6 samples  per GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We use the SGD optimizer with momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The learning rate is  set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='02 and the weight decay is set to 5e-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' To get the labeled subset X from  the noisy test-time training set, we set the clean probability threshold (p) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='8,  which is higher than the default value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='5 in the original DivideMix [5], so as to  achieve more reliable pseudo-labeled samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  6  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Testing phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' During testing, the TenCrop inference are utilized which crops  the given image into four corners and the central crop plus the horizontal flipped  version of these crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  4  Experimental Results  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='1  Ablation Study  Methods  IID  OOD  Shape Pose Context Texture Occlusion Weather Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Swinv2-B  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='90 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='97 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='56  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='67  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='25  83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='63  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='09  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='19  Swinv2-B + AutoAug + CutMix  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='94 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='99 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='46  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='73  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='77  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='32  92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='38  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='11  Strong Baseline  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='80 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='09 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='56  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='26  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='25  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='74  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='34  91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='21  Strong Baseline + TTT  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='10 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='99 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='77  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='30  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='81  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='84  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='18  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='15  Strong Baseline + TTT + TenCrop 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='10 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='40 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='23  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='18  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='35  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='96  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='55  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='78  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Some ablation results on phase 1 of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' “TTT” is short for Test-  Time Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' “Avg.” is the average accuracy of the six domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Table 1 shows the ablation results on phase 1 of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The techniques  can improve the OOD accuracy from 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='19% to 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='78%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' It is much higher than  the performance (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='65%) on the leaderboard of phase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Since there are more  OOD data than IID data in the test set, the IID performance degenerates after  test time training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='2  Ensembles And Fusion Strategies  Settings  IID  OOD  Shape Pose Context Texture Occlusion Weather Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='85  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='40 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='09 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='81  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='39  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='93  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='02  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='74  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='74  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='85 w/ a different seed 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='22 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='77 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='93  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='08  73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='96  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='32  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='60  85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='28  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='8  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='23 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='09 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='62  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='46  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='21  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='73  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='60  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='29  p=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='8 w/ a different epoch 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='88 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='27 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='89  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='46  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='10  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='70  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='40  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='30  Model ensemble  90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='68 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='11 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='48  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='28  77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='50  97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='99  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='80  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='03  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Final results on phase 2 of the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The 2nd-row model is set as the  anchor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The last-row (6th-row) result is the ensemble from the models in 2nd-  5th rows via an entropy-guided model adaptive ensemble strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Note that p is a  hyper-parameter in DivideMix to split noisy data into a labeled set and an unlabeled  set for MixMatch training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  We ensemble 4 models trained by different hyper-parameters via an entropy-  guided model adaptive ensemble strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Specifically, we choose a model as an  ECCV 2022 OOD-CV Challenge Image Classification Track  7  anchor model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' For each image, we trust the result if the entropy of the prediction  from the anchor model is small enough (less than 2/3 mean entropy of the test  set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Otherwise, we use the results from 4 models for ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In this way, the  inference time can be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' The 4 models are trained by different random  seeds or hyper-parameters, different p and different training epochs in DivideMix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  After ensemble, the OOD accuracy is improved from 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='30% to 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='03%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='3  Model Complexity  complexity  Parameters  Train Time  Single model  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='01GFLOPs  86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='9M  12hours * 8GPUs  TenCrop+Ensemble 880.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='4GFLOPs  347.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='6M  48hours * 8GPUs  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Method complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Four models with different training hyper-  parameters are used for model ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Eight Nvidia Tesla V100 GPUs are used  for testing the Train Time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  Please refer to Table 3 for model complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  5  Conclusion  The OOD-CV challenge deepens our understanding of model robustness under  domain shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' We thank the organizers for their great work in the challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Berthelot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Carlini, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Goodfellow, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Papernot, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Oliver, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Raffel,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' : Mixmatch: A holistic approach to semi-supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' In: Wal-  lach, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Beygelzimer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=", d'Alch´e-Buc, F." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Fox, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=', Garnett,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' (eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=') Advances in Neural Information Processing Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' Curran  Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' (2019), https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
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+page_content=': Robin: A benchmark for robustness to individual nuisances in real-world out-  of-distribution shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
+page_content='14341 (2021)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntE3T4oBgHgl3EQf7AuT/content/2301.04795v1.pdf'}
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+Equivariant and Steerable Neural Networks
+– A review with special emphasis on the symmetric group –
+Patrick Krüger1,2* and Hanno Gottschalk1,2
+1*Department of Mathematics and Science, University of
+Wuppertal, Gaussstr. 20, Wuppertal, 42110, Germany.
+2IZMD, University of Wuppertal, Liese Meitner Str. 27-31,
+Wuppertal, 42110, Germany.
+*Corresponding author(s). E-mail(s):
+patrick.krueger@uni-wuppertal.de;
+Contributing authors: hanno.gottschalk@uni-wuppertal.de;
+Abstract
+Convolutional neural networks revolutionized computer vision and na-
+trual language processing. Their efficiency, as compared to fully con-
+nected neural networks, has its origin in the architecture, where convo-
+lutions reflect the translation invariance in space and time in pattern
+or speech recognition tasks. Recently, Cohen and Welling have put this
+in the broader perspective of invariance under symmetry groups, which
+leads to the concept of group equivaiant neural networks and more
+generally steerable neural networks. In this article, we review the archi-
+tecture of such networks including equivariant layers and filter banks,
+activation with capsules and group pooling. We apply this formalism
+to the symmetric group, for which we work out a number of details
+on representations and capsules that are not found in the literature.
+Keywords: Group equivariant neural networks, steerable neural networks,
+symmetic group
+MSC(2020). 68T07, 68T45.
+1
+arXiv:2301.03019v1  [cs.LG]  8 Jan 2023
+
+2
+Equivariant and Steerable Neural Networks
+1 Introduction
+Neural networks are machine learning algorithms that are used in a wide vari-
+ety of applications, for example in image recognition and language processing.
+There are many applications, among them automated communication in the
+service sector, perception for self driving cars and the interpretation of medi-
+cal images in healthcare. In many machine learning problems, it is desirable to
+make the predictions of a network invariant to certain transformations of the
+input. This means that, for instance in image classification, we want an image
+that was transformed by a rotation by some angle to still be classified with the
+same label: A picture of a cat rotated by 90 degrees is still a picture of a cat.
+An important step towards learning these invariant representations was made
+by the introduction of convolutional neural networks by LeCun et al. for
+image classification of handwritten digits [1, 2]. These networks rely on the
+mathematical convolution operation to achieve higher efficiency by parameter
+sharing, as well as equivariance to translations. Equivariance means that a
+shift in the input data in some direction will be carried through all but the
+ultimate fully connected layers of the network and result in a similar shift in
+further deep layers, which can then be used to achieve translation invariant
+representations.
+While convolutional neural networks thus yield some of the desired invari-
+ance, their performance still decreases when the data is transformed by other
+symmetries, e.g. rotations or reflections. While this can be avoided by data
+augmentation, another compelling approach is to extend the translation
+equivariance of convolutional networks to a wider class of transformations.
+This is realized by group equivariant neural networks, or, in short, G-CNNs,
+which were introduced by Cohen & Welling in [3]. G-CNNs use group theory
+to generalize the convolution operation of conventional CNNs to group convo-
+lution, which then yields equivariance not only to translations, but to a wider
+group of transformations G of e.g. compositions of translations, reflections
+and rotations. These G-CNNs are again generalized by steerable CNNs (Co-
+hen & Welling, [4]) which, instead of modifying the convolution operation,
+instantiate equivariant filter banks and steerable feature spaces by making use
+of the theory of group representations. They thereby achieve a more general
+and versatile concept of group equivariance.
+The aim of this article is to give an overview of the theory of equivariant
+networks. For this, some understanding of the mathematical theory of groups
+and group representations is needed, which we provide in chapter 3.
+Chapter 4 will then be concerned with the theory of equivariant networks.
+After a short introduction on conventional, fully connected networks in section
+4.1, basic knowledge about convolutional neural networks is provided in section
+4.2 by explaining the core concepts of feature maps, filters, the convolution op-
+eration and the translation equivariance resulting from it, as well as parameter
+sharing. In section 4.3, we will explain how a modification of the conventional
+
+Equivariant and Steerable Neural Networks
+3
+convolution operation leads to G-CNNs and how these thereby achieve equiv-
+ariance to groups of transformations. The more general theory of steerable
+CNNs will then be presented in section 4.4. We will furthermore recapitulate
+how steerable CNNs are in fact a direct generalization of G-CNNs by explain-
+ing the equivalence of the latter to a special case of the former. The second
+aim of this article is to give some contributions by applications of G-CNNs and
+steerable CNNs for the symmetric group. This will be the focus of chapter 5.
+2 Related Work
+Steerable CNNs have besides [4] been investigated by several other publi-
+cations. Weiler et al. ([5]) discuss 3D-steerable CNNs, i.e. networks which
+process volumetric data with domain R3. In [6], Cohen et al. give a more ab-
+stract approach to intertwiners in steerable CNNs and it is shown that layers
+of an equivariant network in fact need to transform according to an induced
+group representation. A more general theory of steerable CNNs on homoge-
+neos spaces is discussed by the same authors in [7], showing that linear maps
+between feature spaces are in one-to-one correspondence to convolutions with
+equivariant kernels. E2-equivariant steerable CNNs that are equivariant to
+isometries of the plane R2 in the form of continuous rotations, reflections and
+translations are discussed in [8]. Furthermore, gauge equivariant networks,
+which enable equivariance not just to global symmetries, but also to local
+transformations, are discussed by Cohen et al. in [9].
+G-CNNs, i.e. networks that implicitly or explicitly rely on the regular repre-
+sentation of their respective group have also been studied several times. In
+[10], Gens & Domingos present symnets, which are a framework for networks
+with feature maps over arbitrary symmetry groups. Kanazawa et al. present
+in [11] a way to learn scale invariant representations while keeping parameter
+cost low. Dielemann et al. discuss exploiting rotation symmetry to predict
+galaxy morphology in [12] and expand their work to cyclic symmetries in
+[13]. Another network architecture that relies on regular representations is
+given by scattering networks, which were defined by Mallat et al. in [14] for
+compact groups, as well as for the Euclidean group in [15] and [16].
+Another important part of research concerning neural networks in general and
+equivariant networks specifically is the development of universal approxima-
+tion theorems. Many of these have been proved for different kinds of networks
+in varying levels of generality, with some of the first being [17] and [18]. Gen-
+erally speaking, all approximation theorems state that neural networks can,
+under certain conditions, approximate any function from a predefined function
+space with arbitrary precision. In particular, it was proved by Leshno et al.
+in [19] that multilayer feedforward networks can approximate any continuous
+function as long as non-polynomial activation functions are used.
+Petersen and Voigtlaender showed in [20] that fully connected feedforward
+
+4
+Equivariant and Steerable Neural Networks
+networks can under minimal conditions be translated into convolutional (i.e.
+translation equivariant) networks, thereby enabling the application of most
+approximation theorems for the former to the latter. For group equivariant
+maps, Kumagai & Sannai ([21]) also employ a conversion theorem from
+feedforward networks to CNNs to establish a method for obtaining universal
+approximation theorems for group equivariant convolutional networks.
+3 Mathematical Preliminaries
+In this section, some algebraic concepts that will be used and referenced
+throughout this thesis will be explained.
+Secondly, we briefly recall the representation theory of finite groups.
+Lastly, a short summary of the explicit representation theory of Sn, the
+symmetric group of n letters, is presented.
+3.1 Semidirect Products
+We define the core mathematical concepts needed for the theory of equivariant
+networks, starting with definitions of semi-direct products. While this section
+is kept quite short, there exist many introductions to this topic, see g.g. [22]
+or [23].
+Definition 1
+Let (G, ◦) be a group and X be some set. A (left) action of G on X is a binary
+operation
+· : G × X → X, (g, x) �→ g · x,
+(1)
+such that the following requirements are met:
+i) e · x = x ∀x ∈ X, with e being the neutral element of G.
+ii) (g ◦ h) · x = g · (h · x) ∀g, h ∈ G, x ∈ X.
+A group action is also often interpreted as a map
+φG : G → Aut(X) = {f : X → X | f is bijective }, g �→ φg.
+(2)
+Then, the two conditions above translate to the following:
+i) φe = id
+ii) φg ◦ φh = φgh ∀g, h ∈ G.
+Definition 2
+Let (H, ◦H) and (N, ◦N) be two groups and let furthermore φH : H → Aut(N) be
+a group action of H on N. Then, we can construct the outer semidirect product
+(N ⋊ H, •) by setting the cartesian product N × H as the group’s underlying set and
+defining the group operation as follows:
+• : N ⋊ H → N ⋊ H,
+(n1, h1) • (n2, h2) = (n1 ◦N φH(h1)n2, h1 ◦H h2).
+(3)
+
+Equivariant and Steerable Neural Networks
+5
+With this operation, H ∼= {(eN, h) | h ∈ H} and N ∼= {(n, eH) | n ∈ N} become
+subgroups of (N ⋊ H), with the latter being a normal subgroup as
+(n, h) • (n′, e) = (n ◦N φ(h)n′ ◦N n−1, e) • (n, h) ∀(n, h) ∈ N ⋊ H, (n′, e) ∈ N. (4)
+The neutral element is given by (eN, eH) and we have (n, h)−1 = (φH(h−1)n−1, h−1)
+for any (n, h) ∈ H⋊N. Furthermore, using above isomorphisms, any element (n, h) ∈
+N ⋊ H can be written as a unique product
+(n, h) = (n, eH) • (eN, h).
+(5)
+As N ∩ H = (eN, eH) holds trivially, the outer semidirect product thus has all
+properties of the inner semidirect product with respective isomorphic subgroups.
+Example 1
+I shall name two explicit examples for semidirect product groups and their respective
+actions here. These examples will be referenced in later parts of this article.
+i) The group p4 of compositions of rotations and translations of the square grid
+Z2 is the semidirect product of Z2 and C4, the group of 90-degree-rotations
+around any origin. Elements of p4 can be parametrized by 3 × 3 matrices
+depending on a rotational coordinate r ∈ {0, 1, 2, 3} and two translational
+coordinates (u, v) ∈ Z2:
+g(r, u, v) =
+�
+�
+cos( rπ
+2 ) − sin( rπ
+2 ) u
+sin( rπ
+2 )
+cos( rπ
+2 )
+v
+0
+0
+1
+�
+�
+(6)
+p4 now acts on points x = (u′, v′) ∈ Z2 by matrix multiplication from the
+left, after adding a homogenous coordinate to x, i.e. x = (u′, v′) �→ (u′, v′, 1).
+ii) The group p4m of compositions of rotations, mirror reflections and trans-
+lations of Z2 is the semidirect product of Z2 and D4, which is the group
+of 90-degree-rotations and reflections about any origin. Elements of this
+group can be parametrized similarly to p4, with the difference of adding a
+reflection coordinate m:
+g(m, r, u, v) =
+�
+�
+(−1m) cos( rπ
+2 ) −(−1m) sin( rπ
+2 ) u
+sin( rπ
+2 )
+cos( rπ
+2 )
+v
+0
+0
+1
+�
+�
+(7)
+p4m then acts on Z2 analogously to p4.
+In section 5, we will investigate steerable CNNs that rely on the semidi-
+rect product of the symmetric group Sn and Zn. For this, some preliminary
+definitions shall also be given here.
+
+6
+Equivariant and Steerable Neural Networks
+Definition 3
+i) The symmetric group Sn is the group of permutations of n letters, i.e.
+Sn = {σ : {1, .., n} → {1, .., n} | σ is bijective},
+with the group operation being defined by composition of elements, i.e.
+functions.
+ii) For r ≤ n, a r-cycle is an element of σ ∈ Sn, such that there exist
+x1, .., xr ∈ {1, .., n} with
+σ(x1) = x2, σ(x2) = x3, .., σ(xr−1) = xr, σ(xr) = x1,
+σ(xk) = xk ∀ xk /∈ {x1, .., xr}.
+An r-cycle as above is often denoted (x1 x2 .. xr). We call this the cycle
+notation, and we call r the cycle length of σ.
+iii) We obtain an action of Sn on Zn for any n ∈ N by letting σ ∈ Sn permute
+the coordinates of x = (x1, .., xn) ∈ Zn:
+σ ◦ (x1, .., xn) = (xσ(1), .., xσ(n))
+(8)
+Example 2
+With above action, we can construct the outer semidirect product Zn ⋊ Sn of prod-
+ucts tσ of translations t and coordinate permutations σ. This group can now be
+parameterized and made to act on Zn in analogy to example 1 by n + 1 × n + 1 ma-
+trices with the upper left n×n block being a permutation matrix representing σ and
+a translation vector t ∈ Zn in the first n entries of the last column. Two examples
+for n = 2 and n = 3 shall be given here:
+g2(((12), (3, 3))) =
+�
+�
+0 1 3
+1 0 3
+0 0 1
+�
+� , g3(((123), (1, 2, 2))) =
+�
+�
+�
+�
+0 0 1 1
+1 0 0 2
+0 1 0 2
+0 0 0 1
+�
+�
+�
+�
+(9)
+3.2 An Introduction to Representation Theory of Finite
+Groups
+Definition 4
+Let G be a group.
+i) A representation (V, ρ) of G consists of a vector space V , together with a
+group homomorphism ρ : G → GL(V ) from G to the general linear group
+of V , i.e. the group of invertible linear maps from V to V , such that:
+ρ(gh) = ρ(g)ρ(h) ∀g, h ∈ G.
+(10)
+The dimension or degree of the representation is defined as the dimension
+of its vector space V .
+
+Equivariant and Steerable Neural Networks
+7
+ii) A subrepresentation of a representation (V, ρ) is a subspace W
+⊆ V
+which is G-invariant, meaning ρ(g)w ∈ W for all g ∈ G, w ∈ W. The
+subrepresentation is then denoted by (W, ρ ↾W ), and we have
+ρ ↾W (g) = ρ(g) ↾W .
+(11)
+Any representation (V, ρ) always has at least two subrepresentations: Itself
+and {0}.
+iii) If a representation (V, ρ) with V
+̸= {0} has no subrepresentations ex-
+cept itself and {0}, it is called irreducible. Otherwise, it is called reducible.
+Throughout this thesis, we will often refer to irreducible representations as
+irreps.
+iv) For two representations (V1, ρ1) and (V2, ρ2) of G of respective degrees n1
+and n2 we can define the direct sum of representations, (V1 ⊕ V2, (ρ1 ⊕ ρ2)),
+with
+(ρ1 ⊕ ρ2)(g)(v, w) = (ρ1(v), ρ2(w)). This is again a representation of G and
+has degree n1 + n2.
+Example 3 (The Quotient Representation)
+Given the quotient space (or quotient group) G/H of a group G and some subgroup
+H ⊆ G, we define the quotient representation (Vquot, ρquot) by associating a basis
+vector with every coset in G/H:
+Vquot = ⟨{egH | gH ∈ G/H}⟩
+(12)
+and define ρquot using the action of G on cosets:
+ρquot(g′)egH = eg′gH ∀g′ ∈ G, ∀egH ∈ Vquot.
+(13)
+As this representation just permutes the basis vectors, this time corresponding to
+cosets, this can also be realized by permutation matrices. If we choose H = e, and
+thus receive G/H = G, this yields the so-called regular representation which permutes
+basis vectors eg for each g ∈ G.
+Definition 5
+Let (V1, ρ1) and (V2, ρ2) be two representations of some group G and let f : V1 → V2
+be a linear map.
+i) f is called an intertwiner between ρ1 and ρ2, if
+f(ρ1(g)(v)) = ρ2(g)(f(v)), ∀g ∈ G, ∀v ∈ V1.
+ii) ρ1 and ρ2 are called equivalent or isomorphic, if there exists an intertwiner
+f : V1 → V2 between them, which also is a vector space isomorphism, i.e. an
+invertible linear map. We then often write V1 ≃ V2 or ρ1 ≃ ρ2.
+iii) The condition in i) is linear in f, thus any linear combination of intertwiners
+between representations ρ1 and ρ2 is again an intertwiner. We hence receive
+a vector space of intertwiners between ρ1 and ρ2, denoted HomG(ρ1, ρ2).
+
+8
+Equivariant and Steerable Neural Networks
+Theorem 1 (Schur’s Lemma)
+Let (V1, ρ1) and (V2, ρ2) be irreducible representations of G.
+i) Iff V1 and V2 are not isomorphic, then dim HomG(ρ1, ρ2) = 0, ie the only
+intertwiner between ρ1 and ρ2 is the zero map.
+ii) Iff V1 and V2 are isomorphic, then dim HomG(ρ1, ρ2) = 1, and all maps
+intertwining ρ1 and ρ2 are scalar multiples of the identity map.
+Theorem 2
+i) Any finite dimensional representation (V, ρ) of a finite group can be decom-
+posed into a direct sum of irreducible representations:
+(V, ρ) ∼ (⊕Vi, ⊕ρi),
+where (Vi, ρi) are irreps of G.
+ii) The irreps of G are uniquely determined up to isomorphism, and there are
+finitely many of them.
+Proof See [24], section 1.4, Theorem 2 for i) and section 2.5, Theorem 7 for ii).
+□
+Definition 6
+Let Irr(G) be the set of nonisomorphic irreducible representations of G. Then, any
+(Vi, ρi) ∈ Irr(G) can occur in the decomposition of some finite dimensional repre-
+sentation (V, ρ) any number of times. A list of integers mρi(ρ) ≥ 0 corresponding to
+the multiplicity of each irrep (Vi, ρi) in (V, ρ) is called the type of ρ. Representations
+are uniquely determined up to isomorphism by their types, meaning that if two
+representations have the same type, then they are isomorphic and that isomorphic
+representations always have the same type.
+4 The Theory of Equivariant Networks
+In this section, the general theory behind convolutional neural networks is ex-
+plained. We cover three types of convolutional networks which are subsequent
+generalizations of each other. In section 4.1, We give a brief overview on how
+non-convolutional, fully connected networks work. Section 4.2 will then cover
+the basics on standard translation equivariant convolutional networks. Key
+concepts, such as feature maps, filters, convolutional layers and the convolution
+operation, as well as activation and pooling layers is discussed. In section 4.3,
+we elaborate the generalization of convolutional networks to group equivariant
+networks (G-CNNs). All concepts from the previous section are suitably gener-
+alized, such as e.g. G-feature maps and group convolution will be touched on,
+and it will be shown how the resulting networks have a “more general” form of
+
+Equivariant and Steerable Neural Networks
+9
+equivariance. Lastly, in section 4.4, we introduce steerable convolutional net-
+works, which use representation theory to yield a more efficient and versatile
+way to define group equivariant networks where the group acts on fibers of
+feature maps, also called filter banks. It should be noted that while G-CNNs
+and steerable CNNs can be realized for many kinds of groups, the sections 4.3
+and 4.4 will focus on networks that use split groups, i.e. groups that are con-
+structed as a semidirect product. Explicitly, we will consider networks on p4
+and p4m, which were defined in example 1, iv) and v). However, the concepts
+can easily be generalized to other split groups. A G-CNN that does rely on a
+non-split group will be covered in section 5.
+4.1 Fully Connected Neural Networks
+Deep neural networks describe a wide range of computing systems in the do-
+main of machine learning that are meant to loosely resemble the human brain
+by using interconnected layers, indexed in the following by l = 1, ..., L, of
+neurons. Each neuron N l
+i, i = 1, ..., Kl at each layer l of the network can be
+thought of as a simple unit that holds a number, usually referred to as its ac-
+tivation, and which is connected to all of neurons of the next layer. Therefore,
+this kind of network is often called fully connected. Each of these connections
+is determined by a weight and a bias. We denote by ωl
+i,j and bl
+i,j the weight
+and bias that connect the ith neuron of layer l, N l
+i, to the jth neuron N l+1
+j
+of
+layer l + 1. In the so-called forward propagation step of a network, starting at
+l = 1, each neuron is updated in the following way:
+N l
+i = σl
+�
+�
+Kl−1
+�
+j=1
+ωl−1
+j,i N l−1
+j
++ bl−1
+j,i
+�
+� ,
+(14)
+where σ is some non-linear activation function. These will be discussed in
+4.2.5 This process is then repeated sequentially for all remaining l = 2, ..., L,
+until the final layer is reached. This output layer consists of one neuron for
+each possible label of the classification problem at hand. The output neuron
+with the highest activation is returned as the networks predicted label for the
+given input.
+To train a deep neural network, a training dataset with inputs that are already
+labeled with their respective classifications is needed. During the training
+phase, after each forward pass, the (initially large) error of the network, i.e.
+the difference between the model’s prediction and the actual labels of the
+training data is quantified by a loss function which takes as inputs the weights
+and biases of the network. A backpropagation algorithm then produces the
+gradients of this loss function with respect to its variables (i.e. the weights
+and biases), which are often called the parameters of the networks. After
+obtaining the gradients, an optimization algorithm, the most used one being
+
+10
+Equivariant and Steerable Neural Networks
+stochastic gradient descent ([25]), is used to modify the networks parameters,
+thereby slightly optimizing the loss function. This training process of forward
+pass and backpropagation is now repeated until the error is sufficiently small.
+Then, the backpropagation step is no longer needed and the network can be
+used to classify unlabeled data. The reader interested in a more thorough
+introduction to feedforward networks is referred to chapter 6 of [26].
+4.2 Basics on Convolutional Neural Networks
+Convolutional Neural Networks (CNNs) are a powerful tool in machine learn-
+ing, mainly used for pattern recognition and classification of certain input
+data. They can be used on a variety of data, for example sound signatures
+(one-dimensional data) and 2D and/or 3D Images. Just as fully connected net-
+works, CNNs work in layers, in each of which a set of filters representing certain
+features to look for in the input data is convolved with stacks of so-called fea-
+ture maps. After application of certain operations known as nonlinearities and
+pooling, this yields a new set of feature maps, which is then convolved with a
+new set of filters in the next layer. A core feature of CNNs is the equivariance
+to translation of their convolutional layers. This means that shifting the input
+data of a convolutional Layer will result in a similar shift that layer’s out-
+put, allowing us to detect features independent of their location. In the next
+few paragraphs we will explain these terms in more detail. We use a Network
+operating on 2D black and white images, i.e. an input feature map with one
+channel as example.
+4.2.1 Feature Maps and Filters
+Feature maps
+Feature maps are mathematical functions used to describe the input data,
+as well as the outputs of each convolutional layer. Depending on the type
+of the input, their domain can vary. In image recognition, the domain of a
+feature map usually is the two-dimensional pixel grid Z2. In our example
+of a black and white image, an input-level feature map f : Z2 → R simply
+returns the grey-value at each pixel x ∈ Z2. In a coloured image, the function
+would instead be of the form f : Z2 → R3, returning a vector consisting of
+the color channel values at each pixel coordinate. Since images are bound in
+size, a feature map is usually said to just return zero everywhere outside of a
+certain subdomain of pixels. Since layers of convolutional networks contain
+more of one feature map most of the time, it is often also spoken of ’stacks’
+of feature maps f j : Z2 → RK, j = 1, .., n, whereby the index is often omitted
+for simplicity.
+Filters
+Filters are used to extract certain characteristic patterns in our data by con-
+volving them with feature maps, which will be described in the next paragraph.
+Mathematically, they are also described as functions. For convolution to work,
+
+Equivariant and Steerable Neural Networks
+11
+it is important that these functions share the domain and image space of the
+feature maps that they are to be convolved with. In our example, a one-channel
+filter looking for diagonal lines (top left to bottom right) could look like this:
+ψ : Z2 → R;
+ψ(x) =
+�
+1
+x ∈ {(−1, 1), (0, 0), (1, −1)}
+0
+elsewhere.
+(15)
+The support, i.e. the non-zero domain of filters is usually much smaller than
+that of the feature maps, with usual widths being 3 × 3 or 5 × 5 squares cen-
+tered at the origin. While classical computer vision used handcrafted filters
+as in the above example, CNN based computer vision uses learnable filters
+ψ = (ψi,j))i,j=−s,...,s, s = 1, 2. The values ψi,j are the learned parameters of
+the network. As will elaborated in 4.2.4, this drastically reduces the parame-
+ter cost of CNNs in comparison to fully connected networks.
+4.2.2 The Convolution Operation
+The core building block of regular CNNs are the so called convolutional layers.
+In each of these layers, a stack of feature maps f : Z2 → RK is convolved with
+a set of filters ψ : Z2 → RK, producing a new feature map which will then be
+used in the next layer. The convolution operation is mathematically defined
+as follows:
+f ⋆ ψ : Z2 → R;
+[f ⋆ ψ](x) =
+�
+y∈Z2
+K
+�
+k=1
+fk(y)ψk(y − x)
+(16)
+While this may look complicated at first glance, it can be interpreted as sim-
+ply sliding our small filter window over the domain of the image or feature
+map and summing up the element-wise products of the filter’s values and the
+feature map’s values lying “under” them at each output channel k and each
+respective position x of the filter. To demonstrate this further, we can inter-
+pret our example filter ψ from (15), as well as a feature map f as matrices.
+The feature map and filter, as well as the result of the convolution operation
+of said elements is illustrated in figure 1.
+Here, as in all following figures throughout this thesis, black pixels represent
+one, grey pixels zero and white pixels represent minus one.
+Recall that, even though the domain of f and ψ is infinite, both functions just
+return 0 everywhere outside of the depicted areas.
+Convolving a feature map with a filter yields a new feature map
+f ⋆ ψ : Z2 → R, describing how well the feature described by the filter fits at
+different positions x ∈ Z2 in the image.
+As announced, we will from now on often omit the summation over channels
+
+12
+Equivariant and Steerable Neural Networks
+2
+3
+3
+0
+1
+1
+0 0 0 0 0
+0 0 0
+0
+1
+0
+1
+0
+0
+0
+0
+⋆
+=
+f
+ψ
+=
+f ⋆ ψ
+Figure 1: A feature map f representing the letter “V” is convolved with a
+filter detecting diagonal edges from top left to bottom right, yielding a new
+feature map f ⋆ ψ.
+in (16) for simplicity and pretend to have feature maps and filters with just
+one channel, as this will not harm any arguments made in the following para-
+graphs.
+4.2.3 Translation Equivariance
+An important property that makes CNNs such powerful tools, for example in
+image recognition, is their equivariance to translations of the input at each
+layer. Roughly speaking, this means that a CNN is able to detect features in
+an image (or other data) regardless of their specific position, without requiring
+additional parameters. To be more precise, we define a translation of a feature
+map mathematically:
+[Ttf](x) = f(t−1x) = f(x − t); x, t ∈ Z2
+(17)
+A feature map f is hence transformed by a translation t ∈ Z2 by looking up the
+value of f at the point t−1x = x − t and moving it to the position x, resulting
+in the t-transformed feature map Ttf. Visually, this translates to a shift of the
+patterns of a given input by the respective horizontal and vertical coordinates
+of t. For an illustration, see the left side of figure 2. The underlying concept
+of a group action of Z2 on itself is also used by the convolution operation
+itself: In (16), filters are transformed by Tx when calculating the output of the
+convolution at position x, representing a shift of said filter to that position.
+Equivariance to translations in CNNs now means that applying a translation
+t to a feature map f, followed by a convolution with a filter ψ yields the same
+
+Equivariant and Steerable Neural Networks
+13
+result as convolving f with ψ first and then applying the translation:
+[[Ttf] ⋆ ψ](x) =
+�
+y∈Z2
+f(t−1y)ψ(x−1y)
+=
+�
+y∈Z2
+f(y − t)ψ(y − x)
+=
+�
+y∈Z2
+f(y)ψ(y + t − x)
+=
+�
+y∈Z2
+f(y)ψ(y − (x − t))
+=
+�
+y∈Z2
+f(y)ψ((t−1x)−1y)
+= [Tt[f ⋆ ψ]](x).
+(18)
+⋆ψ
+⋆ψ
+Tt
+Tt
+f
+Tt(f)
+Tt(f ⋆ ψ)
+f ⋆ ψ
+Figure 2: A translation equivariant convolution. Here, the feature maps are
+transformed by t = (−2, 1) and the diagram commutes.
+4.2.4 Parameter Sharing
+In comparison to conventional, fully connected neural networks (section 4.1),
+convolutional neural networks increase the efficiency of their parameters by
+utilizing the convolution operation to achieve parameter sharing across layers.
+In fully connected layers of a conventional neural network, each neuron is
+
+14
+Equivariant and Steerable Neural Networks
+connected to the neurons of the next layer by an individual connection con-
+sisting of a weight and a bias, which together make up the parameters of
+the network. Each of these connections is modified individually to achieve
+the best possible classification result. In image recognition for instance, one
+can view each pixel of the input image as a neuron, and each of these is
+connected via individual parameters to the neurons of the subsequent layer.
+As layers are stacked on top of each other, one can imagine that the number
+of connections, and thus of parameters will grow quite large.
+For convolutional neural networks, consider each channel f k′
+l , k′ = 1, .., Kl
+of a feature map fl : Z2 → RKl with Kl output channels in a given layer l.
+Each of these channels is obtained by convolving a small filter patch ψk′ of
+size s × s × Kl−1 with the stack of feature maps f k
+l−1 of the previous layer.
+Here, k = 1, .., Kl−1 denotes the individual channels and Kl−1 is the number
+of channels of the feature map of layer l − 1. All values f k′
+l (x) of an output
+channel k′ ∈ {1, .., Kl} for each x ∈ Z2 are thus obtained from the same filter,
+evaluated at different spatial coordinates, and hence share the same s2Kl−1
+parameters. For the whole layer l, this then amounts to s2Kl−1Kl parameters.
+For comparison, as each pixel would need its own parameters, a fully con-
+nected layer processing the same data would need around Kl−1KlZl−12Zl2
+parameters with Zl−1 and Zl being the spatial extend of the non-zero do-
+mains of the feature maps fl−1 and fl, which is substantially larger than s2,
+with s usually being 7 or smaller.
+4.2.5 Nonlinearities and Pooling
+Nonlinearities
+The convolutional layers of a CNN are interspersed with non-linear layers.
+Such a layer operates by applying a non-linear function to the input is needed
+to enhance the expressive power of CNNs, since hardly any phenomenon that
+CNNs are meant to observe can be described by linear functions. Various
+universal approximation theorems, for instance in [19] and [20, 27], prove
+that (feedforward and convolutional) networks can approximate almost any
+function with arbitrary levels of precision, as long as non-polynomial nonlin-
+earities are used.
+One of the most prominent examples for a nonlinearity is the rectified linear
+unit ReLU:
+ReLU(x) := max(x, 0)
+(19)
+This function has empirically been proven to be the most efficient choice in
+many cases, as its computation is faster than most other non-linear functions
+and it is less prone to plateaus in the training process of the network.
+Note that ReLU is applied element-wise, meaning that if a multi-channel
+feature map f : Z2 → RK is given, ReLU is evaluated at each output channel
+
+Equivariant and Steerable Neural Networks
+15
+fk(x), k = 1, .., K, individually. There are other forms of fiber-wise nonlinear-
+ities, which will be touched on in section 4.4.
+Pooling
+In CNNs, convolutional layers are often interspersed with so called pooling
+layers, with one of the first examples being [28]. In these layers, a pooling
+operation is performed on a given set of feature maps. The main goal of this
+operation is to cut out superfluous information, reducing data size by slightly
+reducing locational precision. This can be done because a CNN usually does not
+require the exact location of a feature to function properly. There are several
+different pooling operations, one of the most frequently used is max pooling:
+P : Z2 → R;
+Pf(x) :=
+max
+x′∈U(x)f(x′),
+(20)
+where f is a feature map and U(x) is some neighbourhood of x in Z2, in
+this case usually being a small square with center x. The pooling operation
+thus works by finding the highest activation in said neighbourhood. Average
+pooling, which is another frequently used operation, would find the average of
+all activations in the pooling window.
+To actually reduce the size of the feature map, pooling needs to be performed
+with a stride, meaning that P is not evaluated on every point in the base space,
+but only on points of certain distance to each other. For example, a stride of 2
+would mean that in (20), P is only evaluated at points in 2Z2 = {2x : x ∈ Z2},
+while the neighbourhoods U(x) remain subsets of Z2.
+4.3 Group Equivariant CNNs
+4.3.1 Motivation
+We can interpret convolutional networks from section 4.2 as already being
+G-CNNs with G the group of translations, Z2. Our convolution operation
+[f ⋆ ψ](x) basically returns the activation of ψ, transformed under the action
+of the group element x, i.e. a translation. HGFor a general G-CNN, one would
+like to find a way to replace the underlying group of the network with some
+other group G, containing more transformations than just translations, for
+example rotations and reflections, and still have equivariance of convolution
+with respect to every element of G.
+4.3.2 Group Convolution, Feature Maps and Equivariance
+We first establish the action of the group G. The convolution operation from
+equation (16) is modified in two steps. First, we consider the initial convolu-
+tional layer of the network, i.e. where the image f : Z2 → R is convolved with
+
+16
+Equivariant and Steerable Neural Networks
+the first set of filters:
+f ⋆ ψ : G → R;
+[f ⋆ ψ](g) =
+�
+y∈Z2
+f(y)ψ(g−1y)
+(21)
+Two things should be noted: First, g ∈ G again transforms feature maps (or
+filters) via a group action on the domain of said maps:
+[Tgf](x) = f(g−1x); x ∈ Z2, g ∈ G.
+(22)
+Specifically for G = p4 or G = p4m, g ∈ G transforms f(x) by moving its out-
+put from x ∈ Z2 to the pixel at g−1x under the actions defined in example 1.
+Secondly, it should be noted that the convolution operation f ⋆ ψ itself now
+yields a function on the group G instead of Z2, thus requiring filters in the
+subsequent layer to also be functions on G, yielding yet another slightly
+different convolution operation:
+f ⋆ ψ : G → R;
+[f ⋆ ψ](g) =
+�
+h∈G
+f(h)ψ(g−1h)
+(23)
+As one might notice, the transformation of feature maps and filters by G is
+again slightly different:
+[Tgf](h) = f(g−1h); h, g ∈ G.
+(24)
+Hence, the output of f at a point h ∈ G of the domain (which also is just G)
+is moved to the point g−1h, which is obtained by the canonical action of G on
+itself by its group operation.
+While it is easy to imagine feature maps f : Z2 → R simply as images
+sampled on a pixel grid, feature maps with base space G might not be as
+intuitive. For G = p4, a feature map f : G → R can be imagined as a graph
+with four patches, of which each corresponds to one of the four rotations
+C4 = {e, r, r2, r3}. Each pixel now has a rotational coordinate, corresponding
+to the patch in which it appears, and two translational coordinates specifying
+its position in the respective patch. These coordinates do also coincide with
+the semidirect product structure of p4, enabling us to write any element
+g ∈ p4 as a product g = tr, t ∈ Z2, r ∈ C4.
+The rotation r for instance now acts on this graph by moving each patch
+along the arrows indicated in figure 3, and by also rotating each of the patches
+themselves by 90 degrees, as is also shown in figure 3.
+The convolution operation of the first layer (21) and the convolution operations
+of the subsequent layers (23) are now not only equivariant to translations, but
+to all transformations from the group G, expanding translational equivariance
+
+Equivariant and Steerable Neural Networks
+17
+e
+r3
+r2
+r
+e
+r3
+r2
+r
+Figure 3: A p4 feature map (left) and its rotation by r (right).
+by e.g. compositions of translations and rotations for p4 and by compositions
+of translations with rotations and reflections for p4m:
+[[Tuf] ⋆ ψ](g) = [Tu[f ⋆ ψ]](g) ∀ u, g ∈ G.
+(25)
+For higher layer convolutions, this is derived in complete analogy to (18), this
+time using the fact that uG = G ∀u ∈ G and thus substituting uh ∈ G for
+h, again keeping the overall sum the same. At the same point for first layer
+convolutions, we use an analogous argument gZ2 = Z2 ∀g ∈ G, as G acts
+transitively on Z2, meaning that any point in the pixel grid can be reached
+from any other point by a transformation from G.
+4.3.3 Group Pooling and Equivariant Nonlinearities
+Equivariance to element-wise nonlinearities
+Element-wise nonlinearities ν : R → R can be used in G-CNNs without
+restriction, as post-composing them with a feature map preserves the equiv-
+ariance to pre-compositions with group transformations:
+Let ν : R → R be an element-wise nonlinearity such as e.g. ReLU, and define
+the post-composition, i.e. the application of such a function to a feature map
+f:
+Cν(f(g)) = [ν ◦ f](g) = ν(f(g)).
+Let Tg be the left transformation operator from (22) or (24). As Tg is realized
+by pre-composition with f and Cν by post-composition, these actions commute
+and we get
+CνTgf = ν ◦ [f ◦ g−1] = [ν ◦ f] ◦ g−1 = TgCνf.
+(26)
+Group Pooling
+As in Regular CNNs, convolutional layers of a G-CNN are often followed by a
+
+18
+Equivariant and Steerable Neural Networks
+pooling layer. As we no longer need to have the simple pixel grid Z2 as base
+space, we need to define the notions of pooling and stride for the more general
+case of the base space being a group G. To simplify the process, we split it into
+two steps, namely the pooling step, which is performed without stride, and the
+subsampling step which can then realize any notion of stride, if desired. In the
+first step, the max pooling operation for instance becomes
+P : G → R;
+Pf(g) := max
+x∈gU f(x).
+(27)
+Here, gU := {gu | u ∈ U} is some g-transformed neighbourhood U of the
+identity element in G. In Z2, this would correspond to a square around the
+origin that is moved across the images by translations t ∈ Z2. This operation
+is now equivariant to G:
+PThf(g) = max
+k∈gU Thf(k)
+= max
+k∈gU f(h−1k)
+= max
+hk∈gU f(k)
+=
+max
+k∈h−1gU f(k)
+= Pf(h−1g)
+= ThPf(g)
+(28)
+The arguments behind these equations are as follows: The first two equations
+are just the definitions of group pooling (27) and the left transformation of
+feature maps (24), respectively. In the third line, we substitute k for hk using
+the fact that taking the maximum of f(h−1k) over k ∈ gU is the same as
+taking the maximum of f(k) with k such that hk is in gU. In the fourth line
+we use that this is again the same as taking k ∈ h−1gU, which can be seen by
+just multiplying both sides with h−1. The last two lines are then obtained by
+just resubstitutuing the definitions of group pooling and Th.
+Any stride could now be realized in the second step by subsampling the
+pooled feature map over a subgroup H ⊆ G, i.e. evaluating just on points
+h ∈ H instead of all g ∈ G. However, a feature map subsampled in this way
+would not anymore be equivariant to all of G, but only to H. As we wish to
+maintain equivariance to the whole group G, this form of stride is usually not
+used in practical G-CNN applications.
+Instead, to preserve G-equivariance throughout the network, one can use coset
+pooling by choosing the pooling neighbourhood U in the first step to be itself
+a subgroup H of G. The resulting transformed pooling regions then are the
+non-overlapping cosets of H in G which are either disjoint or equal for any gH
+and g′H. Because of this, any element of each of the distinct cosets can then
+
+Equivariant and Steerable Neural Networks
+19
+be chosen as a representative to subsample on. The resulting pooled feature
+map can then be interpreted as a map on the quotient space G/H, which can
+then be acted upon by G similar to (24) by utilizing the general action of G on
+its quotient spaces from example 1, preserving equivariance as shown in (28):
+Tg′f(gH) = f((g′−1g)H), g′ ∈ G, gH ∈ G/H
+(29)
+As an example of this, consider a p4-feature map that is pooled over the group
+of rotations, C4 = {e, r, r2, r3}. Visually (see figure 4), this equates to checking
+the four rotational outputs of each pixel coordinate x ∈ Z2 and choosing the
+one with the highest activation as representative for the coset {x, xr, xr2, xr3}.
+The resulting feature map is then of domain p4/C4 = Z2 and thus transforms
+in the same way as an input feature map in (22).
+e
+r3
+r2
+r
+max
+x∈(−2,2)C4
+f(x)
+(−2, 2)
+(−2, 2)r3
+(−2, 2)r2
+(−2, 2)r
+(−2, 2)C4
+Figure 4: Coset pooling by H = C4 of a p4 feature map. A single coset
+((−2, 2)C4) is highlighted.
+4.3.4 Implementation
+G-Convolution can be implemented rather easily, at least for so-called split
+groups. By exploiting this property, we can just use a standard convolution
+routine with an expanded filter bank, which will be described shortly.
+Recall that a group being split means that any element g ∈ G can be written
+as a product g = ts. For p4 and p4m, t ∈ Z2 would be a translation, and s
+would be a transformation from the stabilizer group H = C4 or H = D4 that
+leaves the origin invariant, i.e. a rotation or roto-reflection around the origin.
+This, together with TtTs = Tts for the action of G allows us to rewrite the
+
+20
+Equivariant and Steerable Neural Networks
+definition of G-convolution as follows:
+f ⋆ ψ(g) = f ⋆ ψ(ts) =
+�
+x∈X
+f(x)Tt[Tsψ(x)]
+(30)
+with X = Z2 in layer one and X = G in subsequent layers, thus allowing us to
+precompute the transformed filters Tsψ for all transformations of the stabilizer
+and then convolve them with the input using a fast planar convolution routine.
+The set of untransformed filters at some layer l can be sorted in an array F
+of shape Kl × Kl−1 × Sl−1 × n × n. Here, Kl−1 denotes the number of input
+channels, Kl is the number of output channels i.e. the number of distinct
+filters, and n×n denotes the spatial extend of the filters. Furthermore, Sl−1 is
+the size of the stabilizer group, i.e. the number transformations of G that fix
+the origin in the base space of the feature maps that F is to be convolved with.
+Each transformation Ts now “acts” on F by permuting the scalar entries
+of each of the Kl × Kl−1 distinct “filter blocks” of shape Sl−1 × n × n. If
+Sl transformations are applied, this leads to an extended array of shape
+Kl × Sl × Kl−1 × Sl−1 × n × n.
+The permutations themselves can be realized by implementing an invertible
+map g which yields the group element corresponding to an index from the
+array of shape Sl−1 × n × n, represented as matrices. For example, for p4 this
+map would be the following, as was described in example 1, iv):
+g(s, u, v) =
+�
+�
+cos( sπ
+2 ) − sin( sπ
+2 ) u
+sin( sπ
+2 )
+cos( sπ
+2 )
+v
+0
+0
+1
+�
+�
+(31)
+We then set
+F +[i, s′, j, s, u, v] = F[i, j, ¯s, ¯u, ¯v]
+(32)
+with
+(¯s, ¯u, ¯v) = g−1(g(s′, 0, 0)−1g(s, u, v)).
+(33)
+To use F + in a planar convolution routine, we exploit the fact that X in (30)
+involves a sum over the stabilizer, again allowing us to to rewrite the equation
+as
+�
+x∈X
+f(x)Tt[Tsψ(x)] =
+�
+x∈Z2
+Sl−1Kl−1
+�
+k=1
+fk(x)Tt[Tsψk(x)].
+(34)
+
+Equivariant and Steerable Neural Networks
+21
+We can now reshape F + into an array of shape SlKl ×Sl−1Kl−1 ×n×n which
+can then be applied to similarly reshaped feature maps in a planar convolution
+routine.
+4.4 Steerable CNNs
+4.4.1 Motivation
+It was shown how G-CNNs achieve group equivariance by expanding the do-
+main of the feature maps and filters. The magnitude of the expansion depends
+on the size of the stabilizer group of the origin. Thus, larger groups lead to
+larger expansions, which in turn lead to a proportionally increasing computing
+cost.
+Steerable CNNs are a generalization of G-CNNs which achieve equivariance by
+defining filter banks as intertwiners, which are morphisms between group rep-
+resentations, through which feature spaces become G-steerable. For a classical,
+unrestricted filter bank to be an intertwiner, it needs to satisfy an equivariance
+constraint, which depends on the representations that it is meant to inter-
+twine. Thus, while G-CNNs achieve equivariance by expanding arbitrary filter
+banks, Steerable CNNs achieve it by restricting the space of available filters,
+thus decoupling the required computational power from the size of the group.
+4.4.2 Feature Spaces, Fibers and Steerability
+We once again consider 2D signals f : Z2 → RK with K channels. These can
+be added, as well as multiplied by scalars and therefore form a vector space,
+often also called feature space, which we will denote by Fl. The layer index l
+will often be omitted for simplicity.
+Given a group G acting on Z2, we are able to transform signals f ∈ F0:
+[π0(g)f](x) = f(g−1x).
+(35)
+π0(g) : F0 → F0 is then a linear map ∀g ∈ G that furthermore satisfies
+π0(gh) = π0(g)π0(h) ∀g, h ∈ G,
+(36)
+yielding a group homomorphism π0 : G → GL(F0). A vector space such as
+F0, together with a map such as π0 that satisfies (36) fulfills the conditions of
+a group representation defined in definition 4. It shall be denoted by (F0, π0).
+Oftentimes we will just call π0 (or πl) a representation, if the nature of F0
+(Fl) is clear. As will be explained later, the representations πl transforming
+higher layer feature spaces Fl might look slightly different than in (35). Nev-
+ertheless, they will always satisfy (36).
+While in most other deep learning publications some f ∈ F is usually consid-
+ered as a stack of K feature maps fk : Z2 → R, k = 1, .., K, it is useful for
+
+22
+Equivariant and Steerable Neural Networks
+the theory of steerable CNNs to consider another decomposition: Instead of
+splitting the feature space “horizontally” into one-dimensional planar feature
+maps, it can be decomposed into fibers Fx, one of which being located at each
+“base point” x ∈ Z2. Each fiber consists of the K-dimensional vector space
+representing all channels at the given position. f ∈ F is therefore comprised
+of feature vectors f(x) ∈ Fx. See figure 5 for an illustration.
+Figure 5: The decomposition of F into stacks of feature maps (left), and into
+fibers (right). A single fiber is highlighted.
+Let F, F′ be two feature spaces, and let π : G → GL(F) be a group homomor-
+phism, such that (F, π) is a group representation. Let furthermore Φ describe
+a convolutional layer of a network, i.e.
+Φ : F → F′;
+Φ(f) = f ⋆ Ψ ∀f ∈ F
+(37)
+for some filter bank Ψ : Z2 → RK.
+We now say F′ is steerable w.r.t. G, if there exists another function
+π′ : G → GL(F′) transforming F′ such that
+Φπ(g)f = π′(g)Φf ∀g ∈ G,
+(38)
+meaning that transforming the input by π(g) yields the same result under Φ
+as transforming the output of Φ by π′(g). The following equation shows that
+this condition implies that π′ : G → GL(F′) is also a group homomorphism,
+and thus (F′, π′) is also a group representation:
+π′(gh)Φf = Φπ(gh)f = Φπ(g)π(h)f = π′(g)Φπ(h)f = π′(g)π′(h)Φf.
+(39)
+Note that (39) only implies the desired property of π′ for the span of the image
+of Φ. However, this is enough for our purposes, as any features in F′ that are
+
+Equivariant and Steerable Neural Networks
+23
+transformed by π′ result from applying a convolutional layer, i.e. lie in said
+image.
+4.4.3 The Equivariance Constraint on Filter Banks
+Filterbanks are arrays of shape K′ × K × s × s, where K′ denotes the number
+of output channels, i.e. the number of distinct filters that are to be convolved
+with the input. Each of those K′ filters then has the shape K × s × s, where
+K is the number of input channels, and s denotes the spatial extent, i.e. the
+support/non-zero domain of the filter, usually being 3 × 3 or 5 × 5, though
+other sizes are possible.
+Assuming inductively that such a filter bank Ψ is applied to a steerable fea-
+ture space (F, π), we need it to be an H-Intertwiner between π and ρ, i.e. to
+satisfy the equivariance constraint with respect to H, in order for the output
+of the convolution to be steerable:
+ρ(h)Ψ = Ψπ(h) ∀h ∈ H.
+(40)
+This means that Ψ applied to a feature map that was transformed by π(h) has
+to yield the same result as applying a fiber representation ρ to the fiber that
+results from applying Ψ to the untransformed image, as illustrated in figure 6.
+π0(r)
+ρ(r)
+Ψ
+Ψ
+Figure 6: A H-equivariant filter bank Ψ. Here, ρ represents the rotation r
+by cyclicly permuting the channels in each fiber. For Ψ to be equivariant, this
+diagram needs to commute for any element of the considered group. Note that
+in this figure, we have K = 1 and K′ = 4.
+Two things should be noted at this point:
+Firstly, ρ does not act on a feature space (or on a filter, equivalently) as π0
+does in 35, but on individual fibers F′
+x ∈ RK′. Formally, a fiber representation
+is thus just a K′-dimensional representation (RK′, ρ) of the stabilizer group
+H. In section 4.4.5, it will be explained how a G-representation acting on a
+
+24
+Equivariant and Steerable Neural Networks
+full feature space can be induced from the H-representation ρ.
+Secondly, the equivariance constraint needs only to be fulfilled for all h ∈ H,
+thus excluding translations. This is because translations would be able to
+move patterns out of the receptive field of single fibers (see figure 7). This
+is relevant, as for the equivariance calculations we interpret the action of
+π(h) on F for h ∈ H as the equivalent action of π(h−1) on the filters. Full
+G-equivariance will then be achieved by the induced representation.
+π0(r)
+π0(t)
+Figure 7: Elements of the stabilizer group H (top) do not shift patterns out
+of the receptive fields of fibers, while translations (bottom) do, thus making
+full G-equivariance of filter banks impossible.
+The equivariance constraint is linear in Ψ, meaning that any linear combination
+� λiΨi of intertwiners again satisfies (40) and thus is an intertwiner itself. We
+thus obtain a vector space of intertwiners, denoted HomH(π, ρ).
+4.4.4 The Space of Intertwiners
+To better understand the construction of HomH(π, ρ), we consider an example:
+Let F0 be the space of 3 × 3 filters with one channel, i.e. functions ψ : Z2 → R
+which return zero outside of the 3 × 3 square around the origin. As these
+functions, just like feature maps, can be added and multiplied by scalars, F0
+is a vector space of dimension 9 (or Ks2 for arbitrary spatial extends and
+numbers of channels). H can act on this space via π0 from (35) in the same
+way as it acts on feature maps:
+π0(h)ψ(x) = ψ(h−1x)
+(41)
+and, as mentioned in the previous section, transforming a patch from a feature
+map that ψ is evaluated on by h ∈ H is the same as transforming ψ with h−1.
+Staying with the example, (F0, π0) is a 9-dimensional group representation of
+H. The canonical Basis for this space is shown in figure 8.
+Furthermore, an example for the transformation of a C-linear combination of
+basis filters is depicted in figure 9.
+
+Equivariant and Steerable Neural Networks
+25
+,
+,
+,
+,
+,
+,
+,
+,
+C = {
+}
+Figure 8: The canonical Basis of (F0, π0).
+Irrep
+Basis Filters
+e
+r
+r2
+r3
+m
+mr
+mr2
+mr3
+A1
+[1]
+[1]
+[1]
+[1]
+[1]
+[1]
+[1]
+[1]
+A2
+[1]
+[1]
+[1]
+[1]
+[−1]
+[−1]
+[−1]
+[−1]
+B1
+[1]
+[−1]
+[1]
+[−1]
+[1]
+[−1]
+[1]
+[−1]
+B2
+[1]
+[−1]
+[1]
+[−1]
+[−1]
+[1]
+[−1]
+[1]
+E
+�
+1 0
+0 1
+�
+�
+0 −1
+1
+0
+� �
+−1
+0
+0
+−1
+��
+0
+1
+−1 0
+� �
+−1 0
+0
+1
+� �
+0 1
+1 0
+�
+�
+1
+0
+0 −1
+� �
+0
+−1
+−1
+0
+�
+Table 1: The irreducible decomposition of (F0, π0)
+However, as explained in section 3.2, F0 can be decomposed into irreducible
+representations, i.e. subspaces V ⊆ F0, that are uniquely determined up to
+isomorphism (i.e. change of basis), and are H-invariant, meaning
+π0(h)V ⊆ V ∀h ∈ H.
+(42)
+For G = p4m the irreducible decomposition of (F0, π0) for the stabilizer group
+H = D4 is shown in table 1.
+As depicted, the group H = D4 has five distinct irreps, which are character-
+ized by the way in which π0(h) acts on them for different h ∈ H. For instance
+(see figure 11), π0(h) acts trivially on A1-filters, as rotating or reflecting has
+no effect on them, while π0(r) acts by multiplication by [−1] on B1-filters.
+To obtain the irreducible decomposition of π0, use a simplified character
+formula ([29],[24]):
+mρi(π0) =
+1
+|H|
+�
+h∈H
+χπ0(h)χρi(h).
+(43)
+The characters, which are just the traces of the representing matrices eval-
+uated at each h ∈ H, can be obtained from table 1. For other groups, such
+π0(r)
+Figure 9: π0(r) acting on a C-linear combination.
+
+26
+Equivariant and Steerable Neural Networks
+character tables are also available in literature. To determine the traces of
+π0(h) for h ∈ H, one can look at the canonical basis of F0 (figure 8) and
+how the individual vectors of it transform under π0. The trace χπ0(h) of any
+representation matrix corresponding to π0(h) is then just the number of basis
+vectors of C that are left unchanged by its action. Thus, for instance, we receive
+χπ0(e) = 9, as any representation evaluated at the neutral element always acts
+as the identity matrix, and χπ0(m) = 3. The latter is visualized in figure 10.
+Figure 10: The canonical basis C of (F0, π0) (top) is transformed by the
+reflection π0(m) (bottom). The invariant vectors are highlighted.
+π0 turns out to have type (3, 0, 1, 1, 2) (see definition 6), meaning that there
+are 3 copies of A1, i.e. 3 H-invariant one-dimensional subspaces with exactly
+the transformation properties of this irrep. Furthermore, there is one copy of
+each of B1 and B2, and two copies of the two-dimensional irrep E.
+Decomposing (F0, π0) also makes π0(h) block diagonal for any h ∈ H, after
+applying a change of basis matrix A that is constructed from the basis filters
+shown column two of table 1. The same matrix A block diagonalizes π0(h)
+for all h ∈ H simultaneously and the elements on the diagonal are the irrep
+matrices shown in the third to last columns of table 1, with their respective
+multiplicities. For instance, for h = r, we have:
+π0(r)
+π0(r)
+ψ1 =
+= [1] · ψ1,
+ψ4 =
+= [−1] · ψ4
+Figure 11: A rotation acting trivially on an A1 basis element (left), and as
+[-1] on a B1 basis element (right).
+
+/0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+To(m) =
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+1
+0
+0
+0
+0
+0
+0
+0
+/0Equivariant and Steerable Neural Networks
+27
+Aπ0(r)A−1 = block_diag([1], [1], [1], [−1], [−1], [0, −1; 1, 0], [0, −1; 1, 0])
+(44)
+To get an intuition for HomH(π, ρ), we also need to look at the fiber represen-
+tation ρ, which itself also has a decomposition into irreps and thus a type. In
+fact, ρ would theoretically just be chosen by picking an arbitrary set of inte-
+gers (m1, .., mn) as multiplicities for the irreps (n = 5 for D4), as well as some
+basis A ∈ RK′×K′. However, the choice of a basis is made obsolete later, when
+nonlinearities and capsules are discussed. The number of output channels, i.e.
+the size of the fiber Fx that ρ can act upon is also determined by the irreps:
+K′ =
+n
+�
+i=1
+mi dim ρi.
+(45)
+For the sake of this example, let ρ be of type (2, 1, 1, 1, 1). Then, K′ =
+�5
+i=1 mi dim ρi = 7, thus ρ acts on seven-dimensional fibers Fx ∈ R7, for
+instance as
+ρ(r) = A−1block_diag([1], [1], [1], [−1], [−1], [0, −1; 1, 0])A,
+(46)
+i.e. a block diagonal matrix (after change of basis) with the corresponding
+matrices from column “r” of table 1 on the diagonal. The first channel now
+transforms by A1, the second by A2, and so on.
+As the number of output channels K′ equals the number of “distinct filters”
+that are convolved with the input, a filter bank Ψ intertwining π0 and ρ must
+consist of seven filter units of shape K × s × s, i.e. 3 × 3 in this case. Schur’s
+Lemma (theorem 1) now determines which basis elements of F0 can be used
+to construct each of the individual filters. According to the lemma, the in-
+tertwiner space of two irreps HomH(ρi, ρj) is either zero, or one-dimensional
+iff ρi and ρj are isomorphic. It follows that each individual filter can only
+be constructed from the basis elements in F0 that transform under the same
+irrep as said filter’s output channel.
+To finish the example, consider again the type of ρ. There are two A1-
+channels in the fiber, i.e. two distinct filters that need to be A1-equivariant.
+These filters can be constructed independently from one another, using the
+three basis elements in F0 that span the three A1-subspaces, yielding six
+parameters in total. There is one A2-channel, yet there are no basis elements
+in F0 that transform according to A2, thus this filter is simply zero, adding
+no additional dimensions. While the B1- and B2-filters are obtained in the
+same way as for A1, each yielding one independent parameter, there also is a
+copy of the two-dimensional irrep E in ρ. While this irrep therefore has two
+
+28
+Equivariant and Steerable Neural Networks
+channels, these do not transform independently from one another, but are
+multiplied by two-dimensional matrices, which are given in the last row of
+table 1. Thus, sets of two E2-basis filters in F0 (of which there also are two)
+have to share one parameter, yielding two parameters in total. An illustration
+of this construction is given by figure 12.
+A1
+A1
+A2
+B1
+B2
+E
+E
+m1 = 3
+m2 = 0
+m3 = 1
+m4 = 1
+m5 = 2
+(F0, π0)
+m′
+1 = 2
+m′
+2 = 1
+m′
+3 = 1
+m′
+4 = 1
+m′
+5 = 1
+Figure 12: The construction of an equivariant filter bank. The basis elements
+connected by the dotted lines must share a parameter to preserve equivariance.
+There are seven output channels.
+The process demonstrated in this example can be generalized to intertwiner
+spaces HomH(π, ρ) for arbitrary representations of respective types (m1, .., mn)
+and (m′
+1, .., m′
+n), where π is the feature space representation that transforms
+feature maps and filters and ρ is the fiber representation for the next layer.
+As mentioned before, π might act on on the feature space (and thus on filters)
+in a slightly different way than π0 (35) used in this example. The reason for
+this will be explained in the next section. We receive the following formula for
+the dimension of of said intertwiner spaces, i.e. for the number of parameters
+required by any given layer:
+dim HomH(π, ρ) =
+�
+mim′
+i.
+(47)
+
+Equivariant and Steerable Neural Networks
+29
+4.4.5 Generating Steerable Feature Spaces
+What is left to be shown is how H-steerability of individual fibers F′
+x leads
+to steerability of the whole feature space F′ with respect to all of G. To
+derive this, we use the induced representation of H on G and we show that
+the transformation law that is imposed on an output space F′. By con-
+volving a transformed signal π(g)f, f ∈ F with an equivariant filter bank
+Ψ ∈ HomH(π, ρ) we obtain the formula for the induced representation. While
+the following paragraphs again give the computations the explicit groups p4
+and p4m, the arguments used can be made in complete analogy for any other
+semidirect product G = N ⋊ H.
+Points x ∈ Z2 can be interpreted either as a point (e.g. when looked at as a
+pixel in the base space of feature maps), or as a translation. To make this
+interpretation explicit, we denote x as ¯x when we interpret it as a translation.
+Translation equivariant convolution of Ψ with f can then be defined as
+[f ⋆ Ψ](x) = Ψ[π(¯x)−1f],
+(48)
+for translations ¯x ∈ Z2, with π acting on F as described in (35).
+We now utilize the semidirect product structure of G (definition 2), which
+enables us to write any element g ∈ G as a product g = th where t ∈ Z2 is
+a translation and h ∈ H is an element from the stabilizer group, i.e. some
+rotation from C4 or rotation-flip from D4 for p4 or p4m, respectively. It is
+useful to look at an explicit matrix representation of G. We can write any
+element of G as
+g = th =
+�
+I T
+0 1
+� �
+R 0
+0 1
+�
+=
+�
+R T
+0 1
+�
+.
+(49)
+Here, R is a matrix representation of h ∈ H, for instance a 2 × 2 rotation or
+roto-reflection matrix for p4 or p4m, and T is the translation vector represent-
+ing t. With this form of representation, a translation ¯x ∈ Z2 then amounts to
+¯x =
+�
+I x
+0 1
+�
+.
+(50)
+It is furthermore useful to make the difference between the action of G on
+itself by group operation and the action of G on Z2 visible, thus we will use
+gh for g, h ∈ G for the former and g · x for g ∈ G, x ∈ Z2 for the latter.
+
+30
+Equivariant and Steerable Neural Networks
+Applying th to f via π before convolving with ψ then yields:
+[Ψ ⋆ [π(th)f]](x) = Ψπ(¯x−1)π(th)f
+= Ψπ(¯x−1th)f
+= Ψπ(hh−1¯x−1th)f
+= Ψπ(h)π(h−1¯x−1th)f
+= ρ(h)Ψπ(h−1t−1¯xh)f
+= ρ(h)Ψπ(((th)−1 · x)
+−1)f
+= ρ(h)[Ψ ⋆ f]((th)−1 · x).
+(51)
+The justifications for these equations are as follows: In the first line, we use the
+definition of convolution (48). In the second to fourth line, we use hh−1 = e,
+thus multiplying by one and the fact that π is a group homomorphism, i.e.
+π(gg′) = π(g)π(g′) ∀g, g′ ∈ G. The equivariance constraint (40) is then used
+in the fifth line, and the argument of π is inverted. In line six we use the fact
+that h−1t−1¯xh = ((th)−1 · x)
+−1. This can be verified via the representation
+matrices presented in (49). Finally, the definition of the convolution operation
+is applied backwards.
+We can now define π′ by
+[π′(th)f](x) := ρ(h)[f((th)−1x)],
+(52)
+yielding Ψ ⋆ π(g)f = π′(g)Ψ ⋆ f, thus giving us the desired steerability of F′
+with respect to all of G.
+Comparing π (35) and π′ (52), one notices that they only differ in the factor
+ρ(r) which permutes the individual fibers after they have been moved from x
+to (th)−1x. This kind of action characterizes π′ as the aforementioned induced
+representation of ρ (of H) on G, often also denoted as IndG
+H(ρ). While there
+exists extensive knowledge about this concept, for instance in [30] and [31], for
+our purpose it suffices to know that induced representations transport fibers
+to new locations and then transform them via ρ, the representation induced
+from π′
+Furthermore, any representation of any subgroup of G can induce a rep-
+resentation on G, allowing us to freely choose fiber representations when
+constructing filter banks. The representation π0 from (35) without any factor
+permuting the fibers can also be seen as being induced from the trivial repre-
+sentation, which acts as the identity matrix for all h ∈ H.
+It should also be noted that the content of section 4.4.4, which is using the
+decomposition of the trivially induced representation π0, can also be applied
+
+Equivariant and Steerable Neural Networks
+31
+to induced representations π′ = IndG
+H(ρ). When investigating the action of
+such a representation π′ on basis filters to determine its trace, one just has to
+include the factor of ρ, i.e. the fiber-permutation that is applied additionally.
+Further layers can now be added to the network by choosing a new repre-
+sentation ρ′ of H to act on the fibers of the next output space and compute
+HomH(π′, ρ′) for π′ restricted to H.
+4.4.6 Commutation with Nonlinearities via Capsules
+We have now seen how the convolutional layers of steerable CNNs are built
+and how their equivariance to group transformations is achieved. In particular,
+it was shown that only the basis independent type of the input and output
+representations is relevant for the parameter count of a filterbank, i.e.
+dim HomH(π, ρ) = dim HomH(π, A−1ρA) ∀A ∈ GL(Rdim ρ).
+However, when considering nonlinearities, the choice of the basis becomes
+relevant, which we shall demonstrate with a small example:
+Consider the classical ReLU function and let g = (12) ∈ S2:
+ρ(g) =
+�
+1 −1
+0 1
+�
+and ρ′(g) = A−1ρ(g)A =
+�
+3
+1
+−4 −1
+�
+for A =
+�
+1 1
+2 1
+�
+a change of basis. ρ(g), and thus ρ′(g) correspond to a representation of S2.
+But, e.g. for x = (−2, 5)T and x′ = A−1x = (7, −9)T we have:
+ReLU(ρ(g)(x)) = ReLU(
+�
+−7
+5
+�
+) =
+�
+0
+5
+�
+̸= ReLU(ρ′(g)(x′)) = ReLU(
+�
+12
+−19
+�
+) =
+�
+12
+0
+�
+.
+Thus, different bases can lead to different results under non linear activation,
+even though the underlying type of the representation is the same. To effi-
+ciently tackle this problem, the concept of capsules is introduced:
+A ρ−capsule (RK, ρ, B) is defined as a representation of the stabilizer group
+(RK, ρ) with a fixed basis B of RK. By doing this, the representation matrices
+ρ(h) also become fixed, which allows us us to realize them as matrices, which
+will help in obtaining equivariance of nonlinearities, as will be explained in the
+following. In practice, capsules are typically held relatively low-dimensional,
+rarely exceeding the size of the stabilizer group, |H|. In the following a capsule
+will often just be denoted by ρ, omitting the specified Basis B and simply
+assuming that it is chosen in a way that yields a certain structure for the
+
+32
+Equivariant and Steerable Neural Networks
+representation matrices.
+Just like a convolutional layer, a nonlinearity layer must be equivariant to the
+group G, which is realized by guaranteeing fiber-wise equivariance first and
+then “inducing” feature space equivariance afterwards.
+We define an element-wise nonlinearity ν : R → R, or a fiber-wise nonlinearity
+ν : RK → RK′ to be admissible for an input representation ρ, iff there exists
+an output capsule ρ, such that
+νρ(h) = ρ′(h)ν ∀h ∈ H,
+(53)
+i.e. transforming input fibers by ρ before applying ν is the same as trans-
+forming the nonlinearity’s output by ρ′. We call ρ′ the post-activation capsule
+corresponding to ρ and denote it by Actνρ.
+Given an admissible nonlinearity ν and corresponding pre- and post activation
+capsules ρ and ρ′ = Actνρ such that (53) holds, feature space equivariance to
+the respective induced representation is derived as follows:
+ν([IndG
+Hρ](th)f)(x) = ν(ρ(r)f((th)−1x))
+= ρ′(r)ν(f((th)−1x))
+= ρ′(r)ν(f)((th)−1x)
+= [IndG
+Hρ′](th)ν(f)(x)
+(54)
+Instead of building a layer by choosing multiplicities of irreps for a fiber rep-
+resentation, one now chooses multiplicities mi ≥ 0, i = 1, .., n corresponding
+to a set of predefined capsules ρ1, .., ρn. The chosen capsules are then con-
+catenated into a fiber of dimension � mi dim ρi, which is then acted upon by
+ρ = � miρi, i.e. the block diagonal representation with mi copies of ρi on the
+diagonal. Due to this block diagonal structure, capsules are disentangled, i.e.
+channels of one capsule do not mix with channels belonging to other capsules
+during transformations.
+When it comes to finding explicit pairs of capsules and corresponding admis-
+sible nonlinearities, one needs not only to look at the irrep type of a capsule’s
+underlying representation, but also at the individual form of the representa-
+tion matrices ρ(g) which, as stated, differs not only depending on the choice
+of irreps, but also on the choice of basis for the individual representation
+spaces. Below, we will name the most prominent choices of capsules. For a
+more extensive list, the reader is referred to [8].
+First, any capsule ρ which can be realized by permutation matrices will be
+compatible with any element-wise nonlinearity. Element-wise means that
+instead of acting on a whole feature vector f(x) = (f1(x), .., fK(x)), the
+
+Equivariant and Steerable Neural Networks
+33
+nonlinearity transforms each element fi(x), i = 1, .., K, individually. The
+most prominent example for such a nonlinearity is the ReLU function (see
+(19)). The most frequently used capsules that are realizable by permutation
+matrices are regular capsules, i.e. feature vectors that transform according
+to the regular representation from example 3. While regular capsules have
+been shown to work very well in practice ([3],[4]), they have the drawback of
+leading to relatively high dimensional feature spaces, as each capsule has to
+span |H| channels.
+A possible fix for the high dimensionality of regular capsules is given by
+quotient capsules, in which the feature vectors transform according to the
+quotient representation (example 3) of H by some (normal) subgroup K.
+Quotient capsules have the advantage of requiring |H|
+|K| instead of |H| channels.
+If K is normal, the features represented by such capsules become not just
+equivariant, but invariant to the symmetries of K:
+ρquot(k)ehK = ekhK = ekKh = eKh = ehK.
+(55)
+Hence, performance can be improved by reducing parameter cost and feature
+size through the use of H/K quotient capsules if the K-invariant patterns
+are prevalent in the input data. On the other hand, quotient capsules can
+also severely harm performance if the patterns are not found in the data or
+irrelevant to the learning task at hand. It should furthermore be noted, that
+quotient representations can also be obtained from non-normal subgroups. In
+that case, the features need not be necessarily invariant under K, but can
+perform differently. An example for this, as well as some visual intuition for
+quotient features in general, is given in appendix C of [8].
+A second class of representation matrices are monomial matrices, which have
+the same structure as permutation matrices, but also allow for entries of
+minus one instead of just one. Given a representation that is fully realized
+by monomial matrices, all concatenated nonlinearities will be admissible. A
+concatenated nonlinearity is defined as evaluating an element-wise nonlinear-
+ity on an element x and also on −x. For instance, the concatenated ReLU
+function is defined as
+CReLU(x) := (ReLU(x), ReLU(−x)).
+(56)
+Trivially, regular and quotient capsules will be compatible with concatenated
+nonlinearities. Also, many irreps can be realized (through a suitable basis) by
+monomial matrices, thus allowing for low dimensional irrep capsules.
+The third class of examples is given by orthogonal matrices characterized
+by the fact that they preserve the norm (length) |x| of vectors that they
+act upon. Capsules of this kind will be compatible with norm nonlinearities,
+
+34
+Equivariant and Steerable Neural Networks
+which only act on the norm of any given feature vector (hence do not act
+element-wise), but not on its orientation. A norm nonlinearity can generally
+be written in the following form:
+νnorm : RK → RK,
+f(x) �→ η(|f(x)|) f(x)
+|f(x)|
+(57)
+Here, η : R≥0 → R≥0 describes a non-linear function that is to act on the
+feature vectors norms. A prominent example of this would be Norm-ReLUs,
+with
+η(|f(x)|) = ReLU(|f(x)| − b),
+(58)
+with b being a learned bias. Amongst others, Norm-ReLUs were used in [32].
+The advantage of these nonlinearities is their broad compatibility, as any
+representation of any finite group can be made orthogonal, by choosing an or-
+thogonal change of basis. Further subclasses of norm nonlinearities, such as
+gated and squashed nonlinearities are mentioned in [33] and [5], respectively.
+4.4.7 A Remark on Pooling and Group Pooling in Steerable
+CNNs
+As far as the literature on steerable CNNs ([4],[5],[8]) is concerned, pooling
+operations are, if at all, applied fiber- or capsule-wise. This means that,
+instead of conventional pooling with stride, i.e. subsampling the feature map
+on a subgroup of Z2, thereby reducing equivariance of the network to that
+subgroup, each copy of a capsule located at each point x ∈ Z2 is searched for
+the maximal activation (in the case of max pooling), which is then set as the
+output of the pooling operation. Thus, pooling operations of this nature in
+steerable CNNs can be interpreted as nonlinearities, which were discussed in
+the previous section, and hence also need to satisfy an equivaraince constraint
+that is analogue to 53.
+Let P : RK → R be a pooling operation such as max pooling or average
+pooling, performed on a K-dimensional capsule ρ. For G-equivariance to be
+maintained, we need the pooling operation to be H-equivariant, i.e.
+Pρ(h) = ρ′(h)P ∀h ∈ H.
+(59)
+The choices of compatible capsules ρ, ρ′ is quite limited here, as they must not
+alter the numeric value of the feature vectors on which the pooling operation is
+performed, or the result of the pooling operation. Therefore, any input capsule
+ρ that is to commute with P has to be realized by permutation matrices, as
+taking the maximum (or average) value of a vector with permuted entries still
+yields the same result. Furthermore, ρ′ needs to be the trivial representation,
+i.e. ρ′(h) = [1] ∀h ∈ H as otherwise equivariance could again be broken, for
+
+Equivariant and Steerable Neural Networks
+35
+instance when the result is multiplied by [-1], as e.g. by the irrep A2 in table
+1. The most common choices for ρ therefore are regular capsules, quotient
+capsules, as well as irrep capsules that are realized by permutation matrices.
+As an exception to the choice of ρ′ which was, to our knowledge, not yet
+discussed in the literature, one can implement an equivalent of coset pooling
+from 4.3.3. This will especially make sense taking into account that regular
+steerable CNNs are equivalent to G-CNNs in section 4.4.9. Suppose a sub-
+group K ⊆ H is given, yielding a quotient space H/K. Setting the input
+capsule as the regular representation of H and the output capsule as the
+quotient representation of H/K, we can define quotient pooling as follows:
+Pquot : R|H| → R|H/K|, Pquot(f(x)) =
+�
+hK∈H/K
+max
+h′∈hKf(xh′).
+(60)
+This means that for each of the outputs max
+h′∈hNf(xh′), Pquot only looks at the
+coordinates f(xh) of f(x) that correspond to the the elements of the respective
+coset that is being pooled over. This process is repeated |H/K| times, i.e. for
+each coset. The individual results are then added into a |H/K|-dimensional
+vector that then transforms according to the quotient representation of |H/K|.
+4.4.8 Reduction of Parameters and Implementation
+By the use of steerable filter banks, the parameters of steerable CNNs are
+utilized several times more efficiently than the parameters of regular con-
+volutional networks. As was shown in section 4.4.4, an equivariant filter
+bank Ψ intertwining representations π and ρ has a parameter cost of dim
+HomH(π, ρ) = � mim′
+i, depending on the irrep multiplicities mi and m′
+i of π
+and ρ, respectively. A non-equivariant filter bank of the same size, i.e. same
+filter width s × s and same number of input/output channels K/K′ (which is
+also determined by the dimensions of π and ρ), would instead have a parame-
+ter cost of dim π · dim ρ = s2 · K · K′.
+Thus, the parameter efficiency of a steerable filter bank in comparison to a
+conventional filter bank is expressed as follows:
+µ =
+dim π · dim ρ
+dim HomH(π, ρ).
+(61)
+For effective network architectures, this value usually lies around |H|, i.e. the
+size of the stabilizer group. For instance, we have µ = 8 for H = D4, meaning
+that a p4m-steerable CNN’s convolutional layer uses its parameters eight
+times more efficiently than a layer of the same size in a conventional CNN.
+Efficiency is further increased by the fact that the representations π and ρ
+in each convolutional layer have a block-diagonal structure, i.e. consisting of
+
+36
+Equivariant and Steerable Neural Networks
+direct sums of distinct, disentangled and lower-dimensional capsules. Because
+of this, an intertwiner Ψ between said representations will also have a certain
+block structure:
+For this, let π = π1 ⊕ ... ⊕ πp and ρ = ρ1 ⊕ ... ⊕ ρr. An intertwiner then is a
+matrix Ψ of shape K′ × Ks2, with K′ = �r
+i= dim ρi and Ks2 = �p
+i=1 dim πi,
+and has the following block structure:
+�
+��
+h11 ∈ HomH(π1, ρ1) · · · hp1 ∈ HomH(πp, ρ1)
+...
+...
+...
+h1r ∈ HomH(π1, ρr) · · · hpr ∈ HomH(πp, ρr)
+�
+��
+(62)
+Here, each subblock hij ∈ HomH(πi, ρj) is itself an intertwiner between the
+capsules πi and ρj. In many implementations, the same capsule is used sev-
+eral, or even all the times, allowing to compute many or all of the hij using
+the same intertwiner basis, thus drastically reducing computational cost.
+Ordering the individual capsules such that equivalent capsules are adjacent to
+each other then leads to superblocks Hij of shape mi dim πi × mj dim ρj, with
+mi, nj being the respective multiplicities of the two capsules of Hij. The su-
+perblock itself is then filled with the subblocks hij of shape dim πi × dim ρj.
+In practice, when a list of capsules ρi and corresponding lists of post-activation
+capsules Actνρi (depending on the nonlinearity) are given, the induced rep-
+resentations πi = IndG
+HActνρi, as well as the bases for the intertwiner spaces
+HomH(πi, ρj) for all pairs i, j are computed offline. The bases are stored as
+matrices ψij of shape dim πi · dim ρj × dim HomH(πi, ρj). After that, a list
+of input multiplicities mi and output multiplicities nj provided by the user
+is put into a parameter matrix Φi,j of shape dim HomH(πi, ρj) × minj and
+the aforementioned superblocks Hij are obtained by matrix multiplication of
+ψijΦi,j. After all superblocks are obtained, Ψ is reshaped to K′ × K × s × s
+and can then be convolved with the input.
+4.4.9 On the Equivalency of G-CNNs and Regular Steerable
+CNNs
+It was mentioned before that steerable CNNs are a direct generalizations of G-
+CNNs from section 4.3. To be precise, this means that G-CNNs are equivalent
+to steerable CNNs with regular capsules. To see this, first consider the feature
+maps of the respective architectures:
+fG : G → RKG,
+fsteer : Z2 → RKs.
+(63)
+In G-CNNs, all feature maps except the input image are functions on a group
+G, which in the cases that were considered here has Z2 as a normal subgroup.
+Equivariance is achieved by modifying the domain of the convolution opera-
+tion, leading to G-convolution (23) as described in section 4.3.2. On the other
+
+Equivariant and Steerable Neural Networks
+37
+hand, the feature maps of steerable CNNs keep the domain of Z2 in all layers
+and achieve equivariance by restricting the space of filter banks, as explained
+in detail in the sections 4.4.2 to 4.4.5.
+To understand the equivalence, we first look at how fG transforms under
+actions of G. For this, we again use the example of a p4-feature map (figure
+3). A generalization to other groups is easily done. The group p4 consists of
+unique tuples x = (t, r) with t ∈ Z2 and r ∈ C4 = {e, r, r2, r3}. We can thus
+interpret a p4-feature map as a map on Z2, which returns four “rotational
+coordinates” at each point t ∈ Z2, corresponding to the elements e, r, r2, r3, as
+is depicted in figure 13, which gives a visualization of this slightly different,
+but equivalent interpretation of a p4-feature map. When being acted upon by
+another element x′ = (t′, r′) ∈ p4, the Z2-pixel coordinate t of x and its rota-
+tional outputs are moved to x′−1t. Simultaneously, the rotational coordinates
+are permuted by r′.
+e
+r2
+r
+r3
+e
+r2
+r
+r3
+Figure 13: The transformation of a single pixel x ∈ Z2 by the rotation r ∈ C4
+(top), and the permutation of that pixel’s four rotational outputs (bottom).
+In steerable CNNs, all feature maps and filters are functions on Z2. The equiv-
+alent of the rotational outputs of G-CNNs is realized by the channels of the
+regular representation of C4, which has a dimensionality of |C4| = 4, thus
+consisting of one channel per element of C4. As the transformation law of
+the regular representation on these four channels is defined by the group’s
+action on itself by composition, they transform in exactly the same way as
+the rotational outputs of a G-CNN. The induced representation Indp4
+C4ρreg of
+this regular representation lastly ensures that the respective pixel coordinates
+t ∈ Z2 transform in the same way in both cases, as can be checked by compar-
+ison of the equations (22) and (24) with (35) and (52). Hence, we receive the
+desired equivalence of G-CNNs and regular steerable CNNs. As stated, these
+arguments can be generalized to any semidirect product group G′ = N ⋊ H
+instead of Z2 ⋊ C4, as the regular representation behaves in the same way for
+
+38
+Equivariant and Steerable Neural Networks
+any finite stabilizer group H. In any case, one channel of a G-feature map in
+a layer of a G-CNN equates to one regular capsule, and thus to |H| channels
+in the equivalent regular steerable CNN.
+5 Steerable CNNs on Zn ⋊ Sn
+In this section, possible applications of the theory of equivariant networks
+to the symmetric group Sn will be explored. As G-CNNs are equivalent to
+a special case of steerable CNNs, this chapter will stay in the more general
+language of the latter.
+As in example 2, one can build semidirect product groups Zn ⋊ Sn, with the
+most practical use cases likely being n = 2, for instance for 2D-images and
+n = 3 for volumetric data, as for instance in [5]. One use case is given by
+interpretation of street scenes by one neural network based on the input of
+two redundant camera sensors that are installed at nearby positions. The
+signal of these sensors can be considered to be equivalent, but not necessarily
+equal, as one camera sensor could be disturbed by dirt, but not the other.
+These networks will be very similar to the ones discussed before, with the
+difference that Sn is used as the stabilizer group instead of C4 or D4.
+5.1 Steerable Feature Spaces and Filter Banks for Zn ⋊Sn
+Zn ⋊ Sn (see example 2) acts on functions on Zn in a similar way to the
+product groups p4 and p4m by compositions of translations and coordinate
+permutations. Hence, the theory of steerable CNNs from section 4.4 can be
+applied for Zn ⋊ Sn in near complete analogy. In this section, we will never-
+theless summarize the process of obtaining steerable feature spaces and filter
+banks. We also highlight the key differences, which lie in representation the-
+ory, such as in basis filters for intertwiner spaces, as these depend on the irreps
+of the underlying group. When discussing concrete examples, we will confine
+to S2 and S3, for visualization purposes. However, the general procedure of
+constructing steerable CNNs on Zn ⋊ Sn for any n ∈ N becomes apparent.
+Feature Spaces
+We again deal with feature spaces Fl of functions f : Zn → RKl with Kl
+output channels in each layer l, which transform by a representation πl
+analogously to (35) for l = 0 and (52) in higher layers:
+[π0(t, σ)f](x) = f((tσ)−1x) = f(σ−1(x − t)), t ∈ Zn, σ ∈ Sn
+(64)
+[πl+1(t, σ)f](x) = ρl(σ)[f((tσ)−1x)], t ∈ Zn, σ ∈ Sn.
+(65)
+In (65), ρl is the fiber representation from the previous layer from which
+πl+1 = IndZn⋊Sn
+Sn
+ρl is induced, and with respect to which filter banks Ψ ∈
+
+Equivariant and Steerable Neural Networks
+39
+Irrep
+Basis Filters
+e
+(12)
+id
+[1]
+[1]
+sgn
+[1]
+[−1]
+Table 2: The irreducible decomposition of (FS2
+0 , π0) for S2.
+HomSn(πl, ρl) of layer l must be equivariant:
+ρl(σ)Ψ = Ψπl(σ) ∀σ ∈ Sn.
+(66)
+Equivariant Filter Banks
+The symmetric group of 2 elements has two distinct irreps: The trivial repre-
+sentation, id, and the sign representation sgn, both of them have dimension
+one. By restricting π0 from (64) to S2 and letting it act on 3×3 filters with one
+channel, we once again receive a nine-dimensional representation (FS2
+0 , π0).
+One now obtains the irrep decomposition of π0 by applying the character
+formula for χid and χsgn:
+mπ0(id) =
+1
+|S2|
+�
+σ∈S2
+χπ0(σ)χid(σ) = 1
+2(χπ0(e)χid(e) + χπ0((12))χid((12))) = 6.
+(67)
+While χid(e) and χid((12)) can be looked up from table 2, it is useful for χπ0
+to consider the canonical basis vectors of (FS2
+0 , π0) in figure 8 and how π0(e)
+and π0((12)) transform them. The representation matrix for the identity
+element is always the identity matrix for any representation, hence we receive
+χπ0(e) = 9. On the other hand, π0((12)) swaps the coordinates of each pixel
+in each basis element, thus only leaving the third, fifth, and seventh filter
+invariant, leading to χπ0((12)) = 3.
+Since there is only one other possible irrep, we immediately get mπ0(sgn) = 3.
+With this, we have the full irreducible decomposition of (FS2
+0 , π0), which is
+depicted with an exemplary set of basis filters in table 2.
+For S3 and Z3, we have one extra dimension for the base of feature spaces and
+filters, hence leading to an increase in dimensionality of the space of filters
+when looking at the S3-equivalent of (FS2
+0 , π0). Instead of 3 × 3 filters, we
+now have 3 × 3 × 3 filters, leading to a representation (FS3
+0 , π0) of degree 27.
+The canonical basis for the space FS3
+0
+is depicted in figure 14.
+
+40
+Equivariant and Steerable Neural Networks
+We next determine the irreducible decomposition of π0 in FS3
+0 . While the
+characters of the three irreducible representations of S3 are given in table 3,
+it might look tedious at first to determine the character of π0. However, this
+becomes quite easy if one again looks at which basis elements from CS3 (figure
+14) are left invariant by each group element of S3. As always, π0(e) leaves
+every element invariant, thus yielding χπ0(e) = 27. For the other elements,
+it is helpful to identify each basis vector with the coordinates of its non-zero
+entry (i.e. the black cube in each filter), e.g. the first vector with (−1, 1, 1),the
+second with (0, 1, 1) and the fourth with (−1, 1, 0).
+Any of the transpositions, i.e. 2-cycles (12), (23), (13) ∈ S3 fix all vectors
+of which the two non-zero coordinates that are permuted are equal, e.g.
+{(x1, x1, x2) | x1, x2 ∈ {−1, 0, 1}} for (12). Thus, each transposition fixes
+9 elements, so we have χπ0((12) = χπ0((23) = χπ0((13) = 9. The same
+argument can be used for the two 3-cycles. Each of them fixes the same 3
+vectors of which all three non-zero coordinates are equal, hence we have
+χπ0((123)) = χπ0((132)) = 3.
+Figure 14: Some canonical basis elements of CS3 of (FS3
+0 , π0). As always,
+entries of one are black and entries of zero are grey.
+Plugging these results into the character formula now yields
+mπ0(id) =
+1
+|S3|
+�
+σ∈S3
+χπ0(σ)χid(σ) = 1
+6(27 + 3 · 9 + 2 · 3) = 10,
+mπ0(sgn) =
+1
+|S3|
+�
+σ∈S3
+χπ0(σ)χsgn(σ) = 1
+6(27 + 3 · (9 · (−1)) + 2 · 3) = 1,
+mπ0(Vs) =
+1
+|S3|
+�
+σ∈S3
+χπ0(σ)χVs(σ) = 1
+6(27 · 2 + 3 · (9 · 0) + 2 · (3 · (−1))) = 8.
+(68)
+Thus, the type (i.e. the multiplicities of irreps) of π0 is (10, 1, 8) for the
+trivial representation id, the sign representation sgn and the two-dimensional
+standard representation Vs, respectively. The basis of this irreducible decom-
+position is depicted in figure 15. As one can verify, these filters transform
+under π0(σ) for σ ∈ S3 by multiplication with the representation matrices for
+the respective group element, which are given in table 3.
+
+Equivariant and Steerable Neural Networks
+41
+id
+sgn
+Vs
+Figure 15: The basis vectors for the irrep decomposition of (FS3
+0 , π0). The
+reader may verify that these vectors transform under π0 by the respective
+matrices given in table 3. Again, grey and transparent entries count as zeros,
+black entries as one and white entries as minus one.
+Irrep
+e
+(12)
+(13)
+(23)
+(123)
+(132)
+id
+[1]
+[1]
+[1]
+[1]
+[1]
+[1]
+sgn
+[1]
+[−1]
+[−1]
+[−1]
+[1]
+[1]
+Vs
+�
+1 0
+0 1
+�
+�
+1
+0
+−1 −1
+�
+�
+−1
+1
+0
+−1
+�
+�
+0 1
+1 0
+�
+�
+−1 1
+−1 0
+�
+�
+0
+1
+−1 −1
+�
+Table 3: The representation matrices of π0 for the basis filters of the irrep
+decomposition of (FS3
+0 , π0).
+Equivariant filter banks and steerable feature spaces are now obtained in
+analogy to the sections 4.4.4 and 4.4.5. Given some list of output multiplicities
+(mid, msgn, mVs) (or just (mid, msgn) for S2) of some fiber representation ρ,
+the filter bank Ψ ∈ HomH(π0, ρ) is constructed by linearly combining the irrep
+basis filters from figure 15 for S3 or from table 2 for S2 in the same way as is
+depicted (for p4m) in figure 12, i.e. by only combining filters that correspond
+to the irrep that each of the respective output channels is transformed by.
+5.2 Capsules, Nonlinearities and Pooling for Zn ⋊ Sn
+As was explained in the sections 4.4.6 and 4.4.7, certain requirements have to
+be fulfilled for a layer (realized by a concatenation of capsules as before) to
+commute with fiber-wise nonlinearities and pooling. A layer is thus not built
+by choosing multiplicities for the irreps directly, but by choosing copies of
+
+42
+BIBLIOGRAPHY
+capsules to be concatenated into a fiber. We now list some relevant capsules
+for steerable CNNs with H = S2 or H = S3
+As for any finite group, we can look at regular capsules which transform
+under the regular representation ρreg. For S2, this is the two-dimensional
+representation of type (1, 1) and for S3, we have the four-dimensional rep-
+resentation of type (1, 1, 2). As all regular capsules, these are realized by
+permutation matrices and thus commute with all nonlinearities discussed in
+4.4.6, including ReLU. Furthermore, fiber-wise max-pooling can be applied.
+To obtain quotient capsules (which have the same compatibility properties as
+regular capsules), subgroups are needed. Of these there are none except the
+group itself and {e} for S2, which would lead to a trivial capsule and a regular
+capsule, respectively. For S3, we have (besides the aforementioned ones) one
+normal subgroup, which is the alternating group containing all even permu-
+tations, A3 = {e, (123), (132)}. With this, we have S3/A3 = S2, hence this
+quotient capsule would behave like a regular capsule of S2. S3 furthermore
+has S2 as non-normal subgroup, allowing for S3/S2-quotient capsules. The
+two subgroups can also be used to implement coset pooling as described in 60.
+6 Conclusion
+In this review article, we have seen several relevant approaches to achieve
+invariant representations in machine learning. After establishing the math-
+ematical preliminaries from group theory and representation theory, we
+discussed translation equivariant convolutional networks. We then saw how
+group equivariant neural networks generalize CNNs and thus allow for repre-
+sentations that are invariant to groups of transformations that are more general
+than just translations. This concept was then generalized once more by intro-
+ducing steerable CNNs, which by the use of group representations allow for
+different, more nuanced ways to express equivariance to a group.
+We furthermore presented an application of the theory to the symmetric group
+resulting in a steerable CNN architecture for the symmetric group.
+Acknowledgement. Helpful discussions with Matthias Rottmann are grate-
+fully acknowledged.
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+[29] Reeder, M.: Notes on representations of finite groups (2014)
+[30] Mackey, G.W.: On induced representations of groups. American Journal
+of Mathematics 73(3), 576–592 (1951). Accessed 2022-11-27
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+[31] tom Dieck, T.: Representation Theory
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+monic networks: Deep translation and rotation equivariance. In: Pro-
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+[33] Sabour, S., Frosst, N., Hinton, G.E.: Dynamic routing between capsules.
+arXiv preprint arXiv:1710.09829 (2017)
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf,len=1016
+page_content='Equivariant and Steerable Neural Networks  – A review with special emphasis on the symmetric group –  Patrick Krüger1,2* and Hanno Gottschalk1,2  1*Department of Mathematics and Science, University of  Wuppertal, Gaussstr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 20, Wuppertal, 42110, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  2IZMD, University of Wuppertal, Liese Meitner Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 27-31,  Wuppertal, 42110, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Corresponding author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' E-mail(s):  patrick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='krueger@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Contributing authors: hanno.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='gottschalk@uni-wuppertal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Abstract  Convolutional neural networks revolutionized computer vision and na-  trual language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Their efficiency, as compared to fully con-  nected neural networks, has its origin in the architecture, where convo-  lutions reflect the translation invariance in space and time in pattern  or speech recognition tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Recently, Cohen and Welling have put this  in the broader perspective of invariance under symmetry groups, which  leads to the concept of group equivaiant neural networks and more  generally steerable neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In this article, we review the archi-  tecture of such networks including equivariant layers and filter banks,  activation with capsules and group pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We apply this formalism  to the symmetric group, for which we work out a number of details  on representations and capsules that are not found in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Keywords: Group equivariant neural networks, steerable neural networks,  symmetic group  MSC(2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 68T07, 68T45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='03019v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='LG]  8 Jan 2023  2  Equivariant and Steerable Neural Networks  1 Introduction  Neural networks are machine learning algorithms that are used in a wide vari-  ety of applications, for example in image recognition and language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  There are many applications, among them automated communication in the  service sector, perception for self driving cars and the interpretation of medi-  cal images in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In many machine learning problems, it is desirable to  make the predictions of a network invariant to certain transformations of the  input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This means that, for instance in image classification, we want an image  that was transformed by a rotation by some angle to still be classified with the  same label: A picture of a cat rotated by 90 degrees is still a picture of a cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  An important step towards learning these invariant representations was made  by the introduction of convolutional neural networks by LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' for  image classification of handwritten digits [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' These networks rely on the  mathematical convolution operation to achieve higher efficiency by parameter  sharing, as well as equivariance to translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Equivariance means that a  shift in the input data in some direction will be carried through all but the  ultimate fully connected layers of the network and result in a similar shift in  further deep layers, which can then be used to achieve translation invariant  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  While convolutional neural networks thus yield some of the desired invari-  ance, their performance still decreases when the data is transformed by other  symmetries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' rotations or reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While this can be avoided by data  augmentation, another compelling approach is to extend the translation  equivariance of convolutional networks to a wider class of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  This is realized by group equivariant neural networks, or, in short, G-CNNs,  which were introduced by Cohen & Welling in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' G-CNNs use group theory  to generalize the convolution operation of conventional CNNs to group convo-  lution, which then yields equivariance not only to translations, but to a wider  group of transformations G of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' compositions of translations, reflections  and rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' These G-CNNs are again generalized by steerable CNNs (Co-  hen & Welling, [4]) which, instead of modifying the convolution operation,  instantiate equivariant filter banks and steerable feature spaces by making use  of the theory of group representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' They thereby achieve a more general  and versatile concept of group equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The aim of this article is to give an overview of the theory of equivariant  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For this, some understanding of the mathematical theory of groups  and group representations is needed, which we provide in chapter 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Chapter 4 will then be concerned with the theory of equivariant networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  After a short introduction on conventional, fully connected networks in section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1, basic knowledge about convolutional neural networks is provided in section  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 by explaining the core concepts of feature maps, filters, the convolution op-  eration and the translation equivariance resulting from it, as well as parameter  sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3, we will explain how a modification of the conventional  Equivariant and Steerable Neural Networks  3  convolution operation leads to G-CNNs and how these thereby achieve equiv-  ariance to groups of transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The more general theory of steerable  CNNs will then be presented in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We will furthermore recapitulate  how steerable CNNs are in fact a direct generalization of G-CNNs by explain-  ing the equivalence of the latter to a special case of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The second  aim of this article is to give some contributions by applications of G-CNNs and  steerable CNNs for the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This will be the focus of chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  2 Related Work  Steerable CNNs have besides [4] been investigated by several other publi-  cations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Weiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' ([5]) discuss 3D-steerable CNNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' networks which  process volumetric data with domain R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In [6], Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' give a more ab-  stract approach to intertwiners in steerable CNNs and it is shown that layers  of an equivariant network in fact need to transform according to an induced  group representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A more general theory of steerable CNNs on homoge-  neos spaces is discussed by the same authors in [7], showing that linear maps  between feature spaces are in one-to-one correspondence to convolutions with  equivariant kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' E2-equivariant steerable CNNs that are equivariant to  isometries of the plane R2 in the form of continuous rotations, reflections and  translations are discussed in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Furthermore, gauge equivariant networks,  which enable equivariance not just to global symmetries, but also to local  transformations, are discussed by Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' in [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  G-CNNs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' networks that implicitly or explicitly rely on the regular repre-  sentation of their respective group have also been studied several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In  [10], Gens & Domingos present symnets, which are a framework for networks  with feature maps over arbitrary symmetry groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Kanazawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' present  in [11] a way to learn scale invariant representations while keeping parameter  cost low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Dielemann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' discuss exploiting rotation symmetry to predict  galaxy morphology in [12] and expand their work to cyclic symmetries in  [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Another network architecture that relies on regular representations is  given by scattering networks, which were defined by Mallat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' in [14] for  compact groups, as well as for the Euclidean group in [15] and [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Another important part of research concerning neural networks in general and  equivariant networks specifically is the development of universal approxima-  tion theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Many of these have been proved for different kinds of networks  in varying levels of generality, with some of the first being [17] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Gen-  erally speaking, all approximation theorems state that neural networks can,  under certain conditions, approximate any function from a predefined function  space with arbitrary precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In particular, it was proved by Leshno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  in [19] that multilayer feedforward networks can approximate any continuous  function as long as non-polynomial activation functions are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Petersen and Voigtlaender showed in [20] that fully connected feedforward  4  Equivariant and Steerable Neural Networks  networks can under minimal conditions be translated into convolutional (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  translation equivariant) networks, thereby enabling the application of most  approximation theorems for the former to the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For group equivariant  maps, Kumagai & Sannai ([21]) also employ a conversion theorem from  feedforward networks to CNNs to establish a method for obtaining universal  approximation theorems for group equivariant convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  3 Mathematical Preliminaries  In this section, some algebraic concepts that will be used and referenced  throughout this thesis will be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Secondly, we briefly recall the representation theory of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Lastly, a short summary of the explicit representation theory of Sn, the  symmetric group of n letters, is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1 Semidirect Products  We define the core mathematical concepts needed for the theory of equivariant  networks, starting with definitions of semi-direct products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While this section  is kept quite short, there exist many introductions to this topic, see g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' [22]  or [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Definition 1  Let (G, ◦) be a group and X be some set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A (left) action of G on X is a binary  operation  : G × X → X, (g, x) �→ g · x,  (1)  such that the following requirements are met:  i) e · x = x ∀x ∈ X, with e being the neutral element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  ii) (g ◦ h) · x = g · (h · x) ∀g, h ∈ G, x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  A group action is also often interpreted as a map  φG : G → Aut(X) = {f : X → X | f is bijective }, g �→ φg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (2)  Then, the two conditions above translate to the following:  i) φe = id  ii) φg ◦ φh = φgh ∀g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Definition 2  Let (H, ◦H) and (N, ◦N) be two groups and let furthermore φH : H → Aut(N) be  a group action of H on N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Then, we can construct the outer semidirect product  (N ⋊ H, •) by setting the cartesian product N × H as the group’s underlying set and  defining the group operation as follows:  : N ⋊ H → N ⋊ H,  (n1, h1) • (n2, h2) = (n1 ◦N φH(h1)n2, h1 ◦H h2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (3)  Equivariant and Steerable Neural Networks  5  With this operation, H ∼= {(eN, h) | h ∈ H} and N ∼= {(n, eH) | n ∈ N} become  subgroups of (N ⋊ H), with the latter being a normal subgroup as  (n, h) • (n′, e) = (n ◦N φ(h)n′ ◦N n−1, e) • (n, h) ∀(n, h) ∈ N ⋊ H, (n′, e) ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' (4)  The neutral element is given by (eN, eH) and we have (n, h)−1 = (φH(h−1)n−1, h−1)  for any (n, h) ∈ H⋊N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Furthermore, using above isomorphisms, any element (n, h) ∈  N ⋊ H can be written as a unique product  (n, h) = (n, eH) • (eN, h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (5)  As N ∩ H = (eN, eH) holds trivially, the outer semidirect product thus has all  properties of the inner semidirect product with respective isomorphic subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Example 1  I shall name two explicit examples for semidirect product groups and their respective  actions here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' These examples will be referenced in later parts of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  i) The group p4 of compositions of rotations and translations of the square grid  Z2 is the semidirect product of Z2 and C4, the group of 90-degree-rotations  around any origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Elements of p4 can be parametrized by 3 × 3 matrices  depending on a rotational coordinate r ∈ {0, 1, 2, 3} and two translational  coordinates (u, v) ∈ Z2:  g(r, u, v) =  �  �  cos( rπ  2 ) − sin( rπ  2 ) u  sin( rπ  2 )  cos( rπ  2 )  v  0  0  1  �  �  (6)  p4 now acts on points x = (u′, v′) ∈ Z2 by matrix multiplication from the  left, after adding a homogenous coordinate to x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' x = (u′, v′) �→ (u′, v′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  ii) The group p4m of compositions of rotations, mirror reflections and trans-  lations of Z2 is the semidirect product of Z2 and D4, which is the group  of 90-degree-rotations and reflections about any origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Elements of this  group can be parametrized similarly to p4, with the difference of adding a  reflection coordinate m:  g(m, r, u, v) =  �  �  (−1m) cos( rπ  2 ) −(−1m) sin( rπ  2 ) u  sin( rπ  2 )  cos( rπ  2 )  v  0  0  1  �  �  (7)  p4m then acts on Z2 analogously to p4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  In section 5, we will investigate steerable CNNs that rely on the semidi-  rect product of the symmetric group Sn and Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For this, some preliminary  definitions shall also be given here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  6  Equivariant and Steerable Neural Networks  Definition 3  i) The symmetric group Sn is the group of permutations of n letters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Sn = {σ : {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., n} → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., n} | σ is bijective},  with the group operation being defined by composition of elements, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  ii) For r ≤ n, a r-cycle is an element of σ ∈ Sn, such that there exist  x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., xr ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., n} with  σ(x1) = x2, σ(x2) = x3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., σ(xr−1) = xr, σ(xr) = x1,  σ(xk) = xk ∀ xk /∈ {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., xr}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  An r-cycle as above is often denoted (x1 x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='. xr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We call this the cycle  notation, and we call r the cycle length of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  iii) We obtain an action of Sn on Zn for any n ∈ N by letting σ ∈ Sn permute  the coordinates of x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., xn) ∈ Zn:  σ ◦ (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., xn) = (xσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., xσ(n))  (8)  Example 2  With above action, we can construct the outer semidirect product Zn ⋊ Sn of prod-  ucts tσ of translations t and coordinate permutations σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This group can now be  parameterized and made to act on Zn in analogy to example 1 by n + 1 × n + 1 ma-  trices with the upper left n×n block being a permutation matrix representing σ and  a translation vector t ∈ Zn in the first n entries of the last column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Two examples  for n = 2 and n = 3 shall be given here:  g2(((12), (3, 3))) =  �  �  0 1 3  1 0 3  0 0 1  �  � , g3(((123), (1, 2, 2))) =  �  �  �  �  0 0 1 1  1 0 0 2  0 1 0 2  0 0 0 1  �  �  �  �  (9)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 An Introduction to Representation Theory of Finite  Groups  Definition 4  Let G be a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  i) A representation (V, ρ) of G consists of a vector space V , together with a  group homomorphism ρ : G → GL(V ) from G to the general linear group  of V , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the group of invertible linear maps from V to V , such that:  ρ(gh) = ρ(g)ρ(h) ∀g, h ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (10)  The dimension or degree of the representation is defined as the dimension  of its vector space V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Equivariant and Steerable Neural Networks  7  ii) A subrepresentation of a representation (V, ρ) is a subspace W  ⊆ V  which is G-invariant, meaning ρ(g)w ∈ W for all g ∈ G, w ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The  subrepresentation is then denoted by (W, ρ ↾W ), and we have  ρ ↾W (g) = ρ(g) ↾W .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (11)  Any representation (V, ρ) always has at least two subrepresentations: Itself  and {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  iii) If a representation (V, ρ) with V  ̸= {0} has no subrepresentations ex-  cept itself and {0}, it is called irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Otherwise, it is called reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Throughout this thesis, we will often refer to irreducible representations as  irreps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  iv) For two representations (V1, ρ1) and (V2, ρ2) of G of respective degrees n1  and n2 we can define the direct sum of representations, (V1 ⊕ V2, (ρ1 ⊕ ρ2)),  with  (ρ1 ⊕ ρ2)(g)(v, w) = (ρ1(v), ρ2(w)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This is again a representation of G and  has degree n1 + n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Example 3 (The Quotient Representation)  Given the quotient space (or quotient group) G/H of a group G and some subgroup  H ⊆ G, we define the quotient representation (Vquot, ρquot) by associating a basis  vector with every coset in G/H:  Vquot = ⟨{egH | gH ∈ G/H}⟩  (12)  and define ρquot using the action of G on cosets:  ρquot(g′)egH = eg′gH ∀g′ ∈ G, ∀egH ∈ Vquot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (13)  As this representation just permutes the basis vectors, this time corresponding to  cosets, this can also be realized by permutation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' If we choose H = e, and  thus receive G/H = G, this yields the so-called regular representation which permutes  basis vectors eg for each g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Definition 5  Let (V1, ρ1) and (V2, ρ2) be two representations of some group G and let f : V1 → V2  be a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  i) f is called an intertwiner between ρ1 and ρ2, if  f(ρ1(g)(v)) = ρ2(g)(f(v)), ∀g ∈ G, ∀v ∈ V1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  ii) ρ1 and ρ2 are called equivalent or isomorphic, if there exists an intertwiner  f : V1 → V2 between them, which also is a vector space isomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' an  invertible linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We then often write V1 ≃ V2 or ρ1 ≃ ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  iii) The condition in i) is linear in f, thus any linear combination of intertwiners  between representations ρ1 and ρ2 is again an intertwiner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We hence receive  a vector space of intertwiners between ρ1 and ρ2, denoted HomG(ρ1, ρ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  8  Equivariant and Steerable Neural Networks  Theorem 1 (Schur’s Lemma)  Let (V1, ρ1) and (V2, ρ2) be irreducible representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  i) Iff V1 and V2 are not isomorphic, then dim HomG(ρ1, ρ2) = 0, ie the only  intertwiner between ρ1 and ρ2 is the zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  ii) Iff V1 and V2 are isomorphic, then dim HomG(ρ1, ρ2) = 1, and all maps  intertwining ρ1 and ρ2 are scalar multiples of the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Theorem 2  i) Any finite dimensional representation (V, ρ) of a finite group can be decom-  posed into a direct sum of irreducible representations:  (V, ρ) ∼ (⊕Vi, ⊕ρi),  where (Vi, ρi) are irreps of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  ii) The irreps of G are uniquely determined up to isomorphism, and there are  finitely many of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Proof See [24], section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4, Theorem 2 for i) and section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='5, Theorem 7 for ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  □  Definition 6  Let Irr(G) be the set of nonisomorphic irreducible representations of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Then, any  (Vi, ρi) ∈ Irr(G) can occur in the decomposition of some finite dimensional repre-  sentation (V, ρ) any number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A list of integers mρi(ρ) ≥ 0 corresponding to  the multiplicity of each irrep (Vi, ρi) in (V, ρ) is called the type of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Representations  are uniquely determined up to isomorphism by their types, meaning that if two  representations have the same type, then they are isomorphic and that isomorphic  representations always have the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4 The Theory of Equivariant Networks  In this section, the general theory behind convolutional neural networks is ex-  plained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We cover three types of convolutional networks which are subsequent  generalizations of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1, We give a brief overview on how  non-convolutional, fully connected networks work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 will then cover  the basics on standard translation equivariant convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Key  concepts, such as feature maps, filters, convolutional layers and the convolution  operation, as well as activation and pooling layers is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3,  we elaborate the generalization of convolutional networks to group equivariant  networks (G-CNNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' All concepts from the previous section are suitably gener-  alized, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' G-feature maps and group convolution will be touched on,  and it will be shown how the resulting networks have a “more general” form of  Equivariant and Steerable Neural Networks  9  equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Lastly, in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4, we introduce steerable convolutional net-  works, which use representation theory to yield a more efficient and versatile  way to define group equivariant networks where the group acts on fibers of  feature maps, also called filter banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' It should be noted that while G-CNNs  and steerable CNNs can be realized for many kinds of groups, the sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4 will focus on networks that use split groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' groups that are con-  structed as a semidirect product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Explicitly, we will consider networks on p4  and p4m, which were defined in example 1, iv) and v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' However, the concepts  can easily be generalized to other split groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A G-CNN that does rely on a  non-split group will be covered in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1 Fully Connected Neural Networks  Deep neural networks describe a wide range of computing systems in the do-  main of machine learning that are meant to loosely resemble the human brain  by using interconnected layers, indexed in the following by l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=', L, of  neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Each neuron N l  i, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=', Kl at each layer l of the network can be  thought of as a simple unit that holds a number, usually referred to as its ac-  tivation, and which is connected to all of neurons of the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Therefore,  this kind of network is often called fully connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Each of these connections  is determined by a weight and a bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We denote by ωl  i,j and bl  i,j the weight  and bias that connect the ith neuron of layer l, N l  i, to the jth neuron N l+1  j  of  layer l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In the so-called forward propagation step of a network, starting at  l = 1, each neuron is updated in the following way:  N l  i = σl  �  �  Kl−1  �  j=1  ωl−1  j,i N l−1  j  + bl−1  j,i  �  � ,  (14)  where σ is some non-linear activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' These will be discussed in  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='5 This process is then repeated sequentially for all remaining l = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=', L,  until the final layer is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This output layer consists of one neuron for  each possible label of the classification problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The output neuron  with the highest activation is returned as the networks predicted label for the  given input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  To train a deep neural network, a training dataset with inputs that are already  labeled with their respective classifications is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' During the training  phase, after each forward pass, the (initially large) error of the network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  the difference between the model’s prediction and the actual labels of the  training data is quantified by a loss function which takes as inputs the weights  and biases of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A backpropagation algorithm then produces the  gradients of this loss function with respect to its variables (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the weights  and biases), which are often called the parameters of the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' After  obtaining the gradients, an optimization algorithm, the most used one being  10  Equivariant and Steerable Neural Networks  stochastic gradient descent ([25]), is used to modify the networks parameters,  thereby slightly optimizing the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This training process of forward  pass and backpropagation is now repeated until the error is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Then, the backpropagation step is no longer needed and the network can be  used to classify unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The reader interested in a more thorough  introduction to feedforward networks is referred to chapter 6 of [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 Basics on Convolutional Neural Networks  Convolutional Neural Networks (CNNs) are a powerful tool in machine learn-  ing, mainly used for pattern recognition and classification of certain input  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' They can be used on a variety of data, for example sound signatures  (one-dimensional data) and 2D and/or 3D Images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Just as fully connected net-  works, CNNs work in layers, in each of which a set of filters representing certain  features to look for in the input data is convolved with stacks of so-called fea-  ture maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' After application of certain operations known as nonlinearities and  pooling, this yields a new set of feature maps, which is then convolved with a  new set of filters in the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A core feature of CNNs is the equivariance  to translation of their convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This means that shifting the input  data of a convolutional Layer will result in a similar shift that layer’s out-  put, allowing us to detect features independent of their location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In the next  few paragraphs we will explain these terms in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We use a Network  operating on 2D black and white images, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' an input feature map with one  channel as example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1 Feature Maps and Filters  Feature maps  Feature maps are mathematical functions used to describe the input data,  as well as the outputs of each convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Depending on the type  of the input, their domain can vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In image recognition, the domain of a  feature map usually is the two-dimensional pixel grid Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In our example  of a black and white image, an input-level feature map f : Z2 → R simply  returns the grey-value at each pixel x ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In a coloured image, the function  would instead be of the form f : Z2 → R3, returning a vector consisting of  the color channel values at each pixel coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Since images are bound in  size, a feature map is usually said to just return zero everywhere outside of a  certain subdomain of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Since layers of convolutional networks contain  more of one feature map most of the time, it is often also spoken of ’stacks’  of feature maps f j : Z2 → RK, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., n, whereby the index is often omitted  for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Filters  Filters are used to extract certain characteristic patterns in our data by con-  volving them with feature maps, which will be described in the next paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Mathematically, they are also described as functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For convolution to work,  Equivariant and Steerable Neural Networks  11  it is important that these functions share the domain and image space of the  feature maps that they are to be convolved with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In our example, a one-channel  filter looking for diagonal lines (top left to bottom right) could look like this:  ψ : Z2 → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  ψ(x) =  �  1  x ∈ {(−1, 1), (0, 0), (1, −1)}  0  elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (15)  The support, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the non-zero domain of filters is usually much smaller than  that of the feature maps, with usual widths being 3 × 3 or 5 × 5 squares cen-  tered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While classical computer vision used handcrafted filters  as in the above example, CNN based computer vision uses learnable filters  ψ = (ψi,j))i,j=−s,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=',s, s = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The values ψi,j are the learned parameters of  the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As will elaborated in 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4, this drastically reduces the parame-  ter cost of CNNs in comparison to fully connected networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 The Convolution Operation  The core building block of regular CNNs are the so called convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  In each of these layers, a stack of feature maps f : Z2 → RK is convolved with  a set of filters ψ : Z2 → RK, producing a new feature map which will then be  used in the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The convolution operation is mathematically defined  as follows:  f ⋆ ψ : Z2 → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  [f ⋆ ψ](x) =  �  y∈Z2  K  �  k=1  fk(y)ψk(y − x)  (16)  While this may look complicated at first glance, it can be interpreted as sim-  ply sliding our small filter window over the domain of the image or feature  map and summing up the element-wise products of the filter’s values and the  feature map’s values lying “under” them at each output channel k and each  respective position x of the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To demonstrate this further, we can inter-  pret our example filter ψ from (15), as well as a feature map f as matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The feature map and filter, as well as the result of the convolution operation  of said elements is illustrated in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Here, as in all following figures throughout this thesis, black pixels represent  one, grey pixels zero and white pixels represent minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Recall that, even though the domain of f and ψ is infinite, both functions just  return 0 everywhere outside of the depicted areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Convolving a feature map with a filter yields a new feature map  f ⋆ ψ : Z2 → R, describing how well the feature described by the filter fits at  different positions x ∈ Z2 in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  As announced, we will from now on often omit the summation over channels  12  Equivariant and Steerable Neural Networks  2  3  3  0  1  1  0 0 0 0 0  0 0 0  0  1  0  1  0  0  0  0  ⋆  =  f  ψ  =  f ⋆ ψ  Figure 1: A feature map f representing the letter “V” is convolved with a  filter detecting diagonal edges from top left to bottom right, yielding a new  feature map f ⋆ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  in (16) for simplicity and pretend to have feature maps and filters with just  one channel, as this will not harm any arguments made in the following para-  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3 Translation Equivariance  An important property that makes CNNs such powerful tools, for example in  image recognition, is their equivariance to translations of the input at each  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Roughly speaking, this means that a CNN is able to detect features in  an image (or other data) regardless of their specific position, without requiring  additional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To be more precise, we define a translation of a feature  map mathematically:  [Ttf](x) = f(t−1x) = f(x − t);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' x, t ∈ Z2  (17)  A feature map f is hence transformed by a translation t ∈ Z2 by looking up the  value of f at the point t−1x = x − t and moving it to the position x, resulting  in the t-transformed feature map Ttf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Visually, this translates to a shift of the  patterns of a given input by the respective horizontal and vertical coordinates  of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For an illustration, see the left side of figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The underlying concept  of a group action of Z2 on itself is also used by the convolution operation  itself: In (16), filters are transformed by Tx when calculating the output of the  convolution at position x, representing a shift of said filter to that position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Equivariance to translations in CNNs now means that applying a translation  t to a feature map f, followed by a convolution with a filter ψ yields the same  Equivariant and Steerable Neural Networks  13  result as convolving f with ψ first and then applying the translation:  [[Ttf] ⋆ ψ](x) =  �  y∈Z2  f(t−1y)ψ(x−1y)  =  �  y∈Z2  f(y − t)ψ(y − x)  =  �  y∈Z2  f(y)ψ(y + t − x)  =  �  y∈Z2  f(y)ψ(y − (x − t))  =  �  y∈Z2  f(y)ψ((t−1x)−1y)  = [Tt[f ⋆ ψ]](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (18)  ⋆ψ  ⋆ψ  Tt  Tt  f  Tt(f)  Tt(f ⋆ ψ)  f ⋆ ψ  Figure 2: A translation equivariant convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Here, the feature maps are  transformed by t = (−2, 1) and the diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4 Parameter Sharing  In comparison to conventional, fully connected neural networks (section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1),  convolutional neural networks increase the efficiency of their parameters by  utilizing the convolution operation to achieve parameter sharing across layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  In fully connected layers of a conventional neural network, each neuron is  14  Equivariant and Steerable Neural Networks  connected to the neurons of the next layer by an individual connection con-  sisting of a weight and a bias, which together make up the parameters of  the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Each of these connections is modified individually to achieve  the best possible classification result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In image recognition for instance, one  can view each pixel of the input image as a neuron, and each of these is  connected via individual parameters to the neurons of the subsequent layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  As layers are stacked on top of each other, one can imagine that the number  of connections, and thus of parameters will grow quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  For convolutional neural networks, consider each channel f k′  l , k′ = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., Kl  of a feature map fl : Z2 → RKl with Kl output channels in a given layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Each of these channels is obtained by convolving a small filter patch ψk′ of  size s × s × Kl−1 with the stack of feature maps f k  l−1 of the previous layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Here, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., Kl−1 denotes the individual channels and Kl−1 is the number  of channels of the feature map of layer l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' All values f k′  l (x) of an output  channel k′ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., Kl} for each x ∈ Z2 are thus obtained from the same filter,  evaluated at different spatial coordinates, and hence share the same s2Kl−1  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For the whole layer l, this then amounts to s2Kl−1Kl parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  For comparison, as each pixel would need its own parameters, a fully con-  nected layer processing the same data would need around Kl−1KlZl−12Zl2  parameters with Zl−1 and Zl being the spatial extend of the non-zero do-  mains of the feature maps fl−1 and fl, which is substantially larger than s2,  with s usually being 7 or smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='5 Nonlinearities and Pooling  Nonlinearities  The convolutional layers of a CNN are interspersed with non-linear layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Such a layer operates by applying a non-linear function to the input is needed  to enhance the expressive power of CNNs, since hardly any phenomenon that  CNNs are meant to observe can be described by linear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Various  universal approximation theorems, for instance in [19] and [20, 27], prove  that (feedforward and convolutional) networks can approximate almost any  function with arbitrary levels of precision, as long as non-polynomial nonlin-  earities are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  One of the most prominent examples for a nonlinearity is the rectified linear  unit ReLU:  ReLU(x) := max(x, 0)  (19)  This function has empirically been proven to be the most efficient choice in  many cases, as its computation is faster than most other non-linear functions  and it is less prone to plateaus in the training process of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Note that ReLU is applied element-wise, meaning that if a multi-channel  feature map f : Z2 → RK is given, ReLU is evaluated at each output channel  Equivariant and Steerable Neural Networks  15  fk(x), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., K, individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' There are other forms of fiber-wise nonlinear-  ities, which will be touched on in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Pooling  In CNNs, convolutional layers are often interspersed with so called pooling  layers, with one of the first examples being [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In these layers, a pooling  operation is performed on a given set of feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The main goal of this  operation is to cut out superfluous information, reducing data size by slightly  reducing locational precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This can be done because a CNN usually does not  require the exact location of a feature to function properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' There are several  different pooling operations, one of the most frequently used is max pooling:  P : Z2 → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Pf(x) :=  max  x′∈U(x)f(x′),  (20)  where f is a feature map and U(x) is some neighbourhood of x in Z2, in  this case usually being a small square with center x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The pooling operation  thus works by finding the highest activation in said neighbourhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Average  pooling, which is another frequently used operation, would find the average of  all activations in the pooling window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  To actually reduce the size of the feature map, pooling needs to be performed  with a stride, meaning that P is not evaluated on every point in the base space,  but only on points of certain distance to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For example, a stride of 2  would mean that in (20), P is only evaluated at points in 2Z2 = {2x : x ∈ Z2},  while the neighbourhoods U(x) remain subsets of Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3 Group Equivariant CNNs  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1 Motivation  We can interpret convolutional networks from section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 as already being  G-CNNs with G the group of translations, Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Our convolution operation  [f ⋆ ψ](x) basically returns the activation of ψ, transformed under the action  of the group element x, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' HGFor a general G-CNN, one would  like to find a way to replace the underlying group of the network with some  other group G, containing more transformations than just translations, for  example rotations and reflections, and still have equivariance of convolution  with respect to every element of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 Group Convolution, Feature Maps and Equivariance  We first establish the action of the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The convolution operation from  equation (16) is modified in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' First, we consider the initial convolu-  tional layer of the network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' where the image f : Z2 → R is convolved with  16  Equivariant and Steerable Neural Networks  the first set of filters:  f ⋆ ψ : G → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  [f ⋆ ψ](g) =  �  y∈Z2  f(y)ψ(g−1y)  (21)  Two things should be noted: First, g ∈ G again transforms feature maps (or  filters) via a group action on the domain of said maps:  [Tgf](x) = f(g−1x);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' x ∈ Z2, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (22)  Specifically for G = p4 or G = p4m, g ∈ G transforms f(x) by moving its out-  put from x ∈ Z2 to the pixel at g−1x under the actions defined in example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Secondly, it should be noted that the convolution operation f ⋆ ψ itself now  yields a function on the group G instead of Z2, thus requiring filters in the  subsequent layer to also be functions on G, yielding yet another slightly  different convolution operation:  f ⋆ ψ : G → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  [f ⋆ ψ](g) =  �  h∈G  f(h)ψ(g−1h)  (23)  As one might notice, the transformation of feature maps and filters by G is  again slightly different:  [Tgf](h) = f(g−1h);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' h, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (24)  Hence, the output of f at a point h ∈ G of the domain (which also is just G)  is moved to the point g−1h, which is obtained by the canonical action of G on  itself by its group operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  While it is easy to imagine feature maps f : Z2 → R simply as images  sampled on a pixel grid, feature maps with base space G might not be as  intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For G = p4, a feature map f : G → R can be imagined as a graph  with four patches, of which each corresponds to one of the four rotations  C4 = {e, r, r2, r3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Each pixel now has a rotational coordinate, corresponding  to the patch in which it appears, and two translational coordinates specifying  its position in the respective patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' These coordinates do also coincide with  the semidirect product structure of p4, enabling us to write any element  g ∈ p4 as a product g = tr, t ∈ Z2, r ∈ C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The rotation r for instance now acts on this graph by moving each patch  along the arrows indicated in figure 3, and by also rotating each of the patches  themselves by 90 degrees, as is also shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The convolution operation of the first layer (21) and the convolution operations  of the subsequent layers (23) are now not only equivariant to translations, but  to all transformations from the group G, expanding translational equivariance  Equivariant and Steerable Neural Networks  17  e  r3  r2  r  e  r3  r2  r  Figure 3: A p4 feature map (left) and its rotation by r (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' compositions of translations and rotations for p4 and by compositions  of translations with rotations and reflections for p4m:  [[Tuf] ⋆ ψ](g) = [Tu[f ⋆ ψ]](g) ∀ u, g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (25)  For higher layer convolutions, this is derived in complete analogy to (18), this  time using the fact that uG = G ∀u ∈ G and thus substituting uh ∈ G for  h, again keeping the overall sum the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' At the same point for first layer  convolutions, we use an analogous argument gZ2 = Z2 ∀g ∈ G, as G acts  transitively on Z2, meaning that any point in the pixel grid can be reached  from any other point by a transformation from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3 Group Pooling and Equivariant Nonlinearities  Equivariance to element-wise nonlinearities  Element-wise nonlinearities ν : R → R can be used in G-CNNs without  restriction, as post-composing them with a feature map preserves the equiv-  ariance to pre-compositions with group transformations:  Let ν : R → R be an element-wise nonlinearity such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' ReLU, and define  the post-composition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the application of such a function to a feature map  f:  Cν(f(g)) = [ν ◦ f](g) = ν(f(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Let Tg be the left transformation operator from (22) or (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As Tg is realized  by pre-composition with f and Cν by post-composition, these actions commute  and we get  CνTgf = ν ◦ [f ◦ g−1] = [ν ◦ f] ◦ g−1 = TgCνf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (26)  Group Pooling  As in Regular CNNs, convolutional layers of a G-CNN are often followed by a  18  Equivariant and Steerable Neural Networks  pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As we no longer need to have the simple pixel grid Z2 as base  space, we need to define the notions of pooling and stride for the more general  case of the base space being a group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To simplify the process, we split it into  two steps, namely the pooling step, which is performed without stride, and the  subsampling step which can then realize any notion of stride, if desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In the  first step, the max pooling operation for instance becomes  P : G → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Pf(g) := max  x∈gU f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (27)  Here, gU := {gu | u ∈ U} is some g-transformed neighbourhood U of the  identity element in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In Z2, this would correspond to a square around the  origin that is moved across the images by translations t ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This operation  is now equivariant to G:  PThf(g) = max  k∈gU Thf(k)  = max  k∈gU f(h−1k)  = max  hk∈gU f(k)  =  max  k∈h−1gU f(k)  = Pf(h−1g)  = ThPf(g)  (28)  The arguments behind these equations are as follows: The first two equations  are just the definitions of group pooling (27) and the left transformation of  feature maps (24), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In the third line, we substitute k for hk using  the fact that taking the maximum of f(h−1k) over k ∈ gU is the same as  taking the maximum of f(k) with k such that hk is in gU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In the fourth line  we use that this is again the same as taking k ∈ h−1gU, which can be seen by  just multiplying both sides with h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The last two lines are then obtained by  just resubstitutuing the definitions of group pooling and Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Any stride could now be realized in the second step by subsampling the  pooled feature map over a subgroup H ⊆ G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' evaluating just on points  h ∈ H instead of all g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' However, a feature map subsampled in this way  would not anymore be equivariant to all of G, but only to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As we wish to  maintain equivariance to the whole group G, this form of stride is usually not  used in practical G-CNN applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Instead, to preserve G-equivariance throughout the network, one can use coset  pooling by choosing the pooling neighbourhood U in the first step to be itself  a subgroup H of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The resulting transformed pooling regions then are the  non-overlapping cosets of H in G which are either disjoint or equal for any gH  and g′H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Because of this, any element of each of the distinct cosets can then  Equivariant and Steerable Neural Networks  19  be chosen as a representative to subsample on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The resulting pooled feature  map can then be interpreted as a map on the quotient space G/H, which can  then be acted upon by G similar to (24) by utilizing the general action of G on  its quotient spaces from example 1, preserving equivariance as shown in (28):  Tg′f(gH) = f((g′−1g)H), g′ ∈ G, gH ∈ G/H  (29)  As an example of this, consider a p4-feature map that is pooled over the group  of rotations, C4 = {e, r, r2, r3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Visually (see figure 4), this equates to checking  the four rotational outputs of each pixel coordinate x ∈ Z2 and choosing the  one with the highest activation as representative for the coset {x, xr, xr2, xr3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The resulting feature map is then of domain p4/C4 = Z2 and thus transforms  in the same way as an input feature map in (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  e  r3  r2  r  max  x∈(−2,2)C4  f(x)  (−2, 2)  (−2, 2)r3  (−2, 2)r2  (−2, 2)r  (−2, 2)C4  Figure 4: Coset pooling by H = C4 of a p4 feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A single coset  ((−2, 2)C4) is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4 Implementation  G-Convolution can be implemented rather easily, at least for so-called split  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' By exploiting this property, we can just use a standard convolution  routine with an expanded filter bank, which will be described shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Recall that a group being split means that any element g ∈ G can be written  as a product g = ts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For p4 and p4m, t ∈ Z2 would be a translation, and s  would be a transformation from the stabilizer group H = C4 or H = D4 that  leaves the origin invariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' a rotation or roto-reflection around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  This, together with TtTs = Tts for the action of G allows us to rewrite the  20  Equivariant and Steerable Neural Networks  definition of G-convolution as follows:  f ⋆ ψ(g) = f ⋆ ψ(ts) =  �  x∈X  f(x)Tt[Tsψ(x)]  (30)  with X = Z2 in layer one and X = G in subsequent layers, thus allowing us to  precompute the transformed filters Tsψ for all transformations of the stabilizer  and then convolve them with the input using a fast planar convolution routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The set of untransformed filters at some layer l can be sorted in an array F  of shape Kl × Kl−1 × Sl−1 × n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Here, Kl−1 denotes the number of input  channels, Kl is the number of output channels i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the number of distinct  filters, and n×n denotes the spatial extend of the filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Furthermore, Sl−1 is  the size of the stabilizer group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the number transformations of G that fix  the origin in the base space of the feature maps that F is to be convolved with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Each transformation Ts now “acts” on F by permuting the scalar entries  of each of the Kl × Kl−1 distinct “filter blocks” of shape Sl−1 × n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' If  Sl transformations are applied, this leads to an extended array of shape  Kl × Sl × Kl−1 × Sl−1 × n × n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The permutations themselves can be realized by implementing an invertible  map g which yields the group element corresponding to an index from the  array of shape Sl−1 × n × n, represented as matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For example, for p4 this  map would be the following, as was described in example 1, iv):  g(s, u, v) =  �  �  cos( sπ  2 ) − sin( sπ  2 ) u  sin( sπ  2 )  cos( sπ  2 )  v  0  0  1  �  �  (31)  We then set  F +[i, s′, j, s, u, v] = F[i, j, ¯s, ¯u, ¯v]  (32)  with  (¯s, ¯u, ¯v) = g−1(g(s′, 0, 0)−1g(s, u, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (33)  To use F + in a planar convolution routine, we exploit the fact that X in (30)  involves a sum over the stabilizer, again allowing us to to rewrite the equation  as  �  x∈X  f(x)Tt[Tsψ(x)] =  �  x∈Z2  Sl−1Kl−1  �  k=1  fk(x)Tt[Tsψk(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (34)  Equivariant and Steerable Neural Networks  21  We can now reshape F + into an array of shape SlKl ×Sl−1Kl−1 ×n×n which  can then be applied to similarly reshaped feature maps in a planar convolution  routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4 Steerable CNNs  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1 Motivation  It was shown how G-CNNs achieve group equivariance by expanding the do-  main of the feature maps and filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The magnitude of the expansion depends  on the size of the stabilizer group of the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Thus, larger groups lead to  larger expansions, which in turn lead to a proportionally increasing computing  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Steerable CNNs are a generalization of G-CNNs which achieve equivariance by  defining filter banks as intertwiners, which are morphisms between group rep-  resentations, through which feature spaces become G-steerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For a classical,  unrestricted filter bank to be an intertwiner, it needs to satisfy an equivariance  constraint, which depends on the representations that it is meant to inter-  twine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Thus, while G-CNNs achieve equivariance by expanding arbitrary filter  banks, Steerable CNNs achieve it by restricting the space of available filters,  thus decoupling the required computational power from the size of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 Feature Spaces, Fibers and Steerability  We once again consider 2D signals f : Z2 → RK with K channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' These can  be added, as well as multiplied by scalars and therefore form a vector space,  often also called feature space, which we will denote by Fl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The layer index l  will often be omitted for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Given a group G acting on Z2, we are able to transform signals f ∈ F0:  [π0(g)f](x) = f(g−1x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (35)  π0(g) : F0 → F0 is then a linear map ∀g ∈ G that furthermore satisfies  π0(gh) = π0(g)π0(h) ∀g, h ∈ G,  (36)  yielding a group homomorphism π0 : G → GL(F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A vector space such as  F0, together with a map such as π0 that satisfies (36) fulfills the conditions of  a group representation defined in definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' It shall be denoted by (F0, π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Oftentimes we will just call π0 (or πl) a representation, if the nature of F0  (Fl) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As will be explained later, the representations πl transforming  higher layer feature spaces Fl might look slightly different than in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Nev-  ertheless, they will always satisfy (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  While in most other deep learning publications some f ∈ F is usually consid-  ered as a stack of K feature maps fk : Z2 → R, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., K, it is useful for  22  Equivariant and Steerable Neural Networks  the theory of steerable CNNs to consider another decomposition: Instead of  splitting the feature space “horizontally” into one-dimensional planar feature  maps, it can be decomposed into fibers Fx, one of which being located at each  “base point” x ∈ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Each fiber consists of the K-dimensional vector space  representing all channels at the given position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' f ∈ F is therefore comprised  of feature vectors f(x) ∈ Fx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' See figure 5 for an illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Figure 5: The decomposition of F into stacks of feature maps (left), and into  fibers (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A single fiber is highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Let F, F′ be two feature spaces, and let π : G → GL(F) be a group homomor-  phism, such that (F, π) is a group representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Let furthermore Φ describe  a convolutional layer of a network, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Φ : F → F′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Φ(f) = f ⋆ Ψ ∀f ∈ F  (37)  for some filter bank Ψ : Z2 → RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  We now say F′ is steerable w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' G, if there exists another function  π′ : G → GL(F′) transforming F′ such that  Φπ(g)f = π′(g)Φf ∀g ∈ G,  (38)  meaning that transforming the input by π(g) yields the same result under Φ  as transforming the output of Φ by π′(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The following equation shows that  this condition implies that π′ : G → GL(F′) is also a group homomorphism,  and thus (F′, π′) is also a group representation:  π′(gh)Φf = Φπ(gh)f = Φπ(g)π(h)f = π′(g)Φπ(h)f = π′(g)π′(h)Φf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (39)  Note that (39) only implies the desired property of π′ for the span of the image  of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' However, this is enough for our purposes, as any features in F′ that are  Equivariant and Steerable Neural Networks  23  transformed by π′ result from applying a convolutional layer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' lie in said  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3 The Equivariance Constraint on Filter Banks  Filterbanks are arrays of shape K′ × K × s × s, where K′ denotes the number  of output channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the number of distinct filters that are to be convolved  with the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Each of those K′ filters then has the shape K × s × s, where  K is the number of input channels, and s denotes the spatial extent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the  support/non-zero domain of the filter, usually being 3 × 3 or 5 × 5, though  other sizes are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Assuming inductively that such a filter bank Ψ is applied to a steerable fea-  ture space (F, π), we need it to be an H-Intertwiner between π and ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' to  satisfy the equivariance constraint with respect to H, in order for the output  of the convolution to be steerable:  ρ(h)Ψ = Ψπ(h) ∀h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (40)  This means that Ψ applied to a feature map that was transformed by π(h) has  to yield the same result as applying a fiber representation ρ to the fiber that  results from applying Ψ to the untransformed image, as illustrated in figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  π0(r)  ρ(r)  Ψ  Ψ  Figure 6: A H-equivariant filter bank Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Here, ρ represents the rotation r  by cyclicly permuting the channels in each fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For Ψ to be equivariant, this  diagram needs to commute for any element of the considered group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Note that  in this figure, we have K = 1 and K′ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Two things should be noted at this point:  Firstly, ρ does not act on a feature space (or on a filter, equivalently) as π0  does in 35, but on individual fibers F′  x ∈ RK′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Formally, a fiber representation  is thus just a K′-dimensional representation (RK′, ρ) of the stabilizer group  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='5, it will be explained how a G-representation acting on a  24  Equivariant and Steerable Neural Networks  full feature space can be induced from the H-representation ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Secondly, the equivariance constraint needs only to be fulfilled for all h ∈ H,  thus excluding translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This is because translations would be able to  move patterns out of the receptive field of single fibers (see figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This  is relevant, as for the equivariance calculations we interpret the action of  π(h) on F for h ∈ H as the equivalent action of π(h−1) on the filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Full  G-equivariance will then be achieved by the induced representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  π0(r)  π0(t)  Figure 7: Elements of the stabilizer group H (top) do not shift patterns out  of the receptive fields of fibers, while translations (bottom) do, thus making  full G-equivariance of filter banks impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The equivariance constraint is linear in Ψ, meaning that any linear combination  � λiΨi of intertwiners again satisfies (40) and thus is an intertwiner itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We  thus obtain a vector space of intertwiners, denoted HomH(π, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4 The Space of Intertwiners  To better understand the construction of HomH(π, ρ), we consider an example:  Let F0 be the space of 3 × 3 filters with one channel, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' functions ψ : Z2 → R  which return zero outside of the 3 × 3 square around the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As these  functions, just like feature maps, can be added and multiplied by scalars, F0  is a vector space of dimension 9 (or Ks2 for arbitrary spatial extends and  numbers of channels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' H can act on this space via π0 from (35) in the same  way as it acts on feature maps:  π0(h)ψ(x) = ψ(h−1x)  (41)  and, as mentioned in the previous section, transforming a patch from a feature  map that ψ is evaluated on by h ∈ H is the same as transforming ψ with h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Staying with the example, (F0, π0) is a 9-dimensional group representation of  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The canonical Basis for this space is shown in figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Furthermore, an example for the transformation of a C-linear combination of  basis filters is depicted in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Equivariant and Steerable Neural Networks  25  ,  ,  ,  ,  ,  ,  ,  ,  C = {  }  Figure 8: The canonical Basis of (F0, π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Irrep  Basis Filters  e  r  r2  r3  m  mr  mr2  mr3  A1  [1]  [1]  [1]  [1]  [1]  [1]  [1]  [1]  A2  [1]  [1]  [1]  [1]  [−1]  [−1]  [−1]  [−1]  B1  [1]  [−1]  [1]  [−1]  [1]  [−1]  [1]  [−1]  B2  [1]  [−1]  [1]  [−1]  [−1]  [1]  [−1]  [1]  E  �  1 0  0 1  �  �  0 −1  1  0  � �  −1  0  0  −1  ��  0  1  −1 0  � �  −1 0  0  1  � �  0 1  1 0  �  �  1  0  0 −1  � �  0  −1  −1  0  �  Table 1: The irreducible decomposition of (F0, π0)  However, as explained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2, F0 can be decomposed into irreducible  representations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' subspaces V ⊆ F0, that are uniquely determined up to  isomorphism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' change of basis), and are H-invariant, meaning  π0(h)V ⊆ V ∀h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (42)  For G = p4m the irreducible decomposition of (F0, π0) for the stabilizer group  H = D4 is shown in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  As depicted, the group H = D4 has five distinct irreps, which are character-  ized by the way in which π0(h) acts on them for different h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For instance  (see figure 11), π0(h) acts trivially on A1-filters, as rotating or reflecting has  no effect on them, while π0(r) acts by multiplication by [−1] on B1-filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  To obtain the irreducible decomposition of π0, use a simplified character  formula ([29],[24]):  mρi(π0) =  1  |H|  �  h∈H  χπ0(h)χρi(h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (43)  The characters, which are just the traces of the representing matrices eval-  uated at each h ∈ H, can be obtained from table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For other groups, such  π0(r)  Figure 9: π0(r) acting on a C-linear combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  26  Equivariant and Steerable Neural Networks  character tables are also available in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To determine the traces of  π0(h) for h ∈ H, one can look at the canonical basis of F0 (figure 8) and  how the individual vectors of it transform under π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The trace χπ0(h) of any  representation matrix corresponding to π0(h) is then just the number of basis  vectors of C that are left unchanged by its action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Thus, for instance, we receive  χπ0(e) = 9, as any representation evaluated at the neutral element always acts  as the identity matrix, and χπ0(m) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The latter is visualized in figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Figure 10: The canonical basis C of (F0, π0) (top) is transformed by the  reflection π0(m) (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The invariant vectors are highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  π0 turns out to have type (3, 0, 1, 1, 2) (see definition 6), meaning that there  are 3 copies of A1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 3 H-invariant one-dimensional subspaces with exactly  the transformation properties of this irrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Furthermore, there is one copy of  each of B1 and B2, and two copies of the two-dimensional irrep E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Decomposing (F0, π0) also makes π0(h) block diagonal for any h ∈ H, after  applying a change of basis matrix A that is constructed from the basis filters  shown column two of table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The same matrix A block diagonalizes π0(h)  for all h ∈ H simultaneously and the elements on the diagonal are the irrep  matrices shown in the third to last columns of table 1, with their respective  multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For instance, for h = r, we have:  π0(r)  π0(r)  ψ1 =  = [1] · ψ1,  ψ4 =  = [−1] · ψ4  Figure 11: A rotation acting trivially on an A1 basis element (left), and as  [-1] on a B1 basis element (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  /0  0  0  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  To(m) =  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  1  0  0  0  0  0  0  0  /0Equivariant and Steerable Neural Networks  27  Aπ0(r)A−1 = block_diag([1], [1], [1], [−1], [−1], [0, −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 1, 0], [0, −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 1, 0])  (44)  To get an intuition for HomH(π, ρ), we also need to look at the fiber represen-  tation ρ, which itself also has a decomposition into irreps and thus a type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In  fact, ρ would theoretically just be chosen by picking an arbitrary set of inte-  gers (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., mn) as multiplicities for the irreps (n = 5 for D4), as well as some  basis A ∈ RK′×K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' However, the choice of a basis is made obsolete later, when  nonlinearities and capsules are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The number of output channels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  the size of the fiber Fx that ρ can act upon is also determined by the irreps:  K′ =  n  �  i=1  mi dim ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (45)  For the sake of this example, let ρ be of type (2, 1, 1, 1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Then, K′ =  �5  i=1 mi dim ρi = 7, thus ρ acts on seven-dimensional fibers Fx ∈ R7, for  instance as  ρ(r) = A−1block_diag([1], [1], [1], [−1], [−1], [0, −1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 1, 0])A,  (46)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' a block diagonal matrix (after change of basis) with the corresponding  matrices from column “r” of table 1 on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The first channel now  transforms by A1, the second by A2, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  As the number of output channels K′ equals the number of “distinct filters”  that are convolved with the input, a filter bank Ψ intertwining π0 and ρ must  consist of seven filter units of shape K × s × s, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 3 × 3 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Schur’s  Lemma (theorem 1) now determines which basis elements of F0 can be used  to construct each of the individual filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' According to the lemma, the in-  tertwiner space of two irreps HomH(ρi, ρj) is either zero, or one-dimensional  iff ρi and ρj are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' It follows that each individual filter can only  be constructed from the basis elements in F0 that transform under the same  irrep as said filter’s output channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  To finish the example, consider again the type of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' There are two A1-  channels in the fiber, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' two distinct filters that need to be A1-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  These filters can be constructed independently from one another, using the  three basis elements in F0 that span the three A1-subspaces, yielding six  parameters in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' There is one A2-channel, yet there are no basis elements  in F0 that transform according to A2, thus this filter is simply zero, adding  no additional dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While the B1- and B2-filters are obtained in the  same way as for A1, each yielding one independent parameter, there also is a  copy of the two-dimensional irrep E in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While this irrep therefore has two  28  Equivariant and Steerable Neural Networks  channels, these do not transform independently from one another, but are  multiplied by two-dimensional matrices, which are given in the last row of  table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Thus, sets of two E2-basis filters in F0 (of which there also are two)  have to share one parameter, yielding two parameters in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' An illustration  of this construction is given by figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  A1  A1  A2  B1  B2  E  E  m1 = 3  m2 = 0  m3 = 1  m4 = 1  m5 = 2  (F0, π0)  m′  1 = 2  m′  2 = 1  m′  3 = 1  m′  4 = 1  m′  5 = 1  Figure 12: The construction of an equivariant filter bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The basis elements  connected by the dotted lines must share a parameter to preserve equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  There are seven output channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The process demonstrated in this example can be generalized to intertwiner  spaces HomH(π, ρ) for arbitrary representations of respective types (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., mn)  and (m′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., m′  n), where π is the feature space representation that transforms  feature maps and filters and ρ is the fiber representation for the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  As mentioned before, π might act on on the feature space (and thus on filters)  in a slightly different way than π0 (35) used in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The reason for  this will be explained in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We receive the following formula for  the dimension of of said intertwiner spaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' for the number of parameters  required by any given layer:  dim HomH(π, ρ) =  �  mim′  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (47)  Equivariant and Steerable Neural Networks  29  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='5 Generating Steerable Feature Spaces  What is left to be shown is how H-steerability of individual fibers F′  x leads  to steerability of the whole feature space F′ with respect to all of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To  derive this, we use the induced representation of H on G and we show that  the transformation law that is imposed on an output space F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' By con-  volving a transformed signal π(g)f, f ∈ F with an equivariant filter bank  Ψ ∈ HomH(π, ρ) we obtain the formula for the induced representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While  the following paragraphs again give the computations the explicit groups p4  and p4m, the arguments used can be made in complete analogy for any other  semidirect product G = N ⋊ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Points x ∈ Z2 can be interpreted either as a point (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' when looked at as a  pixel in the base space of feature maps), or as a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To make this  interpretation explicit, we denote x as ¯x when we interpret it as a translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Translation equivariant convolution of Ψ with f can then be defined as  [f ⋆ Ψ](x) = Ψ[π(¯x)−1f],  (48)  for translations ¯x ∈ Z2, with π acting on F as described in (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  We now utilize the semidirect product structure of G (definition 2), which  enables us to write any element g ∈ G as a product g = th where t ∈ Z2 is  a translation and h ∈ H is an element from the stabilizer group, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' some  rotation from C4 or rotation-flip from D4 for p4 or p4m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' It is  useful to look at an explicit matrix representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We can write any  element of G as  g = th =  �  I T  0 1  � �  R 0  0 1  �  =  �  R T  0 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (49)  Here, R is a matrix representation of h ∈ H, for instance a 2 × 2 rotation or  roto-reflection matrix for p4 or p4m, and T is the translation vector represent-  ing t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' With this form of representation, a translation ¯x ∈ Z2 then amounts to  ¯x =  �  I x  0 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (50)  It is furthermore useful to make the difference between the action of G on  itself by group operation and the action of G on Z2 visible, thus we will use  gh for g, h ∈ G for the former and g · x for g ∈ G, x ∈ Z2 for the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  30  Equivariant and Steerable Neural Networks  Applying th to f via π before convolving with ψ then yields:  [Ψ ⋆ [π(th)f]](x) = Ψπ(¯x−1)π(th)f  = Ψπ(¯x−1th)f  = Ψπ(hh−1¯x−1th)f  = Ψπ(h)π(h−1¯x−1th)f  = ρ(h)Ψπ(h−1t−1¯xh)f  = ρ(h)Ψπ(((th)−1 · x)  −1)f  = ρ(h)[Ψ ⋆ f]((th)−1 · x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (51)  The justifications for these equations are as follows: In the first line, we use the  definition of convolution (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In the second to fourth line, we use hh−1 = e,  thus multiplying by one and the fact that π is a group homomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  π(gg′) = π(g)π(g′) ∀g, g′ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The equivariance constraint (40) is then used  in the fifth line, and the argument of π is inverted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In line six we use the fact  that h−1t−1¯xh = ((th)−1 · x)  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This can be verified via the representation  matrices presented in (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Finally, the definition of the convolution operation  is applied backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  We can now define π′ by  [π′(th)f](x) := ρ(h)[f((th)−1x)],  (52)  yielding Ψ ⋆ π(g)f = π′(g)Ψ ⋆ f, thus giving us the desired steerability of F′  with respect to all of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Comparing π (35) and π′ (52), one notices that they only differ in the factor  ρ(r) which permutes the individual fibers after they have been moved from x  to (th)−1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This kind of action characterizes π′ as the aforementioned induced  representation of ρ (of H) on G, often also denoted as IndG  H(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While there  exists extensive knowledge about this concept, for instance in [30] and [31], for  our purpose it suffices to know that induced representations transport fibers  to new locations and then transform them via ρ, the representation induced  from π′  Furthermore, any representation of any subgroup of G can induce a rep-  resentation on G, allowing us to freely choose fiber representations when  constructing filter banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The representation π0 from (35) without any factor  permuting the fibers can also be seen as being induced from the trivial repre-  sentation, which acts as the identity matrix for all h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  It should also be noted that the content of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4, which is using the  decomposition of the trivially induced representation π0, can also be applied  Equivariant and Steerable Neural Networks  31  to induced representations π′ = IndG  H(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' When investigating the action of  such a representation π′ on basis filters to determine its trace, one just has to  include the factor of ρ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the fiber-permutation that is applied additionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Further layers can now be added to the network by choosing a new repre-  sentation ρ′ of H to act on the fibers of the next output space and compute  HomH(π′, ρ′) for π′ restricted to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='6 Commutation with Nonlinearities via Capsules  We have now seen how the convolutional layers of steerable CNNs are built  and how their equivariance to group transformations is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In particular,  it was shown that only the basis independent type of the input and output  representations is relevant for the parameter count of a filterbank, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  dim HomH(π, ρ) = dim HomH(π, A−1ρA) ∀A ∈ GL(Rdim ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  However, when considering nonlinearities, the choice of the basis becomes  relevant, which we shall demonstrate with a small example:  Consider the classical ReLU function and let g = (12) ∈ S2:  ρ(g) =  �  1 −1  0 1  �  and ρ′(g) = A−1ρ(g)A =  �  3  1  −4 −1  �  for A =  �  1 1  2 1  �  a change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' ρ(g), and thus ρ′(g) correspond to a representation of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  But, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' for x = (−2, 5)T and x′ = A−1x = (7, −9)T we have:  ReLU(ρ(g)(x)) = ReLU(  �  −7  5  �  ) =  �  0  5  �  ̸= ReLU(ρ′(g)(x′)) = ReLU(  �  12  −19  �  ) =  �  12  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Thus, different bases can lead to different results under non linear activation,  even though the underlying type of the representation is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To effi-  ciently tackle this problem, the concept of capsules is introduced:  A ρ−capsule (RK, ρ, B) is defined as a representation of the stabilizer group  (RK, ρ) with a fixed basis B of RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' By doing this, the representation matrices  ρ(h) also become fixed, which allows us us to realize them as matrices, which  will help in obtaining equivariance of nonlinearities, as will be explained in the  following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In practice, capsules are typically held relatively low-dimensional,  rarely exceeding the size of the stabilizer group, |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In the following a capsule  will often just be denoted by ρ, omitting the specified Basis B and simply  assuming that it is chosen in a way that yields a certain structure for the  32  Equivariant and Steerable Neural Networks  representation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Just like a convolutional layer, a nonlinearity layer must be equivariant to the  group G, which is realized by guaranteeing fiber-wise equivariance first and  then “inducing” feature space equivariance afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  We define an element-wise nonlinearity ν : R → R, or a fiber-wise nonlinearity  ν : RK → RK′ to be admissible for an input representation ρ, iff there exists  an output capsule ρ, such that  νρ(h) = ρ′(h)ν ∀h ∈ H,  (53)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' transforming input fibers by ρ before applying ν is the same as trans-  forming the nonlinearity’s output by ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We call ρ′ the post-activation capsule  corresponding to ρ and denote it by Actνρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Given an admissible nonlinearity ν and corresponding pre- and post activation  capsules ρ and ρ′ = Actνρ such that (53) holds, feature space equivariance to  the respective induced representation is derived as follows:  ν([IndG  Hρ](th)f)(x) = ν(ρ(r)f((th)−1x))  = ρ′(r)ν(f((th)−1x))  = ρ′(r)ν(f)((th)−1x)  = [IndG  Hρ′](th)ν(f)(x)  (54)  Instead of building a layer by choosing multiplicities of irreps for a fiber rep-  resentation, one now chooses multiplicities mi ≥ 0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., n corresponding  to a set of predefined capsules ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., ρn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The chosen capsules are then con-  catenated into a fiber of dimension � mi dim ρi, which is then acted upon by  ρ = � miρi, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the block diagonal representation with mi copies of ρi on the  diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Due to this block diagonal structure, capsules are disentangled, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  channels of one capsule do not mix with channels belonging to other capsules  during transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  When it comes to finding explicit pairs of capsules and corresponding admis-  sible nonlinearities, one needs not only to look at the irrep type of a capsule’s  underlying representation, but also at the individual form of the representa-  tion matrices ρ(g) which, as stated, differs not only depending on the choice  of irreps, but also on the choice of basis for the individual representation  spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Below, we will name the most prominent choices of capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For a  more extensive list, the reader is referred to [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  First, any capsule ρ which can be realized by permutation matrices will be  compatible with any element-wise nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Element-wise means that  instead of acting on a whole feature vector f(x) = (f1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., fK(x)), the  Equivariant and Steerable Neural Networks  33  nonlinearity transforms each element fi(x), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='., K, individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The  most prominent example for such a nonlinearity is the ReLU function (see  (19)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The most frequently used capsules that are realizable by permutation  matrices are regular capsules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' feature vectors that transform according  to the regular representation from example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While regular capsules have  been shown to work very well in practice ([3],[4]), they have the drawback of  leading to relatively high dimensional feature spaces, as each capsule has to  span |H| channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  A possible fix for the high dimensionality of regular capsules is given by  quotient capsules, in which the feature vectors transform according to the  quotient representation (example 3) of H by some (normal) subgroup K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Quotient capsules have the advantage of requiring |H|  |K| instead of |H| channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  If K is normal, the features represented by such capsules become not just  equivariant, but invariant to the symmetries of K:  ρquot(k)ehK = ekhK = ekKh = eKh = ehK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (55)  Hence, performance can be improved by reducing parameter cost and feature  size through the use of H/K quotient capsules if the K-invariant patterns  are prevalent in the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' On the other hand, quotient capsules can  also severely harm performance if the patterns are not found in the data or  irrelevant to the learning task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' It should furthermore be noted, that  quotient representations can also be obtained from non-normal subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In  that case, the features need not be necessarily invariant under K, but can  perform differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' An example for this, as well as some visual intuition for  quotient features in general, is given in appendix C of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  A second class of representation matrices are monomial matrices, which have  the same structure as permutation matrices, but also allow for entries of  minus one instead of just one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Given a representation that is fully realized  by monomial matrices, all concatenated nonlinearities will be admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A  concatenated nonlinearity is defined as evaluating an element-wise nonlinear-  ity on an element x and also on −x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For instance, the concatenated ReLU  function is defined as  CReLU(x) := (ReLU(x), ReLU(−x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (56)  Trivially, regular and quotient capsules will be compatible with concatenated  nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Also, many irreps can be realized (through a suitable basis) by  monomial matrices, thus allowing for low dimensional irrep capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The third class of examples is given by orthogonal matrices characterized  by the fact that they preserve the norm (length) |x| of vectors that they  act upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Capsules of this kind will be compatible with norm nonlinearities,  34  Equivariant and Steerable Neural Networks  which only act on the norm of any given feature vector (hence do not act  element-wise), but not on its orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A norm nonlinearity can generally  be written in the following form:  νnorm : RK → RK,  f(x) �→ η(|f(x)|) f(x)  |f(x)|  (57)  Here, η : R≥0 → R≥0 describes a non-linear function that is to act on the  feature vectors norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A prominent example of this would be Norm-ReLUs,  with  η(|f(x)|) = ReLU(|f(x)| − b),  (58)  with b being a learned bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Amongst others, Norm-ReLUs were used in [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The advantage of these nonlinearities is their broad compatibility, as any  representation of any finite group can be made orthogonal, by choosing an or-  thogonal change of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Further subclasses of norm nonlinearities, such as  gated and squashed nonlinearities are mentioned in [33] and [5], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='7 A Remark on Pooling and Group Pooling in Steerable  CNNs  As far as the literature on steerable CNNs ([4],[5],[8]) is concerned, pooling  operations are, if at all, applied fiber- or capsule-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This means that,  instead of conventional pooling with stride, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' subsampling the feature map  on a subgroup of Z2, thereby reducing equivariance of the network to that  subgroup, each copy of a capsule located at each point x ∈ Z2 is searched for  the maximal activation (in the case of max pooling), which is then set as the  output of the pooling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Thus, pooling operations of this nature in  steerable CNNs can be interpreted as nonlinearities, which were discussed in  the previous section, and hence also need to satisfy an equivaraince constraint  that is analogue to 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Let P : RK → R be a pooling operation such as max pooling or average  pooling, performed on a K-dimensional capsule ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For G-equivariance to be  maintained, we need the pooling operation to be H-equivariant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Pρ(h) = ρ′(h)P ∀h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (59)  The choices of compatible capsules ρ, ρ′ is quite limited here, as they must not  alter the numeric value of the feature vectors on which the pooling operation is  performed, or the result of the pooling operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Therefore, any input capsule  ρ that is to commute with P has to be realized by permutation matrices, as  taking the maximum (or average) value of a vector with permuted entries still  yields the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Furthermore, ρ′ needs to be the trivial representation,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' ρ′(h) = [1] ∀h ∈ H as otherwise equivariance could again be broken, for  Equivariant and Steerable Neural Networks  35  instance when the result is multiplied by [-1], as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' by the irrep A2 in table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The most common choices for ρ therefore are regular capsules, quotient  capsules, as well as irrep capsules that are realized by permutation matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  As an exception to the choice of ρ′ which was, to our knowledge, not yet  discussed in the literature, one can implement an equivalent of coset pooling  from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This will especially make sense taking into account that regular  steerable CNNs are equivalent to G-CNNs in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Suppose a sub-  group K ⊆ H is given, yielding a quotient space H/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Setting the input  capsule as the regular representation of H and the output capsule as the  quotient representation of H/K, we can define quotient pooling as follows:  Pquot : R|H| → R|H/K|, Pquot(f(x)) =  �  hK∈H/K  max  h′∈hKf(xh′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (60)  This means that for each of the outputs max  h′∈hNf(xh′), Pquot only looks at the  coordinates f(xh) of f(x) that correspond to the the elements of the respective  coset that is being pooled over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This process is repeated |H/K| times, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' for  each coset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The individual results are then added into a |H/K|-dimensional  vector that then transforms according to the quotient representation of |H/K|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='8 Reduction of Parameters and Implementation  By the use of steerable filter banks, the parameters of steerable CNNs are  utilized several times more efficiently than the parameters of regular con-  volutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As was shown in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4, an equivariant filter  bank Ψ intertwining representations π and ρ has a parameter cost of dim  HomH(π, ρ) = � mim′  i, depending on the irrep multiplicities mi and m′  i of π  and ρ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A non-equivariant filter bank of the same size, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' same  filter width s × s and same number of input/output channels K/K′ (which is  also determined by the dimensions of π and ρ), would instead have a parame-  ter cost of dim π · dim ρ = s2 · K · K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Thus, the parameter efficiency of a steerable filter bank in comparison to a  conventional filter bank is expressed as follows:  µ =  dim π · dim ρ  dim HomH(π, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (61)  For effective network architectures, this value usually lies around |H|, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the  size of the stabilizer group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For instance, we have µ = 8 for H = D4, meaning  that a p4m-steerable CNN’s convolutional layer uses its parameters eight  times more efficiently than a layer of the same size in a conventional CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Efficiency is further increased by the fact that the representations π and ρ  in each convolutional layer have a block-diagonal structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' consisting of  36  Equivariant and Steerable Neural Networks  direct sums of distinct, disentangled and lower-dimensional capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Because  of this, an intertwiner Ψ between said representations will also have a certain  block structure:  For this, let π = π1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' ⊕ πp and ρ = ρ1 ⊕ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' ⊕ ρr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' An intertwiner then is a  matrix Ψ of shape K′ × Ks2, with K′ = �r  i= dim ρi and Ks2 = �p  i=1 dim πi,  and has the following block structure:  �  ��  h11 ∈ HomH(π1, ρ1) · · · hp1 ∈ HomH(πp, ρ1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  h1r ∈ HomH(π1, ρr) · · · hpr ∈ HomH(πp, ρr)  �  ��  (62)  Here, each subblock hij ∈ HomH(πi, ρj) is itself an intertwiner between the  capsules πi and ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In many implementations, the same capsule is used sev-  eral, or even all the times, allowing to compute many or all of the hij using  the same intertwiner basis, thus drastically reducing computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Ordering the individual capsules such that equivalent capsules are adjacent to  each other then leads to superblocks Hij of shape mi dim πi × mj dim ρj, with  mi, nj being the respective multiplicities of the two capsules of Hij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The su-  perblock itself is then filled with the subblocks hij of shape dim πi × dim ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  In practice, when a list of capsules ρi and corresponding lists of post-activation  capsules Actνρi (depending on the nonlinearity) are given, the induced rep-  resentations πi = IndG  HActνρi, as well as the bases for the intertwiner spaces  HomH(πi, ρj) for all pairs i, j are computed offline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The bases are stored as  matrices ψij of shape dim πi · dim ρj × dim HomH(πi, ρj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' After that, a list  of input multiplicities mi and output multiplicities nj provided by the user  is put into a parameter matrix Φi,j of shape dim HomH(πi, ρj) × minj and  the aforementioned superblocks Hij are obtained by matrix multiplication of  ψijΦi,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' After all superblocks are obtained, Ψ is reshaped to K′ × K × s × s  and can then be convolved with the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='9 On the Equivalency of G-CNNs and Regular Steerable  CNNs  It was mentioned before that steerable CNNs are a direct generalizations of G-  CNNs from section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To be precise, this means that G-CNNs are equivalent  to steerable CNNs with regular capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' To see this, first consider the feature  maps of the respective architectures:  fG : G → RKG,  fsteer : Z2 → RKs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (63)  In G-CNNs, all feature maps except the input image are functions on a group  G, which in the cases that were considered here has Z2 as a normal subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Equivariance is achieved by modifying the domain of the convolution opera-  tion, leading to G-convolution (23) as described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' On the other  Equivariant and Steerable Neural Networks  37  hand, the feature maps of steerable CNNs keep the domain of Z2 in all layers  and achieve equivariance by restricting the space of filter banks, as explained  in detail in the sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  To understand the equivalence, we first look at how fG transforms under  actions of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For this, we again use the example of a p4-feature map (figure  3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A generalization to other groups is easily done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The group p4 consists of  unique tuples x = (t, r) with t ∈ Z2 and r ∈ C4 = {e, r, r2, r3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We can thus  interpret a p4-feature map as a map on Z2, which returns four “rotational  coordinates” at each point t ∈ Z2, corresponding to the elements e, r, r2, r3, as  is depicted in figure 13, which gives a visualization of this slightly different,  but equivalent interpretation of a p4-feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' When being acted upon by  another element x′ = (t′, r′) ∈ p4, the Z2-pixel coordinate t of x and its rota-  tional outputs are moved to x′−1t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Simultaneously, the rotational coordinates  are permuted by r′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  e  r2  r  r3  e  r2  r  r3  Figure 13: The transformation of a single pixel x ∈ Z2 by the rotation r ∈ C4  (top), and the permutation of that pixel’s four rotational outputs (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  In steerable CNNs, all feature maps and filters are functions on Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The equiv-  alent of the rotational outputs of G-CNNs is realized by the channels of the  regular representation of C4, which has a dimensionality of |C4| = 4, thus  consisting of one channel per element of C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As the transformation law of  the regular representation on these four channels is defined by the group’s  action on itself by composition, they transform in exactly the same way as  the rotational outputs of a G-CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The induced representation Indp4  C4ρreg of  this regular representation lastly ensures that the respective pixel coordinates  t ∈ Z2 transform in the same way in both cases, as can be checked by compar-  ison of the equations (22) and (24) with (35) and (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Hence, we receive the  desired equivalence of G-CNNs and regular steerable CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As stated, these  arguments can be generalized to any semidirect product group G′ = N ⋊ H  instead of Z2 ⋊ C4, as the regular representation behaves in the same way for  38  Equivariant and Steerable Neural Networks  any finite stabilizer group H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In any case, one channel of a G-feature map in  a layer of a G-CNN equates to one regular capsule, and thus to |H| channels  in the equivalent regular steerable CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  5 Steerable CNNs on Zn ⋊ Sn  In this section, possible applications of the theory of equivariant networks  to the symmetric group Sn will be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As G-CNNs are equivalent to  a special case of steerable CNNs, this chapter will stay in the more general  language of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  As in example 2, one can build semidirect product groups Zn ⋊ Sn, with the  most practical use cases likely being n = 2, for instance for 2D-images and  n = 3 for volumetric data, as for instance in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' One use case is given by  interpretation of street scenes by one neural network based on the input of  two redundant camera sensors that are installed at nearby positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The  signal of these sensors can be considered to be equivalent, but not necessarily  equal, as one camera sensor could be disturbed by dirt, but not the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  These networks will be very similar to the ones discussed before, with the  difference that Sn is used as the stabilizer group instead of C4 or D4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='1 Steerable Feature Spaces and Filter Banks for Zn ⋊Sn  Zn ⋊ Sn (see example 2) acts on functions on Zn in a similar way to the  product groups p4 and p4m by compositions of translations and coordinate  permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Hence, the theory of steerable CNNs from section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4 can be  applied for Zn ⋊ Sn in near complete analogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' In this section, we will never-  theless summarize the process of obtaining steerable feature spaces and filter  banks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We also highlight the key differences, which lie in representation the-  ory, such as in basis filters for intertwiner spaces, as these depend on the irreps  of the underlying group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' When discussing concrete examples, we will confine  to S2 and S3, for visualization purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' However, the general procedure of  constructing steerable CNNs on Zn ⋊ Sn for any n ∈ N becomes apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Feature Spaces  We again deal with feature spaces Fl of functions f : Zn → RKl with Kl  output channels in each layer l, which transform by a representation πl  analogously to (35) for l = 0 and (52) in higher layers:  [π0(t, σ)f](x) = f((tσ)−1x) = f(σ−1(x − t)), t ∈ Zn, σ ∈ Sn  (64)  [πl+1(t, σ)f](x) = ρl(σ)[f((tσ)−1x)], t ∈ Zn, σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (65)  In (65), ρl is the fiber representation from the previous layer from which  πl+1 = IndZn⋊Sn  Sn  ρl is induced, and with respect to which filter banks Ψ ∈  Equivariant and Steerable Neural Networks  39  Irrep  Basis Filters  e  (12)  id  [1]  [1]  sgn  [1]  [−1]  Table 2: The irreducible decomposition of (FS2  0 , π0) for S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  HomSn(πl, ρl) of layer l must be equivariant:  ρl(σ)Ψ = Ψπl(σ) ∀σ ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (66)  Equivariant Filter Banks  The symmetric group of 2 elements has two distinct irreps: The trivial repre-  sentation, id, and the sign representation sgn, both of them have dimension  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' By restricting π0 from (64) to S2 and letting it act on 3×3 filters with one  channel, we once again receive a nine-dimensional representation (FS2  0 , π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  One now obtains the irrep decomposition of π0 by applying the character  formula for χid and χsgn:  mπ0(id) =  1  |S2|  �  σ∈S2  χπ0(σ)χid(σ) = 1  2(χπ0(e)χid(e) + χπ0((12))χid((12))) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (67)  While χid(e) and χid((12)) can be looked up from table 2, it is useful for χπ0  to consider the canonical basis vectors of (FS2  0 , π0) in figure 8 and how π0(e)  and π0((12)) transform them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The representation matrix for the identity  element is always the identity matrix for any representation, hence we receive  χπ0(e) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' On the other hand, π0((12)) swaps the coordinates of each pixel  in each basis element, thus only leaving the third, fifth, and seventh filter  invariant, leading to χπ0((12)) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Since there is only one other possible irrep, we immediately get mπ0(sgn) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  With this, we have the full irreducible decomposition of (FS2  0 , π0), which is  depicted with an exemplary set of basis filters in table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  For S3 and Z3, we have one extra dimension for the base of feature spaces and  filters, hence leading to an increase in dimensionality of the space of filters  when looking at the S3-equivalent of (FS2  0 , π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Instead of 3 × 3 filters, we  now have 3 × 3 × 3 filters, leading to a representation (FS3  0 , π0) of degree 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  The canonical basis for the space FS3  0  is depicted in figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  40  Equivariant and Steerable Neural Networks  We next determine the irreducible decomposition of π0 in FS3  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' While the  characters of the three irreducible representations of S3 are given in table 3,  it might look tedious at first to determine the character of π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' However, this  becomes quite easy if one again looks at which basis elements from CS3 (figure  14) are left invariant by each group element of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As always, π0(e) leaves  every element invariant, thus yielding χπ0(e) = 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For the other elements,  it is helpful to identify each basis vector with the coordinates of its non-zero  entry (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the black cube in each filter), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the first vector with (−1, 1, 1),the  second with (0, 1, 1) and the fourth with (−1, 1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Any of the transpositions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 2-cycles (12), (23), (13) ∈ S3 fix all vectors  of which the two non-zero coordinates that are permuted are equal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  {(x1, x1, x2) | x1, x2 ∈ {−1, 0, 1}} for (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Thus, each transposition fixes  9 elements, so we have χπ0((12) = χπ0((23) = χπ0((13) = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The same  argument can be used for the two 3-cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Each of them fixes the same 3  vectors of which all three non-zero coordinates are equal, hence we have  χπ0((123)) = χπ0((132)) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Figure 14: Some canonical basis elements of CS3 of (FS3  0 , π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As always,  entries of one are black and entries of zero are grey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Plugging these results into the character formula now yields  mπ0(id) =  1  |S3|  �  σ∈S3  χπ0(σ)χid(σ) = 1  6(27 + 3 · 9 + 2 · 3) = 10,  mπ0(sgn) =  1  |S3|  �  σ∈S3  χπ0(σ)χsgn(σ) = 1  6(27 + 3 · (9 · (−1)) + 2 · 3) = 1,  mπ0(Vs) =  1  |S3|  �  σ∈S3  χπ0(σ)χVs(σ) = 1  6(27 · 2 + 3 · (9 · 0) + 2 · (3 · (−1))) = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  (68)  Thus, the type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' the multiplicities of irreps) of π0 is (10, 1, 8) for the  trivial representation id, the sign representation sgn and the two-dimensional  standard representation Vs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The basis of this irreducible decom-  position is depicted in figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As one can verify, these filters transform  under π0(σ) for σ ∈ S3 by multiplication with the representation matrices for  the respective group element, which are given in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Equivariant and Steerable Neural Networks  41  id  sgn  Vs  Figure 15: The basis vectors for the irrep decomposition of (FS3  0 , π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The  reader may verify that these vectors transform under π0 by the respective  matrices given in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Again, grey and transparent entries count as zeros,  black entries as one and white entries as minus one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Irrep  e  (12)  (13)  (23)  (123)  (132)  id  [1]  [1]  [1]  [1]  [1]  [1]  sgn  [1]  [−1]  [−1]  [−1]  [1]  [1]  Vs  �  1 0  0 1  �  �  1  0  −1 −1  �  �  −1  1  0  −1  �  �  0 1  1 0  �  �  −1 1  −1 0  �  �  0  1  −1 −1  �  Table 3: The representation matrices of π0 for the basis filters of the irrep  decomposition of (FS3  0 , π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Equivariant filter banks and steerable feature spaces are now obtained in  analogy to the sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Given some list of output multiplicities  (mid, msgn, mVs) (or just (mid, msgn) for S2) of some fiber representation ρ,  the filter bank Ψ ∈ HomH(π0, ρ) is constructed by linearly combining the irrep  basis filters from figure 15 for S3 or from table 2 for S2 in the same way as is  depicted (for p4m) in figure 12, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' by only combining filters that correspond  to the irrep that each of the respective output channels is transformed by.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='2 Capsules, Nonlinearities and Pooling for Zn ⋊ Sn  As was explained in the sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='7, certain requirements have to  be fulfilled for a layer (realized by a concatenation of capsules as before) to  commute with fiber-wise nonlinearities and pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' A layer is thus not built  by choosing multiplicities for the irreps directly, but by choosing copies of  42  BIBLIOGRAPHY  capsules to be concatenated into a fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We now list some relevant capsules  for steerable CNNs with H = S2 or H = S3  As for any finite group, we can look at regular capsules which transform  under the regular representation ρreg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For S2, this is the two-dimensional  representation of type (1, 1) and for S3, we have the four-dimensional rep-  resentation of type (1, 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' As all regular capsules, these are realized by  permutation matrices and thus commute with all nonlinearities discussed in  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='6, including ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Furthermore, fiber-wise max-pooling can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  To obtain quotient capsules (which have the same compatibility properties as  regular capsules), subgroups are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Of these there are none except the  group itself and {e} for S2, which would lead to a trivial capsule and a regular  capsule, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' For S3, we have (besides the aforementioned ones) one  normal subgroup, which is the alternating group containing all even permu-  tations, A3 = {e, (123), (132)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' With this, we have S3/A3 = S2, hence this  quotient capsule would behave like a regular capsule of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' S3 furthermore  has S2 as non-normal subgroup, allowing for S3/S2-quotient capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' The  two subgroups can also be used to implement coset pooling as described in 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  6 Conclusion  In this review article, we have seen several relevant approaches to achieve  invariant representations in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' After establishing the math-  ematical preliminaries from group theory and representation theory, we  discussed translation equivariant convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' We then saw how  group equivariant neural networks generalize CNNs and thus allow for repre-  sentations that are invariant to groups of transformations that are more general  than just translations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' This concept was then generalized once more by intro-  ducing steerable CNNs, which by the use of group representations allow for  different, more nuanced ways to express equivariance to a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  We furthermore presented an application of the theory to the symmetric group  resulting in a steerable CNN architecture for the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' Helpful discussions with Matthias Rottmann are grate-  fully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
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+page_content=' In: Pro-  ceedings of the IEEE Conference on Computer Vision and Pattern  Recognition, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' 5028–5037 (2017)  [33] Sabour, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=', Frosst, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=', Hinton, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content=' : Dynamic routing between capsules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='  arXiv preprint arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
+page_content='09829 (2017)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/odE1T4oBgHgl3EQfOwOM/content/2301.03019v1.pdf'}
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+Draft version January 12, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+The Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters. XXIV. Differences in
+internal kinematics of multiple stellar populations
+Mattia Libralato,1 Enrico Vesperini,2 Andrea Bellini,3 Antonino P. Milone,4, 5 Roeland P. van der Marel,3, 6
+Giampaolo Piotto,4, 5 Jay Anderson,3 Antonio Aparicio,7, 8, 9 Beatriz Barbuy,10 Luigi R. Bedin,5
+Thomas M. Brown,3 Santi Cassisi,11, 12 Domenico Nardiello,5 Ata Sarajedini,13 and Michele Scalco14, 5, 2
+1AURA for the European Space Agency (ESA), Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+2Department of Astronomy, Indiana University, Bloomington, IN 47405, USA
+3Space Telescope Science Institute 3700 San Martin Drive, Baltimore, MD 21218, USA
+4Dipartimento di Fisica e Astronomia, Universit`a di Padova, Vicolo dell’Osservatorio 3, Padova, I-35122, Italy
+5INAF - Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, Padova, I-35122, Italy
+6Center for Astrophysical Sciences, The William H. Miller III Department of Physics & Astronomy, Johns Hopkins University,
+Baltimore, MD 21218, USA
+7Instituto de Astrof´ısica de Canarias, Calle V´ıa L´actea, La Laguna, Tenerife, Canary Islands, E-38205, Spain
+8Departamento de Astrof´ısica, Universidad de La Laguna, Av. Astrof´sico Francisco S´anchez, La Laguna, Tenerife, Canary Islands,
+E-38206, Spain
+9Department of Aerospace Engineering, Khalifa University of Science and Technology, P.O. Box 127788, Abu Dhabi, United Arab
+Emirates
+10Universidade de S˜ao Paulo, IAG, Rua do Mat˜ao 1226, Cidade Universit´asria, S˜ao Paulo 05508-900, Brazil
+11INAF - Osservatorio Astronomico di Abruzzo, Via M. Maggini, s/n, Teramo, I-64100, Italy
+12INFN - Sezione di Pisa, Largo Pontecorvo 3, Pisa, I-56127, Italy
+13Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA
+14Dipartimento di Fisica e Scienze della Terra, Universit`a di Ferrara, Via Giuseppe Saragat 1, Ferrara, I-44122, Italy
+(Received November 16, 2022; Revised December 19, 2022; Accepted December 20, 2022)
+Submitted to ApJ
+ABSTRACT
+Our understanding of the kinematic properties of multiple stellar populations (mPOPs) in Galactic
+globular clusters (GCs) is still limited compared to what we know about their chemical and photomet-
+ric characteristics. Such limitation arises from the lack of a comprehensive observational investigation
+of this topic. Here we present the first homogeneous kinematic analysis of mPOPs in 56 GCs based on
+high-precision proper motions computed with Hubble Space Telescope data. We focused on red-giant-
+branch stars, for which the mPOP tagging is clearer, and measured the velocity dispersion of stars
+belonging to first (1G) and second generations (2G). We find that 1G stars are generally kinematically
+isotropic even at the half-light radius, whereas 2G stars are isotropic at the center and become radially
+anisotropic before the half-light radius. The radial anisotropy is induced by a lower tangential velocity
+dispersion of 2G stars with respect to the 1G population, while the radial component of the motion
+is comparable. We also show possible evidence that the kinematic properties of mPOPs are affected
+by the Galactic tidal field, corroborating previous observational and theoretical results suggesting a
+relation between the strength of the external tidal field and some properties of mPOPs. Although
+limited to the GCs’ central regions, our analysis leads to new insights into the mPOP phenomenon,
+and provides the motivation for future observational studies of the internal kinematics of mPOPs.
+Corresponding author: Mattia Libralato
+libra@stsci.edu
+arXiv:2301.04148v1  [astro-ph.GA]  10 Jan 2023
+
+2
+Libralato et al.
+Keywords: globular clusters: general – proper motions – stars: kinematics and dynamics
+1. INTRODUCTION
+The puzzle of the origin of the multiple stellar pop-
+ulations (mPOPs) in Galactic globular clusters (GCs)
+has been controversial since their discovery. The large
+amount of spectroscopic and photometric data collected
+so far has provided almost all observational information
+we know about mPOPs in GCs, but no definitive con-
+sensus has been reached yet about the formation and
+evolution of mPOPs (Renzini et al. 2015; Bastian &
+Lardo 2018; Gratton et al. 2012, 2019; Cassisi & Salaris
+2020; Milone & Marino 2022). The interplay between
+theoretical and observational efforts has pushed the
+community to find new ways to constrain the origin
+of mPOPs in GCs. For example, this research field is
+progressively seeking answers by looking at young and
+massive clusters in other galaxies (e.g., Larsen et al.
+2014; Dalessandro et al. 2016; Niederhofer et al. 2017;
+Lagioia et al. 2019; Martocchia et al. 2019; Nardiello
+et al. 2019; Milone et al. 2020). However, there is still
+an almost uncharted wealth of information in Galac-
+tic GCs that can enrich the observational picture of
+mPOPs: their internal kinematics.
+Here, we investigate the kinematic properties of first
+(1G) and second (2G) generation stars hosted in GCs.
+This effort focuses on red-giant branch (RGB) stars,
+for which the separation between different populations
+is clearer. We make use of the homogeneous collection
+of proper motions (PMs) obtained with Hubble Space
+Telescope (HST) data recently published by Libralato
+et al. (2022, hereafter L22) for 56 globular and one open
+clusters, and compare the properties of the velocity
+distributions of 1G and 2G stars.
+The outline of the paper is the following. Section 2
+describes the data sets used for this study, the proce-
+dure to identify the mPOPs, and how we calculated
+their kinematic properties. Section 3 reports our results
+concerning the kinematics of mPOPs; while in Section 4
+we investigate the possible dependence of the velocity
+anisotropy on the Galactic tidal field.
+2. DATA SETS, MULTIPLE-POPULATION
+TAGGING AND KINEMATICS
+We made use of the PM catalogs1 of L22, to which we
+refer for a detailed description of how the PMs were com-
+1 Catalogs
+are
+available
+at
+MAST
+as
+a
+High
+Level
+Sci-
+ence
+Product
+via
+10.17909/jpfd-2m08.
+See
+also:
+https://archive.stsci.edu/hlsp/hacks.
+puted2. In brief, we designed a multi-step reduction to
+exploit crowded regions, like the cores of GCs, and mea-
+sured position and flux of sources in HST exposures via
+effective point-spread function fitting. The geometric-
+distortion-corrected positions of each object as a func-
+tion of time were then fit with a least-squares straight
+line, the slope of which is an estimate of the PM of the
+star. We cross-identified stars in our astro-photometric
+catalogs with those in the (pseudo) two-color diagrams
+known as “chromosome maps” of Milone et al. (2017)
+made for all clusters analyzed in the Treasury GO-13297
+program (Piotto et al. 2015). We then applied the astro-
+photometric quality selections described in both papers
+to obtain samples of well-measured RGB stars for the
+mPOP tagging and their kinematic analysis.
+The reason of our choice to focus on the RGB stars
+is twofold.
+First, the mPOPs along the RGB can be
+identified more easily because the UV data (which pro-
+vide the key filters needed for disentangling the mPOPs
+depending on their CNO contents) used by Milone et al.
+(2017) were designed to have the highest signal-to-noise
+ratio (and hence photometric quality) for the RGB
+stars (see discussion in Piotto et al. 2015). Secondly,
+this choice allows us to compare the kinematics of stars
+with similar masses.
+However, focusing on RGB stars necessarily implies
+small number statistics, and this poses a problem if
+we choose to work with each cluster separately, com-
+puting the velocity dispersions for stars in each mPOP
+first, and then collecting all measurements from all
+GCs to study the average kinematic trends of mPOPs.
+For clusters with different population sizes, the veloc-
+ity dispersion representing the kinematics of stars at
+a given distance from the center of the cluster could
+be obtained by considering stars over a different radial
+interval.
+This could potentially introduce a bias that
+can wash out some of the features we are looking for.
+For this reason, we instead normalized positions and
+PMs of the RGB stars in each GC catalog by the GC’s
+half-light radius3 (rh) and central velocity dispersion
+σµ (from L22), respectively. The errors on the central
+2 Since our focus is on GCs,
+we excluded the open cluster
+NGC 6791 from the investigation.
+3 Cluster parameters (half-light radius, distance, half-mass relax-
+ation time, average perigalactic distance) are taken from the
+GC database of Holger Baumgardt (Sollima & Baumgardt 2017;
+Baumgardt et al. 2020; Vasiliev & Baumgardt 2021; Baumgardt
+& Vasiliev 2021).
+Ages are from Dotter et al. (2010), Milone
+et al. (2014) and Koch & McWilliam (2014). See L22 for details.
+
+The Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters. XXIV.
+3
+Figure 1. Examples of “chromosome” maps and mPOP tagging for a type-I (NGC 104; left panel) and a type-II (NGC 1851;
+right panel) GCs. In each plot, gold squares and blue dots represent 1G and 2G stars on the RGB, respectively. Red crosses in
+the right panel highlight the red-RGB stars in NGC 1851.
+Table 1. Median anisotropy of 1G, 2G and red-RGB stars for stars with r > 0.6 rh and statistical significance of the difference
+between the mPOP anisotropies for the various cases discussed in text. The values between brackets in the second, third and
+fourth columns are the number of stars used in each case. No number is provided when no stars are available, or if the anisotropy
+for the specific case was not computed (see text for details).
+1G (N)
+2G (N)
+red-RGB (N)
+1G vs. 2G
+1G vs. red-RGB
+2G vs. red-RGB
+Median anisotropy
+Comparison
+Entire sample
+1.02 ± 0.02 (5962)
+0.91 ± 0.01 (13884)
+0.90 ± 0.03 (1317)
+4.9σ
+3.3σ
+0.3σ
+age/th≥10
+1.02 ± 0.03 (1138)
+0.95 ± 0.02 (2289)
+/
+1.9σ
+/
+/
+7≤age/th<10
+1.02 ± 0.06 (885)
+0.97 ± 0.02 (1745)
+/
+0.8σ
+/
+/
+age/th<7
+0.98 ± 0.03 (3939)
+0.92 ± 0.02 (9850)
+0.90 ± 0.07 (940)
+1.6σ
+1.0σ
+0.3σ
+Rperi ≤ 3.5 kpc
+1.01 ± 0.03 (2479)
+0.94 ± 0.02 (6929)
+/
+1.9σ
+/
+/
+Rperi > 3.5 kpc
+0.94 ± 0.07 (1460)
+0.91 ± 0.03 (2921)
+/
+0.4σ
+/
+/
+Table 2. Slopes m of the least-squares straight-line fits (in linear units of r/rh) to the anisotropy profiles of 1G, 2G and
+red-RGB stars, and statistical significance of the difference between the mPOP slopes for the various cases discussed in text.
+1G
+2G
+red-RGB
+1G vs. 2G
+1G vs. red-RGB
+2G vs. red-RGB
+Slope
+Comparison
+Entire sample
+-0.02 ± 0.02
+-0.06 ± 0.01
+-0.07 ± 0.02
+1.8σ
+1.8σ
+0.4σ
+age/th≥10
+0.02 ± 0.03
+-0.05 ± 0.01
+/
+2.2σ
+/
+/
+7≤age/th<10
+-0.01 ± 0.04
+-0.03 ± 0.02
+/
+0.4σ
+/
+/
+age/th<7
+-0.02 ± 0.02
+-0.06 ± 0.01
+-0.10 ± 0.03
+1.8σ
+2.2σ
+1.3σ
+Rperi ≤ 3.5 kpc
+-0.01 ± 0.02
+-0.06 ± 0.02
+/
+1.8σ
+/
+/
+Rperi > 3.5 kpc
+-0.05 ± 0.03
+-0.08 ± 0.02
+/
+0.7 σ
+/
+/
+σµ were included in the normalized-PM error budget.
+This normalization allowed us to jointly compare pairs
+
+2G
+0.4
+F275W,F336W,F438W
+0
+0.2
+0.1
+-0.4
+-0.3
+0.2
+01
+0
+0.1
+F275W.F814Wred-RGB
+0.4
+F275W,F336W,F438W
+0
+.3
+2G
+0.2
+XX
+0.1
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+V
+7275W.F814W4
+Libralato et al.
+Figure 2. Anisotropy (left), normalized σrad (center) and σtan (right) as a function of distance from the center of the cluster in
+units of rh. Gold squares (top panel), blue dots (second panel from the top) and red crosses (third panel from the top) represent
+the 1G, 2G and red-RGB populations, respectively. The black, dashed horizontal lines in the left panels mark the isotropic case.
+The solid lines in each panel, color-coded as the corresponding points, are least-squares straight-line fits (in linear units of r/rh)
+to the points forced to have the ordinate equal to 1 at the center (r/rh = 0). The light-color shaded regions correspond to the
+1σ errors of the fits. The comparisons between the trends in each case are shown in the bottom panels (1σ errors of the fits are
+not plotted for clarity).
+of stellar positions and PMs from all clusters at once,
+thus increasing the number of data points that can be
+used to study the kinematics of each mPOP without
+the drawback discussed before.
+Milone et al. (2017) classifies GCs in two main fam-
+ilies.
+Most GCs belong to the type-I family, and are
+characterized by chromosome maps with two distinct
+groups4 made by 1G and 2G stars (left panel of Fig. 1).
+The remaining clusters are instead labelled as type-
+II GCs.
+These systems present: more complex chro-
+mosome maps, where the 1G and 2G stars seem to
+be divided in subgroups (right panel of Fig. 1); and
+split sub-giant branches and RGBs (hereafter red-RGB)
+clearly visible in CMDs with specific color combinations.
+Stars belonging to the red-RGB population are typ-
+ically enriched in the overall CNO content, iron and
+4 Note that 1G stars in type-I GCs present a color spread in
+the chromosome maps likely due to a spread in Fe of ≳0.1 dex
+(Marino et al. 2019; Lardo et al. 2022; Legnardi et al. 2022).
+s-elements abundances, and have their own 1G and
+2G subdivision (hereafter, 1Gr and 2Gr, respectively).
+We refer to Marino et al. (2019) for a comprehensive
+description of the spectro-photometric properties of
+these stars. The origin of these red-RGB stars is not
+clear. For example, Marino et al. (2019) suggested two
+possible options: (i) after the gas from which the “clas-
+sical” 1G and 2G stars formed was almost exhausted,
+the clusters reaccreted pristine gas that was enriched
+in iron by supernovae; or (ii) the type-II GCs formed
+within a dwarf galaxy, with 1G/2G and 1Gr/2Gr born
+at different times and/or places.
+The 1Gr/2Gr distinction is not always as clear as that
+between the 1G/2G stars in our chromosome maps, in
+particular for NGC 1851 and NGC 6715. Nevertheless,
+we arbitrarily divided the red-RGB group is our type-II
+clusters in 1Gr and 2Gr stars similarly to what done for
+1G and 2G stars in Fig. 1. The majority of the red-RGB
+stars belongs to the 2Gr group, and only 219 stars are
+part of the 1Gr group. Because the photometric tagging
+
+ETTTT
+HT
+ETT
+1
+1
+EITTT
+rad
+1.1
+0.9
+0.9
+b6
+pes
+0.8
+0.8
+1
+tan/
+二二
+0.7
+b
+0.7
+0.9
+6
+0.6
+0.6
+0.8
+1.2
+1
+E
+-二
+1
+11
+rad
+1.1
+0.9
+0.9
+6/
+rad
+0.8
+0.8
+2G
+1
+tan/
+9
+0.7
+b
+0.7
+0.9
+b
+0.6
+0.6
+0.8
+1.2
+1
+1
+RGB
+rad
+1.1
+0.9
+0.9
+b
+rad
+0.8
+Z
+0.8
+1
+tan/
+0.7
+b
+6
+0.7
+0.9
+red
+b
+0.6
+0.6
+0.8
+ELL
+?
+1
+E
+1
+1
+rad
+0.9
+0.9
+6
+rad
+tan
+0.8
+0.8
+0.9
+tan/
+6
+0.7
+0.7
+6
+0.8
+0.6
+0.6
+E
+0.1
+1
+0.1
+0.1
+1
+rThe Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters. XXIV.
+5
+of 1Gr and 2Gr stars is not straightforward in our sam-
+ple of type-II GCs, and given the very few 1Gr stars,
+we choose to analyzed the red-RGB stars as a whole5.
+Following this classification, we divided our samples
+in either two (1G and 2G for type-I GCs) or three (1G,
+2G and red-RGB for type-II GCs) sub-populations. The
+mPOP tagging was directly performed on the chromo-
+some map.
+3. GLOBAL KINEMATIC PROPERTIES OF
+MULTIPLE POPULATIONS
+As shown in a number of theoretical studies, the
+velocity anisotropy may provide various fundamental
+insights into the formation and evolution of GCs (e.g.,
+Tiongco et al. 2016a; Breen et al. 2017, 2021; Pavl´ık &
+Vesperini 2021, 2022) and their mPOPs (Tiongco et al.
+2019; Vesperini et al. 2021).
+We computed the normalized radial and tangential
+velocity dispersions (σrad and σtan, respectively) in
+equally-populated radial bins of at least 200 stars each6
+as in Sect. 4 of L22, using a maximum-likelihood ap-
+proach. The left panels of Fig. 2 present the anisotropy
+as a function of radial distance from the center of the
+cluster in units of rh for 1G, 2G and red-RGB stars.
+The solid lines in each panel, color-coded as the corre-
+sponding points, are least-squares straight-line fits to
+the points forced to have the ordinate equal to 1 at the
+center (r/rh = 0), i.e., (σtan/σrad)(r) = 1 + m × r/rh.
+These fits are linear in r/rh, thus they appear curved in
+our plots with a logarithmic scale on the x axis. The 1G
+stars (gold points) in our fields are isotropic even outside
+1 rh, with only a marginal (∼1σ) signature of a radial
+anisotropy in the outermost part of the field. The 2G
+(blue) and red-RGB (red) populations are isotropic in
+the center and become progressively radially anisotropic
+further from the GC’s center. Table 1 collects the me-
+dian values of the anisotropy for each population for r >
+0.6 rh. This threshold was chosen as a compromise be-
+tween having enough points to compute a robust median
+anisotropy value for each mPOP and being sufficiently
+far from the center of the cluster to capture indications
+of anisotropy. Table 2 collects the slopes of the straight-
+line fits.
+There is a clear difference in the median
+anisotropy between 1G and 2G stars at the ∼5σ level,
+and between 1G and red-RGB stars at the ∼3σ level.
+5 The kinematic analysis shown in Fig. 2 provides consistent results
+for 1Gr, 2Gr and 1Gr+2Gr stars.
+6 For the analysis of the kinematics of red-RGB stars in GCs with
+different dynamical age, we made at least one radial bin with all
+stars at disposal when not enough stars were available.
+The slopes of the straight-line fits provide similar results,
+although the statistical significance is smaller (∼2σ).
+These findings are in general agreement with the
+predictions of numerical models of the evolution of
+multiple-population clusters (see, e.g., Vesperini et al.
+2021).
+Specifically, theoretical models explain these
+behaviors as the consequence of the initial spatial dif-
+ferences between 1G and 2G stars. Simulations usually
+start with a 2G population more centrally concentrated
+in the inner regions of a more diffuse 1G system as
+suggested by models of formation of mPOPs (D’Ercole
+et al. 2008; Calura et al. 2019). For systems starting
+with an isotropic velocity distribution, the anisotropy
+of the 2G stars is a consequence of the outward diffu-
+sion of 2G stars (Tiongco et al. 2016b; Vesperini et al.
+2021). For systems starting with an anisotropic velocity
+distribution, this difference is the result of a more rapid
+evolution towards a isotropic velocity distribution of
+1G stars. In such case, it is also possible to find both
+populations to be characterized by anisotropic velocity
+distributions. We point out that although the difference
+between the anisotropy of 1G and 2G stars is small, its
+extent is generally consistent with that found in numer-
+ical models at the distances from the clusters’ centers
+probed by our data.
+The middle and right panels show the radial profiles of
+the normalized σrad and σtan, respectively. All the pop-
+ulations have similar radial velocity dispersions, while
+tangential velocity dispersions are larger for 1G stars
+than for 2G and red-RGB sources. These findings are
+in agreement with the theoretical predictions for which
+the different degrees of radial anisotropy between 1G
+and 2G stars is caused by a difference in the tangential
+component of their motions, rather than in the radial
+component (Bellini et al. 2015; Vesperini et al. 2021).
+In Fig. 3, we further explore the anisotropy of mPOPs
+for GCs with different dynamical ages as measured by
+the ratio of the GCs’ ages to their half-mass relaxation
+time (th). L22 found that dynamically-old (age/th>10)
+and young (age/th<7) GCs are characterized by dif-
+ferent velocity distributions at rh.
+We followed this
+same classification to better highlight differences and
+analogies between 1G and 2G stars. Figure 3 presents
+the anisotropy as a function of distance from the center
+of the cluster in units of rh for dynamically old (top),
+intermediate (second from the top) and young (third
+from the top) GCs. The bottom panels show the com-
+parison between the trends of each mPOP in GCs with
+different dynamical ages, while the rightmost panels
+collect the straight-line fits for each population for GCs
+with the same the dynamical age.
+The trends shown
+in these panels are consistent with the global trends
+
+6
+Libralato et al.
+Figure 3. As in the left panels of Fig. 2, but dividing the sample of GCs according to their dynamical age (age/th ratio). The
+first three columns show the anisotropy for 1G (gold squares), 2G (blue dots) and red-RGB (red crosses), respectively. The
+black, horizontal lines mark the isotropic case. The lines, colored as the point in the same plot, are a straight-line fit to the
+data (see details in Fig 2; no line was fit for the red-RGB samples with only one data point). The first three rows present the
+result for old, intermediate and young GCs, from top to bottom, respectively. The rightmost panels collect the straight-line fits
+for each population for GCs with the same the dynamical age, while the panels at the bottom show the comparison between
+the trends of each mPOP in clusters with different dynamical ages.
+shown in Fig. 2, but the differences between the mPOP
+anisotropies are less evident (see Tables 1 and 2). Most
+of the 1G fits are consistent with an isotropic distri-
+bution, while most of the 2G median anisotropies and
+slopes indicate a statistically-significant anisotropy. Our
+analysis suggests that, for a given mPOP, there might
+be kinematic differences depending on the dynamical
+age of the hosting cluster, but the large error bars do
+not allow us to draw any definitive conclusion. Larger
+differences might be present further from the center of
+the cluster, where the relaxation time is longer and fin-
+gerprints of the initial kinematic properties might still
+be detectable. Finally, no conclusion can be inferred for
+the red-RGB stars in intermediate and old GCs because
+of the low statistics.
+4. DEPENDENCE ON THE GALACTIC TIDAL
+FIELD
+Previous studies (e.g., Vesperini et al. 2014; Tiongco
+et al. 2016b; Bianchini et al. 2018) have shown that
+the external tidal field of the host galaxy may play an
+important role in the early- and long-term evolution of
+the velocity anisotropy. In particular, the loss of stars in
+stronger tidal fields preferentially affects stars on more
+radial orbits, causing a decrease of the radial anisotropy
+in the outer regions, and a gradual evolution towards a
+more isotropic (or even tangential) velocity distribution.
+We explore the possible role of the external tidal field as
+a function of perigalactic distances (Rperi) to quantify
+how much the Galactic tidal field affects the evolution
+of GCs.
+We have divided clusters into two groups with
+Rperi > 3.5 kpc and Rperi ≤ 3.5 kpc, respectively.
+The same value of the pericentric distance was adopted
+by Zennaro et al. (2019) and Milone et al. (2020) who
+explored the possible role of the Galactic tidal field on
+the fraction of 1G stars. Those studies found that the
+main correlation is between the fraction of 1G stars and
+the mass of the cluster and that, for a given value of the
+mass, clusters with larger pericentric distances tend to
+have larger 1G fractions.
+
+red
+RGB
+10
+rad
+1
+IV
+1
+ue
+0.9
+6
++
+0.8
+10
+rad
+1.1
+6
+ge
+1
+tan
+0.9
++
+6
+VII
+0.8
+1.2
+6
+V
+1
+tan!
+0.9
+6
+0.8
+1.2
+0
+1
+b
+0.8
+0.1
+0.1
+0.1The Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters. XXIV.
+7
+Figure 4. Similarly to Fig. 3, we show the anisotropy as a function of distance from the centers of the clusters in unit of rh for
+mPOPs in clusters with different Rperi. The top and middle rows present the 1G (gold) and 2G (blue) anisotropy profiles for
+GCs with Rperi ≤ 3.5 kpc and Rperi > 3.5 kpc, respectively. The black, horizontal line is set to 1 (isotropic case). The colored
+lines are a straight-line fit to the data (see details in Fig 2). The comparison between the straight-line fit for mPOPs in GCs
+with the same Rperi is given in the rightmost panels, while between the same mPOP in GCs with different Rperi is highlighted
+in the bottom panels.
+Two-body encounters progressively erase fingerprints
+of initial kinematic differences between mPOPs. Thus,
+we focused only on dynamically-young GCs (age/th<7).
+This choice is also dictated by the properties of the
+GCs in our sample, given we have no dynamically-old
+and intermediate GCs with Rperi > 3.5 kpc. Red-RGB
+stars were excluded because there is only one type-II
+GC (NGC 6715) with age/th<7 and large perigalactic
+distance.
+We show the anisotropy profiles for 1G and 2G stars
+for different Rperi in Fig. 4. Our analysis suggests that
+the degree of kinematic anisotropy in 1G and 2G stars
+does depend on the perigalactic distance. The 1G stars
+are isotropic at all distances in our fields for Rperi ≤ 3.5
+kpc, suggesting that the tidal field may have erased any
+initial anisotropy; the 2G population for clusters with
+the same pericentric distances, on the other hand, still
+displays some velocity anisotropy, in general agreement
+with what expected in models in which the 2G was ini-
+tially more centrally concentrated than the 1G and thus
+less affected by the tidal field (Vesperini et al. 2021).
+In these clusters, we find that the average anisotropy
+for 1G and 2G groups at r > 0.6 rh are 1.01 ± 0.03 and
+0.94±0.02, respectively (∼2σ difference). In the group of
+clusters with Rperi > 3.5 kpc, both populations are still
+characterized by a radially-anisotropic velocity distribu-
+tion, with an average anisotropy at r > 0.6 rh for 1G and
+2G groups of 0.94 ± 0.07 and 0.91 ± 0.03, respectively.
+While the anisotropy of 2G stars is statistically signifi-
+cant at the 3σ level, that of 1G sources is not as strong
+because the large error bars make the kinematics of this
+group consistent with the isotropic case at the ∼1σ level.
+Similar results can be inferred by comparing the slopes
+of the corresponding least-squares straight-line fits. Our
+data show a marginal difference (<1σ) between 1G and
+2G stars at large r/rh for clusters in this group, but the
+possible larger anisotropy of the 2G population in the
+outer regions of clusters with large Rperi needs to be
+investigated further over broader radial ranges.
+Finally, Table 1 shows that the fraction of 1G stars
+in dynamically-young GCs with Rperi < 3.5 kpc is 0.26,
+while that in GCs with Rperi > 3.5 kpc is 0.33. This is
+
+2G
+1.2
+p
+rad
+1
+b
+5
+1
+3
+tan
+VII
+0.9
+peri
+b
+0.8
+R
+1.2
+kpc
+rad
+1.1
+b
+5
+7
+3
+0.9
+6
+0.8
+1.2
+rad
+1
+tan'
+0.9
+b
+0.8
+0.1
+0.18
+Libralato et al.
+qualitatively in agreement with the findings of Zennaro
+et al. (2019) and Milone et al. (2020), i.e., GCs with
+larger Rperi values tend to have larger 1G fractions.
+5. CONCLUSIONS
+This paper presents the first homogeneous kinematic
+investigation of mPOPs in 56 GCs. We have focused
+on bright RGB stars, for which the mPOP tagging is
+clearer in chromosome maps, and measured the velocity
+dispersion of 1G and 2G stars. While 1G stars are, in
+general, kinematically isotropic at both inner and outer
+radii in our fields, 2G stars are isotropic at the center
+and progressively become more radially anisotropic fur-
+ther from the center of the cluster. This anisotropy is
+a reflection of the fact that the 2G stars have the same
+radial dispersions as the 1G stars, but much lower tan-
+gential dispersions. Our study confirms previous results
+obtained for specific GCs with Gaia (Milone et al. 2018;
+Cordoni et al. 2020a,b) and HST (Anderson & van der
+Marel 2010; Richer et al. 2013; Bellini et al. 2015, 2018;
+Libralato et al. 2018, 2019, 2022; Dalessandro et al.
+2021). Our findings are also in general agreement with
+the theoretical predictions of models that follow the
+dynamical evolution of mPOPs and show that these
+properties are expected in systems in which 2G stars
+formed more centrally concentrated than 1G stars.
+Using our sample, we also find possible indications
+that the Galactic tidal fields affect the kinematic prop-
+erties of 1G and 2G stars. Specifically, we show that
+the anisotropy of 1G and 2G stars depends on the peri-
+galactic distance Rperi of the host cluster. Systems with
+large Rperi experience, on average, a weaker tidal field
+and their stars are able to preserve a (stronger) radial
+anisotropy than GCs with pericentric distances in the
+innermost regions of the Galaxy (see also Zennaro et al.
+2019; Milone et al. 2020, for the possible effect of Rperi
+on the fraction of 1G stars).
+Although these results are not conclusive, due to the
+limited sample and limited radial coverage, our analysis
+is a step forward towards a complete understanding of
+the mPOP phenomenon.
+This initial study provides
+further motivation for new and deeper surveys with
+HST, JWST and, in the future, the Nancy Grace Ro-
+man Space Telescope, which will be essential to extend
+the investigation in the almost-uncharted outskirts of
+GCs (WFIRST Astrometry Working Group et al. 2019;
+Bellini et al. 2019).
+ACKNOWLEDGMENTS
+The authors thank the anonymous referee for the
+detailed suggestions that improved the quality of
+our work.
+ML and AB acknowledges support from
+GO-13297 and GO-15857.
+EV acknowledges sup-
+port from NSF grant AST-2009193.
+APM acknowl-
+edgesfunding from the European Research Council
+(ERC) under the European Union’s Horizon 2020
+research
+innovation
+programme
+(Grant
+Agreement
+ERC-StG 2016, No 716082 ’GALFOR’, PI: Milone,
+http://progetti.dfa.unipd.it/GALFOR). AA acknowl-
+edges support from the Ministerio de Ciencia e In-
+novaci´on
+of
+Spain
+(grant
+PID2020-115981GB-I00),
+from the Instituto de Astrof´ısica de Canarias (grant
+P/309403) and from Khalifa University of Abu Dhabi
+through project CIRA-2021-65 8474000413.
+BB ac-
+knowledges partial financial support from FAPESP,
+CNPq and CAPES - financial code 001. AS is grateful
+to the Bjorn Lamborn Endowment for support while
+some of this work was being done. ML thanks LS, GL
+and LL for their support to complete this project.
+This research was pursued in collaboration with the
+HSTPROMO (High-resolution Space Telescope PROper
+MOtion) collaboration, a set of projects aimed at im-
+proving our dynamical understanding of stars, clusters
+and galaxies in the nearby Universe through measure-
+ment and interpretation of proper motions from HST,
+Gaia, and other space observatories. We thank the col-
+laboration members for the sharing of their ideas and
+software.
+Based on observations with the NASA/ESA HST,
+obtained at the Space Telescope Science Institute,
+which is operated by AURA, Inc., under NASA con-
+tract NAS 5-26555. This research made use of astropy,
+a community-developed core python package for As-
+tronomy (Astropy Collaboration et al. 2013, 2018), and
+emcee (Foreman-Mackey et al. 2013).
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+page_content=' Bloomington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+page_content=' USA  3Space Telescope Science Institute 3700 San Martin Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' MD 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' USA  4Dipartimento di Fisica e Astronomia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Universit`a di Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Vicolo dell’Osservatorio 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' I-35122,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Italy  5INAF - Osservatorio Astronomico di Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Vicolo dell’Osservatorio 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Padova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' I-35122,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Italy  6Center for Astrophysical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The William H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Miller III Department of Physics & Astronomy, Johns Hopkins University,  Baltimore, MD 21218, USA  7Instituto de Astrof´ısica de Canarias, Calle V´ıa L´actea, La Laguna, Tenerife, Canary Islands, E-38205, Spain  8Departamento de Astrof´ısica, Universidad de La Laguna, Av.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Astrof´sico Francisco S´anchez, La Laguna, Tenerife, Canary Islands,  E-38206, Spain  9Department of Aerospace Engineering, Khalifa University of Science and Technology, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Box 127788, Abu Dhabi, United Arab  Emirates  10Universidade de S˜ao Paulo, IAG, Rua do Mat˜ao 1226, Cidade Universit´asria, S˜ao Paulo 05508-900, Brazil  11INAF - Osservatorio Astronomico di Abruzzo, Via M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Maggini, s/n, Teramo, I-64100, Italy  12INFN - Sezione di Pisa, Largo Pontecorvo 3, Pisa, I-56127, Italy  13Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA  14Dipartimento di Fisica e Scienze della Terra, Universit`a di Ferrara, Via Giuseppe Saragat 1, Ferrara, I-44122, Italy  (Received November 16, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Revised December 19, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Accepted December 20, 2022)  Submitted to ApJ  ABSTRACT  Our understanding of the kinematic properties of multiple stellar populations (mPOPs) in Galactic  globular clusters (GCs) is still limited compared to what we know about their chemical and photomet-  ric characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Such limitation arises from the lack of a comprehensive observational investigation  of this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Here we present the first homogeneous kinematic analysis of mPOPs in 56 GCs based on  high-precision proper motions computed with Hubble Space Telescope data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We focused on red-giant-  branch stars, for which the mPOP tagging is clearer, and measured the velocity dispersion of stars  belonging to first (1G) and second generations (2G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We find that 1G stars are generally kinematically  isotropic even at the half-light radius, whereas 2G stars are isotropic at the center and become radially  anisotropic before the half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The radial anisotropy is induced by a lower tangential velocity  dispersion of 2G stars with respect to the 1G population, while the radial component of the motion  is comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We also show possible evidence that the kinematic properties of mPOPs are affected  by the Galactic tidal field, corroborating previous observational and theoretical results suggesting a  relation between the strength of the external tidal field and some properties of mPOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Although  limited to the GCs’ central regions, our analysis leads to new insights into the mPOP phenomenon,  and provides the motivation for future observational studies of the internal kinematics of mPOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Corresponding author: Mattia Libralato  libra@stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='edu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='04148v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='GA]  10 Jan 2023  2  Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Keywords: globular clusters: general – proper motions – stars: kinematics and dynamics  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' INTRODUCTION  The puzzle of the origin of the multiple stellar pop-  ulations (mPOPs) in Galactic globular clusters (GCs)  has been controversial since their discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The large  amount of spectroscopic and photometric data collected  so far has provided almost all observational information  we know about mPOPs in GCs, but no definitive con-  sensus has been reached yet about the formation and  evolution of mPOPs (Renzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Bastian &  Lardo 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Gratton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2012, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Cassisi & Salaris  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Milone & Marino 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The interplay between  theoretical and observational efforts has pushed the  community to find new ways to constrain the origin  of mPOPs in GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' For example, this research field is  progressively seeking answers by looking at young and  massive clusters in other galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=', Larsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Niederhofer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Lagioia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Martocchia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Nardiello  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' However, there is still  an almost uncharted wealth of information in Galac-  tic GCs that can enrich the observational picture of  mPOPs: their internal kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Here, we investigate the kinematic properties of first  (1G) and second (2G) generation stars hosted in GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  This effort focuses on red-giant branch (RGB) stars,  for which the separation between different populations  is clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We make use of the homogeneous collection  of proper motions (PMs) obtained with Hubble Space  Telescope (HST) data recently published by Libralato  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2022, hereafter L22) for 56 globular and one open  clusters, and compare the properties of the velocity  distributions of 1G and 2G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The outline of the paper is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Section 2  describes the data sets used for this study, the proce-  dure to identify the mPOPs, and how we calculated  their kinematic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Section 3 reports our results  concerning the kinematics of mPOPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' while in Section 4  we investigate the possible dependence of the velocity  anisotropy on the Galactic tidal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' DATA SETS, MULTIPLE-POPULATION  TAGGING AND KINEMATICS  We made use of the PM catalogs1 of L22, to which we  refer for a detailed description of how the PMs were com-  1 Catalogs  are  available  at  MAST  as  a  High  Level  Sci-  ence  Product  via  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='17909/jpfd-2m08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  See  also:  https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='edu/hlsp/hacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  puted2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' In brief, we designed a multi-step reduction to  exploit crowded regions, like the cores of GCs, and mea-  sured position and flux of sources in HST exposures via  effective point-spread function fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The geometric-  distortion-corrected positions of each object as a func-  tion of time were then fit with a least-squares straight  line, the slope of which is an estimate of the PM of the  star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We cross-identified stars in our astro-photometric  catalogs with those in the (pseudo) two-color diagrams  known as “chromosome maps” of Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2017)  made for all clusters analyzed in the Treasury GO-13297  program (Piotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We then applied the astro-  photometric quality selections described in both papers  to obtain samples of well-measured RGB stars for the  mPOP tagging and their kinematic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The reason of our choice to focus on the RGB stars  is twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  First, the mPOPs along the RGB can be  identified more easily because the UV data (which pro-  vide the key filters needed for disentangling the mPOPs  depending on their CNO contents) used by Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  (2017) were designed to have the highest signal-to-noise  ratio (and hence photometric quality) for the RGB  stars (see discussion in Piotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Secondly,  this choice allows us to compare the kinematics of stars  with similar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  However, focusing on RGB stars necessarily implies  small number statistics, and this poses a problem if  we choose to work with each cluster separately, com-  puting the velocity dispersions for stars in each mPOP  first, and then collecting all measurements from all  GCs to study the average kinematic trends of mPOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  For clusters with different population sizes, the veloc-  ity dispersion representing the kinematics of stars at  a given distance from the center of the cluster could  be obtained by considering stars over a different radial  interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  This could potentially introduce a bias that  can wash out some of the features we are looking for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  For this reason, we instead normalized positions and  PMs of the RGB stars in each GC catalog by the GC’s  half-light radius3 (rh) and central velocity dispersion  σµ (from L22), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The errors on the central  2 Since our focus is on GCs,  we excluded the open cluster  NGC 6791 from the investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  3 Cluster parameters (half-light radius, distance, half-mass relax-  ation time, average perigalactic distance) are taken from the  GC database of Holger Baumgardt (Sollima & Baumgardt 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Vasiliev & Baumgardt 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Baumgardt  & Vasiliev 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Ages are from Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2010), Milone  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2014) and Koch & McWilliam (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' See L22 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' XXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  3  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Examples of “chromosome” maps and mPOP tagging for a type-I (NGC 104;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' left panel) and a type-II (NGC 1851;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  right panel) GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' In each plot, gold squares and blue dots represent 1G and 2G stars on the RGB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Red crosses in  the right panel highlight the red-RGB stars in NGC 1851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Median anisotropy of 1G, 2G and red-RGB stars for stars with r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='6 rh and statistical significance of the difference  between the mPOP anisotropies for the various cases discussed in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The values between brackets in the second, third and  fourth columns are the number of stars used in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' No number is provided when no stars are available, or if the anisotropy  for the specific case was not computed (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  1G (N)  2G (N)  red-RGB (N)  1G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2G  1G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' red-RGB  2G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' red-RGB  Median anisotropy  Comparison  Entire sample  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 (5962)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01 (13884)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03 (1317)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9σ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='3σ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='3σ  age/th≥10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03 (1138)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 (2289)  /  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9σ  /  /  7≤age/th<10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='06 (885)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 (1745)  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8σ  /  /  age/th<7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03 (3939)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='92 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 (9850)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='07 (940)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='6σ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='0σ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='3σ  Rperi ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03 (2479)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 (6929)  /  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9σ  /  /  Rperi > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='07 (1460)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03 (2921)  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='4σ  /  /  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Slopes m of the least-squares straight-line fits (in linear units of r/rh) to the anisotropy profiles of 1G, 2G and  red-RGB stars, and statistical significance of the difference between the mPOP slopes for the various cases discussed in text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  1G  2G  red-RGB  1G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2G  1G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' red-RGB  2G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' red-RGB  Slope  Comparison  Entire sample  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8σ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8σ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='4σ  age/th≥10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01  /  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2σ  /  /  7≤age/th<10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='4σ  /  /  age/th<7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='10 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8σ  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2σ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='3σ  Rperi ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02  /  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8σ  /  /  Rperi > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02  /  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='7 σ  /  /  σµ were included in the normalized-PM error budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  This normalization allowed us to jointly compare pairs  2G  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='4  F275W,F336W,F438W  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2  01  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  F275W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='F814Wred-RGB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='4  F275W,F336W,F438W  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='3  2G  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+page_content='2  V  7275W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='F814W4  Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Anisotropy (left), normalized σrad (center) and σtan (right) as a function of distance from the center of the cluster in  units of rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Gold squares (top panel), blue dots (second panel from the top) and red crosses (third panel from the top) represent  the 1G, 2G and red-RGB populations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The black, dashed horizontal lines in the left panels mark the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The solid lines in each panel, color-coded as the corresponding points, are least-squares straight-line fits (in linear units of r/rh)  to the points forced to have the ordinate equal to 1 at the center (r/rh = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The light-color shaded regions correspond to the  1σ errors of the fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The comparisons between the trends in each case are shown in the bottom panels (1σ errors of the fits are  not plotted for clarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  of stellar positions and PMs from all clusters at once,  thus increasing the number of data points that can be  used to study the kinematics of each mPOP without  the drawback discussed before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2017) classifies GCs in two main fam-  ilies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Most GCs belong to the type-I family, and are  characterized by chromosome maps with two distinct  groups4 made by 1G and 2G stars (left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The remaining clusters are instead labelled as type-  II GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  These systems present: more complex chro-  mosome maps, where the 1G and 2G stars seem to  be divided in subgroups (right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' and  split sub-giant branches and RGBs (hereafter red-RGB)  clearly visible in CMDs with specific color combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Stars belonging to the red-RGB population are typ-  ically enriched in the overall CNO content, iron and  4 Note that 1G stars in type-I GCs present a color spread in  the chromosome maps likely due to a spread in Fe of ≳0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1 dex  (Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Lardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Legnardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  s-elements abundances, and have their own 1G and  2G subdivision (hereafter, 1Gr and 2Gr, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  We refer to Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2019) for a comprehensive  description of the spectro-photometric properties of  these stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The origin of these red-RGB stars is not  clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' For example, Marino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2019) suggested two  possible options: (i) after the gas from which the “clas-  sical” 1G and 2G stars formed was almost exhausted,  the clusters reaccreted pristine gas that was enriched  in iron by supernovae;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' or (ii) the type-II GCs formed  within a dwarf galaxy, with 1G/2G and 1Gr/2Gr born  at different times and/or places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The 1Gr/2Gr distinction is not always as clear as that  between the 1G/2G stars in our chromosome maps, in  particular for NGC 1851 and NGC 6715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Nevertheless,  we arbitrarily divided the red-RGB group is our type-II  clusters in 1Gr and 2Gr stars similarly to what done for  1G and 2G stars in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The majority of the red-RGB  stars belongs to the 2Gr group, and only 219 stars are  part of the 1Gr group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Because the photometric tagging  ETTTT  HT  ETT  1  1  EITTT  rad  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+page_content='8  1  tan/  二二  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+page_content='  1  E  1  1  rad  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+page_content='6  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  1  rThe Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' XXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  5  of 1Gr and 2Gr stars is not straightforward in our sam-  ple of type-II GCs, and given the very few 1Gr stars,  we choose to analyzed the red-RGB stars as a whole5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Following this classification, we divided our samples  in either two (1G and 2G for type-I GCs) or three (1G,  2G and red-RGB for type-II GCs) sub-populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The  mPOP tagging was directly performed on the chromo-  some map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' GLOBAL KINEMATIC PROPERTIES OF  MULTIPLE POPULATIONS  As shown in a number of theoretical studies, the  velocity anisotropy may provide various fundamental  insights into the formation and evolution of GCs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=',  Tiongco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Breen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2017, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Pavl´ık &  Vesperini 2021, 2022) and their mPOPs (Tiongco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  We computed the normalized radial and tangential  velocity dispersions (σrad and σtan, respectively) in  equally-populated radial bins of at least 200 stars each6  as in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 4 of L22, using a maximum-likelihood ap-  proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2 present the anisotropy  as a function of radial distance from the center of the  cluster in units of rh for 1G, 2G and red-RGB stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The solid lines in each panel, color-coded as the corre-  sponding points, are least-squares straight-line fits to  the points forced to have the ordinate equal to 1 at the  center (r/rh = 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=', (σtan/σrad)(r) = 1 + m × r/rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  These fits are linear in r/rh, thus they appear curved in  our plots with a logarithmic scale on the x axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The 1G  stars (gold points) in our fields are isotropic even outside  1 rh, with only a marginal (∼1σ) signature of a radial  anisotropy in the outermost part of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The 2G  (blue) and red-RGB (red) populations are isotropic in  the center and become progressively radially anisotropic  further from the GC’s center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Table 1 collects the me-  dian values of the anisotropy for each population for r >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='6 rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' This threshold was chosen as a compromise be-  tween having enough points to compute a robust median  anisotropy value for each mPOP and being sufficiently  far from the center of the cluster to capture indications  of anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Table 2 collects the slopes of the straight-  line fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  There is a clear difference in the median  anisotropy between 1G and 2G stars at the ∼5σ level,  and between 1G and red-RGB stars at the ∼3σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  5 The kinematic analysis shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2 provides consistent results  for 1Gr, 2Gr and 1Gr+2Gr stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  6 For the analysis of the kinematics of red-RGB stars in GCs with  different dynamical age, we made at least one radial bin with all  stars at disposal when not enough stars were available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The slopes of the straight-line fits provide similar results,  although the statistical significance is smaller (∼2σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  These findings are in general agreement with the  predictions of numerical models of the evolution of  multiple-population clusters (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=', Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Specifically, theoretical models explain these  behaviors as the consequence of the initial spatial dif-  ferences between 1G and 2G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Simulations usually  start with a 2G population more centrally concentrated  in the inner regions of a more diffuse 1G system as  suggested by models of formation of mPOPs (D’Ercole  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Calura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' For systems starting  with an isotropic velocity distribution, the anisotropy  of the 2G stars is a consequence of the outward diffu-  sion of 2G stars (Tiongco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' For systems starting with an anisotropic velocity  distribution, this difference is the result of a more rapid  evolution towards a isotropic velocity distribution of  1G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' In such case, it is also possible to find both  populations to be characterized by anisotropic velocity  distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We point out that although the difference  between the anisotropy of 1G and 2G stars is small, its  extent is generally consistent with that found in numer-  ical models at the distances from the clusters’ centers  probed by our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The middle and right panels show the radial profiles of  the normalized σrad and σtan, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' All the pop-  ulations have similar radial velocity dispersions, while  tangential velocity dispersions are larger for 1G stars  than for 2G and red-RGB sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' These findings are  in agreement with the theoretical predictions for which  the different degrees of radial anisotropy between 1G  and 2G stars is caused by a difference in the tangential  component of their motions, rather than in the radial  component (Bellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 3, we further explore the anisotropy of mPOPs  for GCs with different dynamical ages as measured by  the ratio of the GCs’ ages to their half-mass relaxation  time (th).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' L22 found that dynamically-old (age/th>10)  and young (age/th<7) GCs are characterized by dif-  ferent velocity distributions at rh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  We followed this  same classification to better highlight differences and  analogies between 1G and 2G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Figure 3 presents  the anisotropy as a function of distance from the center  of the cluster in units of rh for dynamically old (top),  intermediate (second from the top) and young (third  from the top) GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The bottom panels show the com-  parison between the trends of each mPOP in GCs with  different dynamical ages, while the rightmost panels  collect the straight-line fits for each population for GCs  with the same the dynamical age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The trends shown  in these panels are consistent with the global trends  6  Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' As in the left panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2, but dividing the sample of GCs according to their dynamical age (age/th ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The  first three columns show the anisotropy for 1G (gold squares), 2G (blue dots) and red-RGB (red crosses), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The  black, horizontal lines mark the isotropic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The lines, colored as the point in the same plot, are a straight-line fit to the  data (see details in Fig 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' no line was fit for the red-RGB samples with only one data point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The first three rows present the  result for old, intermediate and young GCs, from top to bottom, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The rightmost panels collect the straight-line fits  for each population for GCs with the same the dynamical age, while the panels at the bottom show the comparison between  the trends of each mPOP in clusters with different dynamical ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2, but the differences between the mPOP  anisotropies are less evident (see Tables 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Most  of the 1G fits are consistent with an isotropic distri-  bution, while most of the 2G median anisotropies and  slopes indicate a statistically-significant anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Our  analysis suggests that, for a given mPOP, there might  be kinematic differences depending on the dynamical  age of the hosting cluster, but the large error bars do  not allow us to draw any definitive conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Larger  differences might be present further from the center of  the cluster, where the relaxation time is longer and fin-  gerprints of the initial kinematic properties might still  be detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Finally, no conclusion can be inferred for  the red-RGB stars in intermediate and old GCs because  of the low statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' DEPENDENCE ON THE GALACTIC TIDAL  FIELD  Previous studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=', Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Tiongco  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Bianchini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2018) have shown that  the external tidal field of the host galaxy may play an  important role in the early- and long-term evolution of  the velocity anisotropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' In particular, the loss of stars in  stronger tidal fields preferentially affects stars on more  radial orbits, causing a decrease of the radial anisotropy  in the outer regions, and a gradual evolution towards a  more isotropic (or even tangential) velocity distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  We explore the possible role of the external tidal field as  a function of perigalactic distances (Rperi) to quantify  how much the Galactic tidal field affects the evolution  of GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  We have divided clusters into two groups with  Rperi > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc and Rperi ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  The same value of the pericentric distance was adopted  by Zennaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2019) and Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2020) who  explored the possible role of the Galactic tidal field on  the fraction of 1G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Those studies found that the  main correlation is between the fraction of 1G stars and  the mass of the cluster and that, for a given value of the  mass, clusters with larger pericentric distances tend to  have larger 1G fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  red  RGB  10  rad  1  IV  1  ue  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9  6  +  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8  10  rad  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  6  ge  1  tan  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9  +  6  VII  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2  6  V  1  tan!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2  0  1  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1The Hubble Space Telescope UV Legacy Survey of Galactic Globular Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' XXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  7  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Similarly to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 3, we show the anisotropy as a function of distance from the centers of the clusters in unit of rh for  mPOPs in clusters with different Rperi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The top and middle rows present the 1G (gold) and 2G (blue) anisotropy profiles for  GCs with Rperi ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc and Rperi > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The black, horizontal line is set to 1 (isotropic case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The colored  lines are a straight-line fit to the data (see details in Fig 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The comparison between the straight-line fit for mPOPs in GCs  with the same Rperi is given in the rightmost panels, while between the same mPOP in GCs with different Rperi is highlighted  in the bottom panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Two-body encounters progressively erase fingerprints  of initial kinematic differences between mPOPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Thus,  we focused only on dynamically-young GCs (age/th<7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  This choice is also dictated by the properties of the  GCs in our sample, given we have no dynamically-old  and intermediate GCs with Rperi > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Red-RGB  stars were excluded because there is only one type-II  GC (NGC 6715) with age/th<7 and large perigalactic  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  We show the anisotropy profiles for 1G and 2G stars  for different Rperi in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Our analysis suggests that  the degree of kinematic anisotropy in 1G and 2G stars  does depend on the perigalactic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' The 1G stars  are isotropic at all distances in our fields for Rperi ≤ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5  kpc, suggesting that the tidal field may have erased any  initial anisotropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' the 2G population for clusters with  the same pericentric distances, on the other hand, still  displays some velocity anisotropy, in general agreement  with what expected in models in which the 2G was ini-  tially more centrally concentrated than the 1G and thus  less affected by the tidal field (Vesperini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  In these clusters, we find that the average anisotropy  for 1G and 2G groups at r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='6 rh are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='94±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='02, respectively (∼2σ difference).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' In the group of  clusters with Rperi > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc, both populations are still  characterized by a radially-anisotropic velocity distribu-  tion, with an average anisotropy at r > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='6 rh for 1G and  2G groups of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='94 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='07 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='03, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  While the anisotropy of 2G stars is statistically signifi-  cant at the 3σ level, that of 1G sources is not as strong  because the large error bars make the kinematics of this  group consistent with the isotropic case at the ∼1σ level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Similar results can be inferred by comparing the slopes  of the corresponding least-squares straight-line fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Our  data show a marginal difference (<1σ) between 1G and  2G stars at large r/rh for clusters in this group, but the  possible larger anisotropy of the 2G population in the  outer regions of clusters with large Rperi needs to be  investigated further over broader radial ranges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Finally, Table 1 shows that the fraction of 1G stars  in dynamically-young GCs with Rperi < 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='26,  while that in GCs with Rperi > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='5 kpc is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' This is  2G  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2  p  rad  1  b  5  1  3  tan  VII  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9  peri  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8  R  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='2  kpc  rad  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  b  5  7  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9  6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content="2  rad  1  tan'  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='9  b  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='18  Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  qualitatively in agreement with the findings of Zennaro  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2019) and Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' (2020), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=', GCs with  larger Rperi values tend to have larger 1G fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' CONCLUSIONS  This paper presents the first homogeneous kinematic  investigation of mPOPs in 56 GCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We have focused  on bright RGB stars, for which the mPOP tagging is  clearer in chromosome maps, and measured the velocity  dispersion of 1G and 2G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' While 1G stars are, in  general, kinematically isotropic at both inner and outer  radii in our fields, 2G stars are isotropic at the center  and progressively become more radially anisotropic fur-  ther from the center of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' This anisotropy is  a reflection of the fact that the 2G stars have the same  radial dispersions as the 1G stars, but much lower tan-  gential dispersions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Our study confirms previous results  obtained for specific GCs with Gaia (Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Cordoni et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2020a,b) and HST (Anderson & van der  Marel 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Richer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Bellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2015, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Libralato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2018, 2019, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Dalessandro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Our findings are also in general agreement with  the theoretical predictions of models that follow the  dynamical evolution of mPOPs and show that these  properties are expected in systems in which 2G stars  formed more centrally concentrated than 1G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Using our sample, we also find possible indications  that the Galactic tidal fields affect the kinematic prop-  erties of 1G and 2G stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Specifically, we show that  the anisotropy of 1G and 2G stars depends on the peri-  galactic distance Rperi of the host cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Systems with  large Rperi experience, on average, a weaker tidal field  and their stars are able to preserve a (stronger) radial  anisotropy than GCs with pericentric distances in the  innermost regions of the Galaxy (see also Zennaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2020, for the possible effect of Rperi  on the fraction of 1G stars).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Although these results are not conclusive, due to the  limited sample and limited radial coverage, our analysis  is a step forward towards a complete understanding of  the mPOP phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  This initial study provides  further motivation for new and deeper surveys with  HST, JWST and, in the future, the Nancy Grace Ro-  man Space Telescope, which will be essential to extend  the investigation in the almost-uncharted outskirts of  GCs (WFIRST Astrometry Working Group et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Bellini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The authors thank the anonymous referee for the  detailed suggestions that improved the quality of  our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  ML and AB acknowledges support from  GO-13297 and GO-15857.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  EV acknowledges sup-  port from NSF grant AST-2009193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  APM acknowl-  edgesfunding from the European Research Council  (ERC) under the European Union’s Horizon 2020  research  innovation  programme  (Grant  Agreement  ERC-StG 2016, No 716082 ’GALFOR’, PI: Milone,  http://progetti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='dfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='unipd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='it/GALFOR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' AA acknowl-  edges support from the Ministerio de Ciencia e In-  novaci´on  of  Spain  (grant  PID2020-115981GB-I00),  from the Instituto de Astrof´ısica de Canarias (grant  P/309403) and from Khalifa University of Abu Dhabi  through project CIRA-2021-65 8474000413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  BB ac-  knowledges partial financial support from FAPESP,  CNPq and CAPES - financial code 001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' AS is grateful  to the Bjorn Lamborn Endowment for support while  some of this work was being done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' ML thanks LS, GL  and LL for their support to complete this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  This research was pursued in collaboration with the  HSTPROMO (High-resolution Space Telescope PROper  MOtion) collaboration, a set of projects aimed at im-  proving our dynamical understanding of stars, clusters  and galaxies in the nearby Universe through measure-  ment and interpretation of proper motions from HST,  Gaia, and other space observatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content=' We thank the col-  laboration members for the sharing of their ideas and  software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
+page_content='  Based on observations with the NASA/ESA HST,  obtained at the Space Telescope Science Institute,  which is operated by AURA, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/rtE2T4oBgHgl3EQf1Ajv/content/2301.04148v1.pdf'}
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+RecRecNet: Rectangling Rectified Wide-Angle Images by Thin-Plate Spline
+Model and DoF-based Curriculum Learning
+Kang Liao Lang Nie Chunyu Lin* Zishuo Zheng Yao Zhao
+Institute of Information Science, Beijing Jiaotong University
+Beijing Key Laboratory of Advanced Information Science and Network
+{kang liao, nielang, cylin, yzhao}@bjtu.edu.cn
+Abstract
+The wide-angle lens shows appealing applications in VR
+technologies, but it introduces severe radial distortion into
+its captured image. To recover the realistic scene, previ-
+ous works devote to rectifying the content of the wide-angle
+image. However, such a rectification solution inevitably dis-
+torts the image boundary, which potentially changes related
+geometric distributions and misleads the current vision per-
+ception models. In this work, we explore constructing a
+win-win representation on both content and boundary by
+contributing a new learning model, i.e., Rectangling Recti-
+fication Network (RecRecNet). In particular, we propose a
+thin-plate spline (TPS) module to formulate the non-linear
+and non-rigid transformation for rectangling images. By
+learning the control points on the rectified image, our model
+can flexibly warp the source structure to the target domain
+and achieves an end-to-end unsupervised deformation. To
+relieve the complexity of structure approximation, we then
+inspire our RecRecNet to learn the gradual deformation
+rules with a DoF (Degree of Freedom)-based curriculum
+learning. By increasing the DoF in each curriculum stage,
+namely, from similarity transformation (4-DoF) to homog-
+raphy transformation (8-DoF), the network is capable of
+investigating more detailed deformations, offering fast con-
+vergence on the final rectangling task. Experiments show
+the superiority of our solution over the compared methods
+on both quantitative and qualitative evaluations. The code
+and dataset will be made available.
+1. Introduction
+The wide-angle and fisheye lenses have been widely ap-
+plied in various applications such as computational imag-
+ing, virtual reality (VR), and autonomous driving, due to
+their larger field of view (FoV) than conventional lenses. By
+employing a specific mapping, the wide-angle lens can pro-
+*Corresponding author. The first two authors contribute equally.
+RecRecNet
+Rectified Image
+Rectangling Image
+Vision Perception 
+Model
+Detection and Segmentation 
+Results
+TPS Transformation
+Inference
+Figure 1. Existing rectification methods devote to correcting the
+content of the wide-angle image, but they inevitably distort the
+image boundary. We demonstrate that such a deformed boundary
+can lead to dramatic performance degradation of the popular vi-
+sion perception model [16]. By presenting the Rectangling Recti-
+fication Network (RecRecNet), we reach a win-win representation
+on both the undistorted content and regular boundary with a flexi-
+ble TPS transformation. The arrow marks our perception gain.
+duce convex and non-rectilinear images rather than images
+with straight lines of perspective. However, this mapping
+mechanism violates the pinhole camera assumption, and the
+captured image suffers from severe radial distortions.
+To eliminate the distortion induced by the wide-angle
+lens, the distortion rectification algorithm is presented. For
+example, the traditional approaches calibrate the wide-
+angle lens using the calibration target (e.g., a checker-
+board) [18, 54], specific scene assumption [3, 10, 32], and
+camera motions [12, 31, 35]. In recent years, deep learn-
+ing has brought new inspirations to this field. For example,
+the learning-based approaches can automatically rectify the
+radial distortion in wide-angle images based on learned se-
+mantic features. To pursue better performance, the subse-
+quent works improve the learning framework through di-
+verse features [49, 51], learning paradigm [28], calibration
+representation [24,30], and distortion prior [8,27], etc.
+1
+arXiv:2301.01661v1  [cs.CV]  4 Jan 2023
+
+We notice that the above distortion rectification ap-
+proaches mainly focus on rectifying the image content. By
+shrinking the distorted image towards the principal center,
+the spatial distribution of all pixels has been rearranged
+and thus generates the rectified result. However, the image
+boundaries are inevitably distorted by this correction way,
+making the image exhibited with a visually narrow FoV
+and irregular shape. Moreover, we found such a deformed
+boundary can significantly influence the current mainstream
+vision perception models because most of them are trained
+on the dataset with rectangular images. As shown in Fig-
+ure 1, Mask R-CNN [16] is confused by the rectified im-
+age, leading to a missing segmentation and detection result
+near the boundary. While some works [26] fill the blank
+region beyond deformed boundaries using outpainting strat-
+egy [37,45], the generated contents are usually fictitious and
+twist the original semantic characterization.
+In this work, we explore a win-win rectification rep-
+resentation for the wide-angle image on both content and
+boundary without introducing any extra semantic informa-
+tion. For this new representation, the distorted content has
+been corrected while the image boundary keeps straight, as
+shown in Figure 1. To this end, we propose a Rectangling
+Rectification Network (named RecRecNet). Concretely, a
+thin-plate spline (TPS) motion module is proposed to for-
+mulate the non-linear and non-rigid transformation for rect-
+angling the rectified wide-angle image. It can flexibly warp
+the source structure to the target domain by learning the
+control points on the rectified image.
+Subsequently, we design a DoF-based curriculum learn-
+ing to further inspire our RecRecNet to grasp the progres-
+sive deformation rules, relieving the burden of complex
+structure approximation and initializing a better start point
+for training.
+The general paradigm of curriculum learn-
+ing [4] introduces a heuristic and purposeful strategy for
+training a learning model. The model can converge faster
+by gradually learning various samples or knowledge based
+on different stages in a curriculum, mimicking how humans
+and animals learn. In particular, we increase the degree of
+freedom (DoF) of the devised stages/transformations from
+similarity transformation (4-DoF) to homography transfor-
+mation (8-DoF). Moreover, the procedure of our curriculum
+reveals a simple-to-complex order from the straight line,
+sloped line to curve for the image boundary. As a conse-
+quence, our RecRecNet can discover more geometric de-
+tails and converge faster on the final rectangling stage.
+While aiming for a win-win rectification representation,
+we provide a deep analysis of why the deformed bound-
+ary makes the perception deformed, i.e., why the deformed
+image boundary can significantly influence the visual per-
+ception models. We surprisingly find the deformed bound-
+ary can introduce new features on the original feature maps,
+which further forms new semantics or cause blind spots for
+non-salient features. And we demonstrate that this effect
+commonly exists in other research regions.
+We conclude our major contributions as follows:
+• We propose a Rectangling Rectification Network
+(RecRecNet) for a win-win representation of the recti-
+fied wide-angle image. To our knowledge, it has never
+been investigated in previous literature.
+• A thin-plate spline (TPS) motion module is proposed
+to flexibly formulate the non-linear and non-rigid rect-
+angling transformation.
+We also provide a detailed
+analysis of why the deformed image boundary can sig-
+nificantly influence the visual perception models.
+• To grasp the progressive deformation rules and re-
+lieve the burden of complex structure approximation, a
+DoF-based curriculum learning is designed to inspire
+our RecRecNet.
+2. Related Work
+2.1. Distortion Rectification
+The distortion rectification approaches for wide-angle
+lenses can be classified into traditional and learning-based
+approaches. The majority of traditional techniques concen-
+trate on detecting the hand-crafted features [1,3,10,22,32,
+39]. However, these techniques frequently fail due to par-
+ticular constraints like the plumb line and curve. Recently,
+blind distortion rectification has been studied using deep
+learning. It can rectify the radial distortion of the wide-
+angle image without any scene assumptions. For example,
+the regression-based solutions [5, 27, 30, 36, 47, 51] predict
+the distortion parameters of input, and then the estimated
+parameters are used to rectify the distortion offline or on-
+line. The reconstruction-based solutions [8, 24, 28, 50, 55]
+directly learns the pixel-level mapping or displacement field
+from the distortion image into the rectification output.
+Despite promising flexibility and efficiency, these recti-
+fication approaches inevitably distort the image boundary
+during content correction. As a result, the distorted distri-
+bution at the boundary confuses the vision perception net-
+work and leads to inferior performance. One direct way is to
+crop the rectified image into a rectangular shape, however,
+it discards informative content and disobeys the original in-
+tention of the wide-angle lens.
+2.2. Curriculum Learning
+Curriculum learning techniques have been effectively
+exploited across the board in machine learning.
+El-
+man et al. . [11] provide a conceptual framework for cur-
+riculum learning that emphasizes the value of beginning at
+a low level and then works way up to more challenging
+2
+
+Backbone
+TPS Transformation 
+Module
+Rectified Image
+Wide-Angle Image
+R
+Mesh Grid
+RecRecNet
+DoF-based Curriculum Learning 
+Stage1
+Stage2
+R
+Rectification
+4-DoF
+Transformation
+8-DoF 
+Transformation
+Control Points
+G
+Rectangling Image
+Ground Truth
+G
+Gridding
+W
+Warping
+et1
+et2
+LIG
+Inter-Grid 
+Contrainst
+LAP
+LPE
+Optimization
+Loss
+Figure 2. Overview of the proposed framework. Our Rectangling Rectification Network (RecRecNet) aims to construct a win-win rep-
+resentation on both image content and image boundary. RecRecNet learns a flexible local transformation based on the thin-plate spline
+(TPS) motion model, which predicts the control points on the image boundary and warps the target image by using a mesh grid. It is jointly
+optimized with the appearance constraint LAP , perceptual constraint LP E, and inter-grid constraint LIG. To further relieve the challenges
+of structure approximation, we inspire our RecRecNet to learn the gradual deformation rules with a DoF-based curriculum learning.
+cases. Bengio et al. [4] are the first to formalize the cur-
+riculum learning as a strategy to gradually raise the level
+of complexity of the data samples utilized in the train-
+ing process.
+Subsequently, most researchers follow this
+paradigm and apply the curriculum learning into numerous
+regions [34,40,46,52]. Particularly in computer vision, cur-
+riculum learning has been investigated to inspire the learn-
+ing networks on image classification [7, 14, 33], object de-
+tection [23,43,48], semantic segmentation [13,38,53], and
+image generation [20,25,42], etc.
+In this work, we incorporate curriculum learning into a
+new task.
+By disassembling the rectangling rectification
+into three levels based on the DoF of transformation, the
+proposed curriculum can inspire our network to learn the
+challenging deformation in a simple-to-complex order and
+thus facilitates a faster convergence for training.
+3. Methodology
+3.1. Problem Formulation
+Given a rectified wide-angle image IRec ∈ Rh×w×3 with
+deformed boundaries, our Rectangling Rectification Net-
+work (RecRecNet) aims to construct a win-win representa-
+tion on both image content and image boundary, generating
+a rectangular rectified image IRec2 ∈ Rh×w×3. Figure 2
+shows the overview of the proposed framework. To be more
+specific, the rectangling module takes IRec as the input and
+learns a flexible local transformation T based on the thin-
+plate spline (TPS) motion model. It is jointly optimized
+with the appearance constraint, perceptual constraint, and
+inter-grid constraint. To relieve the challenges of structure
+approximation, we inspire the network to learn the gradual
+deformation rules with a DoF-based curriculum learning.
+The general polynomial camera model [19] is widely
+used in the approximation for the radial distortion of the
+wide-angle image. We formulate the relationship between
+the lens’s projection manner and its distortion parameters
+K = [k1, k2, k3, · · · ] as follows:
+r(θ) =
+N
+�
+i=1
+kiθ2i−1, N = 1, 2, 3, · · · ,
+(1)
+where r is the distance between the principal point and the
+pixels in the image, and θ indicates the angle between the
+optical axis and incident ray of the wide-angle lens. Such a
+projection manner can produce a convex and non-rectilinear
+image, and thus the captured image suffers from severe ge-
+ometric distortions. Suppose that the coordinates of a pixel
+in the wide-angle image and rectified images are p = (x, y)
+and prec = (xrec, yrec), the rectification solution can be
+3
+
+formulated based on Eq. 1 as follows:
+pd = R(prec, K) =
+�
+u0
+v0
+�
++ r(θ)prec
+∥prec∥2
+,
+(2)
+where (u0, v0) represents the principal point.
+With the
+above equation, the wide-angle image can be rectified by
+using bilinear interpolation. By shrinking the wide-angle
+image towards the principal center, the spatial distribution
+of all pixels is rearranged, and thus rectified content is ob-
+tained. However, the image boundaries are inevitably dis-
+torted by this rectification way.
+3.2. Rectangling with TPS Transformation
+As shown in Eq. 2, the rectification of the wide-angle im-
+age is a non-linear and non-rigid mapping function, which
+cannot be represented by simple transformations.
+Be-
+sides, it is challenging to build an accurate parameterization
+model to supervise the rectangling process with estimated
+parameters. From the perspective of energy minimization,
+we can approximate the rectangling rectification by mini-
+mizing the transformation energy with a uniform model:
+ε = εT + λεd,
+T : (xrec, yrec) �→ (xrec2, yrec2),
+(3)
+where ε denotes the total energy of the expected transforma-
+tion. (xrec, yrec) ∈ ΩS and (xrec2, yrec2) ∈ ΩT represent
+the point in the domain of source S and target T, respec-
+tively. εT indicate the data penalty energy and εd indicate
+the distortion energy. The above formula with the lowest
+overall energy is the desired transformation, in which a hy-
+perparameter λ is designed to balance the energies between
+the data penalty and distortion.
+In particular, we propose to exploit TPS transforma-
+tion [6] T ′ to minimize the aforementioned energy func-
+tion, which enables an end-to-end unsupervised deforma-
+tion. TPS transformation provides for the representation of
+more complex motions since it is flexible and non-linear.
+It allows warping from one image to the other with mini-
+mum distortion, given two sets of control points in the cor-
+responding images. Suppose q = [q1, q2, · · · , qN] and q′ =
+[q′
+1, q′
+2, · · · , q′
+N] are source points and the target points, the
+data term εT ′ can be calculated by the difference of control
+points displacement �N
+i=1∥T ′(qi) − q′
+i∥2
+2. Having aligned
+the control points, TPS finds an optimal interpolation trans-
+formation with minimal distortion term εd by:
+min
+��
+R2
+��∂2T ′
+∂x2
+�2
++ 2
+� ∂2T ′
+∂x∂y
+�2
++
+�∂2T ′
+∂y2
+�2�
+dxdy.
+(4)
+Rectified Image
+Rectangling
+Rectangling Flow
+Rectification Flow
+Figure 3. Visualization of the rectangling flow and rectification
+flow for the rectified wide-angle image. We mark the different
+radial centers in different rectangling flows with arrows.
+Then we reach a spatial deformation function parameterized
+by control points as follows:
+T ′(q) = A
+�q
+1
+�
++
+N
+�
+i=1
+wiU (∥q′
+i − q∥2) ,
+(5)
+where q is a point located at the rectified wide-angle image.
+A ∈ R2×3 and wi ∈ R2×1 are the transformation param-
+eters, which can be derived by minimizing Eq. 3. Besides,
+U(r) is a radial basis function that denotes the influence of
+the control point on q: U(r) = r2 log r2.
+In RecRecNet, we define N = (U + 1) × (V + 1) con-
+trol points that can be connected to form a mesh, and the
+network learns to predict the mesh of rectified wide-angle
+image. Then TPS transforms the predicted mesh to the reg-
+ular mesh on the target rectangular image, where the control
+points are set to be evenly distributed on the image.
+For the network, we first choose ResNet50 [17] as the
+backbone of our RecRecNet to extract the high-level seman-
+tic features. Suppose the size of the input rectified image is
+W × H, and the size of the output feature map of the back-
+bone is W
+16 × H
+16. Then these feature maps are fed to a motion
+header to predict (U + 1) × (V + 1) control points on the
+rectified image. 4 convolutional layers with a kernel size
+of 3 and BacthNormalization are stacked to aggregate the
+high-level features, of which all channel dimensions are set
+to 512. Subsequently, 3 fully connected layers with the fol-
+lowing numbers of units 4096, 2048, (U +1)×(V +1)×2
+are used to predict the coordinates of all control points.
+3.3. DoF-based Curriculum Learning
+As mentioned above, we propose to exploit TPS trans-
+formation to flexibly approximate the motion from the rec-
+tified image to rectangling result by a set of control points.
+To further relieve the challenges of this transformation, we
+inspire our RecRecNet with a curriculum learning strategy.
+4
+
+3.3.1
+Image Transformation Formulation
+Image transformation contains different elements such as
+translations, scales, rotations, shearing, etc. Any composi-
+tion of these elements can form a new transformation with
+the degree of freedom (DoF). In general, the degree of free-
+dom is a collection of independent displacements that com-
+pletely describes a system’s displaced or deformed position.
+There are some classical transformations on 2D images:
+translation transformation (2-DoF), Euclidean/Rigid trans-
+formation (translation + rotation, 3-DoF), similarity trans-
+formation (translation + rotation + scale, 4-DoF), affine
+transformation (translation + rotation + scale + shear, 6-
+DoF), and projective transformation (two each for transla-
+tion, rotation, scale, and lines at infinity, 8-DoF).
+The image transformation between the rectified wide-
+angle image and its rectangling image has never been stud-
+ied and formulated. Compared to distortion rectification,
+the rectangling transformation seems to be an inverse asym-
+metric warping due to the more important image bound-
+ary. To understand this transformation intuitively, we vi-
+sualize the pixel displacement field between the rectified
+wide-angle image and its rectangling image. Specifically,
+we leverage the state-of-the-art optical flow estimation net-
+work RAFT [44] to exhibit the motion difference and name
+it as rectangling flow (rectified image �→ rectangling im-
+age). As shown in Figure 3, compared with the rectification
+flow (wide-angle image �→ rectified image), we have the
+following observations: (1) Both rectangling flow and rec-
+tification flow show a radial spatial distribution, in which
+four dominated directions spread towards or backwards four
+image corners. (2) Rectangling flow has an unfixed radial
+center for each sample, which is determined by the content
+and layout of the image. Rectification flow has a relatively
+fixed radial center, which is related to the geometry prior to
+the radial distortion in the wide-angle image.
+Thus, we can conclude the image content highly iden-
+tifies the distribution of rectangling flow. It is painful to
+represent this transformation using classical image transfor-
+mations and their compositions. Nevertheless, such a non-
+linear and non-rigid transformation can be locally disentan-
+gled into different transformation elements such as transla-
+tion, scale, and shearing. It would be meaningful to inspire
+the network to learn some basic transformations and gradu-
+ally transfer them into more challenging cases.
+3.3.2
+Curriculum Selection
+To choose effective curriculums for the rectangling task, we
+should obey two principles: First, all stages in a curriculum
+have closely associated knowledge of the target. Second,
+the stages in a curriculum intrinsically offer a simple-to-
+complex order. To this end, we design a DoF-based curricu-
+lum to inspire RecRecNet, of which we increase the DoF of
+the devised transformations from similarity transformation
+(4-DoF) to homography transformation (8-DoF) and even-
+tually to the rectangling transformation.
+We argue that the proposed curriculum has the following
+strengths: (1) As mentioned in Section 3.3.1, the DoF di-
+rectly expresses the difficulty of the image transformation.
+As the DoF increases, the transformation shows more com-
+plex global mappings and diverse motions. Thus, RecRec-
+Net can learn the gradual deformation rules. (2) For the
+local geometry, i.e., the shape of the image boundary, our
+curriculum shows a progressive shift from a straight line,
+sloped line, to a curve. Such a design can improve the lo-
+calization ability of control points in TPS transformation
+and enable a boundary-aware ability for RecRecNet. More
+details and experiments will be demonstrated in Section 4.4.
+3.4. Training Losses
+Appearance Loss: We constrain the rectangling image
+IRec2 to be close to the ground truth IGT at the pixel level.
+The appearance loss LAP can be calculated by the differ-
+ence between IRec2 and IGT with L1 norm.
+Perceptual Loss: We then minimize the L2 distance be-
+tween IRec2 and IGT in high-level semantic perception to
+guarantee the rectangling results perceptually natural:
+LP E =
+1
+Wi,jHi,j
+Wi,j
+�
+x=1
+Hi,j
+�
+y=1
+||φi,j(Ix,y
+Rec2) − φi,j(Ix,y
+GT )||2,
+(6)
+where the difference of rectangling rectified image and
+ground truth is minimized on the feature map φi,j, which is
+derived from the j-th convolution (after activation) before
+the i-th max-pooling layer in the VGG19 network [41].
+Inter-grid Mesh Loss: To avoid the content distortion after
+our rectangling, the predicted mesh should not be largely
+deformed. Therefore, we design an inter-grid mesh term
+LIG to constrain the shape of the predicted mesh, which
+encourages the neighboring grids to transform consistently.
+The edges of two successive deformed grid {⃗et1,⃗et2} are
+supervised to be co-linear as follows:
+LIG = 1
+M
+�
+{⃗et1,⃗et2}∈mp∪mf
+(1 −
+⟨⃗et1,⃗et2⟩
+∥ ⃗et1 ∥ · ∥ ⃗et2 ∥),
+(7)
+where M is the number of tuples of two successive edges
+in a mesh. ms and mt are the predicted source mesh and
+target mesh, respectively.
+The overall training loss can be obtained by:
+L = λAP LAP + λP ELP E + λIGLIG,
+(8)
+where λAP , λP E, and λIG are the weights to balance the
+appearance loss, perceptual loss, and inter-grid mesh loss,
+which are empirically set to 1, 1e−4, and 1.
+5
+
+Rectified Image
+Cropped Image
+RecRecNet
+He et al.
+Rectified Image
+Cropped Image
+RecRecNet
+He et al.
+Normal Scene
+Simple Scene
+Wide-angle Image
+Wide-angle Image
+Figure 4. Comparison to the image rectangling work He et al. [15]. For a comprehensive evaluation, we split the test dataset into the
+normal scene (left) and the simple scene (right). Our RecRecNet can achieve the best performance on the undistorted content and regular
+boundary, regardless of the scene change.
+Rectified Image
+Cropped Image
+RecRecNet
+Label
+ROP
+Figure 5.
+Comparison to the rectification outpainting work
+ROP [26].
+Our RecRecNet can straighten the irregular image
+boundary in the rectified image without introducing new content.
+Rectified Image
+RecRecNet
+Figure 6.
+RecRecNet helps the downstream vision perception
+tasks. The arrows mark the failure results in rectified images.
+4. Experiments
+4.1. Implementation Details
+Dataset Establishment: Following existing methods [28,
+36,50,51], we first synthesize the wide-angle images by us-
+ing a 4th order polynomial model based on Eq. 1. Then we
+perform the distortion rectification on the wide-angle image
+in terms of Eq. 2. Due to non-linear and non-rigid character-
+istics, forming an accurate transformation model for rectan-
+gling the rectified image is difficult. We notice that there is
+a classical panoramic image rectangling work He et al. [15]
+on computer graphics. It makes the stitched image regular
+by optimizing an energy function with line-preserving mesh
+deformation. Thus, we perform the same energy function
+on the rectified image to fit our task. However, the capabil-
+ity to preserve linear structures [15] is limited by line de-
+tection. Consequently, some rectangling images have non-
+negligible distortions. To overcome this issue, we carefully
+filter all results and repeat the selection process three times,
+resulting in 5,160 training data from 30,000 source images
+and 500 test data from 2,000 source images. Each manual
+operation takes around 10s. More details about the dataset
+are reported in the supplementary material.
+Experimental Configuration: We train RecRecNet with
+an exponentially decaying learning rate with an initial value
+of 10−4 using Adam [21].
+The batch size is set to 10
+and total epoch is set to 260. Particularly, RecRecNet is
+trained with 3 stages in terms of the proposed DoF-based
+curriculum. The training epochs of the 4-DoF curriculum,
+8-DoF curriculum, and final rectangling curriculum are di-
+6
+
+WOMDWOMDCEbGL26UT0066L2001T0001260Caioas612000.8b6L200TO0160U12L9CK6
+D6L20UT00Content Fidelity
+0
+50
+100
+150
+200
+187
+41
+139
+121
+Structure Preserving
+0
+50
+100
+150
+200
+172
+87
+125
+146
+Perception Performance
+0
+50
+100
+150
+200
+177
+98
+143
+135
+RecRecNet
+Cropped Result
+He et al.
+ROP
+Figure 7. User study for the rectangling rectification results.
+vided into 30, 50, and 180. Such a training strategy satisfies
+the difficulty of each curriculum and shows a reasonable
+learning magnitude. All implementations are trained on an
+NVIDIA GeForce RTX 2080 Ti GPU.
+4.2. Rectangling Rectification Results
+RecRecNet is the first effort to rectangling the rectified
+wide-angle image. We compare our method with the clas-
+sical image rectangling work He et al. [15] and the state-
+of-art outpainting work for rectified image [26]. In the first
+comparison (Figure 4), we split the test dataset into the nor-
+mal scene and simple scene based on the image feature. We
+also show the cropped result that directly discards the con-
+tent around the deformed boundary, which is commonly ap-
+plied in previous works. Experimental results demonstrate
+while RecRecNet is trained based on the filtered labels from
+He et al. [15], it can learn a higher upper bound of perfor-
+mance and generalize to more diverse scenes. By contrast,
+He et al. . [15] fail to rectangling the scene with a complex
+layout or monotonous background due to line-preserving
+rules. In the second comparison (Figure 5), we can observe
+ROP [26] makes the deformed boundary of the rectified im-
+age regular by extrapolating the image content.
+But the
+generated results are usually fictitious and twist the original
+semantic characterization, in which unpleasant fracture of
+semantics occurs near the boundary. Without introducing
+new and ambiguous content, our RecRecNet constructs a
+win-win representation for the rectified image using a flexi-
+ble TPS transformation, showing the closest appearances to
+the rectangling label. Besides, Figure 6 shows RecRecNet
+can considerably help the downstream vision tasks.
+Since there is no completely accurate rectangling label,
+quantitatively evaluating the performance of different meth-
+ods is challenging. The aim to yield the rectangular im-
+ages is for the visual sense and vision perception, thus we
+conduct a user study to evaluate the rectangling methods
+based on content fidelity, structure-preserving, and percep-
+tion performance. We collect 200 samples in random order,
+and 10 volunteers with vision expertise are required to vote
+on the results under different contexts. As shown in Fig-
+ure 7, RecRecNet achieves the highest votes in three tests
+and shows a superior capacity for scene faithfulness.
+Rectangling
+Rectified Image
+Feature Maps
+Shallow
+Deep
+Figure 8. Failure case of Mask R-CNN [16] in regards to the
+wrong perception. We find that the deformed boundary can in-
+troduce new features to the original feature maps.
+4.3. Why Deformed Boundary Makes Perception
+Deformed
+While our task is to rectangling the rectified image, we
+are still curious about the secret behind the performance
+degradation of the vision perception models. In this part,
+we provide a deep analysis for this effect. First, two typical
+failure cases of the perception model are formulated: wrong
+perception and missing perception. Concretely, we focus on
+the detection performance of Mask R-CNN [16] and visual-
+ize some samples in Figure 6 and Figure 8. We can observe
+the model fails to detect the objects, especially those located
+near the boundary. By rectangling the deformed boundary,
+the perception performance is surprisingly recovered.
+In experiments, we find the deformed boundary can in-
+troduce new features onto the feature maps, which (i) form
+new semantics (leads to the wrong perception) or (ii) cause
+blind spots for original features (leads to missing percep-
+tion). Especially in Figure 8, we exhibit the feature maps of
+Mask R-CNN with ResNet101 [17] backbone from shallow
+layers to deep layers. The noticeable line artifacts and curve
+artifacts can be observed at the outermost boundaries and
+deformed boundaries, respectively. In particular, the line ar-
+tifacts are essentially generated by zero padding [2]. When
+a convolutional kernel extracts the feature at the boundary
+with zero padding, the sharp transitions between the zero
+values and the original content are wrongly identified as
+an edge. Similarly, this edge effect can be induced by the
+deformed boundary of rectified images. As the convolu-
+tional layer increases, such an effect is gradually enlarged
+and constructs new features on the feature maps. Compared
+to the regular image, the rectified image allows a faster ex-
+tension of the edge effect. Thus, more new features will be
+involved in the inference process.
+For the wrong perception case, the network mistakenly
+recognizes the boundary and its surrounding region as a
+surfboard. Since the edge effect introduces new features
+on the shallow feature maps and new semantics is gradu-
+ally formed in the deep feature maps. On the other hand, if
+7
+
+1600U21600U2Table 1. Ablation study on the size of the control points in TPS
+transformation.
+Size
+PSNR ↑
+SSIM ↑
+FID ↓
+LPIPS ↓
+3 × 3
+17.96
+0.5134
+25.00
+0.1347
+5 × 5
+18.41
+0.5288
+21.49
+0.1221
+7 × 7
+18.59
+0.5385
+19.62
+0.1153
+9 × 9
+18.68
+0.5450
+19.01
+0.1136
+11 × 11
+18.64
+0.5392
+19.16
+0.1139
+3x3
+5x5
+7x7
+9x9
+11x11
+Figure 9. Ablation study on the size of the control points in TPS
+transformation. More control points can improve the localization
+ability of RecRecNet for the deformed boundaries.
+the features near the boundary are non-salient, the newly in-
+troduced features can cover them and generate blind spots
+for the perception model. More experimental results are
+reported in the supplementary material. Furthermore, the
+deformed image boundaries not only exist in the rectified
+wide-angle image, but also in other research regions such as
+image warping, image stitching, and roll shutter correction.
+It can raise into most image transformation tasks. In these
+regions, the boundaries are deformed to trade off different
+requirements on content. Thus, the edge extension effect
+can introduce new features onto feature maps and further
+influence the downstream perception performance.
+4.4. Ablation Study
+Control Points: As shown in Table 1 and Figure 9, the ex-
+periments demonstrate more control points can facilitate the
+structure approximation and localization ability of RecRec-
+Net for the deformed boundaries. And the size of 9 × 9
+achieves the best performance on all evaluation metrics.
+Curriculum Learning: We define RecRecNet without a
+curriculum as Vanilla, and define RecRecNet with different
+curriculums: 2-DoF �→ 4-DoF as Vanilla + C1, 2-DoF �→
+8-DoF as Vanilla + C2, and 4-DoF �→ 8-DoF as Vanilla
++ C3, respectively. In Figure 10 (left), the training loss
+curves show that curriculum learning can facilitate a better
+initialization point for the training and faster convergence
+on the rectangling task. Moreover, a reasonable design for
+the curriculum (such as Vanilla + C3) helps to improve the
+boundary-aware ability of RecRecNet, enhancing the struc-
+ture recovery performance as shown in Figure 10 (right).
+0
+40
+80
+120
+160
+200
+0.45
+0.40
+0.35
+0.30
+0.25
+0.24
+0.22
+0.48
+0.40
+Vanilla
+Vanilla + C1
+Vanilla + C2
+Vanilla + C3
+Vanilla + C3
+Vanilla
+Epoch
+Loss
+Figure 10. Ablation study on the curriculum learning for rectan-
+gling task. C1, C2, and C3 denote different curriculums.
+RecRecNet
+Input
+Mesh
+Figure 11. Cross domain evaluation on real-world wide-angle im-
+ages and other types of synthesized datasets.
+4.5. Cross-domain Evaluation
+To evaluate the generalization ability to other datasets,
+we collect 300 rectified wide-angle image results from the
+state-of-the-art rectification methods [27,50]. Their results
+are derived from various types of datasets and real-world
+wide-angle lenses such as the Rokinon 8mm Cine Lens,
+Opteka 6.5mm Lens, and GoPro. Figure 11 shows RecRec-
+Net can well generalize to other domains and different rec-
+tified structures while it trained using only a synthesized
+image dataset with one type of camera model.
+5. Conclusion
+In this work, we consider a new rectangling rectification
+task for the wide-angle image. While it has never been stud-
+ied in previous literature, we demonstrate that the deformed
+boundary of the prevalent rectified wide-angle image can
+significantly influence the vision perception models.
+To
+eliminate this issue, we present to construct a win-win rep-
+resentation for the rectified wide-angle image and design
+a novel RecRecNet. Equipped with a flexible TPS trans-
+formation motion model, RecRecNet can formulate the lo-
+cal deformation from the deformed boundary to the straight
+one in an unsupervised end-to-end manner. Besides, we
+8
+
+0
+JO0
+500
+300
+400
+200
+eoo
+100
+0'52
+0E.0
+0'32
+0^.0
+0'42Levsketaysleinspire RecRecNet to learn the gradual deformation rules
+with a DoF-based curriculum learning, which can relieve
+the complexity of non-linear and non-rigid transformation.
+Furthermore, a detailed analysis is provided to explain why
+the deformed image boundary makes the current vision per-
+ception deformed. In future work, we plan to extend to a
+general paradigm for rectangling any deformed images and
+further study the relationship between the image boundary
+and vision perception performance.
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+A. Supplemental Material
+In this document, we provide the following supplemen-
+tary contents:
+• Details of the dataset construction (Section A.1).
+• More qualitative results of the comparison methods
+and our approach (Section A.2).
+• More vision perception results (Section A.3).
+• More cross-domain evaluation results (Section A.4).
+A.1. Details of the Dataset Construction
+We construct a rectangling rectification dataset that sev-
+ering the first dataset in the research region, and we
+would like to release it to promote the research develop-
+ment. In particular, our dataset is built using the follow-
+ing four steps: (i) Wide-angle image and rectified im-
+age synthesis.
+Since it is extremely challenging to col-
+lect large-scale paired wide-angle images and their recti-
+fied ground truth, we follow the existing distortion rec-
+tification approaches [5, 28, 36, 50, 51] to synthesize the
+dataset. First, the original images are collected from the
+MS-COCO dataset [29]. We leverage a 4th order polyno-
+mial model to approximate the radial distortion of the wide-
+angle image, which is verified to meet most projection mod-
+els with high accuracy. To be specific, four distortion pa-
+rameters are randomly generated from the following ranges:
+k1 ∈ [−1×10−4, −1×10−8], k2 ∈ [1×10−12, 1×10−8] or
+∈ [−1×10−8, −1×10−12], k3 ∈ [1×10−16, 1×10−12] or ∈
+[−1×10−12, −1×10−16], and k4 ∈ [1×10−20, 1×10−16]
+or ∈ [−1 × 10−16, −1 × 10−20]. Then we perform the
+distortion rectification on the wide-angle image and obtain
+the rectified images. (ii) Rectangling the rectified image.
+Formulating an accurate transformation model for rectan-
+gling the rectified image is difficult due to its non-linear and
+non-rigid characteristics. We notice that there is a classi-
+cal panoramic image rectangling technique He et al. [15]
+on computer graphics, it makes the stitched image regular
+by optimizing an energy function with line-preserving mesh
+deformation. Thus, we perform the same energy function
+on our rectified image dataset to fit the rectangling rectifica-
+tion task. However, the capability to preserve linear struc-
+tures in He et al. [15] is limited by line detection. Conse-
+quently, some rectangling rectified images have nonnegligi-
+ble distortions. To overcome this issue, we carefully filter
+all rectangling results and repeat the selection process three
+times, resulting in 5,160 training data from 30,000 source
+images and 500 test data from 2,000 source images. Each
+manual operation takes around 10s. The size of all images
+is 256 × 256. (iii) Cross-domain validation. In addition to
+the synthesized dataset, we collect 300 rectified wide-angle
+image results from the state-of-the-art rectification meth-
+ods [25, 50]. Their results are derived from other types of
+datasets and real-world wide-angle lenses such as the Roki-
+non 8mm Cine Lens, Opteka 6.5mm Lens, and GoPro. (iv)
+DoF-based curriculum dataset. To relieve the challenge
+of the structure approximation in rectangling task, we pro-
+posed a Degree of Freedom (DoF)-based curriculum learn-
+ing. Specifically, three curriculum stages are leveraged to
+inspire our RecRecNet, namely, from similarity transforma-
+tion (4-DoF) to homography transformation (8-DoF), and
+to rectangling transformation. Thus, we also construct two
+datasets (4-DoF dataset and 8-DoF dataset) to provide the
+basic transformation knowledge, in which each dataset con-
+tains 5,000 image pairs. For the transformation synthesis,
+we randomly perturb four corners of the original image, and
+warp it to the target image following the previous work [9].
+A.2. More Qualitative Comparison Results
+As shown in Figure 12, we exhibit more qualitative com-
+parison results. Our RecRecNet is capable of rectangling
+the rectified wide-angle image in various scenes. We can
+observe the deformed boundary is straightened and the im-
+age content is rearranged to keep undistorted by RecRec-
+Net, contributing a win-win rectification representation for
+the wide-angle image. By contrast, previous methods fail
+to trade-off the image boundary and image content in the
+rectified image. Their results usually show incomplete con-
+tent or distorted distributions. For example, the rectangling
+results produced by He et al. [15] rotate the original scene
+and twist the object due to their line-preserving mesh defor-
+mation.
+A.3. More Vision Perception Results
+As illustrated in Figure 13, we show more vision per-
+ception results, including the object detection and seman-
+tic segmentation results, which are derived by the popular
+perception model Mask R-CNN [16]. As we can observe,
+the learning model is misled by the deformed structure in
+the original rectified image, leading to wrong perception
+and missing perception results. As a benefit of the pro-
+posed flexible TPS transformation module and DoF-based
+curriculum learning, our RecRecNet can significantly help
+the downstream vision tasks by eliminating the deformed
+boundary issue. And the perception performance is recov-
+ered especially near the image boundary. For the perfor-
+mance degradation in rectified images, one straightforward
+reason is that the perception model has never seen the data
+with a deformed image boundary during training, and the
+domain gap between test data and training data confuses
+it on rectified images. Nevertheless, we would like to ex-
+plore deeper reasons and disclose why the deformed geom-
+etry makes the vision perception deformed.
+A sample is shown in Figure 14 to visualize why the
+12
+
+Rectified Image
+Cropped Image
+RecRecNet
+He et al.
+Wide-angle Image
+Rectified Image
+Cropped Image
+RecRecNet
+He et al.
+Wide-angle Image
+Figure 12. More qualitative results compared to He et al. [15]. We show the wide-angle image, rectified image, cropped rectified image,
+and the rectangling result of our RecRecNet, as well as the rectangling result by He et al. [15] from left to right.
+deformed image boundary makes the perception deformed.
+As described in the main manuscript, we find the deformed
+boundary can introduce new features onto the feature maps,
+which (i) form new semantics (leads to the wrong percep-
+tion) or (ii) cause blind spots for original features (leads to
+missing perception). The noticeable line artifacts and curve
+artifacts can be observed at the outermost boundaries and
+deformed boundaries respectively in shallow feature maps.
+In particular, the line artifacts are essentially generated by
+zero padding. Zero padding is a fundamental component of
+prominent CNNs architectures, it serves to maintain the size
+of the feature maps across the network by adding zero val-
+ues to the feature map. For the missing perception case, if
+the features near the boundary are non-salient, the newly in-
+troduced features can cover them and generate blind spots
+for the perception model. As a result, the network cannot
+recognize the elephant near the deformed image boundary.
+A.4. More Cross-Domain Evaluation Results
+As mentioned in Section A.1, we collect 300 rectified
+wide-angle image results from the state-of-the-art rectifica-
+tion methods [25, 50] to conduct the cross-domain evalua-
+tion. Their results are derived from other types of synthe-
+sized datasets and real-world wide-angle lenses with dif-
+ferent camera models. Figure 15 shows the cross-domain
+evaluation results. While the new data space has never been
+seen, our RecRecNet can achieve a promising generaliza-
+tion ability by learning a flexible TPS transformation. And
+13
+
+4seRectified 
+Image
+RecRecNet
+Detection and 
+Segmentation
+Detection and 
+Segmentation
+Figure 13. More detection and segmentation results by Mask R-CNN [16]. We show the rectified wide-angle image and its vision perception
+result, and our rectangling result and its vision perception result, from top to bottom.
+Rectangling
+Rectified Image
+Feature Maps
+Shallow
+Deep
+Figure 14. Failure case of Mask R-CNN [16] in regards to the
+missing perception. We find that the deformed boundary can in-
+troduce new features to the original feature maps. As a result, the
+network cannot recognize the elephant near the deformed image
+boundary as the original features are blinded in the deep feature
+maps.
+we can observe that the rectangling results display reason-
+able global distributions and visually pleasing local details.
+Moreover, the predicted mesh locates at most spatial range
+of the rectified image, demonstrating the effectiveness of
+the learned rectangling transformation.
+14
+
+cribloey
+cnbloa3
+pomiioes
+pomilose
+cnbl
+pOScnblo.al
+Ee.olsldj.pninib
+b6l20ulob6l2oul08cnbloe3
+b6l20ulo8
+06L20U03porrieloap
+tp6lole
+06L20W0COCKTOOD6L20Ub6120b6L20ul0.80coc9ocklo82ee.ojn6rqsl9
+lbugurlee.ojn6rqsl9
+lbugurl151151ee.0n6rigalsRecRecNet
+Rectified 
+Image
+Mesh
+Wide-angle 
+Image
+Figure 15. More results for the cross-domain evaluation. We show the wide-angle image, rectified image, predicted mesh, and rectangling
+result of our RecRecNet from top to bottom.
+15
+
+ieuele
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+page_content='RecRecNet: Rectangling Rectified Wide-Angle Images by Thin-Plate Spline  Model and DoF-based Curriculum Learning  Kang Liao Lang Nie Chunyu Lin* Zishuo Zheng Yao Zhao  Institute of Information Science, Beijing Jiaotong University  Beijing Key Laboratory of Advanced Information Science and Network  {kang liao, nielang, cylin, yzhao}@bjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='cn  Abstract  The wide-angle lens shows appealing applications in VR  technologies, but it introduces severe radial distortion into  its captured image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To recover the realistic scene, previ-  ous works devote to rectifying the content of the wide-angle  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' However, such a rectification solution inevitably dis-  torts the image boundary, which potentially changes related  geometric distributions and misleads the current vision per-  ception models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In this work, we explore constructing a  win-win representation on both content and boundary by  contributing a new learning model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=', Rectangling Recti-  fication Network (RecRecNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In particular, we propose a  thin-plate spline (TPS) module to formulate the non-linear  and non-rigid transformation for rectangling images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By  learning the control points on the rectified image, our model  can flexibly warp the source structure to the target domain  and achieves an end-to-end unsupervised deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To  relieve the complexity of structure approximation, we then  inspire our RecRecNet to learn the gradual deformation  rules with a DoF (Degree of Freedom)-based curriculum  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By increasing the DoF in each curriculum stage,  namely, from similarity transformation (4-DoF) to homog-  raphy transformation (8-DoF), the network is capable of  investigating more detailed deformations, offering fast con-  vergence on the final rectangling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Experiments show  the superiority of our solution over the compared methods  on both quantitative and qualitative evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The code  and dataset will be made available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Introduction  The wide-angle and fisheye lenses have been widely ap-  plied in various applications such as computational imag-  ing, virtual reality (VR), and autonomous driving, due to  their larger field of view (FoV) than conventional lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By  employing a specific mapping, the wide-angle lens can pro-  Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The first two authors contribute equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  RecRecNet  Rectified Image  Rectangling Image  Vision Perception  Model  Detection and Segmentation  Results  TPS Transformation  Inference  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Existing rectification methods devote to correcting the  content of the wide-angle image, but they inevitably distort the  image boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We demonstrate that such a deformed boundary  can lead to dramatic performance degradation of the popular vi-  sion perception model [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By presenting the Rectangling Recti-  fication Network (RecRecNet), we reach a win-win representation  on both the undistorted content and regular boundary with a flexi-  ble TPS transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The arrow marks our perception gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  duce convex and non-rectilinear images rather than images  with straight lines of perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' However, this mapping  mechanism violates the pinhole camera assumption, and the  captured image suffers from severe radial distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  To eliminate the distortion induced by the wide-angle  lens, the distortion rectification algorithm is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For  example, the traditional approaches calibrate the wide-  angle lens using the calibration target (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=', a checker-  board) [18, 54], specific scene assumption [3, 10, 32], and  camera motions [12, 31, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In recent years, deep learn-  ing has brought new inspirations to this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For example,  the learning-based approaches can automatically rectify the  radial distortion in wide-angle images based on learned se-  mantic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To pursue better performance, the subse-  quent works improve the learning framework through di-  verse features [49, 51], learning paradigm [28], calibration  representation [24,30], and distortion prior [8,27], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='01661v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='CV]  4 Jan 2023  We notice that the above distortion rectification ap-  proaches mainly focus on rectifying the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By  shrinking the distorted image towards the principal center,  the spatial distribution of all pixels has been rearranged  and thus generates the rectified result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' However, the image  boundaries are inevitably distorted by this correction way,  making the image exhibited with a visually narrow FoV  and irregular shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Moreover, we found such a deformed  boundary can significantly influence the current mainstream  vision perception models because most of them are trained  on the dataset with rectangular images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As shown in Fig-  ure 1, Mask R-CNN [16] is confused by the rectified im-  age, leading to a missing segmentation and detection result  near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' While some works [26] fill the blank  region beyond deformed boundaries using outpainting strat-  egy [37,45], the generated contents are usually fictitious and  twist the original semantic characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  In this work, we explore a win-win rectification rep-  resentation for the wide-angle image on both content and  boundary without introducing any extra semantic informa-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For this new representation, the distorted content has  been corrected while the image boundary keeps straight, as  shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To this end, we propose a Rectangling  Rectification Network (named RecRecNet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Concretely, a  thin-plate spline (TPS) motion module is proposed to for-  mulate the non-linear and non-rigid transformation for rect-  angling the rectified wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' It can flexibly warp  the source structure to the target domain by learning the  control points on the rectified image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Subsequently, we design a DoF-based curriculum learn-  ing to further inspire our RecRecNet to grasp the progres-  sive deformation rules, relieving the burden of complex  structure approximation and initializing a better start point  for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  The general paradigm of curriculum learn-  ing [4] introduces a heuristic and purposeful strategy for  training a learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The model can converge faster  by gradually learning various samples or knowledge based  on different stages in a curriculum, mimicking how humans  and animals learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In particular, we increase the degree of  freedom (DoF) of the devised stages/transformations from  similarity transformation (4-DoF) to homography transfor-  mation (8-DoF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Moreover, the procedure of our curriculum  reveals a simple-to-complex order from the straight line,  sloped line to curve for the image boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As a conse-  quence, our RecRecNet can discover more geometric de-  tails and converge faster on the final rectangling stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  While aiming for a win-win rectification representation,  we provide a deep analysis of why the deformed bound-  ary makes the perception deformed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=', why the deformed  image boundary can significantly influence the visual per-  ception models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We surprisingly find the deformed bound-  ary can introduce new features on the original feature maps,  which further forms new semantics or cause blind spots for  non-salient features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' And we demonstrate that this effect  commonly exists in other research regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  We conclude our major contributions as follows:  We propose a Rectangling Rectification Network  (RecRecNet) for a win-win representation of the recti-  fied wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To our knowledge, it has never  been investigated in previous literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  A thin-plate spline (TPS) motion module is proposed  to flexibly formulate the non-linear and non-rigid rect-  angling transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  We also provide a detailed  analysis of why the deformed image boundary can sig-  nificantly influence the visual perception models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  To grasp the progressive deformation rules and re-  lieve the burden of complex structure approximation, a  DoF-based curriculum learning is designed to inspire  our RecRecNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Related Work  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Distortion Rectification  The distortion rectification approaches for wide-angle  lenses can be classified into traditional and learning-based  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The majority of traditional techniques concen-  trate on detecting the hand-crafted features [1,3,10,22,32,  39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' However, these techniques frequently fail due to par-  ticular constraints like the plumb line and curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Recently,  blind distortion rectification has been studied using deep  learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' It can rectify the radial distortion of the wide-  angle image without any scene assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For example,  the regression-based solutions [5, 27, 30, 36, 47, 51] predict  the distortion parameters of input, and then the estimated  parameters are used to rectify the distortion offline or on-  line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The reconstruction-based solutions [8, 24, 28, 50, 55]  directly learns the pixel-level mapping or displacement field  from the distortion image into the rectification output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Despite promising flexibility and efficiency, these recti-  fication approaches inevitably distort the image boundary  during content correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As a result, the distorted distri-  bution at the boundary confuses the vision perception net-  work and leads to inferior performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' One direct way is to  crop the rectified image into a rectangular shape, however,  it discards informative content and disobeys the original in-  tention of the wide-angle lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Curriculum Learning  Curriculum learning techniques have been effectively  exploited across the board in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  El-  man et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [11] provide a conceptual framework for cur-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='riculum learning that emphasizes the value of beginning at  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='a low level and then works way up to more challenging  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Backbone  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='TPS Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Module  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Rectified Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Wide-Angle Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Mesh Grid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='RecRecNet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='DoF-based Curriculum Learning  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Stage1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Stage2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='R  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Rectification  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='4-DoF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='8-DoF  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Transformation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Control Points  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Rectangling Image  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Ground Truth  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='G  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Gridding  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='W  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Warping  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='et1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='et2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='LIG  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Inter-Grid  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Contrainst  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='LAP  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='LPE  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Optimization  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Loss  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Overview of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Our Rectangling Rectification Network (RecRecNet) aims to construct a win-win rep-  resentation on both image content and image boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' RecRecNet learns a flexible local transformation based on the thin-plate spline  (TPS) motion model, which predicts the control points on the image boundary and warps the target image by using a mesh grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' It is jointly  optimized with the appearance constraint LAP , perceptual constraint LP E, and inter-grid constraint LIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To further relieve the challenges  of structure approximation, we inspire our RecRecNet to learn the gradual deformation rules with a DoF-based curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [4] are the first to formalize the cur-  riculum learning as a strategy to gradually raise the level  of complexity of the data samples utilized in the train-  ing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Subsequently, most researchers follow this  paradigm and apply the curriculum learning into numerous  regions [34,40,46,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Particularly in computer vision, cur-  riculum learning has been investigated to inspire the learn-  ing networks on image classification [7, 14, 33], object de-  tection [23,43,48], semantic segmentation [13,38,53], and  image generation [20,25,42], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  In this work, we incorporate curriculum learning into a  new task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  By disassembling the rectangling rectification  into three levels based on the DoF of transformation, the  proposed curriculum can inspire our network to learn the  challenging deformation in a simple-to-complex order and  thus facilitates a faster convergence for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Methodology  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Problem Formulation  Given a rectified wide-angle image IRec ∈ Rh×w×3 with  deformed boundaries, our Rectangling Rectification Net-  work (RecRecNet) aims to construct a win-win representa-  tion on both image content and image boundary, generating  a rectangular rectified image IRec2 ∈ Rh×w×3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Figure 2  shows the overview of the proposed framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To be more  specific, the rectangling module takes IRec as the input and  learns a flexible local transformation T based on the thin-  plate spline (TPS) motion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' It is jointly optimized  with the appearance constraint, perceptual constraint, and  inter-grid constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To relieve the challenges of structure  approximation, we inspire the network to learn the gradual  deformation rules with a DoF-based curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  The general polynomial camera model [19] is widely  used in the approximation for the radial distortion of the  wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We formulate the relationship between  the lens’s projection manner and its distortion parameters  K = [k1, k2, k3, · · · ] as follows:  r(θ) =  N  �  i=1  kiθ2i−1, N = 1, 2, 3, · · · ,  (1)  where r is the distance between the principal point and the  pixels in the image, and θ indicates the angle between the  optical axis and incident ray of the wide-angle lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Such a  projection manner can produce a convex and non-rectilinear  image, and thus the captured image suffers from severe ge-  ometric distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Suppose that the coordinates of a pixel  in the wide-angle image and rectified images are p = (x, y)  and prec = (xrec, yrec), the rectification solution can be  3  formulated based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 1 as follows:  pd = R(prec, K) =  �  u0  v0  �  + r(θ)prec  ∥prec∥2  ,  (2)  where (u0, v0) represents the principal point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  With the  above equation, the wide-angle image can be rectified by  using bilinear interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By shrinking the wide-angle  image towards the principal center, the spatial distribution  of all pixels is rearranged, and thus rectified content is ob-  tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' However, the image boundaries are inevitably dis-  torted by this rectification way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Rectangling with TPS Transformation  As shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 2, the rectification of the wide-angle im-  age is a non-linear and non-rigid mapping function, which  cannot be represented by simple transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Be-  sides, it is challenging to build an accurate parameterization  model to supervise the rectangling process with estimated  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' From the perspective of energy minimization,  we can approximate the rectangling rectification by mini-  mizing the transformation energy with a uniform model:  ε = εT + λεd,  T : (xrec, yrec) �→ (xrec2, yrec2),  (3)  where ε denotes the total energy of the expected transforma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' (xrec, yrec) ∈ ΩS and (xrec2, yrec2) ∈ ΩT represent  the point in the domain of source S and target T, respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' εT indicate the data penalty energy and εd indicate  the distortion energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The above formula with the lowest  overall energy is the desired transformation, in which a hy-  perparameter λ is designed to balance the energies between  the data penalty and distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  In particular, we propose to exploit TPS transforma-  tion [6] T ′ to minimize the aforementioned energy func-  tion, which enables an end-to-end unsupervised deforma-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' TPS transformation provides for the representation of  more complex motions since it is flexible and non-linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  It allows warping from one image to the other with mini-  mum distortion, given two sets of control points in the cor-  responding images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Suppose q = [q1, q2, · · · , qN] and q′ =  [q′  1, q′  2, · · · , q′  N] are source points and the target points, the  data term εT ′ can be calculated by the difference of control  points displacement �N  i=1∥T ′(qi) − q′  i∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Having aligned  the control points, TPS finds an optimal interpolation trans-  formation with minimal distortion term εd by:  min  ��  R2  ��∂2T ′  ∂x2  �2  + 2  � ∂2T ′  ∂x∂y  �2  +  �∂2T ′  ∂y2  �2�  dxdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  (4)  Rectified Image  Rectangling  Rectangling Flow  Rectification Flow  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Visualization of the rectangling flow and rectification  flow for the rectified wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We mark the different  radial centers in different rectangling flows with arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Then we reach a spatial deformation function parameterized  by control points as follows:  T ′(q) = A  �q  1  �  +  N  �  i=1  wiU (∥q′  i − q∥2) ,  (5)  where q is a point located at the rectified wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  A ∈ R2×3 and wi ∈ R2×1 are the transformation param-  eters, which can be derived by minimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Besides,  U(r) is a radial basis function that denotes the influence of  the control point on q: U(r) = r2 log r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  In RecRecNet, we define N = (U + 1) × (V + 1) con-  trol points that can be connected to form a mesh, and the  network learns to predict the mesh of rectified wide-angle  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Then TPS transforms the predicted mesh to the reg-  ular mesh on the target rectangular image, where the control  points are set to be evenly distributed on the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  For the network, we first choose ResNet50 [17] as the  backbone of our RecRecNet to extract the high-level seman-  tic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Suppose the size of the input rectified image is  W × H, and the size of the output feature map of the back-  bone is W  16 × H  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Then these feature maps are fed to a motion  header to predict (U + 1) × (V + 1) control points on the  rectified image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 4 convolutional layers with a kernel size  of 3 and BacthNormalization are stacked to aggregate the  high-level features, of which all channel dimensions are set  to 512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Subsequently, 3 fully connected layers with the fol-  lowing numbers of units 4096, 2048, (U +1)×(V +1)×2  are used to predict the coordinates of all control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' DoF-based Curriculum Learning  As mentioned above, we propose to exploit TPS trans-  formation to flexibly approximate the motion from the rec-  tified image to rectangling result by a set of control points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  To further relieve the challenges of this transformation, we  inspire our RecRecNet with a curriculum learning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1  Image Transformation Formulation  Image transformation contains different elements such as  translations, scales, rotations, shearing, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Any composi-  tion of these elements can form a new transformation with  the degree of freedom (DoF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In general, the degree of free-  dom is a collection of independent displacements that com-  pletely describes a system’s displaced or deformed position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  There are some classical transformations on 2D images:  translation transformation (2-DoF), Euclidean/Rigid trans-  formation (translation + rotation, 3-DoF), similarity trans-  formation (translation + rotation + scale, 4-DoF), affine  transformation (translation + rotation + scale + shear, 6-  DoF), and projective transformation (two each for transla-  tion, rotation, scale, and lines at infinity, 8-DoF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  The image transformation between the rectified wide-  angle image and its rectangling image has never been stud-  ied and formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Compared to distortion rectification,  the rectangling transformation seems to be an inverse asym-  metric warping due to the more important image bound-  ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To understand this transformation intuitively, we vi-  sualize the pixel displacement field between the rectified  wide-angle image and its rectangling image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Specifically,  we leverage the state-of-the-art optical flow estimation net-  work RAFT [44] to exhibit the motion difference and name  it as rectangling flow (rectified image �→ rectangling im-  age).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As shown in Figure 3, compared with the rectification  flow (wide-angle image �→ rectified image), we have the  following observations: (1) Both rectangling flow and rec-  tification flow show a radial spatial distribution, in which  four dominated directions spread towards or backwards four  image corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' (2) Rectangling flow has an unfixed radial  center for each sample, which is determined by the content  and layout of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Rectification flow has a relatively  fixed radial center, which is related to the geometry prior to  the radial distortion in the wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Thus, we can conclude the image content highly iden-  tifies the distribution of rectangling flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' It is painful to  represent this transformation using classical image transfor-  mations and their compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Nevertheless, such a non-  linear and non-rigid transformation can be locally disentan-  gled into different transformation elements such as transla-  tion, scale, and shearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' It would be meaningful to inspire  the network to learn some basic transformations and gradu-  ally transfer them into more challenging cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='2  Curriculum Selection  To choose effective curriculums for the rectangling task, we  should obey two principles: First, all stages in a curriculum  have closely associated knowledge of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Second,  the stages in a curriculum intrinsically offer a simple-to-  complex order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To this end, we design a DoF-based curricu-  lum to inspire RecRecNet, of which we increase the DoF of  the devised transformations from similarity transformation  (4-DoF) to homography transformation (8-DoF) and even-  tually to the rectangling transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  We argue that the proposed curriculum has the following  strengths: (1) As mentioned in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1, the DoF di-  rectly expresses the difficulty of the image transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  As the DoF increases, the transformation shows more com-  plex global mappings and diverse motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Thus, RecRec-  Net can learn the gradual deformation rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' (2) For the  local geometry, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=', the shape of the image boundary, our  curriculum shows a progressive shift from a straight line,  sloped line, to a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Such a design can improve the lo-  calization ability of control points in TPS transformation  and enable a boundary-aware ability for RecRecNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More  details and experiments will be demonstrated in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Training Losses  Appearance Loss: We constrain the rectangling image  IRec2 to be close to the ground truth IGT at the pixel level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  The appearance loss LAP can be calculated by the differ-  ence between IRec2 and IGT with L1 norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Perceptual Loss: We then minimize the L2 distance be-  tween IRec2 and IGT in high-level semantic perception to  guarantee the rectangling results perceptually natural:  LP E =  1  Wi,jHi,j  Wi,j  �  x=1  Hi,j  �  y=1  ||φi,j(Ix,y  Rec2) − φi,j(Ix,y  GT )||2,  (6)  where the difference of rectangling rectified image and  ground truth is minimized on the feature map φi,j, which is  derived from the j-th convolution (after activation) before  the i-th max-pooling layer in the VGG19 network [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Inter-grid Mesh Loss: To avoid the content distortion after  our rectangling, the predicted mesh should not be largely  deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Therefore, we design an inter-grid mesh term  LIG to constrain the shape of the predicted mesh, which  encourages the neighboring grids to transform consistently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  The edges of two successive deformed grid {⃗et1,⃗et2} are  supervised to be co-linear as follows:  LIG = 1  M  �  {⃗et1,⃗et2}∈mp∪mf  (1 −  ⟨⃗et1,⃗et2⟩  ∥ ⃗et1 ∥ · ∥ ⃗et2 ∥),  (7)  where M is the number of tuples of two successive edges  in a mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' ms and mt are the predicted source mesh and  target mesh, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  The overall training loss can be obtained by:  L = λAP LAP + λP ELP E + λIGLIG,  (8)  where λAP , λP E, and λIG are the weights to balance the  appearance loss, perceptual loss, and inter-grid mesh loss,  which are empirically set to 1, 1e−4, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  5  Rectified Image  Cropped Image  RecRecNet  He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Rectified Image  Cropped Image  RecRecNet  He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Normal Scene  Simple Scene  Wide-angle Image  Wide-angle Image  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Comparison to the image rectangling work He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For a comprehensive evaluation, we split the test dataset into the  normal scene (left) and the simple scene (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Our RecRecNet can achieve the best performance on the undistorted content and regular  boundary, regardless of the scene change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Rectified Image  Cropped Image  RecRecNet  Label  ROP  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Comparison to the rectification outpainting work  ROP [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Our RecRecNet can straighten the irregular image  boundary in the rectified image without introducing new content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Rectified Image  RecRecNet  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  RecRecNet helps the downstream vision perception  tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The arrows mark the failure results in rectified images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Experiments  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Implementation Details  Dataset Establishment: Following existing methods [28,  36,50,51], we first synthesize the wide-angle images by us-  ing a 4th order polynomial model based on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Then we  perform the distortion rectification on the wide-angle image  in terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Due to non-linear and non-rigid character-  istics, forming an accurate transformation model for rectan-  gling the rectified image is difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We notice that there is  a classical panoramic image rectangling work He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15]  on computer graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' It makes the stitched image regular  by optimizing an energy function with line-preserving mesh  deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Thus, we perform the same energy function  on the rectified image to fit our task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' However, the capabil-  ity to preserve linear structures [15] is limited by line de-  tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Consequently, some rectangling images have non-  negligible distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To overcome this issue, we carefully  filter all results and repeat the selection process three times,  resulting in 5,160 training data from 30,000 source images  and 500 test data from 2,000 source images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Each manual  operation takes around 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More details about the dataset  are reported in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Experimental Configuration: We train RecRecNet with  an exponentially decaying learning rate with an initial value  of 10−4 using Adam [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  The batch size is set to 10  and total epoch is set to 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Particularly, RecRecNet is  trained with 3 stages in terms of the proposed DoF-based  curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The training epochs of the 4-DoF curriculum,  8-DoF curriculum, and final rectangling curriculum are di-  6  WOMDWOMDCEbGL26UT0066L2001T0001260Caioas612000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='8b6L200TO0160U12L9CK6  D6L20UT00Content Fidelity  0  50  100  150  200  187  41  139  121  Structure Preserving  0  50  100  150  200  172  87  125  146  Perception Performance  0  50  100  150  200  177  98  143  135  RecRecNet  Cropped Result  He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  ROP  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' User study for the rectangling rectification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  vided into 30, 50, and 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Such a training strategy satisfies  the difficulty of each curriculum and shows a reasonable  learning magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' All implementations are trained on an  NVIDIA GeForce RTX 2080 Ti GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Rectangling Rectification Results  RecRecNet is the first effort to rectangling the rectified  wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We compare our method with the clas-  sical image rectangling work He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15] and the state-  of-art outpainting work for rectified image [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In the first  comparison (Figure 4), we split the test dataset into the nor-  mal scene and simple scene based on the image feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We  also show the cropped result that directly discards the con-  tent around the deformed boundary, which is commonly ap-  plied in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Experimental results demonstrate  while RecRecNet is trained based on the filtered labels from  He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15], it can learn a higher upper bound of perfor-  mance and generalize to more diverse scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By contrast,  He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15] fail to rectangling the scene with a complex  layout or monotonous background due to line-preserving  rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In the second comparison (Figure 5), we can observe  ROP [26] makes the deformed boundary of the rectified im-  age regular by extrapolating the image content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  But the  generated results are usually fictitious and twist the original  semantic characterization, in which unpleasant fracture of  semantics occurs near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Without introducing  new and ambiguous content, our RecRecNet constructs a  win-win representation for the rectified image using a flexi-  ble TPS transformation, showing the closest appearances to  the rectangling label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Besides, Figure 6 shows RecRecNet  can considerably help the downstream vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Since there is no completely accurate rectangling label,  quantitatively evaluating the performance of different meth-  ods is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The aim to yield the rectangular im-  ages is for the visual sense and vision perception, thus we  conduct a user study to evaluate the rectangling methods  based on content fidelity, structure-preserving, and percep-  tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We collect 200 samples in random order,  and 10 volunteers with vision expertise are required to vote  on the results under different contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As shown in Fig-  ure 7, RecRecNet achieves the highest votes in three tests  and shows a superior capacity for scene faithfulness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Rectangling  Rectified Image  Feature Maps  Shallow  Deep  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Failure case of Mask R-CNN [16] in regards to the  wrong perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We find that the deformed boundary can in-  troduce new features to the original feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Why Deformed Boundary Makes Perception  Deformed  While our task is to rectangling the rectified image, we  are still curious about the secret behind the performance  degradation of the vision perception models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In this part,  we provide a deep analysis for this effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' First, two typical  failure cases of the perception model are formulated: wrong  perception and missing perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Concretely, we focus on  the detection performance of Mask R-CNN [16] and visual-  ize some samples in Figure 6 and Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We can observe  the model fails to detect the objects, especially those located  near the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By rectangling the deformed boundary,  the perception performance is surprisingly recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  In experiments, we find the deformed boundary can in-  troduce new features onto the feature maps, which (i) form  new semantics (leads to the wrong perception) or (ii) cause  blind spots for original features (leads to missing percep-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Especially in Figure 8, we exhibit the feature maps of  Mask R-CNN with ResNet101 [17] backbone from shallow  layers to deep layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The noticeable line artifacts and curve  artifacts can be observed at the outermost boundaries and  deformed boundaries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In particular, the line ar-  tifacts are essentially generated by zero padding [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' When  a convolutional kernel extracts the feature at the boundary  with zero padding, the sharp transitions between the zero  values and the original content are wrongly identified as  an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Similarly, this edge effect can be induced by the  deformed boundary of rectified images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As the convolu-  tional layer increases, such an effect is gradually enlarged  and constructs new features on the feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Compared  to the regular image, the rectified image allows a faster ex-  tension of the edge effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Thus, more new features will be  involved in the inference process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  For the wrong perception case, the network mistakenly  recognizes the boundary and its surrounding region as a  surfboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Since the edge effect introduces new features  on the shallow feature maps and new semantics is gradu-  ally formed in the deep feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' On the other hand, if  7  1600U21600U2Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Ablation study on the size of the control points in TPS  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Size  PSNR ↑  SSIM ↑  FID ↓  LPIPS ↓  3 × 3  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5134  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1347  5 × 5  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='41  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5288  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='49  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1221  7 × 7  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='59  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5385  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='62  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1153  9 × 9  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='68  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5450  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1136  11 × 11  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='64  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5392  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1139  3x3  5x5  7x7  9x9  11x11  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Ablation study on the size of the control points in TPS  transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More control points can improve the localization  ability of RecRecNet for the deformed boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  the features near the boundary are non-salient, the newly in-  troduced features can cover them and generate blind spots  for the perception model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More experimental results are  reported in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Furthermore, the  deformed image boundaries not only exist in the rectified  wide-angle image, but also in other research regions such as  image warping, image stitching, and roll shutter correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  It can raise into most image transformation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In these  regions, the boundaries are deformed to trade off different  requirements on content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Thus, the edge extension effect  can introduce new features onto feature maps and further  influence the downstream perception performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Ablation Study  Control Points: As shown in Table 1 and Figure 9, the ex-  periments demonstrate more control points can facilitate the  structure approximation and localization ability of RecRec-  Net for the deformed boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' And the size of 9 × 9  achieves the best performance on all evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Curriculum Learning: We define RecRecNet without a  curriculum as Vanilla, and define RecRecNet with different  curriculums: 2-DoF �→ 4-DoF as Vanilla + C1, 2-DoF �→  8-DoF as Vanilla + C2, and 4-DoF �→ 8-DoF as Vanilla  + C3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In Figure 10 (left), the training loss  curves show that curriculum learning can facilitate a better  initialization point for the training and faster convergence  on the rectangling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Moreover, a reasonable design for  the curriculum (such as Vanilla + C3) helps to improve the  boundary-aware ability of RecRecNet, enhancing the struc-  ture recovery performance as shown in Figure 10 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  0  40  80  120  160  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='24  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='22  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='40  Vanilla  Vanilla + C1  Vanilla + C2  Vanilla + C3  Vanilla + C3  Vanilla  Epoch  Loss  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Ablation study on the curriculum learning for rectan-  gling task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' C1, C2, and C3 denote different curriculums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  RecRecNet  Input  Mesh  Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Cross domain evaluation on real-world wide-angle im-  ages and other types of synthesized datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Cross-domain Evaluation  To evaluate the generalization ability to other datasets,  we collect 300 rectified wide-angle image results from the  state-of-the-art rectification methods [27,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Their results  are derived from various types of datasets and real-world  wide-angle lenses such as the Rokinon 8mm Cine Lens,  Opteka 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5mm Lens, and GoPro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Figure 11 shows RecRec-  Net can well generalize to other domains and different rec-  tified structures while it trained using only a synthesized  image dataset with one type of camera model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Conclusion  In this work, we consider a new rectangling rectification  task for the wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' While it has never been stud-  ied in previous literature, we demonstrate that the deformed  boundary of the prevalent rectified wide-angle image can  significantly influence the vision perception models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  To  eliminate this issue, we present to construct a win-win rep-  resentation for the rectified wide-angle image and design  a novel RecRecNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Equipped with a flexible TPS trans-  formation motion model, RecRecNet can formulate the lo-  cal deformation from the deformed boundary to the straight  one in an unsupervised end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=" Besides, we  8  0  JO0  500  300  400  200  eoo  100  0'52  0E." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content="0  0'32  0^." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content="0  0'42Levsketaysleinspire RecRecNet to learn the gradual deformation rules  with a DoF-based curriculum learning, which can relieve  the complexity of non-linear and non-rigid transformation." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Furthermore, a detailed analysis is provided to explain why  the deformed image boundary makes the current vision per-  ception deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In future work, we plan to extend to a  general paradigm for rectangling any deformed images and  further study the relationship between the image boundary  and vision perception performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  References  [1] Miguel Alem´an-Flores, Luis Alvarez, Luis Gomez, and  Daniel Santana-Cedr´es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Automatic lens distortion correction  using one-parameter division models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Image Processing On  Line, 4:327–343, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 2  [2] Bilal Alsallakh, Narine Kokhlikyan, Vivek Miglani, Jun  Yuan, and Orion Reblitz-Richardson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Mind the pad–cnns can  develop blind spots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In International Conference on Learn-  ing Representations, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 7  [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Barreto and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Araujo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Geometric properties of central  catadioptric line images and their application in calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  IEEE Transactions on Pattern Analysis and Machine Intelli-  gence, 27(8):1327–1333, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' 1, 2  [4] Yoshua Bengio, J´erˆome Louradour, Ronan Collobert, and Ja-  son Weston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Curriculum learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In Proceedings of Interna-  tional Conference on Machine Learning, pages 41–48, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  2, 3  [5] Oleksandr Bogdan, Viktor Eckstein, Francois Rameau, and  Jean-Charles Bazin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Deepcalib: a deep learning approach  for automatic intrinsic calibration of wide field-of-view cam-  eras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
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+page_content=' Semi-supervised wide-angle portraits  correction by multi-scale transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
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+page_content=' 2  11  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Supplemental Material  In this document, we provide the following supplemen-  tary contents:  Details of the dataset construction (Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  More qualitative results of the comparison methods  and our approach (Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  More vision perception results (Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  More cross-domain evaluation results (Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Details of the Dataset Construction  We construct a rectangling rectification dataset that sev-  ering the first dataset in the research region, and we  would like to release it to promote the research develop-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In particular, our dataset is built using the follow-  ing four steps: (i) Wide-angle image and rectified im-  age synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Since it is extremely challenging to col-  lect large-scale paired wide-angle images and their recti-  fied ground truth, we follow the existing distortion rec-  tification approaches [5, 28, 36, 50, 51] to synthesize the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' First, the original images are collected from the  MS-COCO dataset [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We leverage a 4th order polyno-  mial model to approximate the radial distortion of the wide-  angle image, which is verified to meet most projection mod-  els with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To be specific, four distortion pa-  rameters are randomly generated from the following ranges:  k1 ∈ [−1×10−4, −1×10−8], k2 ∈ [1×10−12, 1×10−8] or  ∈ [−1×10−8, −1×10−12], k3 ∈ [1×10−16, 1×10−12] or ∈  [−1×10−12, −1×10−16], and k4 ∈ [1×10−20, 1×10−16]  or ∈ [−1 × 10−16, −1 × 10−20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Then we perform the  distortion rectification on the wide-angle image and obtain  the rectified images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' (ii) Rectangling the rectified image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Formulating an accurate transformation model for rectan-  gling the rectified image is difficult due to its non-linear and  non-rigid characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We notice that there is a classi-  cal panoramic image rectangling technique He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15]  on computer graphics, it makes the stitched image regular  by optimizing an energy function with line-preserving mesh  deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Thus, we perform the same energy function  on our rectified image dataset to fit the rectangling rectifica-  tion task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' However, the capability to preserve linear struc-  tures in He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15] is limited by line detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Conse-  quently, some rectangling rectified images have nonnegligi-  ble distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To overcome this issue, we carefully filter  all rectangling results and repeat the selection process three  times, resulting in 5,160 training data from 30,000 source  images and 500 test data from 2,000 source images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Each  manual operation takes around 10s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The size of all images  is 256 × 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' (iii) Cross-domain validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' In addition to  the synthesized dataset, we collect 300 rectified wide-angle  image results from the state-of-the-art rectification meth-  ods [25, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Their results are derived from other types of  datasets and real-world wide-angle lenses such as the Roki-  non 8mm Cine Lens, Opteka 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='5mm Lens, and GoPro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' (iv)  DoF-based curriculum dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' To relieve the challenge  of the structure approximation in rectangling task, we pro-  posed a Degree of Freedom (DoF)-based curriculum learn-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Specifically, three curriculum stages are leveraged to  inspire our RecRecNet, namely, from similarity transforma-  tion (4-DoF) to homography transformation (8-DoF), and  to rectangling transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Thus, we also construct two  datasets (4-DoF dataset and 8-DoF dataset) to provide the  basic transformation knowledge, in which each dataset con-  tains 5,000 image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For the transformation synthesis,  we randomly perturb four corners of the original image, and  warp it to the target image following the previous work [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More Qualitative Comparison Results  As shown in Figure 12, we exhibit more qualitative com-  parison results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Our RecRecNet is capable of rectangling  the rectified wide-angle image in various scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We can  observe the deformed boundary is straightened and the im-  age content is rearranged to keep undistorted by RecRec-  Net, contributing a win-win rectification representation for  the wide-angle image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' By contrast, previous methods fail  to trade-off the image boundary and image content in the  rectified image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Their results usually show incomplete con-  tent or distorted distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For example, the rectangling  results produced by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15] rotate the original scene  and twist the object due to their line-preserving mesh defor-  mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More Vision Perception Results  As illustrated in Figure 13, we show more vision per-  ception results, including the object detection and seman-  tic segmentation results, which are derived by the popular  perception model Mask R-CNN [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As we can observe,  the learning model is misled by the deformed structure in  the original rectified image, leading to wrong perception  and missing perception results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As a benefit of the pro-  posed flexible TPS transformation module and DoF-based  curriculum learning, our RecRecNet can significantly help  the downstream vision tasks by eliminating the deformed  boundary issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' And the perception performance is recov-  ered especially near the image boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For the perfor-  mance degradation in rectified images, one straightforward  reason is that the perception model has never seen the data  with a deformed image boundary during training, and the  domain gap between test data and training data confuses  it on rectified images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Nevertheless, we would like to ex-  plore deeper reasons and disclose why the deformed geom-  etry makes the vision perception deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  A sample is shown in Figure 14 to visualize why the  12  Rectified Image  Cropped Image  RecRecNet  He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Wide-angle Image  Rectified Image  Cropped Image  RecRecNet  He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Wide-angle Image  Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More qualitative results compared to He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We show the wide-angle image, rectified image, cropped rectified image,  and the rectangling result of our RecRecNet, as well as the rectangling result by He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' [15] from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  deformed image boundary makes the perception deformed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  As described in the main manuscript, we find the deformed  boundary can introduce new features onto the feature maps,  which (i) form new semantics (leads to the wrong percep-  tion) or (ii) cause blind spots for original features (leads to  missing perception).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' The noticeable line artifacts and curve  artifacts can be observed at the outermost boundaries and  deformed boundaries respectively in shallow feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  In particular, the line artifacts are essentially generated by  zero padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Zero padding is a fundamental component of  prominent CNNs architectures, it serves to maintain the size  of the feature maps across the network by adding zero val-  ues to the feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' For the missing perception case, if  the features near the boundary are non-salient, the newly in-  troduced features can cover them and generate blind spots  for the perception model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As a result, the network cannot  recognize the elephant near the deformed image boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More Cross-Domain Evaluation Results  As mentioned in Section A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='1, we collect 300 rectified  wide-angle image results from the state-of-the-art rectifica-  tion methods [25, 50] to conduct the cross-domain evalua-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Their results are derived from other types of synthe-  sized datasets and real-world wide-angle lenses with dif-  ferent camera models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Figure 15 shows the cross-domain  evaluation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' While the new data space has never been  seen, our RecRecNet can achieve a promising generaliza-  tion ability by learning a flexible TPS transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' And  13  4seRectified  Image  RecRecNet  Detection and  Segmentation  Detection and  Segmentation  Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More detection and segmentation results by Mask R-CNN [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We show the rectified wide-angle image and its vision perception  result, and our rectangling result and its vision perception result, from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Rectangling  Rectified Image  Feature Maps  Shallow  Deep  Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' Failure case of Mask R-CNN [16] in regards to the  missing perception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We find that the deformed boundary can in-  troduce new features to the original feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' As a result, the  network cannot recognize the elephant near the deformed image  boundary as the original features are blinded in the deep feature  maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  we can observe that the rectangling results display reason-  able global distributions and visually pleasing local details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  Moreover, the predicted mesh locates at most spatial range  of the rectified image, demonstrating the effectiveness of  the learned rectangling transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  14  cribloey  cnbloa3  pomiioes  pomilose  cnbl  pOScnblo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='al  Ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='olsldj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='pninib  b6l20ulob6l2oul08cnbloe3  b6l20ulo8  06L20U03porrieloap  tp6lole  06L20W0COCKTOOD6L20Ub6120b6L20ul0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='80coc9ocklo82ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='ojn6rqsl9  lbugurlee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='ojn6rqsl9  lbugurl151151ee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='0n6rigalsRecRecNet  Rectified  Image  Mesh  Wide-angle  Image  Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' More results for the cross-domain evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content=' We show the wide-angle image, rectified image, predicted mesh, and rectangling  result of our RecRecNet from top to bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
+page_content='  15  ieuele' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtAzT4oBgHgl3EQfsf2x/content/2301.01661v1.pdf'}
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+DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained
+NFTs using Classifiers Learned on Graphs
+Hanna Kim
+gkssk3654@kaist.ac.kr
+KAIST
+Republic of Korea
+Jian Cui
+cj19960819@s2w.inc
+S2W
+Republic of Korea
+Eugene Jang
+genesith@s2w.inc
+S2W
+Republic of Korea
+Chanhee Lee
+leemember@s2w.inc
+S2W
+Republic of Korea
+Yongjae Lee
+lee@s2w.inc
+S2W
+Republic of Korea
+Jin-Woo Chung
+jwchung@s2w.inc
+S2W
+Republic of Korea
+Seungwon Shin
+claude@kaist.ac.kr
+KAIST
+Republic of Korea
+ABSTRACT
+As Non-Fungible Tokens (NFTs) continue to grow in popularity, NFT
+users have become targets of phishing attacks by cybercriminals,
+called NFT drainers. Over the last year, $100 million worth of NFTs
+were stolen by drainers, and their presence remains as a serious
+threat to the NFT trading space. Since NFTs are different from
+cryptocurrencies, existing work on detecting Ethereum phishers
+is unsuitable to detect NFT drainers. Moreover, no work has yet
+comprehensively investigated the behaviors of drainers in the NFT
+ecosystem.
+In this paper, we present the first study on trading behavior of
+NFT drainers and present the first dedicated NFT drainer detec-
+tion system. We extract data of 83M NFT transactions from the
+Ethereum blockchain and collect 742 drainer accounts from five
+sources. We find drainers have significantly different transaction
+context and social context compared to regular users. With the
+insights gained from our analysis, we design an automatic drainer
+detection system, DRAINCLoG, that uses graph neural networks to
+capture the complex relationships in the NFT ecosystem. Our model
+effectively captures NFT transaction contexts and social contexts
+using an NFT-User graph and a User graph, respectively. Evalu-
+ated on real-world NFT transaction data, we prove the model’s
+effectiveness and robustness.
+ACM Reference Format:
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo
+Chung, and Seungwon Shin. 2023. DRAINCLoG: Detecting Rogue Accounts
+Permission to make digital or hard copies of all or part of this work for personal or
+classroom use is granted without fee provided that copies are not made or distributed
+for profit or commercial advantage and that copies bear this notice and the full citation
+on the first page. Copyrights for components of this work owned by others than ACM
+must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
+to post on servers or to redistribute to lists, requires prior specific permission and/or a
+fee. Request permissions from permissions@acm.org.
+Conference’17, July 2017, Washington, DC, USA
+© 2023 Association for Computing Machinery.
+ACM ISBN 978-x-xxxx-xxxx-x/YY/MM...$15.00
+https://doi.org/10.1145/nnnnnnn.nnnnnnn
+with Illegally-obtained NFTs using Classifiers Learned on Graphs. In Pro-
+ceedings of ACM Conference (Conference’17). ACM, New York, NY, USA,
+17 pages. https://doi.org/10.1145/nnnnnnn.nnnnnnn
+1
+INTRODUCTION
+With the emergence of blockchain technology, Non-Fungible Tokens
+(NFTs) have revolutionized the digital creator economy. NFTs are
+digital assets, such as art or collectibles, with unique identification
+codes and metadata [20]. NFTs have attracted numerous content cre-
+ators and investors, and by 2021 the NFT market exploded, growing
+to around $22 billion [4].
+As a result, phishing scams have also emerged in the NFT ecosys-
+tem [27, 43]. According to blockchain analysis provider Elliptic, over
+$100 million worth of NFTs were stolen by NFT scammers, called
+drainers, in a one-year period from July 2021 to July 2022 [27]. NFT
+phishing scams continue to make headlines, causing significant
+damage to users [9, 10, 38]. For instance, a sophisticated phishing
+attack on Uniswap1 (the largest Ethereum-based decentralized ex-
+change) caused $8 million worth of damages to NFT users [25, 48].
+Efforts to combat NFT drainers have had limited effectiveness.
+The Opensea NFT marketplace implemented the policy of marking
+stolen NFTs as untradeable [40]. However, this is only effective
+when victims are able to notice and report the theft. The cryptowal-
+let Metamask implemented phishing warnings [11, 19], but were
+able to be bypassed by certain drainers [6].
+Even before NFT drainers, phishing attacks targeting cryptocur-
+rency were already considered a significant threat to trading secu-
+rity of Ethereum [12]. As a response to such attacks, researchers
+have proposed several methods to detect phishers. One line of
+research relies on using hand-crafted user features to detect scam-
+mers [14], while other approaches employ network representation
+learning by utilizing Node2Vec [54] or Graph Neural Networks
+(GNN) [13, 31, 55] to capture scammers.
+1uniswap.com
+arXiv:2301.13577v1  [cs.CR]  31 Jan 2023
+
+Conference’17, July 2017, Washington, DC, USA
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin
+However, these methods of detecting cryptocurrency phishers
+are unsuitable for detecting NFT drainers for the following rea-
+sons. First, features that are essential for cryptocurrency phisher
+detection, such as those that describe the liquidation process, are
+not shown in NFT drainers. In addition, cryptocurrency-focused
+detection systems cannot leverage individual NFTs and fail to ac-
+count for the multiple transaction types of NFTs. Thus, they cannot
+fully capture the differences between NFT drainers from regular
+users. This raises the need for an automatic detection system that
+captures various factors of the NFT ecosystem. This also motivates
+a comprehensive investigation on the transaction patterns of NFT
+drainers.
+To the best of our knowledge, the existing literature has not
+explored phishing scams in the NFT ecosystem. To fill this gap, we
+aim to investigate the trading traits of NFT drainers and propose an
+automatic detector that identifies suspicious trading behaviors. To
+this end, we collect over 83 million NFT transactions (between Jan-
+uary 2022 and August 2022) directly from the Ethereum blockchain.
+We also collect information of 742 reported drainers from five chan-
+nels, including Twitter[49] and Etherscan[21]. Then, we investigate
+the activity of drainer accounts in the NFT market and identify
+how their NFT trading patterns differ from those of regular users.
+We identify two key contexts that are critical in detecting drain-
+ers: their distinct NFT transaction context (rapid liquidation of
+NFTs at significantly lower prices) and their distinct social context
+(connections with affiliated users who also exhibit distinct trading
+patterns).
+However, capturing the intricate relationships between users
+and NFTs for drainer detection remains a challenging task. This is
+because the transaction history of millions of NFTs and the various
+types of interactions between millions of users must be considered.
+To address this challenge, we propose a novel GNN-based drainer
+detection system, DRAINCLoG. We use GNNs as they are well-suited
+for modeling relationships between users and NFTs in a graph
+structure. GNNs can incorporate both the features of nodes and
+edges, making them effective at identifying patterns of anomalous
+behavior among interconnected entities [7, 15, 46, 56]. The GNN-
+based structure allows DRAINCLoG to effectively capture the NFT-
+to-user and user-to-user relationships.
+Our proposed system utilizes two types of graphs: an NFT-User
+graph and a User graph. The NFT-User graph models interactions
+between users and NFTs, with two types of nodes and attributed
+edges, allowing us to obtain a representation of users’ transaction
+context by considering each NFT’s transaction history. The User
+graph models interactions between users, with attributed nodes
+and two types of edges, enabling us to capture the social context
+by integrating information on the users and their relationships.
+We combine these two contexts to leverage information from both
+graphs. The designed model significantly outperforms existing base-
+lines in detecting drainers. We also demonstrate the robustness of
+our model by simulating evasion attacks. Overall, we believe that
+our study will inspire further research and practical efforts to im-
+prove NFT trading security.
+In summary, the contributions of the work are listed below:
+Figure 1: Summary of NFT transaction types: a○ Mint, b○
+Sale, c○ Gift, and d○ Burn
+• We collect 83 million NFT transaction data from the Ethereum
+blockchain and 742 drainer accounts from five different chan-
+nels. Drainer accounts are publicly available for future re-
+search 2.
+• We present the first empirical study on NFT drainers and find
+that drainers have distinct characteristics and transaction
+patterns from regular users.
+• We propose a graph-based automatic NFT drainer detector,
+DRAINCLoG, that can capture NFT transaction context and
+user social context, and provide experimental results that
+show our detector outperforms existing detection methods.
+• We verify the robustness of DRAINCLoG by simulating eva-
+sion attacks.
+2
+BACKGROUND AND MOTIVATION
+2.1
+Background
+In this section, we outline the required background related to NFTs
+and phishing attacks in the NFT ecosystem. Then, we detail the
+motivation of our study.
+Non-Fungible Token (NFT) is a cryptographic asset on the blockchain
+with unique identification codes and metadata that distinguish them
+from each other. NFTs guarantee ownership of unique digital assets,
+such as images, video files, and even physical assets. By 2017, NFTs
+on the Ethereum blockchain gained attention with the creation of
+collectibles such as CryptoPunks [8]. The Ethereum blockchain is
+favored among NFT traders and accounts for around 80% of NFT
+trading volume [42]. An NFT collection is a set of NFTs. NFTs from
+the same collection share similar characteristics, with unique varia-
+tions from each other. These variations can make some NFTs much
+more valuable than others, even within a collection.
+NFT Transaction Types Here, we introduce the four types of NFT
+transactions: mint, sale, gift, and burn, as illustrated in Figure 1.
+a○ Mint. An NFT is created by minting, the process of inscribing
+a digital asset to the blockchain. Minted NFTs can be listed and
+traded on NFT marketplaces, such as OpenSea [39] and Rarible [41].
+b○ Sale. A sale is a process of transferring an NFT ownership to
+another account for payment. NFTs are typically traded with Ether,
+the native cryptocurrency of Ethereum, and sometimes fungible
+tokens. Users can partake in sales in two ways: buying and selling.
+c○ Gift. On the other hand, a gift is a process of handing over
+ownership without any payment. Gifts usually occur when the
+sender is related to the recipient. For instance, an individual can
+create multiple accounts and gift NFTs to move them between
+accounts. Sometimes it can be used between users to avoid moni-
+toring when manipulating markets by wash trading [17]. Users can
+partake in transfers in two ways: gifting-in and gifting-out.
+2will be made available on acceptance
+
+mint
+sale
+gift
+d
+burn
+NULL
+NULLDRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+d○ Burn. An NFT cannot be deleted from the blockchain, but it
+can be ‘burned’. Burning is the process of sending NFTs to an inac-
+cessible address (NULL), which will remove them from circulation.
+Burning is used for various purposes, such as adjusting the supply
+of NFTs, operating a collection’s community, etc.
+Phishing attacks in the NFT ecosystem is commonly called
+draining [32, 47], and we will use this term in this paper. The main
+goal of NFT drainers is to “drain” (steal) NFTs from victims, although
+cryptocurrencies and private information could also be targeted.
+We detail the procedures for how NFT phishing scams operate by
+dividing them into three steps: (1) spreading phishing websites, (2)
+draining NFTs, and (3) cashing out drained NFTs. Figure 2 illustrates
+the process.
+(1) Spread phishing websites. NFT drainers mainly use two meth-
+ods to spread phishing websites to victims. First, they 1.1) use social
+media, such as Twitter. They can make social media accounts mas-
+querading as official accounts of popular NFTs, sometimes even
+compromising them. The drainers upload posts linking to scam
+sites on these channels. Another method is to 1.2) use phishing token
+airdrops. Airdrops are commonly used in marketing campaigns to
+promote creators’ projects by sending free tokens [50]. However,
+it can also be abused by drainers to trick victims. A drainer can
+send fake tokens to target wallets, tricking victims into clicking a
+phishing website link in the token’s description.
+(2) Drain NFTs from victims. The drainers steal NFTs from vic-
+tims who entered phishing websites by either 2.1) stealing login
+credentials of crypto wallets or 2.2) abusing smart contracts. The first
+method is similar to the traditional phishing methods that target
+victims’ passwords. When victims type in login credentials to their
+crypto wallet, drainers can record the information using keyloggers,
+etc [45]. With the stolen credentials, drainers can access victims’
+accounts and transfer NFTs to their own wallets. Alternatively,
+drainers can abuse smart contracts to deceive victims. Drainers can
+create malicious smart contracts, that include functions such as
+setApprovalForAll, and lure victims with valuable tokens to sign a
+transaction [53]. SetApprovalForAll is an important method used
+for NFT trading. When a user sells NFTs on a marketplace, the user
+needs to call this function to authorize the marketplace in trans-
+ferring the NFTs from the user’s account to the buyer’s. However,
+by calling the function on a phishing website, a victim grants the
+drainer permissions to transfer all the victim’s tokens [35].
+(3) Cash out drained NFTs. Scammers who target cryptocurrencies
+can cash out stolen assets by using crypto exchanges, and some-
+times through mixing services [18, 55]. Conversely, NFT drainers
+must first sell the stolen NFTs by listing them on different market-
+places [18]. To combat NFT scams, OpenSea [39], the largest NFT
+marketplace, made a policy of disabling the buying, selling, and
+gifting of stolen items [40].
+2.2
+Motivation
+NFTs have attracted the attention of investors and criminals alike.
+Phishing scams targeting NFT users continue to make headlines [9,
+10, 38]. For instance, a sophisticated phishing attack on Uniswap
+NFT holders caused damages of $8 million when users were tricked
+to approve malicious transactions [25, 48]. In another instance,
+(1) Spread phishing websites
+setApproval
+ForAll
+(2) Drain NFTs 
+from victims
+www.phish.com
+(3) Cash out drained NFTs
+Figure 2: The process of phishing attacks in the NFT ecosys-
+tem: (1) spreading phishing websites, (2) draining NFTs from
+victims, and (3) cashing out drained NFTs.
+the official Instagram account of the most valuable NFT, Bored
+Ape Yacht Club, was compromised and used in a phishing attack
+in April [22, 29]. The drainers stole NFTs by luring victims with
+free tokens through the Instagram account. The total damage was
+valued at $2.7 million, including a token worth $354,500 (at that
+time of exchange rate).
+Alarmingly, there have been a wealth of tools made available to
+assist NFT draining. Software packages for NFT draining are being
+sold between $29 – $149 on dedicated websites [33] and the dark-
+web [16]. Some groups even publish a free version of their draining
+code on GitHub in order to advertise their online stores [33]. The
+accessibility of draining tools lowers the barriers to entry for future
+NFT drainers. To prevent their tools, MetaMask, a famous crypto
+wallet, updated a feature to warn users when transactions request
+setApprovalForAll [11, 19]. However, NFT drainers also developed
+their tools to bypass this update [6].
+Although the damage caused by NFT drainers is increasing, no
+previous work has yet conducted an in-depth measurement study
+or proposed a detection method for NFT draining. Several methods
+have been proposed to detect phishers targeting cryptocurrency [13,
+14, 31], but they are unsuitable to apply to the NFT ecosystem for
+the following reasons.
+First, essential features to detect cryptocurrency phishers do
+not apply for NFT drainers. For instance, NFT drainers do not
+follow the liquidation process of cryptocurrency phishers. Stolen
+cryptocurrencies are generally cashed out through exchanges after
+a mixing process, which is one of the most decisive features in
+detecting cryptocurrency phishers [14]. On the other hand, stolen
+NFTs cannot be laundered because each NFT is trackable. Also,
+NFTs cannot be directly cashed out in exchanges. Therefore, NFT
+drainers must list NFTs on the market for sale in order to cash
+out. The ineffectiveness of previous features raises the need for a
+measurement study on NFT drainers to gain insights on how to
+detect phished NFTs. In Section 4, we investigate the activity of
+NFT drainers to identify trading traits that can detect NFT draining.
+Second, existing methods cannot consider complex contexts
+within the NFT ecosystem. Unlike cryptocurrency, where coins
+are interchangeable and have equal value, NFTs are of distinct iden-
+tities with dynamic values, making complex transaction patterns
+(factoring price and frequency). To fully understand a user’s trad-
+ing habits, the transaction history of each NFT traded by the user
+should be taken into account. In Section 5.2, we describe our NFT
+drainer detection system, DRAINCLoG, which includes a module
+
+NFT
+NFT
+NFT
+NFT
+NF
+NFT
+NFT
+NFT
+alamy
+ImageID:2J6YXM5
+www.alamy.comConference’17, July 2017, Washington, DC, USA
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin
+Internal Transaction (Token Transfer Log)
+External  Transaction
+A
+B
+s
+ⓐ Sale NFT for ETH
+A
+B
+ⓑ Sale NFT for FT
+FT
+A
+B
+ⓒ Gift NFT
+Figure 3: Summary of collecting NFT transaction data from
+the Ethereum blockchain. A purple line indicates an inter-
+nal transaction, and a green line indicates an external trans-
+action. We classify each NFT transaction record between
+users into sales ( a○ and b○) or gifts ( c○) based on payment
+method.
+Table 1: Summary of collected NFT transaction records.
+*w/o spam: a summary without spam NFTs which were
+minted multiple times for fraud, etc.
+Type
+Collection
+NFT (*w/o spam)
+52,889,522 (52,514,506)
+Account
+3,421,654
+Transaction
+Mint (*w/o spam)
+53,653,562 (47,655,998)
+Sale
+16,642,569
+Gift
+12,234,533
+Burn
+799,546
+Total transaction records
+83,329,082
+Period
+January 1,2022 ∼ August 31, 2022
+specifically designed to leverage a user’s NFT transaction context
+(NFT Transaction Context Extractor).
+Third, previous studies on detecting cryptocurrency phishing
+cannot fully capture relationship between NFT users. While cryp-
+tocurrency transactions are only limited to transfers, NFT transac-
+tions can be further classified into four distinct categories. Particu-
+larly, there are two transactions types between NFT users (Figure
+1): sales and gifts. Our investigation shows that this distinction can
+be an significant indicator when interpreting relationships between
+users. Section 5.3 explains how DRAINCLoG uses a module (Social
+Context Extractor) to capture user relationships in the NFT trading
+network.
+3
+DATASET CONSTRUCTION
+This section summarizes our data collection approach and the
+datasets used in this study. We collect two datasets in our paper:
+NFT transaction data and NFT drainer accounts.
+Fetching NFT transaction records from Ethereum blockchain:
+To understand a token’s transmission history on the Ethereum
+blockchain, we can use transfer logs, which are written by smart
+contracts on the blockchain when token transfers occur. With this
+information, we can identify changes in ownership of tokens, such
+as who sends the tokens to whom.
+There are two types of NFT transactions between users: sales
+and gifts. In the case of sales, most NFTs are traded on the market-
+place for Ether or various Fungible Tokens (FTs). However, because
+the NFT transfer logs do not have payment information, we need
+additional reference data depending on the payment types. If an
+NFT buyer pays in Ether, it is recorded as an external transaction,
+which is used to send Ether. If the buyer paid in FT, it is recorded
+Table 2: Summary of collected drainer accounts from vari-
+ous channels.
+Channel
+# accounts
+Channel
+# accounts
+ScamSniffer
+462
+Chainabuse
+150
+Etherscan
+127
+Twitter
+105
+CryptoScamDB
+49
+Total
+742
+Period
+January 1,2022 ∼ August 31, 2022
+in an FT transfer log. Thus, we collect external transactions and
+transfer logs related to NFTs and FTs.
+Figure 3 shows our logic for collecting NFT transaction data
+from the Ethereum blockchain. If there is an NFT transfer log from
+user 𝐴 to user 𝐵 and an external transaction in which 𝐵 sends Ether
+to 𝐴, we decide the NFT was sold for Ether ( a○). Instead, when an
+FT transfer log exists in which 𝐵 sends FTs to 𝐴, we decide the
+NFT was sold for FTs ( b○). Otherwise, we regard that there was
+no payment for the NFT, and the NFT transaction from 𝐴 to 𝐵 is
+considered as gifting ( c○).
+According to Elliptic[3], NFT draining grew rapidly and has been
+on the rise since January 2022 [27]. As a result, we focus on data
+with block_timestamp between the first day of 2022 to August 31,
+2022. We obtain more than 83 million transactions from this period
+for 53 million NFTs. The data includes transactions from over 3
+million unique accounts. Our NFT transaction data is summarized
+in Table 1. In our analysis, we only consider mints, sales, and gifts
+because transaction records of burns account for less than 1% of
+total transactions.
+Crawling NFT drainer accounts: We crawled drainer accounts
+reported for NFT phishing scams from five websites: Twitter, Scam-
+Sniffer, Etherscan, CryptoScamDB, and Chainabuse. (1) Twitter [49]
+is utilized by NFT users as a channel to share information on drainer
+accounts. We collect tweets that mention the phrase “NFT” and one
+of the following keywords: phish, hack, drain, stole. From the col-
+lected tweets, we manually extract drainer Ethereum accounts. (2)
+ScamSniffer [5] collects malicious accounts by visiting suspicious
+sites that trick users into making dangerous transactions. (3) Ether-
+scan [21], an Ethereum block explorer, provides a list of Ethereum
+addresses reported for phishing/hacking. (4) CryptoScamDB [2] col-
+lects information on scams in the crypto space, including malicious
+accounts, URLs, and descriptions. We only consider reports using
+the same criteria in (1) Twitter. (5) Chainabuse [1] provides scam
+reports with descriptions across multiple blockchains. We collected
+Ethereum addresses reported under the NFT Scam category.
+We crawled all the reports from the above channels based on
+each criteria before September 1, 2022. Note that there are duplicate
+reported accounts on different sites. From the reported accounts,
+we define the accounts that gifted NFTs at least once as drainers. Ac-
+cording to this definition, we label 742 unique accounts as drainers
+(summarized in Table 2).
+4
+DRAINER ACTIVITY CHARACTERIZATION
+Since drainers have malicious intent, they are likely to exhibit
+different behaviour patterns in NFT trading from regular users.
+In this section, we look into distinguishable traits of drainers that
+
+DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+(d) Holding times - Sell
+(e) Holding times - Gift-out
+(c) Out-in ratio
+(a) Active timespan (days)
+(b) Gift-in ratio
+(c) Out-in ratio
+Figure 4: CDFs of (a) active timespan, (b) gift-in ratio, and (c) out-in ratio, where the y-axis denotes proportion of users. CDFs
+of holding times of different user types (regular users, affiliated users, and drainers) based on out-transaction types: (d) sell
+and (e) gift-out, where the y-axis denotes proportion of NFTs. We include an enlarged graph for holding times smaller than a
+day.
+motivate the design of our NFT drainer detection model. We conduct
+a measurement study with NFT transaction records from the first
+day of 2022 to July 31, 2022, including 645 drainer accounts. First, by
+comparing primary trading features with regular users, we verify
+that most drainer accounts are used only for draining. Then, we
+focus on their liquidation patterns and how they are distinct from
+regular transactions. We track drained NFTs and find that they
+have different NFT transaction context and social context from
+regular users.
+4.1
+Trading behavior of drainers
+As mentioned in 2.1, there are many efforts to prevent phishing
+scams in the NFT space, such as sharing information about drainers,
+phishing websites, and drained NFTs. If drainer accounts become
+known to be suspicious, it is nearly impossible for them to trade
+NFTs with benign users. Thus, we expect that drainers are more
+likely to 1) have short-lived accounts and 2) use their accounts only
+for draining.
+In order to investigate these two assumptions, we analyze the
+trading activity of 645 NFT drainers. For comparison, we randomly
+extract 10,000 users from our data of 3 million regular users. From
+this set, users with only one transaction record were excluded since
+their active timespan cannot be calculated. This left 6,658 regular
+users to be used in the analysis. We compare these two groups
+along three dimensions.
+First, we measure how long they traded by calculating the active
+timespan: the time difference (in days) between the first and last
+transaction recorded for each user. Figure 4(a) shows that drainers
+trade for a shorter period than regular users. This phenomenon is
+also similarly observed in Ether phishing accounts [14].
+Second, we look into how NFT drainers utilize their accounts.
+Previous work to detect cryptocurrency phishers [14, 31] only fo-
+cused on the amount and frequency of user transactions. However,
+multiple transaction types exist in NFT trading. This is an impor-
+tant consideration since we assume drainers will be gifted NFTs
+from victims (as opposed to buying or minting NFTs). Thus, we
+calculate the ratio of gifting-in to all in-transaction types (gifting-in,
+buying, and minting). In Figure 4(b), the majority of drainers have
+a relatively high proportion of NFTs gifted; 75.2% of drainers obtain
+NFTs only through gifting-in in the three in-transaction types.
+Table 3: Statistics on transactions about drained NFTs ac-
+cording to behaviors right after draining: sell, gift-out, and
+none(just holding). # gifting is the number of giftings before
+the first sale after draining, and % sold is the percentage of
+NFTs eventually sold.
+Type
+# gifting
+# drained NFTs
+% sold
+Sell
+0
+11,195 (41.8%)
+100%
+Gift-out
+1
+6,426 (24.0%)
+75.1%
+≥ 2
+1,125 (4.21%)
+39.3%
+None
+0
+8,065 (30.1%)
+0%
+Total
+26,811 (100%)
+61.4%
+In addition, we analyze how likely drainers are to transfer out.
+Drainers looking to liquidate do not wish to hold their NFTs. There-
+fore, drainers are likely to transfer their NFTs out. We measure
+out-in ratio, the ratio of the number of out-transactions to in-
+transactions. In Figure 4(c), we observe drainers have a higher
+out-in ratio than regular users. Regular users generally have a
+lower out-in ratio: 38.1% of them did not make any out-transactions
+at all. On the other hand, drainers have higher out-in ratios: 75.9%
+of them make out transactions on more than half of their NFTs.
+This suggests drainers have different intentions from regular users,
+who are more likely to use NFTs for collecting or investing.
+By combining the observations above, we conclude that most
+drainer accounts are for draining purposes only. Additionally, we
+further analyze differences between drainers and regular users
+along 19 dimensions (more details in Appendix D) and utilize them
+in our detection method.
+4.2
+Liquidation behavior of drainers
+In this section, we dive deep into the liquidation process of drained
+NFTs. The uniqueness of NFTs enables us to track the transaction
+history of each NFT. Since drainer accounts are only used for drain-
+ing, we assume all gifted-in NFTs were stolen from victims.
+Alternate accounts. We find a total of 26,811 NFTs that were
+gifted to drainers (Table 3). Of these, 42.2% were sold directly, 27.3%
+were gifted-out to other users, and the rest (30.1%) remained in the
+drainer’s wallets. We define the accounts that were gifted NFTs
+from drainers as Affiliated users.
+Only 17.5% of drainers do not have affiliated users. In other
+words, most drainers (82.5%) have one or more affiliated users and
+
+1.00.20.60.40.0-- Known Drainer AccountsRandom UsersAffiliated Users-- Known Drainer AccountsRandom Users0.00.20.60.40.01.00.00.8Conference’17, July 2017, Washington, DC, USA
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin
+Table 4: Statistics of holding times of drained NFTs depend-
+ing on user type along out-transaction types: sell and gift-
+out. Percent decrease is the measure of the decrease from
+𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔 to 𝐻𝑇_𝑑𝑟𝑎𝑖𝑛 as a percentage of 𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔.
+# drained NFTs
+Case
+Sell
+Gift-out
+𝐻𝑇 _𝑑𝑟𝑎𝑖𝑛 < 𝐻𝑇 _𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔
+8,077 (88.9%)
+6,210 (90.9%)
+𝐻𝑇 _𝑑𝑟𝑎𝑖𝑛 = 𝐻𝑇 _𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑚𝑖𝑛
+6,498 (71.4%)
+5,034 (73.7%)
+Total
+9,085 (100%)
+6,832 (100%)
+Stats of percent decrease
+Mean
+87.7%
+87.5%
+Standard deviation
+29.5
+29.8
+use them to liquidate drained NFTs indirectly. We find 637 affiliated
+users including 15.4% that are related to two or more drainers. The
+most overlapped affiliated user is connected with 16 drainers (more
+details are in Appendix A). The fact that many drainers choose
+to liquidate through the same affiliated accounts suggests a close
+relationship between them. 60% of affiliated users get NFTs only
+through gifting-in, while the rest of them (40%) participate in buying
+and minting like regular users (Appendix D.1). From the result,
+we observe that most affiliated users exhibit similar behaviors to
+drainer accounts, but there are also a significant number of affiliated
+accounts that engage in general NFT trading.
+Rapid liquidation: We now analyze how quickly drainers liqui-
+date NFTs by measuring the holding times of each NFT. We define
+holding time as the timespan a user held ownership of an NFT, and
+measure it by taking the difference (in days) of the in-transaction
+and out-transaction. Holding times can vary greatly depending on
+the user’s investment strategy and characteristics of the NFT, such
+as market price and rarity. We measure the holding times of all
+users that owned the NFT for the each of the 18,746 drained NFTs
+with out-transactions. Note that the holding time can be longer
+than seven months (our collection period) because we refer to NFT
+transaction data before 2022 to minimize the bias. We compare
+drainer holding times with those of affiliated users and those of
+regular users along two out-transaction types: sell and gift-out.
+Drainers and affiliated users show similar distributions which are
+noticeably different from that of regular users. Figure 4 (d) shows
+they sell more than 80% of NFTs within a day, which is twice more
+likely than regular users. They also have short holding times before
+gifting, but drainers gift NFTs much faster compared to affiliated
+users (Figure 4 (e)). By integrating these observations with the fact
+that up to 75% of NFTs sent to affiliated users are sold (Table 3), we
+notice that drainers alternate between two strategies of liquidation:
+(1) directly selling NFTs quickly or (2) quickly gifting-out NFTs to
+affiliated users for selling.
+To make comparisons with regular transactions, we calculate the
+average holding time (𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔) for each NFT as a reference
+to a drainer’s holding time(𝐻𝑇_𝑑𝑟𝑎𝑖𝑛). As shown in Table 4, 90%
+of 𝐻𝑇_𝑑𝑟𝑎𝑖𝑛s are shorter than 𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔s, with an average of
+87.7% percent decrease regardless of the out-transaction types. We
+also observe that 𝐻𝑇_𝑑𝑟𝑎𝑖𝑛 is the minimum holding time for 70%
+of NFTs. From these results, we identify that drainers deviate from
+regular transaction patterns in terms of holding time.
+Table 5: Statistics on sale price of drained NFTs compared
+to their market price (Price𝑎𝑣𝑔, Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡). Only consider
+NFTs sold after draining with other sale records of more
+than one. p.d. denotes the measure of decrease from each
+market price to Price𝑑𝑟𝑎𝑖𝑛 as a percentage of the market
+price.
+Stats of p.d.
+Case
+# drained NFTs
+Mean
+Std
+Price𝑎𝑣𝑔 > Price𝑑𝑟𝑎𝑖𝑛
+8,214 (74.0%)
+37.3%
+24.9%
+Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡 > Price𝑑𝑟𝑎𝑖𝑛
+8,490 (76.5%)
+39.1%
+24.2%
+Total
+11,100 (100%)
+Bargain prices: The measurements made above raise a question;
+how do drainers liquidate drained NFTs so quickly? To answer this
+question, we compare the sales prices of drainers/affiliated users
+(Price𝑑𝑟𝑎𝑖𝑛) with the market prices. However, unlike stocks or cryp-
+tocurrencies, each NFT has its own unique value. NFTs are also
+prone to price changes in response to market supply and demand
+[34]. Thus it is impossible to define its market price except when
+there is a sale transaction. Therefore, to use as a market price, we
+use two baselines: Price𝑎𝑣𝑔 and Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡. Price𝑎𝑣𝑔 is the average
+sale price, and Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡 is the sale price of the sale made closest
+to when the drainer made the sale. We observe that drainers sell 74%
+and 76% of their NFTs cheaper than the two baselines respectively
+with an average price decrease of 37% and 39% (Table 5).
+In summary, our analysis has revealed that NFT drainers ex-
+hibit distinct patterns of NFT transactions and social relationships
+compared to regular users. Specifically, drainers tend to have ir-
+regular NFT transaction contexts, such as quickly liquidating
+NFTs at prices lower than the market value. Additionally, drainers
+often have unique social contexts, such as making sales through
+affiliated users and connections to the same affiliated users.
+5
+DESIGN OF NFT DRAINER DETECTOR
+Our findings in Section 4 suggest that understanding both the trans-
+action context and social context of users is crucial for detecting
+NFT drainers. However, capturing the intricate relationships be-
+tween users and NFTs for drainer detection remains a challenging
+task. To fully grasp a user’s transaction context, it is necessary to
+take into account the transaction history of each NFT traded by the
+user. Additionally, to fully understand a user’s social context, it is
+important to consider the different types of interactions between
+users.
+To address this complexity, we designed a graph-based NFT
+drainer detection model, DRAINCLoG. Figure 5 illustrates the over-
+all architecture. The model creates a comprehensive representation
+of each user by using a transaction context extractor and a social
+context extractor. Each extractor is trained to learn the relevant
+context on a NFT-User graph and a User graph, respectively. The
+NFT-User graph models NFT-to-user interactions using NFT owner-
+ship edge attributes, and the user transaction contexts can be fully
+captured by aggregating the interaction history of all of their owned
+NFTs. The User graph models comprehensive user-to-user interac-
+tions using user node attributes, which detail users’ trading behavior,
+
+DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+Drainer 
+Classifier
+NFT-User Graph
+User Graph
+Feature 
+Engineering
+Transaction context 
+representation 
+⋯
+Transaction 
+Context 
+Extractor
+Social
+Context 
+Extractor
+Social context 
+representation
+⋯
+Attributed 
+Node   
+Attributed
+Edge    
+User node
+NFT node
+NFT gift edge
+NFT sale edge
+NFT ownership edge
+Drainer
+Figure 5: The overall architecture of DRAINCLoG. First, NFT transaction records and user accounts obtain attributes through
+feature engineering and are used to construct a NFT-User graph and a User graph. The transaction context extractor and social
+context extractor are trained to extract information from each graph, respectively. The user features and the representations
+from two extractors are concatenated together and fed into the classifier.
+and two types of edges. The two contexts are then combined and
+fed into a classifier to integrate the information.
+5.1
+Feature Engineering
+Before learning high-level user representations, we perform fea-
+ture engineering based on observations in Section 4 to obtain NFT
+ownership edge attributes and user node attributes.
+5.1.1
+NFT ownership edge attributes. To capture different transac-
+tion behaviors, we create representations of how users interact with
+NFTs, which is used in the NFT-User graph. We use the following 7
+features to represent NFT ownership:
+(1) Holding time: Holding time is the timespan a user held
+ownership of the NFT. If there is no out-transaction, we
+calculate it by taking the difference (in days) between the
+in-transaction and the last day of our collection period. As
+drainers tend to sell or gift NFTs faster than regular users,
+their holding times are shorter than those of regular users.
+(2) In-transaction type & Out-transaction type: Each trans-
+action type is a categorical feature of how the user received
+the NFT (buy or gift-in) and what the user did with the NFT
+(sell, gift-out, or hold), respectively. Drained NFTs have the
+gift-in transaction type.
+(3) In-price & Out-price: Each is the price (in Ether) the user
+sent to receive the NFT and the price the user received when
+sending the NFT out, respectively. It is zero if the transaction
+type is gifting, and -1 if no out-transaction was made. The
+in-price and out-price are significant because drainers sell
+their NFTs cheaper than regular transactions.
+(4) Average holding time & Average sale price: For compar-
+ison, we include the NFT’s average holding time and sales
+price across all owners. This can be used as a reference in
+finding anomalies in holding times and pricing.
+5.1.2
+User node attributes. To comprehensively understand the
+social relationships between users, we created detailed represen-
+tations of their trading behavior, which is used in the User graph.
+We introduce 19-dimensional user node attributes that take into
+account NFT characteristics such as collections and transaction
+types, which helps the model to identify distinct behavior patterns
+of drainers.
+(1) Active timespan: The time difference between the first and
+the last transaction. Drainers are more likely to have a short
+active timespan than regular users.
+(2) Gift-in ratio: The ratio of gifting-in to all in-transactions
+(minting, buying, gifting-in). Most drainers obtain NFTs only
+through gifting.
+(3) Out-in ratio: The ratio of out-transactions to in-transactions.
+Drainers have a higher out-in ratio than regular users.
+(4) Number of each transaction type: Transaction types con-
+sidered are minting, buying, gifting-in, selling, and gifting-
+out. Drainers participate in selling and gifting-in more than
+other types. (Appendix D.1)
+(5) Number of collections for each transaction type: NFT
+users commonly form communities based on collections [37]
+and trade NFTs from similar collections. Unlike regular users,
+drainers tend to trade more kinds of collections than regular
+users. The transaction types are the same as above. (Appen-
+dix D.2)
+(6) Number of neighbors for each transaction type: Ac-
+counts that made a transaction with each other are con-
+sidered neighbors. Drainers tend to have more neighbors
+from gift-in transactions. The transaction types are the same
+as above, excluding minting (which produces no neighbors).
+(Appendix D.3)
+(7) Frequency of gift-ins & sales: The number of gifting-in/selling
+transactions divided by the active timespan. Gift-ins and
+sales occur more frequently for drainers. (Appendix D.4)
+5.2
+NFT Transaction Context Extractor
+To fully understand a user’s NFT ownership, we leverage the trans-
+action histories of all the user’s NFTs. We construct a NFT-User
+graph to model ownership changes in NFTs and train an extractor
+for the graph to extract NFT transaction context of each user. From
+the NFT-User graph, we first obtain the transaction context of each
+
+NFT
+NFT
+NFT
+NFT
+NF
+NFT
+NFT
+NFT
+alamy
+ImageID:2J6YXM5
+www.alamy.comConference’17, July 2017, Washington, DC, USA
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin
+NFT by aggregating its transaction history. Finally, to get a repre-
+sentation of the user’s NFT transaction context, we aggregate the
+transaction contexts of all of their owned NFTs.
+5.2.1
+NFT-User graph construction. We construct an undirected
+graph 𝐺(𝑈, 𝑁, 𝐸𝑇 ). There are two types of nodes: user nodes 𝑈 ,
+NFT nodes 𝑁, and the 𝑢 nodes and 𝑛 nodes can be connected
+with an attributed edge 𝑒 ∈ 𝐸𝑇 , called NFT ownership attributes.
+Additionally, 𝑁𝑒 (𝑛) is the node 𝑛’s one-hop neighbors (users who
+traded NFT 𝑛), 𝑁𝑒 (𝑛) = {𝑢′ ∈ 𝑈 |(𝑢′,𝑛) ∈ 𝐸𝑇 }, and 𝑁𝑒 (𝑢) is the
+set of NFT nodes in node 𝑢’s one-hop neighbors, 𝑁𝑒 (𝑢) = {𝑛′ ∈
+𝑁 |(𝑛′,𝑢) ∈ 𝐸𝑇 }.
+5.2.2
+NFT transaction context extraction. A transaction context ℎ𝑁
+of an NFT node 𝑛 is updated based on attributed edges connected
+to, denoted as 𝑇𝑛 = {𝑡𝑢1,𝑡𝑢2, . . . ,𝑡𝑢𝑚 }, where 𝑢𝑖 (1 ≤ 𝑖 ≤ 𝑚) is one
+of 𝑁𝑒 (𝑛), and 𝑚 is the size of 𝑁𝑒 (𝑛). Then, we aggregate them with
+convolution layer as follows :
+ℎ𝑁
+𝑛 = 𝜎
+�
+𝑊 𝑁 · 𝐴𝐺𝐺𝑁 (𝑇𝑛)
+�
+where 𝜎 is an activation function, and 𝑊 𝑁 is the trainable matrix
+for learning. We use mean-pooling as an aggregation function to
+represent the transaction context of each NFT.
+Next, a user representation ℎ𝑈 is updated by all neighboring NFT
+nodes’ transaction context vectors. For a user 𝑢, the representation
+ℎ𝑈𝑢𝑛 of each NFT 𝑛 ∈ 𝑁𝑒 (𝑢) is obtained as follows.
+ℎ𝑈
+𝑢𝑛 = 𝑊 𝑈 · 𝑐𝑜𝑛𝑐𝑎𝑡(𝑡𝑢𝑛,ℎ𝑁
+𝑛 )
+where𝑡𝑢𝑛 is𝑢’s ownership of𝑛, andℎ𝑁𝑛 is the transaction context
+vector of 𝑛. We concatenate the two and apply a linear transforma-
+tion with the learnable matrix 𝑊 𝑈 .
+Finally, we integrate all of the NFT transaction context vectors
+{ℎ𝑁𝑛1,ℎ𝑁𝑛2, . . . ,ℎ𝑁𝑛𝑣 } (𝑣 is the size of 𝑁𝑒 (𝑢)). To be sensitive to irregu-
+lar NFT transaction patterns, we implement attention mechanisms
+to combine each NFT ownership vector. Specifically, we adapt a
+multi-head graph attention operation to reduce the effects of noise.
+𝑧𝑢𝑛 = LeakyReLU(𝑎 · ℎ𝑈
+𝑢𝑛)
+ℎ𝑈
+𝑢 =
+∑︁
+𝑛∈𝑁𝑒 (𝑢)
+𝛼𝑢𝑛𝑧𝑢𝑛
+where 𝑧𝑢𝑛 is an attention score calculated by taking the dot
+product of learnable weight 𝑎 and applying LeakyReLU. 𝛼𝑢𝑛 is the
+normalized attention score using the softmax function. Lastly, we
+compute the final representation by averaging the attention head
+outputs.
+Training. The final representation feeds into the classification layer
+for classification. The extractor updates the trainable parameters
+to better learn features that distinguish drainers from regular users.
+We use cross-entropy loss as the loss function.
+5.3
+Social Context Extractor
+Most drainers choose to liquidate through affiliated accounts. More-
+over, some affiliated accounts are used by multiple drainers, which
+suggests a close relationship between the co-users. Thus, the rela-
+tionships between users are essential to detect drainers. To model
+Table 6: Dataset statistics for training and evaluation. Ratio
+refers to the ratio of drainers to sampled regular users. For
+evaluation datasets, each number is the average value over
+10 runs.
+Dataset
+Ratio
+# central nodes
+# total nodes
+# transactions
+Training
+𝐷0
+1:29
+19,350
+1,680,371.0
+19,269,001.0
+Evaluation
+𝐷1
+1:10
+1,067
+1,326,162.2
+15,340,788.1
+𝐷2
+1:20
+2,037
+1,520,345.9
+18,918,056.6
+𝐷3
+1:50
+4,947
+1,759,077.9
+23,672,245.3
+𝐷4
+1:100
+9,797
+1,922,109.6
+26,504,190.9
+user interactions, we construct a User graph and use it to train an
+extractor to learn the social context of users.
+5.3.1
+User graph construction. Trading between users can be con-
+structed as a graph 𝐺(𝑈, 𝐸, 𝑅,𝑋𝑈 ), where 𝑈 is the set of user nodes,
+and 𝐸 is labeled edges (𝑢𝑖,𝑟,𝑢𝑗), where 𝑟 ∈ 𝑅 = {𝑠𝑎𝑙𝑒,𝑔𝑖𝑓 𝑡} is a re-
+lation type. Each user node has 19-dimensional user node attributes,
+𝑋𝑈 . If 𝑢𝑖 transfers an NFT to 𝑢𝑗, then 𝑢𝑖 and 𝑢𝑗 are connected with
+an edge 𝑒 ∈ 𝐸 with a label gift.
+5.3.2
+Social context extraction. Transaction types are important to
+understand user relationships in the NFT ecosystem. To consider
+transaction types between users, we use the R-GCN [44] model,
+where edges can represent different relations.
+From the user graph, we obtain a representation vector of a user
+node 𝑢 updated by its neighboring user nodes. The propagation at
+(𝑙 + 1)-th layer of R-GCN with 𝐿 layers is as follows:
+ℎ𝑙+1
+𝑢
+= 𝜎
+�
+𝑊 𝑙ℎ𝑙
+𝑢 ·
+∑︁
+𝑟 ∈𝑅
+AGGU( 1
+𝑐𝑢,𝑟
+𝑊 𝑙
+𝑟 ℎ𝑙
+𝑣), ∀𝑣 ∈ 𝑁 (𝑢)𝑟
+�
+where 𝜎 is an activation function, and 𝑊 𝑙 is a learnable matrix
+shared among all nodes at 𝑙-th sub-layer. 𝑁 (𝑢)𝑟 is neighboring user
+nodes under relation𝑟 of𝑢. We use mean-pooling as the aggregation
+function AGGU.
+Training. We train this module in the same process as the NFT
+transaction context extractor by feeding the outputs into the classi-
+fication layer and using cross-entropy loss.
+5.4
+Drainer Classifier
+Following the above operations, we obtain two types of features
+to represent users: (1) NFT transaction context representation, and
+(2) social context representation. We concatenate the two repre-
+sentations together to create our final representation, integrating
+comprehensive information learned from our graphs. In order to
+learn the differences between drainers and regular users, we feed
+the final representation to a classifier layer.
+We choose a support vector machine (SVM) [26] as our classi-
+fier layer. SVM is a supervised machine learning model that uses
+classification algorithms. SVM has the advantage of reducing the
+chances of model overfitting, making the model highly stable. SVM
+is also powerful to deal with high dimensional features.
+
+DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+6
+EXPERIMENTS
+To validate our model in different aspects, we present several em-
+pirical evaluations in this section. Specifically, we seek to answer
+the following research questions:
+(RQ1) How effective is our model in detecting NFT drainers?
+(RQ2) How does each component of our model affect performance?
+(RQ3) How robust is our model against evasion attacks?
+6.1
+Datasets
+Data Filtering: In order to conduct experiments, we used a dataset
+consisting of NFT transaction data and accounts identified as NFT
+drainers (Section 3). However, we found that the ratio of regular
+users to drainers in the dataset was highly imbalanced, with only
+742 drainers out of 34 million total accounts. To address this, we
+performed data filtering to remove accounts that were unlikely
+to be drainers. This included removing accounts that had never
+received NFTs from other accounts and accounts with zero active
+time. After filtering, 1,577,200 accounts remained.
+Datasets Construction: To create our training dataset, we used
+accounts that made transactions between the first day of 2022 and
+July 31, 2022, including 645 identified drainer accounts. We sampled
+12,900 regular user accounts to create a ratio of 1:20 between regular
+users and drainers. However, we found that regular users with more
+transactions, defined as heavy users, were more likely to be targeted
+by drainers. Thus, we added an additional 5,805 heavy users (1:9
+ratio with drainer accounts) to our training data, which ended up
+with 645 drainers and 18,705 regular users. We will discuss the
+effect of adding extra heavy users to the training data on model
+performance in Section 7.
+For the evaluation dataset, we used accounts that made at least
+one transaction in August 2022, including 97 drainers. To address
+the imbalance in the ratio of drainers to regular users, we evaluated
+the model multiple times with varying numbers of regular users.
+Specifically, we set the number of regular users to be 10, 20, 50, and
+100 times the number of drainers. This allows us to measure the
+performance of the model under different levels of class imbalance.
+While using higher ratios of regular users more closely represents
+real-world scenarios, it increases the influence of false negatives
+(drainer accounts that were not reported during this time). The
+statistics of the datasets are summarized in Table 6. For each dataset,
+we also add nodes representing first and second-order neighbor
+accounts. The neighbor accounts were used only to enrich the graph
+and were not used in training or evaluation.
+6.2
+Experimental Setup
+Baseline Models: There is no previous work for NFT drainer de-
+tection, so as baselines, we use methods that effectively detect
+Ethereum phishing accounts. In addition, we also use other widely-
+used graph-based models, because DRAINCLoG is a graph-based
+model. The baselines can be divided into two categories: Feature-
+based and graph-based. Explanations of each method are summa-
+rized below:
+Feature-based:
+• Ethereum features [14] are 119-dimensional statistical fea-
+tures previously used for Ethereum phishing account de-
+tection. The features mainly consist of first-order neighbor
+information.
+• E-GCN features [13] are the initial node features used in
+E-GCN, a method proposed to detect Ethereum phishing
+accounts.
+• DRAINCLoG user features are the initial node features
+used in our User-graph.
+Graph-based:
+• Node2Vec [23] is a graph node embedding method based
+on random walks.
+• E-GCN [13] applies a Graph Convolutional Network (GCN)
+to detect Ethereum phishing accounts.
+• GAT [51] is a widely used GNN model. It learns node repre-
+sentations by aggregating neighbor nodes with an attention
+mechanism.
+• GraphSAGE [24] is another GNN variant. It learns node
+representations by aggregating sampled neighbor node fea-
+tures.
+To evaluate the effectiveness of our features, we implement each
+graph-based baseline twice: once using E-GCN features (denoted by
+the "E-" prefix) and once using DRAINCLoG user features (denoted
+by "N-" prefix).
+Implementation Details: The embedding size of both NFT Trans-
+action Context Extractor and Social Context Extractor is set to 64,
+and the learning rate is set to 6e-4 and 5e-4, respectively. In NFT
+Transaction Context Extractor, the number of attention heads is
+set to 8. The regularization parameter and gamma of SVM are set
+to 60 and 1, respectively. As the classifier for each baseline, we
+choose the one with better performance between SVM and light-
+GBM [28], another machine learning algorithm widely used for
+classification problems. Feature-based methods, E-GCN, E-GAT,
+and E-GraghSAGE use lightGBM, and the rest use SVM.
+6.3
+Experimental Results (RQ1)
+Table 7 shows the detection performances of our model and base-
+lines. (The ROC and PR curves for the top performing methods are
+illustrated in Appendix C.) Our model outperforms the baselines
+in all evaluation metrics, which proves the effectiveness of DRAIN-
+CLoG for NFT drainer detection. As the dataset size increases, the
+performance gap between DRAINCLoG and the baselines becomes
+more pronounced.
+DRAINCLoG outperforms feature-based models. This is because
+DRAINCLoG benefits from the NFT-specific features and high-level
+representations obtained from two extractors. Many NFT-specific
+features, such as those related to NFT collections and transaction
+types, can be instrumental in distinguishing NFT drainers from
+regular users but are ignored in the existing feature-based methods.
+Moreover, one of the most important features used in Ethereum
+phishing account detection [14] comes from how the phishers cash
+out: 1-hop neighboring nodes of phishers are usually mixing service
+accounts. However, these features are not effective in detecting
+stolen NFTs, which cannot go through mixing services. Previously
+used features do not fully apply to NFT trading and can miss other
+important features that are leveraged by DRAINCLoG.
+
+Conference’17, July 2017, Washington, DC, USA
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin
+Table 7: The results of experiments averaged over 10 runs on datasets 𝐷1 ∼ 𝐷4. Pre., Rec., F1, and Avg. FPR mean precision,
+recall, F1 score, and average false positive rate, respectively.
+Model
+Dataset (ratio)
+𝐷1 (1:10)
+𝐷2 (1:20)
+𝐷3 (1:50)
+𝐷4 (1:100)
+Metrics
+Pre.
+Rec.
+F1
+Pre.
+Rec.
+F1
+Pre.
+Rec.
+F1
+Pre.
+Rec.
+F1
+Avg. FPR
+Feature-based
+Ethereum features
+0.772
+0.598
+0.673
+0.639
+0.598
+0.617
+0.414
+0.598
+0.488
+0.272
+0.598
+0.374
+0.017
+E-GCN features
+0.730
+0.567
+0.638
+0.571
+0.567
+0.569
+0.347
+0.567
+0.430
+0.208
+0.567
+0.304
+0.022
+DRAINCLoG user features
+0.821
+0.753
+0.785
+0.687
+0.753
+0.718
+0.473
+0.753
+0.581
+0.299
+0.753
+0.428
+0.017
+Graph-based
+Node2Vec
+0.135
+0.022
+0.038
+0.072
+0.022
+0.033
+0.028
+0.022
+0.024
+0.014
+0.022
+0.017
+0.048
+E-GCN
+0.252
+0.010
+0.019
+0.097
+0.010
+0.019
+0.034
+0.010
+0.016
+0.016
+0.010
+0.013
+0.006
+N-GCN
+0.538
+0.134
+0.214
+0.386
+0.134
+0.198
+0.186
+0.134
+0.156
+0.098
+0.134
+0.113
+0.012
+E-GAT
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+0.000
+-
+N-GAT
+0.909
+0.423
+0.577
+0.873
+0.423
+0.569
+0.703
+0.423
+0.528
+0.541
+0.423
+0.474
+0.004
+E-GraphSAGE
+0.899
+0.412
+0.565
+0.781
+0.412
+0.540
+0.585
+0.412
+0.483
+0.411
+0.412
+0.411
+0.006
+N-GraphSAGE
+0.911
+0.784
+0.842
+0.841
+0.784
+0.811
+0.682
+0.784
+0.729
+0.515
+0.784
+0.621
+0.007
+DRAINCLoG
+0.981
+0.825
+0.896
+0.952
+0.825
+0.884
+0.877
+0.825
+0.850
+0.797
+0.825
+0.810
+0.002
+Table 8: The results of ablation experiments averaged over 10 runs. The most impactful features of each module are highlighted
+in color. (*) indicates a set of features involving in and out directions, and (**) indicates a set of features involving all transaction
+types.
+Module
+Dataset (ratio)
+𝐷1 (1:10)
+𝐷2 (1:20)
+𝐷3 (1:50)
+𝐷4 (1:100)
+Removed feature group
+Pre.
+Rec.
+F1
+Pre.
+Rec.
+F1
+Pre.
+Rec.
+F1
+Pre.
+Rec.
+F1
+Holding time
+0.976
+0.825
+0.894
+0.954
+0.825
+0.885
+0.876
+0.825
+0.849
+0.788
+0.825
+0.806
+Transaction type*
+0.967
+0.814
+0.884
+0.933
+0.814
+0.869
+0.823
+0.814
+0.818
+0.708
+0.814
+0.757
+Price*
+0.976
+0.794
+0.876
+0.945
+0.794
+0.863
+0.867
+0.794
+0.829
+0.767
+0.794
+0.780
+Avg. price & holding time
+0.974
+0.794
+0.875
+0.943
+0.794
+0.862
+0.860
+0.794
+0.825
+0.756
+0.794
+0.774
+In-transaction type & In-price
+0.981
+0.794
+0.878
+0.956
+0.794
+0.867
+0.899
+0.794
+0.843
+0.819
+0.794
+0.806
+𝑇𝐶𝐸−feature + 𝑆𝐶𝐸
+Out-transaction type & Out-price
+0.974
+0.825
+0.893
+0.942
+0.825
+0.879
+0.855
+0.825
+0.839
+0.758
+0.825
+0.789
+𝑇𝐶𝐸 + 𝑆𝐶𝐸−feature
+Active timespan
+0.969
+0.835
+0.897
+0.935
+0.835
+0.882
+0.844
+0.835
+0.839
+0.750
+0.835
+0.790
+Gift-in ratio
+0.969
+0.835
+0.897
+0.932
+0.835
+0.881
+0.846
+0.835
+0.840
+0.750
+0.835
+0.790
+Out-in ratio
+0.973
+0.804
+0.880
+0.935
+0.804
+0.864
+0.859
+0.804
+0.831
+0.764
+0.804
+0.783
+# transactions**
+0.974
+0.845
+0.905
+0.937
+0.845
+0.889
+0.858
+0.845
+0.852
+0.763
+0.845
+0.802
+# collections**
+0.966
+0.784
+0.865
+0.938
+0.784
+0.854
+0.846
+0.784
+0.813
+0.750
+0.784
+0.766
+# neighbors**
+0.988
+0.691
+0.813
+0.970
+0.691
+0.807
+0.896
+0.691
+0.780
+0.828
+0.691
+0.753
+Freq. of gift-in & sell
+0.969
+0.825
+0.891
+0.934
+0.825
+0.876
+0.852
+0.825
+0.838
+0.765
+0.825
+0.794
+DRAINCLoG
+0.981
+0.825
+0.896
+0.952
+0.825
+0.884
+0.877
+0.825
+0.850
+0.797
+0.825
+0.810
+One surprising finding is that graph-based methods (E-GCN,
+E-GAT, and E-GraphSAGE) without our NFT-specific attributes per-
+form poorly compared to the feature-based methods. For instance,
+the E-GCN method performs worse than using only the E-GCN
+features. However, by incorporating NFT-specific attributes in the
+graph-based methods (N-GCN, N-GAT, and N-GraphSAGE), their
+performances improve significantly, highlighting the importance of
+our feature engineering. In addition to the NFT-specific attributes,
+DRAINCLoG takes into account transaction types and utilizes NFT
+transaction context extracted from the NFT-User graph. This results
+in a multi-faceted representation that allows it to outperform other
+graph-based methods.
+6.4
+Ablation Study (RQ2)
+In this section, we aim to understand the impact of each compo-
+nent of DRAINCLoG on its performance. First, we analyze how
+each structural component affects performance. We conducted the
+same detection task after eliminating each representation produced
+(a) F1- score
+(b) Precision
+Figure 6: The results of ablation experiments averaged
+over 10 runs. 𝑆𝐶𝐸 and 𝑇𝐶𝐸 denote social context extrac-
+tor and NFT transaction context extractor, respectively.
+𝐷𝑅𝐴𝐼𝑁𝐶𝐿𝑜𝐺−𝑟𝑒𝑙 is a modified DRAINCLoG which does not
+consider transaction types between users in the user graph.
+by NFT Transaction Context Extractor (TCE) and Social Context Ex-
+tractor (SCE). In addition, to evaluate the importance of consider-
+ing transaction types between users in the User graph, we tested
+
+DRAINCLOG
+SCE
+TCE
+DRAINCLoG-rel
++-DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+Table 9: The results of the evasion attack simulation aver-
+aged over 5 runs with varying attack levels. The values listed
+below for Attack 2 were simulated with X = 1%.
+Attack1
+Attack2
+Level (L%)
+𝐷1
+𝐷2
+𝐷3
+𝐷4
+𝐷1
+𝐷2
+𝐷3
+𝐷4
+w/o attack
+0.896
+0.884
+0.850
+0.810
+0.896
+0.884
+0.850
+0.810
+𝐿1 (10%)
+0.853
+0.841
+0.802
+0.760
+0.860
+0.848
+0.810
+0.768
+𝐿2 (30%)
+0.775
+0.764
+0.726
+0.686
+0.790
+0.779
+0.742
+0.701
+𝐿3 (50%)
+0.745
+0.733
+0.696
+0.657
+0.752
+0.741
+0.705
+0.666
+𝐿4 (70%)
+0.737
+0.725
+0.688
+0.649
+0.651
+0.640
+0.607
+0.571
+𝐷𝑅𝐴𝐼𝐶𝐿𝑜𝐺−𝑟𝑒𝑙, which uses GCN instead of R-GCN for social con-
+text extraction.
+As shown in Figure 6, the best performance is achieved when all
+the components are used. SCE has similar precision as DRAINCLoG
+but its recall (0.784) is lower than DRAINCLoG’s (0.825), resulting
+in a larger gap in the F1-score. Although using TCE alone per-
+forms poorly, the NFT transaction context helps DRAINCLoG find
+drainers not captured by SCE. 𝐷𝑅𝐴𝐼𝐶𝐿𝑜𝐺−𝑟𝑒𝑙 also shows poor per-
+formance compared to DRAINCLoG and suffers from significant
+performance degradation as the number of regular users increases.
+This highlights the importance of transaction types between users
+for identifying NFT drainers.
+We also examine the effect of elements of NFT transaction edge
+attributes and user node attributes, which are used in graph construc-
+tion. We grouped features that represent similar concepts together.
+For example, the number of transactions for each transaction type
+(5) was grouped into one, the number of transactions. Then, we
+trained and evaluated DRAINCLoG on the graph constructed with-
+out each feature group.
+In Table 8, the most influential factors used in the TCE com-
+ponent are transaction type, average information, and price. When
+the evaluation dataset size is small, price is more important than
+transaction type, but as the dataset size increases, the transaction
+type becomes more crucial. Removing the average information of
+each NFT decreases the performance, which shows they are useful
+for capturing transaction context.
+For user node attributes used in the SCE component, the most
+influential factors are the number of neighbors, the number of col-
+lections, and out-in ratio. This result aligns with the drainers’ goal
+of stealing NFTs from as many users as possible and selling them
+for profit. Removing any of three groups: active timespan, gift-in
+ratio, and the number of each transaction type results in a higher
+recall but lower precision compared to DRAINCLoG. This suggests
+that drainers have unique characteristics not present in regular
+users in these three features, and they help DRAINCLoG avoid
+misclassifying regular users as drainers.
+Overall, these results confirm the importance of each representa-
+tion and the effectiveness of our feature engineering in enhancing
+the detection ability of DRAINCLoG.
+6.5
+Robustness (RQ3)
+If drainers notice that they are monitored by DRAINCLoG, they
+may purposely change their trading patterns to avoid detection. In
+(a) Attack 1
+(b) Attack 2
+(10%)
+(30%)
+(50%)
+(70%)
+Figure 7: F1-score after re-updating on different cutoff dates
+against (a) Attack 1 and (b) Attack 2. Each color represents a
+different attack level, denoted by 𝐿1 to 𝐿4. In Attack 2, differ-
+ent line types represent the varying amounts of Ether sent
+by the attackers. For each date, the ratio of attackers to reg-
+ular users was 1:100 in evaluation.
+this section, we evaluate the robustness of our model by simulating
+evasion attacks.
+6.5.1
+Evasion attack. Drainers can adopt various strategies to con-
+ceal themselves as regular users. We introduce two types of evasion
+attacks. We assume that drainers cannot afford to change their liq-
+uidation strategy while avoiding marketplace bans, and therefore
+focus on the NFT acquisition step.
+Attack 1. Mint NFTs. The easiest way that drainers can engage
+in innocuous-appearing NFT transactions is by minting NFTs. By
+increasing the number of minted NFTs, drainers can decrease their
+gift-in ratio, out-in ratio, etc.
+Attack 2. Send Ether to the victim. Drainers can hide their drain-
+ing activities by sending Ether to victims after stealing NFTs. This
+way, their behavior is recorded as a sale instead of a gift.
+6.5.2
+Evaluation. To evaluate how well DRAINCLoG can detect
+drainers under different evasion attack strategies, we modified
+previous evaluation datasets (𝐷1 ∼ 𝐷4) by altering the transaction
+records of drainers based on the attack strategy and attack level 𝐿.
+For Attack 1, we increased the number of minted NFTs by 𝐿% of
+gifted-in NFTs, where 𝐿 ∈ {10, 30, 50, 70}. For Attack 2, we changed
+𝐿% of gifting-in transactions to buying transactions by sending
+X% of the average sale price of each NFT to those victims, where
+𝐿 ∈ {10, 30, 50, 70} and 𝑋 ∈ {1, 10, 60}. We set 𝑋 to be less than
+60 because drainers typically sell stolen NFTs at 40% below their
+average sale price.
+The results in Table 9 show that DRAINCLoG’s performance
+decreases as the attack level increases. Attack 1 is less effective at
+evading detection than Attack 2 because it does not change the
+relationship between users. Making significant changes to user
+attributes by increasing the number of minted NFTs is not enough
+to fool the detection system. On the other hand, Attack 2 is much
+more effective as it changes the relationship between users as well
+as the attributes. This comes at a cost for the attacker, since each
+drained will incur an extra cost. Despite this, this method still fails to
+fully deceive the system, which means DRAINCLoG can effectively
+capture their transaction context and social context from other
+behaviors such as those during liquidation.
+
+* X%)
+1% --
+10%
+60%L2
+L3
+L4L3
+L4L4L4w/o attack L1  L2 
+w/o attack
+L4AttacklevelConference’17, July 2017, Washington, DC, USA
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin
+6.5.3
+Defense. One of the defenses against evasive attacks is to
+periodically update DRAINCLoG to learn new types of drainers. We
+update DRAINCLoG in a simplified manner, re-training only the
+last layer (SVM), while leaving two extractors intact. To prevent
+evaluating on users seen in training, we train only on users that
+went inactive before a cutoff date. We re-trained the SVM for 5
+different cutoff dates spaced apart by 2 days, and evaluated on the
+remaining data for each date. The number of active attackers at
+the end of the last day (August 10th) was 18(18.6% of 97 drainers).
+We trained and evaluated using a fixed ratio of drainers to regular
+users as previously described in Section 6.1. Figure 7 shows the
+evaluation results (F1-score) of experiments averaged over 5 runs.
+After just one update, with seven added attackers (7.2%), DRAIN-
+CLoG’s performance significantly improved. As the model contin-
+ues to update, its performance approaches the original level, and
+the performance difference between attack levels becomes small.
+Attack 2 is noticeably more effective in evading the detector when
+more Ether is sent to victims. In the extreme scenario when 70%
+of gifting is changed to buying (𝐿4) and 60% of 𝑃𝑟𝑖𝑐𝑒𝑎𝑣𝑔 is sent, it
+becomes difficult to detect attackers even after updating. However,
+this is not likely to happen as it gives up a significant amount of
+profit from NFT draining. In the next section, we will discuss ways
+to make DRAINCLoG more robust against new types of drainers.
+7
+DISCUSSION
+Future work for evasion attack: In Section 6.5, to improve the
+robustness of DRAINCLoG, we updated the drainer classifier (SVM)
+with evasion attackers. Additionally, we can improve the model
+by updating the transaction context extractor and social context
+extractor to account for new types of drainers. Furthermore, addi-
+tional adversarial examples could be created to augment training
+data. In Section 6.5.3, we observe that the performance increases
+significantly after the detector is updated with only a few adver-
+sarial attackers. By updating DRAINCLoG with examples similar to
+new drainers, the detection performance recovers quickly.
+Adoption in the real-world: In this study, we focused on under-
+standing the behavior of NFT drainers and designing a specialized
+model for identifying drainers. To ensure that DRAINCLoG remains
+effective in the real world, it’s important to consider how it responds
+to new NFT transactions.
+It is crucial for the detection system to quickly identify if a user is
+a drainer. Our model is viable in scanning many different targets at
+once since it requires only the two-hop neighbors, rather than the
+entire graph. Therefore, we believe that our model can be effectively
+utilized for real-time detection in the real world.
+Effect of heavy users on model performance: We find that
+drainers and regular users show different distributions in the num-
+ber of transactions (Appendix B). This could cause our model,
+DRAINCLoG, to focus on the number of transactions when iden-
+tifying drainers, potentially misclassifying legitimate users who
+make many transactions. To prevent this problem, we added regular
+users with more than 16 transactions, defined as heavy users, to our
+training dataset.
+In order to decide how many heavy users will be added to the
+training dataset, we conduct experiments with varying number of
+heavy users. Figure 8 (a) shows that training the model without
+(a) F1-score
+(b) Distribution of false positives
+only random users
+random users + heavy users
+F.Ps when the training set consists of 
+Figure 8: (a) The performance when varying the number
+of heavy users included in the training dataset. (b) Regu-
+lar users misclassified as drainers (F.Ps) in dataset 𝐷4 when
+heavy users are added/not added.
+extra heavy users suffers from severe performance degradation as
+the dataset size increases. We found that when the number of heavy
+users is 9 times the number of drainers, the model performs well
+across all dataset sizes. Thus, we fix the ratio of heavy users to
+drainers as 9 in Section 6. Figure 8 (b) shows that using heavy users
+in the training phase prevents the model from classifying users
+based on their number of transactions made. Without heavy users,
+the model often misclassifies regular users with many transactions
+as drainers. But with heavy users, the model is less likely to make
+this mistake.
+8
+RELATED WORK
+Suspicious behaviors in NFT markets: With the increasing pop-
+ularity of NFTs, suspicious activities targeting NFTs are also rising.
+A few studies exist for analyzing security issues in the NFT ecosys-
+tem, such as wash tradings and shill bidding. However, to the best
+of our knowledge, we are the first to perform an in-depth study of
+NFT drainers and propose an NFT drainer detection system.
+Das et al. [17] conducted a comprehensive study of design weak-
+nesses originating from the NFT marketplaces and external entities.
+Also, they investigated various types of fraudulent user activities
+occurring in NFT marketplaces, such as counterfeit NFT creation,
+wash trading, and shill bidding. Von et al. [52] quantified market
+abuse in the NFT ecosystem with their proposed NFT wash trading
+detection algorithm.
+Ethereum phishing scam detection: Ethereum phishing scam
+detection can be categorized into two main types: feature-based
+and graph-based approaches.
+Chen et al. [14] extracted 119-dimensional statistical features to
+consider the 1-order neighbors of the node as well as the node itself.
+They used a LightGBM-based dual-sampling ensemble algorithm
+to classify phishing nodes.
+Another line of research focuses on network representation.
+Wu et al. [54] proposed Trans2Vec, which is a modified random
+walk-based network embedding method with biases of transaction
+amount and timestamp for neighbor sampling. Chen et al. [13]
+introduced E-GCN, the first Ethereum phishing scam detection
+method based on Graph Neural Networks (GNN). They extracted
+8-dimensional statistical features, such as in/out-degree, number
+of neighbors, etc., and fed them into a GCN for embedding. Li et
+al. [31] constructed edge representations from transaction records
+
+DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+to capture the temporal relationship between users. The edge rep-
+resentations are aggregated into node representations and used to
+obtain structural features using GCN. Unlike the above works that
+approached using node classification, Zhang et al. [55] regarded
+the problem as a graph classification. They used hierarchical graph
+pooling layers to extract node-level representations, which were
+then aggregated to form graph-level representations.
+However, the works discussed above are difficult to apply to NFT
+drainers detection due to the characteristics of the NFT ecosystem,
+as mentioned in Section 2.1.
+Graph Neural Network: In recent years, deep learning methods
+have achieved remarkable performance in various fields. Deep neu-
+ral networks have also been applied to graph data to leverage the
+structural properties of graphs.
+Graph Convolution Networks (GCNs) [30] is one of the most
+prominent graph neural network models. GCNs perform convo-
+lution operations on graph data and learn embeddings of nodes
+by aggregating features from neighboring nodes. Unlike GCNs,
+which uses information from adjacent nodes as is, Graph Attention
+Networks (GATs) [51] utilize information from neighbors by us-
+ing node attention. Multi-head attention is used to learn a number
+of attentions, and the node features obtained from each attention
+are concatenated to form a single feature. GraphSAGE [24] is an
+inductive GNN model, which generalizes the unobserved nodes.
+By incorporating node features into the learning algorithm and
+aggregator functions, it can simultaneously learn the distribution
+of neighboring node features and the topological structure for the
+neighbors of each node.
+9
+CONCLUSION
+Phishing attacks are a significant threat to the NFT trading ecosys-
+tem. Despite the increasing damages caused by NFT drainers, their
+behaviors are not well studied. To conduct an in-depth study on
+NFT drainers, we extract data from 83M NFT transaction records
+and collect 742 NFT drainer accounts. We verify that they have dif-
+ferent transaction contexts and social contexts compared to regular
+users. Based on our measurement results, we propose a detection
+model, DRAINCLoG, tailored to detect drainers in the NFT environ-
+ment. DRAINCLoG is able to generate a user representation that
+considers NFT transaction context and social context. Evaluated
+on real-world NFT transaction data, we verify our model’s effec-
+tiveness and robustness. We believe that our findings and detection
+method will contribute to the security NFT ecosystem.
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+
+DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+Appendices
+A
+ANALYSIS OF CONNECTIONS BETWEEN
+DRAINERS AND AFFILIATED USERS
+(a) Number of affiliated users connected to a drainer
+(b) Number of drainers connected to an affiliated user
+Figure 9: Cumulative distribution functions (CDFs) of (a)
+drainer and (b) affiliated users
+There is no limit of creating Ethereum accounts, so an agent
+can create multiple accounts for the purpose of transferring as-
+sets between them. In certain cases, a gift can be used between
+users to avoid monitoring when manipulating markets by wash
+trading [17, 36]. A gift is the transfer of NFT ownership without
+any financial compensation. As gifts are commonly observed to
+occur between users who have a relationship, we can infer that the
+accounts that gift drained NFTs drainers are likely to be affiliated
+with said drainers. Therefore, we define these accounts as affiliated
+users.
+In this section, we provide detailed information about the number
+of affiliated users of drainers and the number of drainers connected
+to affiliated users. Figure 9(a) illustrates how many affiliated users
+each drainer has. Most drainers (82.5%) have one or more affiliated
+users. They have 2.1 affiliated users on average. One drainer gift
+his drained NFTs to 21 affiliated users.
+Figure 9(b) focuses on how many drainers are connected to each
+affiliated user. We find 637 affiliated users, and 15.4% are related
+to two or more drainers. The most overlapped affiliated user is
+connected with 16 drainers.
+B
+DISTRIBUTION OF TRANSACTIONS
+40~41
+41~42
+42~43
+43~44
+44~45
+45~46
+Number of transactions
+0
+10
+20
+30
+40
+Percentage
+Random users
+Known drainer accounts
+Figure 10: Distribution of random users’ transactions and
+drainers’ transactions.
+Figure 10 shows the distribution of random users’ transactions
+and drainers’ transactions. We analyze drainers and random users
+who were active between 1, January 2022 to 31, July 2022. Drainers
+and sampled random users show noticeably different distributions
+of transactions. Most drainers made transactions of more than 16,
+while most random regular users made transactions of smaller than
+16. Thus, we define random regular users with transactions of more
+than 16 as heavy users.
+C
+ROC CURVE & PR CURVE
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+DRAINCLoG: 0.9894
+N-GraphSAGE: 0.9894
+E-GraphSAGE: 0.9521
+N-GAT: 0.9303
+(a) ROC (Receiver Operating Characteristic) curve
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+DRAINCLoG: 0.8457
+N-GraphSAGE: 0.7301
+E-GraphSAGE: 0.3843
+N-GAT: 0.4336
+(b) PR (Precision-Recall) curve
+Figure 11: ROC and PR curve for the top four models on 𝐷4:
+DRAINCLoG, N-GraphSAGE, E-GraphSAGE, and N-GAT.
+
+1.0
+0.8
+ion
+0.6
+Proportic
+0.4
+0.2
+0.0
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+# drainers1.0
+0.8
+ion
+0.6
+roportic
+0.4
+0.2
+0.0
+0
+5
+10
+15
+20
+# affiliated usersConference’17, July 2017, Washington, DC, USA
+Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin
+D
+FEATURE ANALYSIS
+To better understand the trading behavior of different types of users,
+we analyze the activities of 645 drainers, 637 affiliated users, and a
+sample of 10,000 regular users who were active between January 1st
+and August 31st, 2022. We compare these three groups by plotting
+the cumulative distribution functions (CDFs) for 19 different dimen-
+sions. To make these visualizations easier to interpret, we limit the
+x-axis of each graph to 100. We can observe that drainers exhibit
+distinct behavior patterns compared to regular users. Furthermore,
+we find that affiliated users also show different patterns of behavior,
+which sets them apart from both regular users and drainers.
+D.1
+Number of transactions
+(c) Number of gifting-in
+(d) Number of gifting-out
+(a) Number of buying
+(b) Number of selling
+(e) Number of minting
+Figure 12: CDFs of the number of transactions according to
+the user types.
+In this section, we focus on the number of transactions users
+made for each transaction type. Figure 12 illustrates CDFs of the
+number of each transaction type: (a) buying, (b) selling, (c) gifting-
+in, (d) gifting-out, and (e) minting.
+It is apparent that drainers are more active in selling, gifting-in,
+and gifting-out transactions. However, they rarely participate in
+buying and minting transactions when compared to regular users.
+These results suggest that the primary focus of drainers in the NFT
+ecosystem is on draining NFTs.
+Upon closer examination, trends between drainers and affiliated
+users show little difference in selling, gifting-in, and gifting-out
+transactions. Interestingly, affiliated users are more active in buying
+and minting NFTs, unlike drainers. This behavior sets them apart
+from both drainers and regular users.
+D.2
+Number of collections
+(c) Number of collections gifted-in
+(d) Frequency of gifted-out
+(a) Number of collections bought
+(b) Number of collections sold
+(e) Number of collections minted
+Figure 13: CDFs of the number of collections according to
+the user types.
+Figure 13 illustrates CDFs of the number of collections for each
+transaction type: (a) buying, (b) selling, (c) gifting-in, (d) gifting-out,
+and (e) minting.
+We find that the number of collections for each transaction type
+is generally smaller than the number of transactions. It is well
+known that NFT users tend to form communities based on specific
+collections and trade within those communities [37]. Although
+drainers have different intentions from regular users, they also trade
+a smaller number of collections than transactions. This is because
+they steal NFTs that are collected by regular users. However, due to
+their high levels of gifting-in, gifting-out, and selling transactions,
+
+-- Known Drainer Accounts
+  Affiliated Users
+s  Random Users-- Known Drainer Accounts
+  Affiliated Users
+s  Random UsersDRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs
+Conference’17, July 2017, Washington, DC, USA
+drainers have a greater diversity of collections in those three types
+of transactions. As a result, they exhibit a significant difference from
+regular users in the number of NFT collections gifted-in, gifted-out,
+and sold.
+In contrast, affiliated users are observed to actively participate in
+trading a wide range of collections across all types of transactions.
+This is because they are actively engaged in all types of transactions.
+D.3
+Number of neighbors
+(c) Number of neighbors gifted-in
+(d) Number of neighbors gifted-out
+(a) Number of neighbors bought
+(b) Number of neighbors sold
+Figure 14: CDFs of the number of neighbors according to the
+user types.
+Figure 14 illustrates CDFs of the number of neighbors for each
+transaction type: (a) buying, (b) selling, (c) gifting-in, and (d) gifting-
+out.
+We analyze the number of neighbors a user has for each transac-
+tion type by considering the accounts with which a user has made a
+transaction as their neighbors. For buying and selling transactions,
+the distribution of neighbors is similar to that of the transactions
+themselves. However, for gifting-in and gifting-out transactions,
+the distribution of neighbors is significantly different. Most users
+gift NFTs to only a few neighbors, while they sell or buy NFTs
+with many neighbors. This suggests that NFT users have specific
+relationships through gifting NFTs, given that gifting is a process
+of transferring ownership without any payment.
+This phenomenon is more pronounced in drainers than in reg-
+ular users. When drainers steal NFTs, they may acquire all the
+tokens from each victim, resulting in a smaller number of gifting-in
+neighbors than the number of NFTs gifted-in. Additionally, drain-
+ers only gift NFTs to a few affiliated users, resulting in a much
+smaller number of gifting-out neighbors than the number of NFTs
+gifted-out.
+In the case of affiliated users, their distributions of gifting-in and
+gifting-out are as expected, similar to those of drainers.
+D.4
+Ratio & Frequency & Active timespan
+(c) Frequency of selling
+(d) Frequency of gifting-in
+(a) Out-in ratio
+(b) Gift-in ratio
+(e) Active timespan (days)
+Figure 15: CDFs of gift-in ratio, out-in ratio, frequency of
+sale, frequency of gift-in, and active timespan according to
+the user types.
+Figure 15 illustrates CDFs of (a) out-in ratio, (b) gift-in ratio,
+(c) frequency of selling, (d) frequency of gifting-in, and (e) active
+timespan.
+Drainers have a higher gift-in ratio than regular users and af-
+filiated users. This is because they do not engage in buying and
+minting NFTs, but rather steal a large number of NFTs in a short
+period of time. This results in a high frequency of gifting-in trans-
+actions. Also, drainers are more likely to transfer out their NFTs
+than regular users. They also sell their NFT much more frequently
+in a short active timespan.
+The behavior of affiliated users falls between that of drainers
+and regular users. They are active in all types of transactions, par-
+ticularly gifting-in, which results in a high gift-in ratio similar to
+drainers. However, their active period is not as short as drainers.
+
+- Known Drainer Accounts
+s  Affiliated Users 
+s  Random Users-- Known Drainer Accounts
+  Affiliated Users
+s  Random Users
\ No newline at end of file
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf,len=1522
+page_content='DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained  NFTs using Classifiers Learned on Graphs  Hanna Kim  gkssk3654@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='kr  KAIST  Republic of Korea  Jian Cui  cj19960819@s2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='inc  S2W  Republic of Korea  Eugene Jang  genesith@s2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='inc  S2W  Republic of Korea  Chanhee Lee  leemember@s2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='inc  S2W  Republic of Korea  Yongjae Lee  lee@s2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='inc  S2W  Republic of Korea  Jin-Woo Chung  jwchung@s2w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='inc  S2W  Republic of Korea  Seungwon Shin  claude@kaist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='kr  KAIST  Republic of Korea  ABSTRACT  As Non-Fungible Tokens (NFTs) continue to grow in popularity, NFT  users have become targets of phishing attacks by cybercriminals,  called NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Over the last year, $100 million worth of NFTs  were stolen by drainers, and their presence remains as a serious  threat to the NFT trading space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Since NFTs are different from  cryptocurrencies, existing work on detecting Ethereum phishers  is unsuitable to detect NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Moreover, no work has yet  comprehensively investigated the behaviors of drainers in the NFT  ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In this paper, we present the first study on trading behavior of  NFT drainers and present the first dedicated NFT drainer detec-  tion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We extract data of 83M NFT transactions from the  Ethereum blockchain and collect 742 drainer accounts from five  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We find drainers have significantly different transaction  context and social context compared to regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' With the  insights gained from our analysis, we design an automatic drainer  detection system, DRAINCLoG, that uses graph neural networks to  capture the complex relationships in the NFT ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Our model  effectively captures NFT transaction contexts and social contexts  using an NFT-User graph and a User graph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Evalu-  ated on real-world NFT transaction data, we prove the model’s  effectiveness and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  ACM Reference Format:  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo  Chung, and Seungwon Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' DRAINCLoG: Detecting Rogue Accounts  Permission to make digital or hard copies of all or part of this work for personal or  classroom use is granted without fee provided that copies are not made or distributed  for profit or commercial advantage and that copies bear this notice and the full citation  on the first page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='  Conference’17, July 2017, Washington, DC, USA  © 2023 Association for Computing Machinery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='$15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='00  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='nnnnnnn  with Illegally-obtained NFTs using Classifiers Learned on Graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Pro-  ceedings of ACM Conference (Conference’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' ACM, New York, NY, USA,  17 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1145/nnnnnnn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='nnnnnnn  1  INTRODUCTION  With the emergence of blockchain technology, Non-Fungible Tokens  (NFTs) have revolutionized the digital creator economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFTs are  digital assets, such as art or collectibles, with unique identification  codes and metadata [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFTs have attracted numerous content cre-  ators and investors, and by 2021 the NFT market exploded, growing  to around $22 billion [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  As a result, phishing scams have also emerged in the NFT ecosys-  tem [27, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' According to blockchain analysis provider Elliptic, over  $100 million worth of NFTs were stolen by NFT scammers, called  drainers, in a one-year period from July 2021 to July 2022 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFT  phishing scams continue to make headlines, causing significant  damage to users [9, 10, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For instance, a sophisticated phishing  attack on Uniswap1 (the largest Ethereum-based decentralized ex-  change) caused $8 million worth of damages to NFT users [25, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Efforts to combat NFT drainers have had limited effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  The Opensea NFT marketplace implemented the policy of marking  stolen NFTs as untradeable [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, this is only effective  when victims are able to notice and report the theft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The cryptowal-  let Metamask implemented phishing warnings [11, 19], but were  able to be bypassed by certain drainers [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Even before NFT drainers, phishing attacks targeting cryptocur-  rency were already considered a significant threat to trading secu-  rity of Ethereum [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As a response to such attacks, researchers  have proposed several methods to detect phishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' One line of  research relies on using hand-crafted user features to detect scam-  mers [14], while other approaches employ network representation  learning by utilizing Node2Vec [54] or Graph Neural Networks  (GNN) [13, 31, 55] to capture scammers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  1uniswap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='13577v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='CR]  31 Jan 2023  Conference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  However, these methods of detecting cryptocurrency phishers  are unsuitable for detecting NFT drainers for the following rea-  sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' First, features that are essential for cryptocurrency phisher  detection, such as those that describe the liquidation process, are  not shown in NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In addition, cryptocurrency-focused  detection systems cannot leverage individual NFTs and fail to ac-  count for the multiple transaction types of NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, they cannot  fully capture the differences between NFT drainers from regular  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This raises the need for an automatic detection system that  captures various factors of the NFT ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This also motivates  a comprehensive investigation on the transaction patterns of NFT  drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  To the best of our knowledge, the existing literature has not  explored phishing scams in the NFT ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To fill this gap, we  aim to investigate the trading traits of NFT drainers and propose an  automatic detector that identifies suspicious trading behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To  this end, we collect over 83 million NFT transactions (between Jan-  uary 2022 and August 2022) directly from the Ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We also collect information of 742 reported drainers from five chan-  nels, including Twitter[49] and Etherscan[21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Then, we investigate  the activity of drainer accounts in the NFT market and identify  how their NFT trading patterns differ from those of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We identify two key contexts that are critical in detecting drain-  ers: their distinct NFT transaction context (rapid liquidation of  NFTs at significantly lower prices) and their distinct social context  (connections with affiliated users who also exhibit distinct trading  patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  However, capturing the intricate relationships between users  and NFTs for drainer detection remains a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This is  because the transaction history of millions of NFTs and the various  types of interactions between millions of users must be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  To address this challenge, we propose a novel GNN-based drainer  detection system, DRAINCLoG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We use GNNs as they are well-suited  for modeling relationships between users and NFTs in a graph  structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' GNNs can incorporate both the features of nodes and  edges, making them effective at identifying patterns of anomalous  behavior among interconnected entities [7, 15, 46, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The GNN-  based structure allows DRAINCLoG to effectively capture the NFT-  to-user and user-to-user relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Our proposed system utilizes two types of graphs: an NFT-User  graph and a User graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The NFT-User graph models interactions  between users and NFTs, with two types of nodes and attributed  edges, allowing us to obtain a representation of users’ transaction  context by considering each NFT’s transaction history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The User  graph models interactions between users, with attributed nodes  and two types of edges, enabling us to capture the social context  by integrating information on the users and their relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We combine these two contexts to leverage information from both  graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The designed model significantly outperforms existing base-  lines in detecting drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We also demonstrate the robustness of  our model by simulating evasion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Overall, we believe that  our study will inspire further research and practical efforts to im-  prove NFT trading security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In summary, the contributions of the work are listed below:  Figure 1: Summary of NFT transaction types: a○ Mint, b○  Sale, c○ Gift, and d○ Burn  We collect 83 million NFT transaction data from the Ethereum  blockchain and 742 drainer accounts from five different chan-  nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainer accounts are publicly available for future re-  search 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We present the first empirical study on NFT drainers and find  that drainers have distinct characteristics and transaction  patterns from regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We propose a graph-based automatic NFT drainer detector,  DRAINCLoG, that can capture NFT transaction context and  user social context, and provide experimental results that  show our detector outperforms existing detection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We verify the robustness of DRAINCLoG by simulating eva-  sion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  2  BACKGROUND AND MOTIVATION  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  Background  In this section, we outline the required background related to NFTs  and phishing attacks in the NFT ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Then, we detail the  motivation of our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Non-Fungible Token (NFT) is a cryptographic asset on the blockchain  with unique identification codes and metadata that distinguish them  from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFTs guarantee ownership of unique digital assets,  such as images, video files, and even physical assets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' By 2017, NFTs  on the Ethereum blockchain gained attention with the creation of  collectibles such as CryptoPunks [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The Ethereum blockchain is  favored among NFT traders and accounts for around 80% of NFT  trading volume [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' An NFT collection is a set of NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFTs from  the same collection share similar characteristics, with unique varia-  tions from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' These variations can make some NFTs much  more valuable than others, even within a collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  NFT Transaction Types Here, we introduce the four types of NFT  transactions: mint, sale, gift, and burn, as illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  a○ Mint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' An NFT is created by minting, the process of inscribing  a digital asset to the blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Minted NFTs can be listed and  traded on NFT marketplaces, such as OpenSea [39] and Rarible [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  b○ Sale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' A sale is a process of transferring an NFT ownership to  another account for payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFTs are typically traded with Ether,  the native cryptocurrency of Ethereum, and sometimes fungible  tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Users can partake in sales in two ways: buying and selling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  c○ Gift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' On the other hand, a gift is a process of handing over  ownership without any payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Gifts usually occur when the  sender is related to the recipient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For instance, an individual can  create multiple accounts and gift NFTs to move them between  accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Sometimes it can be used between users to avoid moni-  toring when manipulating markets by wash trading [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Users can  partake in transfers in two ways: gifting-in and gifting-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  2will be made available on acceptance  mint  sale  gift  d  burn  NULL  NULLDRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17, July 2017, Washington, DC, USA  d○ Burn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' An NFT cannot be deleted from the blockchain, but it  can be ‘burned’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Burning is the process of sending NFTs to an inac-  cessible address (NULL), which will remove them from circulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Burning is used for various purposes, such as adjusting the supply  of NFTs, operating a collection’s community, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Phishing attacks in the NFT ecosystem is commonly called  draining [32, 47], and we will use this term in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The main  goal of NFT drainers is to “drain” (steal) NFTs from victims, although  cryptocurrencies and private information could also be targeted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We detail the procedures for how NFT phishing scams operate by  dividing them into three steps: (1) spreading phishing websites, (2)  draining NFTs, and (3) cashing out drained NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 2 illustrates  the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (1) Spread phishing websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFT drainers mainly use two meth-  ods to spread phishing websites to victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' First, they 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1) use social  media, such as Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' They can make social media accounts mas-  querading as official accounts of popular NFTs, sometimes even  compromising them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The drainers upload posts linking to scam  sites on these channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Another method is to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2) use phishing token  airdrops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Airdrops are commonly used in marketing campaigns to  promote creators’ projects by sending free tokens [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However,  it can also be abused by drainers to trick victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' A drainer can  send fake tokens to target wallets, tricking victims into clicking a  phishing website link in the token’s description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (2) Drain NFTs from victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The drainers steal NFTs from vic-  tims who entered phishing websites by either 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1) stealing login  credentials of crypto wallets or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2) abusing smart contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The first  method is similar to the traditional phishing methods that target  victims’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' When victims type in login credentials to their  crypto wallet, drainers can record the information using keyloggers,  etc [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' With the stolen credentials, drainers can access victims’  accounts and transfer NFTs to their own wallets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Alternatively,  drainers can abuse smart contracts to deceive victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainers can  create malicious smart contracts, that include functions such as  setApprovalForAll, and lure victims with valuable tokens to sign a  transaction [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' SetApprovalForAll is an important method used  for NFT trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' When a user sells NFTs on a marketplace, the user  needs to call this function to authorize the marketplace in trans-  ferring the NFTs from the user’s account to the buyer’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However,  by calling the function on a phishing website, a victim grants the  drainer permissions to transfer all the victim’s tokens [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (3) Cash out drained NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Scammers who target cryptocurrencies  can cash out stolen assets by using crypto exchanges, and some-  times through mixing services [18, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Conversely, NFT drainers  must first sell the stolen NFTs by listing them on different market-  places [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To combat NFT scams, OpenSea [39], the largest NFT  marketplace, made a policy of disabling the buying, selling, and  gifting of stolen items [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  Motivation  NFTs have attracted the attention of investors and criminals alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Phishing scams targeting NFT users continue to make headlines [9,  10, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For instance, a sophisticated phishing attack on Uniswap  NFT holders caused damages of $8 million when users were tricked  to approve malicious transactions [25, 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In another instance,  (1) Spread phishing websites  setApproval  ForAll  (2) Drain NFTs  from victims  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='phish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com  (3) Cash out drained NFTs  Figure 2: The process of phishing attacks in the NFT ecosys-  tem: (1) spreading phishing websites, (2) draining NFTs from  victims, and (3) cashing out drained NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  the official Instagram account of the most valuable NFT, Bored  Ape Yacht Club, was compromised and used in a phishing attack  in April [22, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The drainers stole NFTs by luring victims with  free tokens through the Instagram account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The total damage was  valued at $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='7 million, including a token worth $354,500 (at that  time of exchange rate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Alarmingly, there have been a wealth of tools made available to  assist NFT draining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Software packages for NFT draining are being  sold between $29 – $149 on dedicated websites [33] and the dark-  web [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Some groups even publish a free version of their draining  code on GitHub in order to advertise their online stores [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The  accessibility of draining tools lowers the barriers to entry for future  NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To prevent their tools, MetaMask, a famous crypto  wallet, updated a feature to warn users when transactions request  setApprovalForAll [11, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, NFT drainers also developed  their tools to bypass this update [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Although the damage caused by NFT drainers is increasing, no  previous work has yet conducted an in-depth measurement study  or proposed a detection method for NFT draining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Several methods  have been proposed to detect phishers targeting cryptocurrency [13,  14, 31], but they are unsuitable to apply to the NFT ecosystem for  the following reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  First, essential features to detect cryptocurrency phishers do  not apply for NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For instance, NFT drainers do not  follow the liquidation process of cryptocurrency phishers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Stolen  cryptocurrencies are generally cashed out through exchanges after  a mixing process, which is one of the most decisive features in  detecting cryptocurrency phishers [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' On the other hand, stolen  NFTs cannot be laundered because each NFT is trackable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Also,  NFTs cannot be directly cashed out in exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Therefore, NFT  drainers must list NFTs on the market for sale in order to cash  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The ineffectiveness of previous features raises the need for a  measurement study on NFT drainers to gain insights on how to  detect phished NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Section 4, we investigate the activity of  NFT drainers to identify trading traits that can detect NFT draining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Second, existing methods cannot consider complex contexts  within the NFT ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Unlike cryptocurrency, where coins  are interchangeable and have equal value, NFTs are of distinct iden-  tities with dynamic values, making complex transaction patterns  (factoring price and frequency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To fully understand a user’s trad-  ing habits, the transaction history of each NFT traded by the user  should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2, we describe our NFT  drainer detection system, DRAINCLoG, which includes a module  NFT  NFT  NFT  NFT  NF  NFT  NFT  NFT  alamy  ImageID:2J6YXM5  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='alamy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='comConference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  Internal Transaction (Token Transfer Log)  External  Transaction  A  B  s  ⓐ Sale NFT for ETH  A  B  ⓑ Sale NFT for FT  FT  A  B  ⓒ Gift NFT  Figure 3: Summary of collecting NFT transaction data from  the Ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' A purple line indicates an inter-  nal transaction, and a green line indicates an external trans-  action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We classify each NFT transaction record between  users into sales ( a○ and b○) or gifts ( c○) based on payment  method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Table 1: Summary of collected NFT transaction records.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  w/o spam: a summary without spam NFTs which were  minted multiple times for fraud, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Type  Collection  NFT (*w/o spam)  52,889,522 (52,514,506)  Account  3,421,654  Transaction  Mint (*w/o spam)  53,653,562 (47,655,998)  Sale  16,642,569  Gift  12,234,533  Burn  799,546  Total transaction records  83,329,082  Period  January 1,2022 ∼ August 31, 2022  specifically designed to leverage a user’s NFT transaction context  (NFT Transaction Context Extractor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Third, previous studies on detecting cryptocurrency phishing  cannot fully capture relationship between NFT users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' While cryp-  tocurrency transactions are only limited to transfers, NFT transac-  tions can be further classified into four distinct categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Particu-  larly, there are two transactions types between NFT users (Figure  1): sales and gifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Our investigation shows that this distinction can  be an significant indicator when interpreting relationships between  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3 explains how DRAINCLoG uses a module (Social  Context Extractor) to capture user relationships in the NFT trading  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  3  DATASET CONSTRUCTION  This section summarizes our data collection approach and the  datasets used in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We collect two datasets in our paper:  NFT transaction data and NFT drainer accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Fetching NFT transaction records from Ethereum blockchain:  To understand a token’s transmission history on the Ethereum  blockchain, we can use transfer logs, which are written by smart  contracts on the blockchain when token transfers occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' With this  information, we can identify changes in ownership of tokens, such  as who sends the tokens to whom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  There are two types of NFT transactions between users: sales  and gifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In the case of sales, most NFTs are traded on the market-  place for Ether or various Fungible Tokens (FTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, because  the NFT transfer logs do not have payment information, we need  additional reference data depending on the payment types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' If an  NFT buyer pays in Ether, it is recorded as an external transaction,  which is used to send Ether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' If the buyer paid in FT, it is recorded  Table 2: Summary of collected drainer accounts from vari-  ous channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Channel  # accounts  Channel  # accounts  ScamSniffer  462  Chainabuse  150  Etherscan  127  Twitter  105  CryptoScamDB  49  Total  742  Period  January 1,2022 ∼ August 31, 2022  in an FT transfer log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, we collect external transactions and  transfer logs related to NFTs and FTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Figure 3 shows our logic for collecting NFT transaction data  from the Ethereum blockchain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' If there is an NFT transfer log from  user 𝐴 to user 𝐵 and an external transaction in which 𝐵 sends Ether  to 𝐴, we decide the NFT was sold for Ether ( a○).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Instead, when an  FT transfer log exists in which 𝐵 sends FTs to 𝐴, we decide the  NFT was sold for FTs ( b○).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Otherwise, we regard that there was  no payment for the NFT, and the NFT transaction from 𝐴 to 𝐵 is  considered as gifting ( c○).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  According to Elliptic[3], NFT draining grew rapidly and has been  on the rise since January 2022 [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As a result, we focus on data  with block_timestamp between the first day of 2022 to August 31,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We obtain more than 83 million transactions from this period  for 53 million NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The data includes transactions from over 3  million unique accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Our NFT transaction data is summarized  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In our analysis, we only consider mints, sales, and gifts  because transaction records of burns account for less than 1% of  total transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Crawling NFT drainer accounts: We crawled drainer accounts  reported for NFT phishing scams from five websites: Twitter, Scam-  Sniffer, Etherscan, CryptoScamDB, and Chainabuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (1) Twitter [49]  is utilized by NFT users as a channel to share information on drainer  accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We collect tweets that mention the phrase “NFT” and one  of the following keywords: phish, hack, drain, stole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' From the col-  lected tweets, we manually extract drainer Ethereum accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (2)  ScamSniffer [5] collects malicious accounts by visiting suspicious  sites that trick users into making dangerous transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (3) Ether-  scan [21], an Ethereum block explorer, provides a list of Ethereum  addresses reported for phishing/hacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (4) CryptoScamDB [2] col-  lects information on scams in the crypto space, including malicious  accounts, URLs, and descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We only consider reports using  the same criteria in (1) Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (5) Chainabuse [1] provides scam  reports with descriptions across multiple blockchains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We collected  Ethereum addresses reported under the NFT Scam category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We crawled all the reports from the above channels based on  each criteria before September 1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Note that there are duplicate  reported accounts on different sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' From the reported accounts,  we define the accounts that gifted NFTs at least once as drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Ac-  cording to this definition, we label 742 unique accounts as drainers  (summarized in Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  4  DRAINER ACTIVITY CHARACTERIZATION  Since drainers have malicious intent, they are likely to exhibit  different behaviour patterns in NFT trading from regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In this section, we look into distinguishable traits of drainers that  DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17, July 2017, Washington, DC, USA  (d) Holding times - Sell  (e) Holding times - Gift-out  (c) Out-in ratio  (a) Active timespan (days)  (b) Gift-in ratio  (c) Out-in ratio  Figure 4: CDFs of (a) active timespan, (b) gift-in ratio, and (c) out-in ratio, where the y-axis denotes proportion of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' CDFs  of holding times of different user types (regular users, affiliated users, and drainers) based on out-transaction types: (d) sell  and (e) gift-out, where the y-axis denotes proportion of NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We include an enlarged graph for holding times smaller than a  day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  motivate the design of our NFT drainer detection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We conduct  a measurement study with NFT transaction records from the first  day of 2022 to July 31, 2022, including 645 drainer accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' First, by  comparing primary trading features with regular users, we verify  that most drainer accounts are used only for draining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Then, we  focus on their liquidation patterns and how they are distinct from  regular transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We track drained NFTs and find that they  have different NFT transaction context and social context from  regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  Trading behavior of drainers  As mentioned in 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1, there are many efforts to prevent phishing  scams in the NFT space, such as sharing information about drainers,  phishing websites, and drained NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' If drainer accounts become  known to be suspicious, it is nearly impossible for them to trade  NFTs with benign users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, we expect that drainers are more  likely to 1) have short-lived accounts and 2) use their accounts only  for draining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In order to investigate these two assumptions, we analyze the  trading activity of 645 NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For comparison, we randomly  extract 10,000 users from our data of 3 million regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' From  this set, users with only one transaction record were excluded since  their active timespan cannot be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This left 6,658 regular  users to be used in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We compare these two groups  along three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  First, we measure how long they traded by calculating the active  timespan: the time difference (in days) between the first and last  transaction recorded for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 4(a) shows that drainers  trade for a shorter period than regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This phenomenon is  also similarly observed in Ether phishing accounts [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Second, we look into how NFT drainers utilize their accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Previous work to detect cryptocurrency phishers [14, 31] only fo-  cused on the amount and frequency of user transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However,  multiple transaction types exist in NFT trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This is an impor-  tant consideration since we assume drainers will be gifted NFTs  from victims (as opposed to buying or minting NFTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, we  calculate the ratio of gifting-in to all in-transaction types (gifting-in,  buying, and minting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Figure 4(b), the majority of drainers have  a relatively high proportion of NFTs gifted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2% of drainers obtain  NFTs only through gifting-in in the three in-transaction types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Table 3: Statistics on transactions about drained NFTs ac-  cording to behaviors right after draining: sell, gift-out, and  none(just holding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' # gifting is the number of giftings before  the first sale after draining, and % sold is the percentage of  NFTs eventually sold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Type  # gifting  # drained NFTs  % sold  Sell  0  11,195 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8%)  100%  Gift-out  1  6,426 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0%)  75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1%  ≥ 2  1,125 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='21%)  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3%  None  0  8,065 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1%)  0%  Total  26,811 (100%)  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4%  In addition, we analyze how likely drainers are to transfer out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Drainers looking to liquidate do not wish to hold their NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' There-  fore, drainers are likely to transfer their NFTs out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We measure  out-in ratio, the ratio of the number of out-transactions to in-  transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Figure 4(c), we observe drainers have a higher  out-in ratio than regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Regular users generally have a  lower out-in ratio: 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1% of them did not make any out-transactions  at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' On the other hand, drainers have higher out-in ratios: 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9%  of them make out transactions on more than half of their NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  This suggests drainers have different intentions from regular users,  who are more likely to use NFTs for collecting or investing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  By combining the observations above, we conclude that most  drainer accounts are for draining purposes only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Additionally, we  further analyze differences between drainers and regular users  along 19 dimensions (more details in Appendix D) and utilize them  in our detection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  Liquidation behavior of drainers  In this section, we dive deep into the liquidation process of drained  NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The uniqueness of NFTs enables us to track the transaction  history of each NFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Since drainer accounts are only used for drain-  ing, we assume all gifted-in NFTs were stolen from victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Alternate accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We find a total of 26,811 NFTs that were  gifted to drainers (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Of these, 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2% were sold directly, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3%  were gifted-out to other users, and the rest (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1%) remained in the  drainer’s wallets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We define the accounts that were gifted NFTs  from drainers as Affiliated users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Only 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5% of drainers do not have affiliated users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In other  words, most drainers (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5%) have one or more affiliated users and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0-- Known Drainer AccountsRandom UsersAffiliated Users-- Known Drainer AccountsRandom Users0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8Conference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  Table 4: Statistics of holding times of drained NFTs depend-  ing on user type along out-transaction types: sell and gift-  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Percent decrease is the measure of the decrease from  𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔 to 𝐻𝑇_𝑑𝑟𝑎𝑖𝑛 as a percentage of 𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  # drained NFTs  Case  Sell  Gift-out  𝐻𝑇 _𝑑𝑟𝑎𝑖𝑛 < 𝐻𝑇 _𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔  8,077 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9%)  6,210 (90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9%)  𝐻𝑇 _𝑑𝑟𝑎𝑖𝑛 = 𝐻𝑇 _𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑚𝑖𝑛  6,498 (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4%)  5,034 (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='7%)  Total  9,085 (100%)  6,832 (100%)  Stats of percent decrease  Mean  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='7%  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5%  Standard deviation  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8  use them to liquidate drained NFTs indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We find 637 affiliated  users including 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4% that are related to two or more drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The  most overlapped affiliated user is connected with 16 drainers (more  details are in Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The fact that many drainers choose  to liquidate through the same affiliated accounts suggests a close  relationship between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' 60% of affiliated users get NFTs only  through gifting-in, while the rest of them (40%) participate in buying  and minting like regular users (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' From the result,  we observe that most affiliated users exhibit similar behaviors to  drainer accounts, but there are also a significant number of affiliated  accounts that engage in general NFT trading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Rapid liquidation: We now analyze how quickly drainers liqui-  date NFTs by measuring the holding times of each NFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We define  holding time as the timespan a user held ownership of an NFT, and  measure it by taking the difference (in days) of the in-transaction  and out-transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Holding times can vary greatly depending on  the user’s investment strategy and characteristics of the NFT, such  as market price and rarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We measure the holding times of all  users that owned the NFT for the each of the 18,746 drained NFTs  with out-transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Note that the holding time can be longer  than seven months (our collection period) because we refer to NFT  transaction data before 2022 to minimize the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We compare  drainer holding times with those of affiliated users and those of  regular users along two out-transaction types: sell and gift-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Drainers and affiliated users show similar distributions which are  noticeably different from that of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 4 (d) shows  they sell more than 80% of NFTs within a day, which is twice more  likely than regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' They also have short holding times before  gifting, but drainers gift NFTs much faster compared to affiliated  users (Figure 4 (e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' By integrating these observations with the fact  that up to 75% of NFTs sent to affiliated users are sold (Table 3), we  notice that drainers alternate between two strategies of liquidation:  (1) directly selling NFTs quickly or (2) quickly gifting-out NFTs to  affiliated users for selling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  To make comparisons with regular transactions, we calculate the  average holding time (𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔) for each NFT as a reference  to a drainer’s holding time(𝐻𝑇_𝑑𝑟𝑎𝑖𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As shown in Table 4, 90%  of 𝐻𝑇_𝑑𝑟𝑎𝑖𝑛s are shorter than 𝐻𝑇_𝑟𝑒𝑔𝑢𝑙𝑎𝑟𝑎𝑣𝑔s, with an average of  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='7% percent decrease regardless of the out-transaction types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We  also observe that 𝐻𝑇_𝑑𝑟𝑎𝑖𝑛 is the minimum holding time for 70%  of NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' From these results, we identify that drainers deviate from  regular transaction patterns in terms of holding time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Table 5: Statistics on sale price of drained NFTs compared  to their market price (Price𝑎𝑣𝑔, Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Only consider  NFTs sold after draining with other sale records of more  than one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' denotes the measure of decrease from each  market price to Price𝑑𝑟𝑎𝑖𝑛 as a percentage of the market  price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Stats of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Case  # drained NFTs  Mean  Std  Price𝑎𝑣𝑔 > Price𝑑𝑟𝑎𝑖𝑛  8,214 (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0%)  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3%  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9%  Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡 > Price𝑑𝑟𝑎𝑖𝑛  8,490 (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5%)  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1%  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2%  Total  11,100 (100%)  Bargain prices: The measurements made above raise a question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  how do drainers liquidate drained NFTs so quickly?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To answer this  question, we compare the sales prices of drainers/affiliated users  (Price𝑑𝑟𝑎𝑖𝑛) with the market prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, unlike stocks or cryp-  tocurrencies, each NFT has its own unique value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' NFTs are also  prone to price changes in response to market supply and demand  [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus it is impossible to define its market price except when  there is a sale transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Therefore, to use as a market price, we  use two baselines: Price𝑎𝑣𝑔 and Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Price𝑎𝑣𝑔 is the average  sale price, and Price𝑐𝑙𝑜𝑠𝑒𝑠𝑡 is the sale price of the sale made closest  to when the drainer made the sale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We observe that drainers sell 74%  and 76% of their NFTs cheaper than the two baselines respectively  with an average price decrease of 37% and 39% (Table 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In summary, our analysis has revealed that NFT drainers ex-  hibit distinct patterns of NFT transactions and social relationships  compared to regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Specifically, drainers tend to have ir-  regular NFT transaction contexts, such as quickly liquidating  NFTs at prices lower than the market value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Additionally, drainers  often have unique social contexts, such as making sales through  affiliated users and connections to the same affiliated users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5  DESIGN OF NFT DRAINER DETECTOR  Our findings in Section 4 suggest that understanding both the trans-  action context and social context of users is crucial for detecting  NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, capturing the intricate relationships be-  tween users and NFTs for drainer detection remains a challenging  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To fully grasp a user’s transaction context, it is necessary to  take into account the transaction history of each NFT traded by the  user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Additionally, to fully understand a user’s social context, it is  important to consider the different types of interactions between  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  To address this complexity, we designed a graph-based NFT  drainer detection model, DRAINCLoG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 5 illustrates the over-  all architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The model creates a comprehensive representation  of each user by using a transaction context extractor and a social  context extractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Each extractor is trained to learn the relevant  context on a NFT-User graph and a User graph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The  NFT-User graph models NFT-to-user interactions using NFT owner-  ship edge attributes, and the user transaction contexts can be fully  captured by aggregating the interaction history of all of their owned  NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The User graph models comprehensive user-to-user interac-  tions using user node attributes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' which detail users’ trading behavior,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' USA  Drainer  Classifier  NFT-User Graph  User Graph  Feature  Engineering  Transaction context  representation  ⋯  Transaction  Context  Extractor  Social  Context  Extractor  Social context  representation  ⋯  Attributed  Node  Attributed  Edge  User node  NFT node  NFT gift edge  NFT sale edge  NFT ownership edge  Drainer  Figure 5: The overall architecture of DRAINCLoG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' First, NFT transaction records and user accounts obtain attributes through  feature engineering and are used to construct a NFT-User graph and a User graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The transaction context extractor and social  context extractor are trained to extract information from each graph, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The user features and the representations  from two extractors are concatenated together and fed into the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  and two types of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The two contexts are then combined and  fed into a classifier to integrate the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  Feature Engineering  Before learning high-level user representations, we perform fea-  ture engineering based on observations in Section 4 to obtain NFT  ownership edge attributes and user node attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  NFT ownership edge attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To capture different transac-  tion behaviors, we create representations of how users interact with  NFTs, which is used in the NFT-User graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We use the following 7  features to represent NFT ownership:  (1) Holding time: Holding time is the timespan a user held  ownership of the NFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' If there is no out-transaction, we  calculate it by taking the difference (in days) between the  in-transaction and the last day of our collection period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As  drainers tend to sell or gift NFTs faster than regular users,  their holding times are shorter than those of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (2) In-transaction type & Out-transaction type: Each trans-  action type is a categorical feature of how the user received  the NFT (buy or gift-in) and what the user did with the NFT  (sell, gift-out, or hold), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drained NFTs have the  gift-in transaction type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (3) In-price & Out-price: Each is the price (in Ether) the user  sent to receive the NFT and the price the user received when  sending the NFT out, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' It is zero if the transaction  type is gifting, and -1 if no out-transaction was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The  in-price and out-price are significant because drainers sell  their NFTs cheaper than regular transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (4) Average holding time & Average sale price: For compar-  ison, we include the NFT’s average holding time and sales  price across all owners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This can be used as a reference in  finding anomalies in holding times and pricing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  User node attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To comprehensively understand the  social relationships between users, we created detailed represen-  tations of their trading behavior, which is used in the User graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We introduce 19-dimensional user node attributes that take into  account NFT characteristics such as collections and transaction  types, which helps the model to identify distinct behavior patterns  of drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (1) Active timespan: The time difference between the first and  the last transaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainers are more likely to have a short  active timespan than regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (2) Gift-in ratio: The ratio of gifting-in to all in-transactions  (minting, buying, gifting-in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Most drainers obtain NFTs only  through gifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (3) Out-in ratio: The ratio of out-transactions to in-transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Drainers have a higher out-in ratio than regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (4) Number of each transaction type: Transaction types con-  sidered are minting, buying, gifting-in, selling, and gifting-  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainers participate in selling and gifting-in more than  other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1)  (5) Number of collections for each transaction type: NFT  users commonly form communities based on collections [37]  and trade NFTs from similar collections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Unlike regular users,  drainers tend to trade more kinds of collections than regular  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The transaction types are the same as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (Appen-  dix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2)  (6) Number of neighbors for each transaction type: Ac-  counts that made a transaction with each other are con-  sidered neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainers tend to have more neighbors  from gift-in transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The transaction types are the same  as above, excluding minting (which produces no neighbors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3)  (7) Frequency of gift-ins & sales: The number of gifting-in/selling  transactions divided by the active timespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Gift-ins and  sales occur more frequently for drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  NFT Transaction Context Extractor  To fully understand a user’s NFT ownership, we leverage the trans-  action histories of all the user’s NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We construct a NFT-User  graph to model ownership changes in NFTs and train an extractor  for the graph to extract NFT transaction context of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' From  the NFT-User graph, we first obtain the transaction context of each  NFT  NFT  NFT  NFT  NF  NFT  NFT  NFT  alamy  ImageID:2J6YXM5  www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='alamy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='comConference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  NFT by aggregating its transaction history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Finally, to get a repre-  sentation of the user’s NFT transaction context, we aggregate the  transaction contexts of all of their owned NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  NFT-User graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We construct an undirected  graph 𝐺(𝑈, 𝑁, 𝐸𝑇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' There are two types of nodes: user nodes 𝑈 ,  NFT nodes 𝑁, and the 𝑢 nodes and 𝑛 nodes can be connected  with an attributed edge 𝑒 ∈ 𝐸𝑇 , called NFT ownership attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Additionally, 𝑁𝑒 (𝑛) is the node 𝑛’s one-hop neighbors (users who  traded NFT 𝑛), 𝑁𝑒 (𝑛) = {𝑢′ ∈ 𝑈 |(𝑢′,𝑛) ∈ 𝐸𝑇 }, and 𝑁𝑒 (𝑢) is the  set of NFT nodes in node 𝑢’s one-hop neighbors, 𝑁𝑒 (𝑢) = {𝑛′ ∈  𝑁 |(𝑛′,𝑢) ∈ 𝐸𝑇 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  NFT transaction context extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' A transaction context ℎ𝑁  of an NFT node 𝑛 is updated based on attributed edges connected  to, denoted as 𝑇𝑛 = {𝑡𝑢1,𝑡𝑢2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' ,𝑡𝑢𝑚 }, where 𝑢𝑖 (1 ≤ 𝑖 ≤ 𝑚) is one  of 𝑁𝑒 (𝑛), and 𝑚 is the size of 𝑁𝑒 (𝑛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Then, we aggregate them with  convolution layer as follows :  ℎ𝑁  𝑛 = 𝜎  �  𝑊 𝑁 · 𝐴𝐺𝐺𝑁 (𝑇𝑛)  �  where 𝜎 is an activation function, and 𝑊 𝑁 is the trainable matrix  for learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We use mean-pooling as an aggregation function to  represent the transaction context of each NFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Next, a user representation ℎ𝑈 is updated by all neighboring NFT  nodes’ transaction context vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For a user 𝑢, the representation  ℎ𝑈𝑢𝑛 of each NFT 𝑛 ∈ 𝑁𝑒 (𝑢) is obtained as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  ℎ𝑈  𝑢𝑛 = 𝑊 𝑈 · 𝑐𝑜𝑛𝑐𝑎𝑡(𝑡𝑢𝑛,ℎ𝑁  𝑛 )  where𝑡𝑢𝑛 is𝑢’s ownership of𝑛, andℎ𝑁𝑛 is the transaction context  vector of 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We concatenate the two and apply a linear transforma-  tion with the learnable matrix 𝑊 𝑈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Finally, we integrate all of the NFT transaction context vectors  {ℎ𝑁𝑛1,ℎ𝑁𝑛2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' ,ℎ𝑁𝑛𝑣 } (𝑣 is the size of 𝑁𝑒 (𝑢)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To be sensitive to irregu-  lar NFT transaction patterns, we implement attention mechanisms  to combine each NFT ownership vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Specifically, we adapt a  multi-head graph attention operation to reduce the effects of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  𝑧𝑢𝑛 = LeakyReLU(𝑎 · ℎ𝑈  𝑢𝑛)  ℎ𝑈  𝑢 =  ∑︁  𝑛∈𝑁𝑒 (𝑢)  𝛼𝑢𝑛𝑧𝑢𝑛  where 𝑧𝑢𝑛 is an attention score calculated by taking the dot  product of learnable weight 𝑎 and applying LeakyReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' 𝛼𝑢𝑛 is the  normalized attention score using the softmax function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Lastly, we  compute the final representation by averaging the attention head  outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The final representation feeds into the classification layer  for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The extractor updates the trainable parameters  to better learn features that distinguish drainers from regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We use cross-entropy loss as the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3  Social Context Extractor  Most drainers choose to liquidate through affiliated accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' More-  over, some affiliated accounts are used by multiple drainers, which  suggests a close relationship between the co-users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, the rela-  tionships between users are essential to detect drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To model  Table 6: Dataset statistics for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Ratio  refers to the ratio of drainers to sampled regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For  evaluation datasets, each number is the average value over  10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Dataset  Ratio  # central nodes  # total nodes  # transactions  Training  𝐷0  1:29  19,350  1,680,371.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  19,269,001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  Evaluation  𝐷1  1:10  1,067  1,326,162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  15,340,788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  𝐷2  1:20  2,037  1,520,345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9  18,918,056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  𝐷3  1:50  4,947  1,759,077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9  23,672,245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3  𝐷4  1:100  9,797  1,922,109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  26,504,190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9  user interactions, we construct a User graph and use it to train an  extractor to learn the social context of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  User graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Trading between users can be con-  structed as a graph 𝐺(𝑈, 𝐸, 𝑅,𝑋𝑈 ), where 𝑈 is the set of user nodes,  and 𝐸 is labeled edges (𝑢𝑖,𝑟,𝑢𝑗), where 𝑟 ∈ 𝑅 = {𝑠𝑎𝑙𝑒,𝑔𝑖𝑓 𝑡} is a re-  lation type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Each user node has 19-dimensional user node attributes,  𝑋𝑈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' If 𝑢𝑖 transfers an NFT to 𝑢𝑗, then 𝑢𝑖 and 𝑢𝑗 are connected with  an edge 𝑒 ∈ 𝐸 with a label gift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  Social context extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Transaction types are important to  understand user relationships in the NFT ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To consider  transaction types between users, we use the R-GCN [44] model,  where edges can represent different relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  From the user graph, we obtain a representation vector of a user  node 𝑢 updated by its neighboring user nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The propagation at  (𝑙 + 1)-th layer of R-GCN with 𝐿 layers is as follows:  ℎ𝑙+1  𝑢  = 𝜎  �  𝑊 𝑙ℎ𝑙  𝑢 ·  ∑︁  𝑟 ∈𝑅  AGGU( 1  𝑐𝑢,𝑟  𝑊 𝑙  𝑟 ℎ𝑙  𝑣), ∀𝑣 ∈ 𝑁 (𝑢)𝑟  �  where 𝜎 is an activation function, and 𝑊 𝑙 is a learnable matrix  shared among all nodes at 𝑙-th sub-layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' 𝑁 (𝑢)𝑟 is neighboring user  nodes under relation𝑟 of𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We use mean-pooling as the aggregation  function AGGU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We train this module in the same process as the NFT  transaction context extractor by feeding the outputs into the classi-  fication layer and using cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  Drainer Classifier  Following the above operations, we obtain two types of features  to represent users: (1) NFT transaction context representation, and  (2) social context representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We concatenate the two repre-  sentations together to create our final representation, integrating  comprehensive information learned from our graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In order to  learn the differences between drainers and regular users, we feed  the final representation to a classifier layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We choose a support vector machine (SVM) [26] as our classi-  fier layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' SVM is a supervised machine learning model that uses  classification algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' SVM has the advantage of reducing the  chances of model overfitting, making the model highly stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' SVM  is also powerful to deal with high dimensional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17, July 2017, Washington, DC, USA  6  EXPERIMENTS  To validate our model in different aspects, we present several em-  pirical evaluations in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Specifically, we seek to answer  the following research questions:  (RQ1) How effective is our model in detecting NFT drainers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (RQ2) How does each component of our model affect performance?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  (RQ3) How robust is our model against evasion attacks?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  Datasets  Data Filtering: In order to conduct experiments, we used a dataset  consisting of NFT transaction data and accounts identified as NFT  drainers (Section 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, we found that the ratio of regular  users to drainers in the dataset was highly imbalanced, with only  742 drainers out of 34 million total accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To address this, we  performed data filtering to remove accounts that were unlikely  to be drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This included removing accounts that had never  received NFTs from other accounts and accounts with zero active  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' After filtering, 1,577,200 accounts remained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Datasets Construction: To create our training dataset, we used  accounts that made transactions between the first day of 2022 and  July 31, 2022, including 645 identified drainer accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We sampled  12,900 regular user accounts to create a ratio of 1:20 between regular  users and drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, we found that regular users with more  transactions, defined as heavy users, were more likely to be targeted  by drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, we added an additional 5,805 heavy users (1:9  ratio with drainer accounts) to our training data, which ended up  with 645 drainers and 18,705 regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We will discuss the  effect of adding extra heavy users to the training data on model  performance in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  For the evaluation dataset, we used accounts that made at least  one transaction in August 2022, including 97 drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To address  the imbalance in the ratio of drainers to regular users, we evaluated  the model multiple times with varying numbers of regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Specifically, we set the number of regular users to be 10, 20, 50, and  100 times the number of drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This allows us to measure the  performance of the model under different levels of class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  While using higher ratios of regular users more closely represents  real-world scenarios, it increases the influence of false negatives  (drainer accounts that were not reported during this time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The  statistics of the datasets are summarized in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For each dataset,  we also add nodes representing first and second-order neighbor  accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The neighbor accounts were used only to enrich the graph  and were not used in training or evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  Experimental Setup  Baseline Models: There is no previous work for NFT drainer de-  tection, so as baselines, we use methods that effectively detect  Ethereum phishing accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In addition, we also use other widely-  used graph-based models, because DRAINCLoG is a graph-based  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The baselines can be divided into two categories: Feature-  based and graph-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Explanations of each method are summa-  rized below:  Feature-based:  Ethereum features [14] are 119-dimensional statistical fea-  tures previously used for Ethereum phishing account de-  tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The features mainly consist of first-order neighbor  information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  E-GCN features [13] are the initial node features used in  E-GCN, a method proposed to detect Ethereum phishing  accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  DRAINCLoG user features are the initial node features  used in our User-graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Graph-based:  Node2Vec [23] is a graph node embedding method based  on random walks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  E-GCN [13] applies a Graph Convolutional Network (GCN)  to detect Ethereum phishing accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  GAT [51] is a widely used GNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' It learns node repre-  sentations by aggregating neighbor nodes with an attention  mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  GraphSAGE [24] is another GNN variant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' It learns node  representations by aggregating sampled neighbor node fea-  tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  To evaluate the effectiveness of our features, we implement each  graph-based baseline twice: once using E-GCN features (denoted by  the "E-" prefix) and once using DRAINCLoG user features (denoted  by "N-" prefix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Implementation Details: The embedding size of both NFT Trans-  action Context Extractor and Social Context Extractor is set to 64,  and the learning rate is set to 6e-4 and 5e-4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In NFT  Transaction Context Extractor, the number of attention heads is  set to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The regularization parameter and gamma of SVM are set  to 60 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As the classifier for each baseline, we  choose the one with better performance between SVM and light-  GBM [28], another machine learning algorithm widely used for  classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Feature-based methods, E-GCN, E-GAT,  and E-GraghSAGE use lightGBM, and the rest use SVM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3  Experimental Results (RQ1)  Table 7 shows the detection performances of our model and base-  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (The ROC and PR curves for the top performing methods are  illustrated in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=') Our model outperforms the baselines  in all evaluation metrics, which proves the effectiveness of DRAIN-  CLoG for NFT drainer detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As the dataset size increases, the  performance gap between DRAINCLoG and the baselines becomes  more pronounced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  DRAINCLoG outperforms feature-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This is because  DRAINCLoG benefits from the NFT-specific features and high-level  representations obtained from two extractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Many NFT-specific  features, such as those related to NFT collections and transaction  types, can be instrumental in distinguishing NFT drainers from  regular users but are ignored in the existing feature-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Moreover, one of the most important features used in Ethereum  phishing account detection [14] comes from how the phishers cash  out: 1-hop neighboring nodes of phishers are usually mixing service  accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, these features are not effective in detecting  stolen NFTs, which cannot go through mixing services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Previously  used features do not fully apply to NFT trading and can miss other  important features that are leveraged by DRAINCLoG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Conference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  Table 7: The results of experiments averaged over 10 runs on datasets 𝐷1 ∼ 𝐷4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=', Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=', F1, and Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' FPR mean precision,  recall, F1 score, and average false positive rate, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Model  Dataset (ratio)  𝐷1 (1:10)  𝐷2 (1:20)  𝐷3 (1:50)  𝐷4 (1:100)  Metrics  Pre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Rec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='  F1  Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' FPR  Feature-based  Ethereum features  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='794  DRAINCLoG  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='810  One surprising finding is that graph-based methods (E-GCN,  E-GAT, and E-GraphSAGE) without our NFT-specific attributes per-  form poorly compared to the feature-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For instance,  the E-GCN method performs worse than using only the E-GCN  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, by incorporating NFT-specific attributes in the  graph-based methods (N-GCN, N-GAT, and N-GraphSAGE), their  performances improve significantly, highlighting the importance of  our feature engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In addition to the NFT-specific attributes,  DRAINCLoG takes into account transaction types and utilizes NFT  transaction context extracted from the NFT-User graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This results  in a multi-faceted representation that allows it to outperform other  graph-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  Ablation Study (RQ2)  In this section, we aim to understand the impact of each compo-  nent of DRAINCLoG on its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' First, we analyze how  each structural component affects performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We conducted the  same detection task after eliminating each representation produced  (a) F1- score  (b) Precision  Figure 6: The results of ablation experiments averaged  over 10 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' 𝑆𝐶𝐸 and 𝑇𝐶𝐸 denote social context extrac-  tor and NFT transaction context extractor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  𝐷𝑅𝐴𝐼𝑁𝐶𝐿𝑜𝐺−𝑟𝑒𝑙 is a modified DRAINCLoG which does not  consider transaction types between users in the user graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  by NFT Transaction Context Extractor (TCE) and Social Context Ex-  tractor (SCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In addition, to evaluate the importance of consider-  ing transaction types between users in the User graph, we tested  DRAINCLOG  SCE  TCE  DRAINCLoG-rel  +-DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17, July 2017, Washington, DC, USA  Table 9: The results of the evasion attack simulation aver-  aged over 5 runs with varying attack levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The values listed  below for Attack 2 were simulated with X = 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Attack1  Attack2  Level (L%)  𝐷1  𝐷2  𝐷3  𝐷4  𝐷1  𝐷2  𝐷3  𝐷4  w/o attack  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='737  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='651  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='571  𝐷𝑅𝐴𝐼𝐶𝐿𝑜𝐺−𝑟𝑒𝑙, which uses GCN instead of R-GCN for social con-  text extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  As shown in Figure 6, the best performance is achieved when all  the components are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' SCE has similar precision as DRAINCLoG  but its recall (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='784) is lower than DRAINCLoG’s (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='825), resulting  in a larger gap in the F1-score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Although using TCE alone per-  forms poorly, the NFT transaction context helps DRAINCLoG find  drainers not captured by SCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' 𝐷𝑅𝐴𝐼𝐶𝐿𝑜𝐺−𝑟𝑒𝑙 also shows poor per-  formance compared to DRAINCLoG and suffers from significant  performance degradation as the number of regular users increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  This highlights the importance of transaction types between users  for identifying NFT drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We also examine the effect of elements of NFT transaction edge  attributes and user node attributes, which are used in graph construc-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We grouped features that represent similar concepts together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  For example, the number of transactions for each transaction type  (5) was grouped into one, the number of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Then, we  trained and evaluated DRAINCLoG on the graph constructed with-  out each feature group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In Table 8, the most influential factors used in the TCE com-  ponent are transaction type, average information, and price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' When  the evaluation dataset size is small, price is more important than  transaction type, but as the dataset size increases, the transaction  type becomes more crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Removing the average information of  each NFT decreases the performance, which shows they are useful  for capturing transaction context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  For user node attributes used in the SCE component, the most  influential factors are the number of neighbors, the number of col-  lections, and out-in ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This result aligns with the drainers’ goal  of stealing NFTs from as many users as possible and selling them  for profit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Removing any of three groups: active timespan, gift-in  ratio, and the number of each transaction type results in a higher  recall but lower precision compared to DRAINCLoG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This suggests  that drainers have unique characteristics not present in regular  users in these three features, and they help DRAINCLoG avoid  misclassifying regular users as drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Overall, these results confirm the importance of each representa-  tion and the effectiveness of our feature engineering in enhancing  the detection ability of DRAINCLoG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5  Robustness (RQ3)  If drainers notice that they are monitored by DRAINCLoG, they  may purposely change their trading patterns to avoid detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In  (a) Attack 1  (b) Attack 2  (10%)  (30%)  (50%)  (70%)  Figure 7: F1-score after re-updating on different cutoff dates  against (a) Attack 1 and (b) Attack 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Each color represents a  different attack level, denoted by 𝐿1 to 𝐿4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Attack 2, differ-  ent line types represent the varying amounts of Ether sent  by the attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For each date, the ratio of attackers to reg-  ular users was 1:100 in evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  this section, we evaluate the robustness of our model by simulating  evasion attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  Evasion attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainers can adopt various strategies to con-  ceal themselves as regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We introduce two types of evasion  attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We assume that drainers cannot afford to change their liq-  uidation strategy while avoiding marketplace bans, and therefore  focus on the NFT acquisition step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Attack 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Mint NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The easiest way that drainers can engage  in innocuous-appearing NFT transactions is by minting NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' By  increasing the number of minted NFTs, drainers can decrease their  gift-in ratio, out-in ratio, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Attack 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Send Ether to the victim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainers can hide their drain-  ing activities by sending Ether to victims after stealing NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This  way, their behavior is recorded as a sale instead of a gift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To evaluate how well DRAINCLoG can detect  drainers under different evasion attack strategies, we modified  previous evaluation datasets (𝐷1 ∼ 𝐷4) by altering the transaction  records of drainers based on the attack strategy and attack level 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  For Attack 1, we increased the number of minted NFTs by 𝐿% of  gifted-in NFTs, where 𝐿 ∈ {10, 30, 50, 70}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For Attack 2, we changed  𝐿% of gifting-in transactions to buying transactions by sending  X% of the average sale price of each NFT to those victims, where  𝐿 ∈ {10, 30, 50, 70} and 𝑋 ∈ {1, 10, 60}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We set 𝑋 to be less than  60 because drainers typically sell stolen NFTs at 40% below their  average sale price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  The results in Table 9 show that DRAINCLoG’s performance  decreases as the attack level increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Attack 1 is less effective at  evading detection than Attack 2 because it does not change the  relationship between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Making significant changes to user  attributes by increasing the number of minted NFTs is not enough  to fool the detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' On the other hand, Attack 2 is much  more effective as it changes the relationship between users as well  as the attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This comes at a cost for the attacker, since each  drained will incur an extra cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Despite this, this method still fails to  fully deceive the system, which means DRAINCLoG can effectively  capture their transaction context and social context from other  behaviors such as those during liquidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  X%)  1% --  10%  60%L2  L3  L4L3  L4L4L4w/o attack L1  L2  w/o attack  L4AttacklevelConference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3  Defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' One of the defenses against evasive attacks is to  periodically update DRAINCLoG to learn new types of drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We  update DRAINCLoG in a simplified manner, re-training only the  last layer (SVM), while leaving two extractors intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To prevent  evaluating on users seen in training, we train only on users that  went inactive before a cutoff date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We re-trained the SVM for 5  different cutoff dates spaced apart by 2 days, and evaluated on the  remaining data for each date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The number of active attackers at  the end of the last day (August 10th) was 18(18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6% of 97 drainers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We trained and evaluated using a fixed ratio of drainers to regular  users as previously described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 7 shows the  evaluation results (F1-score) of experiments averaged over 5 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  After just one update, with seven added attackers (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2%), DRAIN-  CLoG’s performance significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As the model contin-  ues to update, its performance approaches the original level, and  the performance difference between attack levels becomes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Attack 2 is noticeably more effective in evading the detector when  more Ether is sent to victims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In the extreme scenario when 70%  of gifting is changed to buying (𝐿4) and 60% of 𝑃𝑟𝑖𝑐𝑒𝑎𝑣𝑔 is sent, it  becomes difficult to detect attackers even after updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However,  this is not likely to happen as it gives up a significant amount of  profit from NFT draining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In the next section, we will discuss ways  to make DRAINCLoG more robust against new types of drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  7  DISCUSSION  Future work for evasion attack: In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5, to improve the  robustness of DRAINCLoG, we updated the drainer classifier (SVM)  with evasion attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Additionally, we can improve the model  by updating the transaction context extractor and social context  extractor to account for new types of drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Furthermore, addi-  tional adversarial examples could be created to augment training  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3, we observe that the performance increases  significantly after the detector is updated with only a few adver-  sarial attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' By updating DRAINCLoG with examples similar to  new drainers, the detection performance recovers quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Adoption in the real-world: In this study, we focused on under-  standing the behavior of NFT drainers and designing a specialized  model for identifying drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To ensure that DRAINCLoG remains  effective in the real world, it’s important to consider how it responds  to new NFT transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  It is crucial for the detection system to quickly identify if a user is  a drainer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Our model is viable in scanning many different targets at  once since it requires only the two-hop neighbors, rather than the  entire graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Therefore, we believe that our model can be effectively  utilized for real-time detection in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Effect of heavy users on model performance: We find that  drainers and regular users show different distributions in the num-  ber of transactions (Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This could cause our model,  DRAINCLoG, to focus on the number of transactions when iden-  tifying drainers, potentially misclassifying legitimate users who  make many transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To prevent this problem, we added regular  users with more than 16 transactions, defined as heavy users, to our  training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In order to decide how many heavy users will be added to the  training dataset, we conduct experiments with varying number of  heavy users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 8 (a) shows that training the model without  (a) F1-score  (b) Distribution of false positives  only random users  random users + heavy users  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='Ps when the training set consists of  Figure 8: (a) The performance when varying the number  of heavy users included in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' (b) Regu-  lar users misclassified as drainers (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='Ps) in dataset 𝐷4 when  heavy users are added/not added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  extra heavy users suffers from severe performance degradation as  the dataset size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We found that when the number of heavy  users is 9 times the number of drainers, the model performs well  across all dataset sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, we fix the ratio of heavy users to  drainers as 9 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 8 (b) shows that using heavy users  in the training phase prevents the model from classifying users  based on their number of transactions made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Without heavy users,  the model often misclassifies regular users with many transactions  as drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' But with heavy users, the model is less likely to make  this mistake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  8  RELATED WORK  Suspicious behaviors in NFT markets: With the increasing pop-  ularity of NFTs, suspicious activities targeting NFTs are also rising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  A few studies exist for analyzing security issues in the NFT ecosys-  tem, such as wash tradings and shill bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, to the best  of our knowledge, we are the first to perform an in-depth study of  NFT drainers and propose an NFT drainer detection system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Das et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [17] conducted a comprehensive study of design weak-  nesses originating from the NFT marketplaces and external entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Also, they investigated various types of fraudulent user activities  occurring in NFT marketplaces, such as counterfeit NFT creation,  wash trading, and shill bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Von et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [52] quantified market  abuse in the NFT ecosystem with their proposed NFT wash trading  detection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Ethereum phishing scam detection: Ethereum phishing scam  detection can be categorized into two main types: feature-based  and graph-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [14] extracted 119-dimensional statistical features to  consider the 1-order neighbors of the node as well as the node itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  They used a LightGBM-based dual-sampling ensemble algorithm  to classify phishing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Another line of research focuses on network representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [54] proposed Trans2Vec, which is a modified random  walk-based network embedding method with biases of transaction  amount and timestamp for neighbor sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [13]  introduced E-GCN, the first Ethereum phishing scam detection  method based on Graph Neural Networks (GNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' They extracted  8-dimensional statistical features, such as in/out-degree, number  of neighbors, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=', and fed them into a GCN for embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Li et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [31] constructed edge representations from transaction records  DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17, July 2017, Washington, DC, USA  to capture the temporal relationship between users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The edge rep-  resentations are aggregated into node representations and used to  obtain structural features using GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Unlike the above works that  approached using node classification, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [55] regarded  the problem as a graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' They used hierarchical graph  pooling layers to extract node-level representations, which were  then aggregated to form graph-level representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  However, the works discussed above are difficult to apply to NFT  drainers detection due to the characteristics of the NFT ecosystem,  as mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Graph Neural Network: In recent years, deep learning methods  have achieved remarkable performance in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Deep neu-  ral networks have also been applied to graph data to leverage the  structural properties of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Graph Convolution Networks (GCNs) [30] is one of the most  prominent graph neural network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' GCNs perform convo-  lution operations on graph data and learn embeddings of nodes  by aggregating features from neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Unlike GCNs,  which uses information from adjacent nodes as is, Graph Attention  Networks (GATs) [51] utilize information from neighbors by us-  ing node attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Multi-head attention is used to learn a number  of attentions, and the node features obtained from each attention  are concatenated to form a single feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' GraphSAGE [24] is an  inductive GNN model, which generalizes the unobserved nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  By incorporating node features into the learning algorithm and  aggregator functions, it can simultaneously learn the distribution  of neighboring node features and the topological structure for the  neighbors of each node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  9  CONCLUSION  Phishing attacks are a significant threat to the NFT trading ecosys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Despite the increasing damages caused by NFT drainers, their  behaviors are not well studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To conduct an in-depth study on  NFT drainers, we extract data from 83M NFT transaction records  and collect 742 NFT drainer accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We verify that they have dif-  ferent transaction contexts and social contexts compared to regular  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Based on our measurement results, we propose a detection  model, DRAINCLoG, tailored to detect drainers in the NFT environ-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' DRAINCLoG is able to generate a user representation that  considers NFT transaction context and social context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Evaluated  on real-world NFT transaction data, we verify our model’s effec-  tiveness and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We believe that our findings and detection  method will contribute to the security NFT ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  REFERENCES  [1] Chainabuse webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='chainabuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-  September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [2] Cryptoscamdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://cryptoscamdb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='org/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [3] Elliptic webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='co/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [4] Nftgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://nftgo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [5] Scamsniffer webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://scamsniffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [6] OLGA  POLISHCHUK  ADAM  DARRAH  and  STEWART  MITCHELL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Flash report: Nft drainer claims to bypass cryptocurrency wallet update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='zerofox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/blog/flash-report-nft-drainer-claims-to-bypass-  cryptocurrency-wallet-update/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 22-November-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [7] Leman Akoglu and Christos Faloutsos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Anomaly, event, and fraud detection in  large network datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Proceedings of the sixth ACM international conference  on Web search and data mining, pages 773–774, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [8] Robin Barber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Nft statistics, facts & trends in 2022: All you need to know about  non-fungible tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='cloudwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='net/nft-statistics/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  accessed 21-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [9] Daniel Van Boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Nft investors lose $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='7m in opensea phishing at-  tack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  https://threatpost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/nft-investors-lose-1-7m-in-opensea-phishing-  attack/178558/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [10] Daniel Van Boom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Seth green loses $200k bored ape yacht club nft in phishing  scam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='cnet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/personal-finance/seth-green-loses-200k-bored-  ape-yacht-club-nft-in-phishing-scam/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [11] Jack Brassell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Metamask gets new ‘set approval for all’ controls to boost secu-  rity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='beyondgames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='biz/25286/metamask-gets-new-set-approval-  for-all-controls-to-boost-security/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 22-November-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [12] Huashan Chen, Marcus Pendleton, Laurent Njilla, and Shouhuai Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' A sur-  vey on ethereum systems security: Vulnerabilities, attacks, and defenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' ACM  Computing Surveys (CSUR), 53(3):1–43, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [13] Liang Chen, Jiaying Peng, Yang Liu, Jintang Li, Fenfang Xie, and Zibin Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Phishing scams detection in ethereum transaction network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' ACM Transactions  on Internet Technology (TOIT), 21(1):1–16, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [14] Weili Chen, Xiongfeng Guo, Zhiguang Chen, Zibin Zheng, and Yutong Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Phish-  ing scam detection on ethereum: Towards financial security for blockchain ecosys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In IJCAI, pages 4506–4512, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [15] Jian Cui, Kwanwoo Kim, Seung Ho Na, and Seungwon Shin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Meta-path-based fake  news detection leveraging multi-level social context information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Proceedings  of the 31st ACM International Conference on Information & Knowledge Management,  pages 325–334, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [16] ADAM DARRAH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Flash report: Nft drainer claims to bypass cryptocurrency  wallet update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='zerofox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/blog/flash-report-nft-drainer-claims-  to-bypass-cryptocurrency-wallet-update/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [17] Dipanjan Das, Priyanka Bose, Nicola Ruaro, Christopher Kruegel, and Giovanni  Vigna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Understanding security issues in the nft ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' arXiv preprint  arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='08893, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [18] Tech Desk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Here’s what you can do if your nfts are stolen on opensea nft  marketplace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://indianexpress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/article/technology/crypto/heres-what-  you-can-do-if-your-nfts-are-stolen-on-opensea-nft-marketplace-8085992, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [19] James Ellis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Metamask add feature to stop wallet drainer nft scams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://  nftevening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/metamask-add-feature-to-stop-wallet-drainer-nft-scams/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 22-November-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [20] Ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Non-fungible tokens (nft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://ethereum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='org/en/nft/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [21] Etherscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Etherscan webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://etherscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='io/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-  September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [22] Jonathan Greig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Bored ape yacht club says its instagram was hacked to funnel  users to nft phishing sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://therecord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='media/bored-ape-yacht-club-says-  instagram-hacked-nfts-stolen/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [23] Aditya Grover and Jure Leskovec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' node2vec: Scalable feature learning for net-  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content=' Inductive representation learning  on large graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content=' Uniswap users fall victim to a usd 8m nft phishing attack,  binance pulls false alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://cryptonews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/news/uniswap-users-fall-  victim-to-a-usd-8m-nft-phishing-attack-binance-pulls-false-alarm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='htm, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content=' Support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' IEEE Intelligent Systems and their ap-  plications, 13(4):18–28, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [27] Steve Kaarus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Elliptic: $100m in nfts stolen via scams in the past year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https:  //coingeek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/elliptic-100m-in-nfts-stolen-via-scams-in-the-past-year/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [28] Guolin Ke, Qi Meng, Thomas Finley, Taifeng Wang, Wei Chen, Weidong Ma,  Qiwei Ye, and Tie-Yan Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Lightgbm: A highly efficient gradient boosting decision  tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Advances in neural information processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [29] Dylan Kelly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Bored ape yacht club’s instagram was hacked in $2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='7 million usd nft  phishing attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://hypebeast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/2022/4/bored-ape-yacht-club-instagram-  hacked-phising-attack-nft, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='02907, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [31] Sijia Li, Gaopeng Gou, Chang Liu, Chengshang Hou, Zhenzhen Li, and Gang  Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Ttagn: Temporal transaction aggregation graph network for ethereum  phishing scams detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In Proceedings of the ACM Web Conference 2022, pages  661–669, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [32] lunaray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' New nft wallet-draining exploit-degen meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/  coinmonks/new-nft-wallet-draining-exploit-degen-meta-f81da02adb6f, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Conference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  [33] lunaray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' New nft wallet-draining exploit-degen meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='  Nft collection meaning and how many nfts are in a collec-  tion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://bitkan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='  [35] Metamask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' What is a token approval?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://metamask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='zendesk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='  accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content=' What is nft wash trading?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content=' British army’s youtube and twitter hacked to promote crypto  scams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content='com/article/british-army-youtube-twitter-account-  hack, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [39] OpenSea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Opensea webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://opensea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='io/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-  September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [40] OpenSea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' What is opensea’s stolen item policy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='opensea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='io/  hc/en-us/articles/4815371492499-What-is-OpenSea-s-stolen-item-policy, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [41] Rarible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Rarible webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://rarible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-  September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [42] Harry  Robertson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Pmorgan  says  ethereum  is  losing  ground  to  rivals  in  the  nft  market,  which  could  boost  the  likes  of  solana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  https://markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='businessinsider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/news/currencies/ethereum-eth-killers-  nfts-defi-solana-cardano-wax-crypto-investing-2022-1, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed  10-October-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [43] NIC SALHUANA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Nft theft: Here’s how the dark side of web3 gets away with  it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  https://nftnow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/features/nft-theft-heres-how-the-dark-side-of-web3-  gets-away-with-it/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [44] Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne van den Berg, Ivan  Titov, and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Modeling relational data with graph convolutional  networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In European semantic web conference, pages 593–607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Springer, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [45] Clare Stouffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Nft scams: 10 types + how to avoid nft fraud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='norton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  com/internetsecurity-online-scams-nft-scams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='html#, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed  14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [46] Runnan Tan, Qingfeng Tan, Peng Zhang, and Zhao Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Graph neural network for  ethereum fraud detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In 2021 IEEE International Conference on Big Knowledge  (ICBK), pages 78–85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' IEEE, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [47] Bryan Teoh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Hackers target duppies nft and successfully drain multiple wal-  lets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://nftevening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/hackers-target-duppies-nft-and-successfully-drain-  multiple-wallets/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [48] Bill Toulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  $8 million stolen in large-scale uniswap airdrop phishing at-  tack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='bleepingcomputer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/news/security/8-million-stolen-in-  large-scale-uniswap-airdrop-phishing-attack/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-  September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [49] Twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Twitter webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://twitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-  2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [50] Iulia Vasile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Crypto and nft airdrops: What are they, and how do they work?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  https://beincrypto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/learn/crypto-and-nft-airdrop/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed  14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [51] Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero,  Pietro Lio, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Graph attention networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  arXiv preprint  arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='10903, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [52] Victor von Wachter, Johannes Rude Jensen, Ferdinand Regner, and Omri Ross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Nft  wash trading: Quantifying suspicious behaviour in nft markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' arXiv preprint  arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='03866, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [53] Sophie Waldman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Metamask looks to reduce nft scams with updated "set approval  for all" featurem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' https://hypemoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='com/2022/7/metamask-set-approval-for-all-  update-nft-scams/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' [Online;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' accessed 14-September-2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [54] Jiajing Wu, Qi Yuan, Dan Lin, Wei You, Weili Chen, Chuan Chen, and Zibin  Zheng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Who are the phishers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' phishing scam detection on ethereum via network  embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [55] Dunjie Zhang, Jinyin Chen, and Xiaosong Lu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Blockchain phishing scam detection  via multi-channel graph classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In International Conference on Blockchain  and Trustworthy Systems, pages 241–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Springer, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  [56] Ge Zhang, Zhao Li, Jiaming Huang, Jia Wu, Chuan Zhou, Jian Yang, and Jianliang  Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' efraudcom: An e-commerce fraud detection system via competitive graph  neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' ACM Transactions on Information Systems (TOIS), 40(3):1–29,  2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  DRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' July 2017,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' DC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' USA  Appendices  A  ANALYSIS OF CONNECTIONS BETWEEN  DRAINERS AND AFFILIATED USERS  (a) Number of affiliated users connected to a drainer  (b) Number of drainers connected to an affiliated user  Figure 9: Cumulative distribution functions (CDFs) of (a)  drainer and (b) affiliated users  There is no limit of creating Ethereum accounts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' so an agent  can create multiple accounts for the purpose of transferring as-  sets between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' In certain cases, a gift can be used between  users to avoid monitoring when manipulating markets by wash  trading [17, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' A gift is the transfer of NFT ownership without  any financial compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As gifts are commonly observed to  occur between users who have a relationship, we can infer that the  accounts that gift drained NFTs drainers are likely to be affiliated  with said drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Therefore, we define these accounts as affiliated  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In this section, we provide detailed information about the number  of affiliated users of drainers and the number of drainers connected  to affiliated users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 9(a) illustrates how many affiliated users  each drainer has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Most drainers (82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5%) have one or more affiliated  users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' They have 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1 affiliated users on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' One drainer gift  his drained NFTs to 21 affiliated users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Figure 9(b) focuses on how many drainers are connected to each  affiliated user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We find 637 affiliated users, and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4% are related  to two or more drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' The most overlapped affiliated user is  connected with 16 drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  B  DISTRIBUTION OF TRANSACTIONS  40~41  41~42  42~43  43~44  44~45  45~46  Number of transactions  0  10  20  30  40  Percentage  Random users  Known drainer accounts  Figure 10: Distribution of random users’ transactions and  drainers’ transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Figure 10 shows the distribution of random users’ transactions  and drainers’ transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We analyze drainers and random users  who were active between 1, January 2022 to 31, July 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Drainers  and sampled random users show noticeably different distributions  of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Most drainers made transactions of more than 16,  while most random regular users made transactions of smaller than  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Thus, we define random regular users with transactions of more  than 16 as heavy users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  C  ROC CURVE & PR CURVE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  DRAINCLoG: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9894  N-GraphSAGE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9894  E-GraphSAGE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9521  N-GAT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='9303  (a) ROC (Receiver Operating Characteristic) curve  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  DRAINCLoG: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8457  N-GraphSAGE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='7301  E-GraphSAGE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3843  N-GAT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4336  (b) PR (Precision-Recall) curve  Figure 11: ROC and PR curve for the top four models on 𝐷4:  DRAINCLoG, N-GraphSAGE, E-GraphSAGE, and N-GAT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8  ion  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  Proportic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='5  # drainers1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='8  ion  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='6  roportic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='0  0  5  10  15  20  # affiliated usersConference’17, July 2017, Washington, DC, USA  Hanna Kim, Jian Cui, Eugene Jang, Chanhee Lee, Yongjae Lee, Jin-Woo Chung, and Seungwon Shin  D  FEATURE ANALYSIS  To better understand the trading behavior of different types of users,  we analyze the activities of 645 drainers, 637 affiliated users, and a  sample of 10,000 regular users who were active between January 1st  and August 31st, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We compare these three groups by plotting  the cumulative distribution functions (CDFs) for 19 different dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' To make these visualizations easier to interpret, we limit the  x-axis of each graph to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' We can observe that drainers exhibit  distinct behavior patterns compared to regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Furthermore,  we find that affiliated users also show different patterns of behavior,  which sets them apart from both regular users and drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='1  Number of transactions  (c) Number of gifting-in  (d) Number of gifting-out  (a) Number of buying  (b) Number of selling  (e) Number of minting  Figure 12: CDFs of the number of transactions according to  the user types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In this section, we focus on the number of transactions users  made for each transaction type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Figure 12 illustrates CDFs of the  number of each transaction type: (a) buying, (b) selling, (c) gifting-  in, (d) gifting-out, and (e) minting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  It is apparent that drainers are more active in selling, gifting-in,  and gifting-out transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, they rarely participate in  buying and minting transactions when compared to regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  These results suggest that the primary focus of drainers in the NFT  ecosystem is on draining NFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Upon closer examination, trends between drainers and affiliated  users show little difference in selling, gifting-in, and gifting-out  transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Interestingly, affiliated users are more active in buying  and minting NFTs, unlike drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This behavior sets them apart  from both drainers and regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='2  Number of collections  (c) Number of collections gifted-in  (d) Frequency of gifted-out  (a) Number of collections bought  (b) Number of collections sold  (e) Number of collections minted  Figure 13: CDFs of the number of collections according to  the user types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Figure 13 illustrates CDFs of the number of collections for each  transaction type: (a) buying, (b) selling, (c) gifting-in, (d) gifting-out,  and (e) minting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We find that the number of collections for each transaction type  is generally smaller than the number of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' It is well  known that NFT users tend to form communities based on specific  collections and trade within those communities [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Although  drainers have different intentions from regular users, they also trade  a smaller number of collections than transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This is because  they steal NFTs that are collected by regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, due to  their high levels of gifting-in, gifting-out, and selling transactions,  -- Known Drainer Accounts  Affiliated Users  s  Random Users-- Known Drainer Accounts  Affiliated Users  s  Random UsersDRAINCLoG: Detecting Rogue Accounts with Illegally-obtained NFTs using Classifiers Learned on Graphs  Conference’17, July 2017, Washington, DC, USA  drainers have a greater diversity of collections in those three types  of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' As a result, they exhibit a significant difference from  regular users in the number of NFT collections gifted-in, gifted-out,  and sold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In contrast, affiliated users are observed to actively participate in  trading a wide range of collections across all types of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  This is because they are actively engaged in all types of transactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='3  Number of neighbors  (c) Number of neighbors gifted-in  (d) Number of neighbors gifted-out  (a) Number of neighbors bought  (b) Number of neighbors sold  Figure 14: CDFs of the number of neighbors according to the  user types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Figure 14 illustrates CDFs of the number of neighbors for each  transaction type: (a) buying, (b) selling, (c) gifting-in, and (d) gifting-  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  We analyze the number of neighbors a user has for each transac-  tion type by considering the accounts with which a user has made a  transaction as their neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' For buying and selling transactions,  the distribution of neighbors is similar to that of the transactions  themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, for gifting-in and gifting-out transactions,  the distribution of neighbors is significantly different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Most users  gift NFTs to only a few neighbors, while they sell or buy NFTs  with many neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This suggests that NFT users have specific  relationships through gifting NFTs, given that gifting is a process  of transferring ownership without any payment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  This phenomenon is more pronounced in drainers than in reg-  ular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' When drainers steal NFTs, they may acquire all the  tokens from each victim, resulting in a smaller number of gifting-in  neighbors than the number of NFTs gifted-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Additionally, drain-  ers only gift NFTs to a few affiliated users, resulting in a much  smaller number of gifting-out neighbors than the number of NFTs  gifted-out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  In the case of affiliated users, their distributions of gifting-in and  gifting-out are as expected, similar to those of drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='4  Ratio & Frequency & Active timespan  (c) Frequency of selling  (d) Frequency of gifting-in  (a) Out-in ratio  (b) Gift-in ratio  (e) Active timespan (days)  Figure 15: CDFs of gift-in ratio, out-in ratio, frequency of  sale, frequency of gift-in, and active timespan according to  the user types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Figure 15 illustrates CDFs of (a) out-in ratio, (b) gift-in ratio,  (c) frequency of selling, (d) frequency of gifting-in, and (e) active  timespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Drainers have a higher gift-in ratio than regular users and af-  filiated users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This is because they do not engage in buying and  minting NFTs, but rather steal a large number of NFTs in a short  period of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' This results in a high frequency of gifting-in trans-  actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' Also, drainers are more likely to transfer out their NFTs  than regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' They also sell their NFT much more frequently  in a short active timespan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  The behavior of affiliated users falls between that of drainers  and regular users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' They are active in all types of transactions, par-  ticularly gifting-in, which results in a high gift-in ratio similar to  drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content=' However, their active period is not as short as drainers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
+page_content='  Known Drainer Accounts  s  Affiliated Users  s  Random Users-- Known Drainer Accounts  Affiliated Users  s  Random Users' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vtFRT4oBgHgl3EQffzca/content/2301.13577v1.pdf'}
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+ 
+1 
+ 
+ 
+  
+MR Elastography with Optimization-Based 
+Phase Unwrapping and Traveling Wave 
+Expansion-based Neural Network (TWENN) 
+ 
+Shengyuan Ma, Runke Wang, Suhao Qiu, Ruokun Li, Qi Yue, Qingfang Sun, Liang Chen, Fuhua Yan, Guang-
+Zhong Yang, Yuan Feng 
+ 
+Abstract— Magnetic Resonance Elastography (MRE) can 
+characterize biomechanical properties of soft tissue for 
+disease diagnosis and treatment planning. However, 
+complicated wavefields acquired from MRE coupled with 
+noise pose challenges for accurate displacement extraction 
+and modulus estimation. Here we propose a pipeline for 
+processing 
+MRE 
+images 
+using 
+optimization-based 
+displacement extraction and Traveling Wave Expansion-
+based Neural Network (TWENN) modulus estimation. Phase 
+unwrapping and displacement extraction were achieved by 
+optimization of an objective function with Dual Data 
+Consistency (Dual-DC). A complex-valued neural network 
+using displacement covariance as input has been 
+constructed for the estimation of complex wavenumbers. A 
+model of traveling wave expansion is used to generate 
+training datasets with different levels of noise for the 
+network. The complex shear modulus map is obtained by a 
+fusion of multifrequency and multidirectional data. 
+Validation using images of brain and liver simulation 
+demonstrates the practical value of the proposed pipeline, 
+which can estimate the biomechanical properties with 
+minimum root-mean-square-errors compared with state-of-
+the-art methods. Applications of the proposed method for 
+processing MRE images of phantom, brain, and liver show 
+clear anatomical features and that the pipeline is robust to 
+noise and has a good generalization capability.   
+ 
+Index 
+Terms—Magnetic 
+resonance 
+elastography, 
+Modulus estimation, Neural network, Traveling waves, 
+Phase unwrapping 
+ 
+I. INTRODUCTION 
+agnetic 
+Resonance 
+Elastography 
+(MRE) 
+can 
+noninvasively 
+measure 
+viscoelastic 
+mechanical 
+parameters of soft tissues [1], [2]. In recent years, studies 
+have investigated the potential of using MRE for the diagnosis 
+of liver cirrhosis [3], brain tumors [4], liver tumors [5], and 
+Parkinson’s diseases [6]. Post-processing of MRE images 
+includes two major steps: 1) extraction of wavefield from the 
+original wrapped phase; and 2) estimation of biomechanical 
+properties based on the measured wavefield [7]. However, it is 
+ 
+ 
+This work was supported by grant 32271359 from the National Natural Science Foundation of China, grant 22ZR1429600 from the Natural 
+Science Foundation of Shanghai, and grant 2022YFB4702700 from the National Key R&D Program of China. (Corresponding author: Yuan Feng) 
+S. Ma, R. Wang, S. Qiu, G. Z. Yang and Y. Feng are with the School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai, 
+200030, China (e-mail: fengyuan@sjtu.edu.cn).  
+R. Li, F. Yan, and Y. Feng are with the Department of Radiology, Ruijin Hospital, School of Medicine, Shanghai Jiao Tong University, Shanghai, 
+China. 
+Q. Sun is with the Department of Neurosurgery, Ruijin Hospital, School of Medicine, Shanghai Jiao Tong University, Shanghai, China. 
+Q. Yue and Liang Chen are with the Department of Neurosurgery, Huashan Hospital, Fudan University, Shanghai, China. 
+G. Z. Yang and Y. Feng are with the Institute of Medical Robotics, Shanghai Jiao Tong University, Shanghai, China. 
+still a challenge to extract accurate displacement from wrapped 
+phase images with noise [8].  In vivo wavefields of soft tissues 
+with complicated anatomical structures usually contain 
+deflections and reflections with acquisition noise. These bring 
+challenges to the accurate estimation of the biomechanical 
+properties of soft tissues [9].  
+As a first step of processing MRE images, the quality of the 
+wavefield extracted determines the overall performance of the 
+derived biomechanical properties. Since wave information is 
+encoded with motion encoding gradient in phase images, the 
+magnitude of the motion can introduce a wrapping effect. Thus, 
+phase unwrapping is necessary. Conventional approaches 
+involve unwrapping a single image frame before performing  
+Fast Fourier transform (FFT) in the time domain to extract the 
+principal component [10]. In addition, spatiotemporal 
+information can also be used for phase unwrapping [11]. 
+Unwrapping algorithms such as Sorting by Reliability (SG) 
+algorithm  [12] and Dilate-Erode-Propagate (DE) [8] work in a 
+search-like or discrete greedy update manner, which can fail in 
+complicated 
+wrapped 
+scenarios 
+[8]. 
+Laplacian-based 
+Estimation (LBE) carries out phase unwrapping and FFT in the 
+frequency domain simultaneously [13]. LBE is robust to noise 
+but is lossy can induce new background offset noise. This can 
+produce offsets to the estimated biomechanical properties. The 
+phase gradient method (PG) did not need direct phase 
+unwrapping, but was sensitive to noise and had to be used 
+together with specially designed modulus estimation algorithms 
+[14]. Therefore, an accurate wavefield extraction method that 
+can perform both phase unwrapping and extraction of principle 
+component, and is robust to noise is needed. 
+MRE is a shear wave based elastography, where the complex 
+shear modulus is to be estimated. Essentially, it is to find an 
+operator that can estimate the complex wavenumber with 
+acquired wave images. Many methods have been proposed so 
+far [9]. Typical algorithms include Direction Inversion (DI) 
+using Helmholtz equation [15] and local frequency estimation 
+(LFE) [16]. However, DI is sensitive to noise and easy to bring 
+edge artifacts due to the use of Laplace operators. Iterative 
+algorithms with prior information such as Multifrequency 
+M 
+
+2 
+ 
+Elasticity Reconstruction using Structured Sparsity and 
+ADMM (MERSA) [17] and MRE Inversion by Compressive 
+Recovery (MICRo) [18] were proposed to use DI as an 
+estimation kernel. But these methods had similar problems due 
+to the differentiation operation. LFE is more resilient to noise 
+but is limited in recovering anatomical features. To provide 
+viscosity information, Enhanced Complex Local Frequency 
+(EC-LFE) was proposed but still had the structural resilience 
+problem [19].  
+To overcome the limitations of the DI-based method, 
+Multifrequency Dual Elasto-Visco inversion (MDEV) was 
+proposed to improve noise robustness using multi-frequency 
+data averaging [20]. However, differentiation operation could 
+still introduce structural edge artifacts, making it sensitive to 
+noise. To improve the capability of distinguishing anatomical 
+features, k-MDEV applied directional filter banks to suppress 
+noise and new estimation kernels with a single-wave 
+assumption [21]. However, the use of the Laplace operator may 
+over-enhance the object boundary and the assumption of single-
+wave may be invalid when there are strong reflections. 
+Elastography Software Pipeline (ESP)  [22] used a series of 
+filters for denoising and Gabor wavelets for inversion. ESP 
+could recover refined anatomical details but required many 
+parameter-tuning steps.  
+Finite Element (FE) based Non-Linear Inversion (NLI) 
+algorithms can provide good inversion [9]. However, the 
+computation cost was high and the boundary condition settings 
+could greatly influence the results. Recently, data-driven 
+algorithms using Deep Learning (DL) were proposed using the 
+training set from FE simulation [23], [24]. These methods also 
+relied on specific FE models for specific application scenarios, 
+hampering their generalization performance. Traveling Wave 
+Expansion (TWE) was first introduced to solve the inversion 
+problem by using large fitting windows [25]. However, the 
+inverse of the ill-conditioned matrix limited the spatial 
+resolution that can be achieved. Therefore, an algorithm that is 
+noise-resilient, low in computation cost, good at distinguishing 
+anatomical features but with strong generalizability is needed. 
+A. Paper Contribution 
+In this study, we proposed an optimization-based phase 
+unwrapping method with Dual Data Consistency (Dual-DC) to 
+achieve phase unwrapping and FFT simultaneously. A TWE-
+based Neural Network (TWENN) algorithm was proposed for 
+the inversion of complex shear modulus.  Simulation, phantom, 
+and human studies were carried out to demonstrate the accuracy 
+and robustness of the proposed pipeline.   
+B. Paper Organization 
+The rest of the paper is organized as follows: The proposed 
+pipeline including Dual-DC and TWENN is presented in 
+Section 
+II. 
+The 
+test 
+datasets, 
+evaluation 
+metrics, 
+implementation details of the proposed pipeline, and the 
+algorithms used for comparison are introduced in Section III. 
+The results are described in Section IV, and a detailed 
+discussion of the results is presented in Section V. 
+II. METHODS 
+The proposed pipeline for processing MRE images includes 
+an optimization-based phase unwrapping method with Dual-
+DC, and a Traveling Wave Expansion-based Neural Network 
+(TWENN) modulus estimation to estimate complex shear 
+modulus (Fig. 1). 
+A. Optimization-based Phase Unwrapping with Dual 
+Data Consistency  
+The displacement extraction of phase images acquired from 
+MRE generally consists of two steps: phase unwrapping and 
+computation of the principal components using FFT [7]. Here, 
+an optimization-based method was proposed to accomplish 
+these two tasks simultaneously.  
+1) Problem Formulation 
+If 𝑼∗ = 𝑼′ + 𝑖 ⋅ 𝑼′′  is the complex displacement field 
+rotating at a frequency 𝜔, representing the oscillating motion. 
+In the process of image acquisition, at a specific time point 
+𝑡"(𝑗 = 1, … , 𝐽) within the motion cycle where the phase offset 
+is 𝜑" = 𝜔𝑡" , the image phase recorded by motion encoding 
+gradient is 𝝓": 
+ 
+𝜙" = Re5𝑈∗ ⋅ e#$!7 = 𝑼% ⋅ cos5𝜑"7 − 𝑼%% ⋅ sin5𝜑"7, (1) 
+Therefore, the image 𝑰" acquired at 𝑡" is 
+ 
+𝐼" = |𝐴| ⋅ e#& ⋅ e#'!, 
+(2) 
+where |𝑨| is the magnitude of image and 𝚽 is the background 
+phase of 𝑰". The purpose of displacement extraction is to obtain 
+𝑼∗ from 𝑰"(𝑗 = 1 … 𝐽), which includes two tasks: computation 
+of principal component and phase unwrapping. Two loss 
+functions are proposed to solve them, respectively. 
+2) Loss Function for Principal Components 
+Cross differences among the acquired images are used to 
+remove the background phase. For images acquired at two 
+temporal 
+points 
+𝑡(
+ and 
+𝑡)
+, 
+(𝑝, 𝑞) ∈
+{(𝑥, 𝑦)|𝑥 < 𝑦, 1 ≤ 𝑥, 𝑦 ≤ 𝐽}: 
+ 
+𝑰(
+𝑰)
+= 𝑒#*𝝓",𝝓#-
+= 𝑒#.𝑼$⋅1234*$"-,234*$#-56𝑼$$⋅1,478*$"-6478*$#-59 
+(3) 
+Define the constant coefficients 𝐸() and 𝐹(), 
+𝐸() = cos5𝜑(7 − cos5𝜑)7	
+𝐹() = −sin5𝜑(7 + sin5𝜑)7 
+(4) 
+The relationship between unknown 𝑼∗  and known 𝑰"  is 
+established so that the principal components are obtained. Thus, 
+the first data consistency (DC) term of phase can be written as: 
+Loss:;< = Q R
+𝑰(
+𝑰)
+− 𝑒#*𝑼$⋅="#6𝑼$$⋅>"#-R
+?
+(,)
+ 
+(5) 
+3) Loss Function for Unwrapping 
+With Eq. (5) only, 𝑼∗ may still be wrapped. The absolute 
+value of the wrapped phase gradient is much higher than that 
+after unwrapping. Using the chain rule, the gradient of phase 
+after unwrapping (true phase gradient) can be obtained directly 
+from the wrapped phase [7]. Thus, data consistency is achieved 
+between the true phase gradient and the gradient of 𝑼∗. 
+If the background phase is ignored, 
+𝑰!
+B𝑰!B	≈ 𝑒#𝝓! , using the 
+chain rule, the phase gradient is calculated from the wrapped 
+phase, 
+
+3 
+ 
+𝜕𝑒#𝝓!
+𝜕𝑥
+= 𝑖 ⋅ 𝑒#𝝓! ⋅
+𝜕 V𝑼% ⋅ cos5𝜑"7 − 𝑼%% ⋅ sin5𝜑"7W
+𝜕𝑥
+ 
+cos5𝜑"7
+C𝑼$
+CD − sin5𝜑"7
+C𝑼$$
+CD = −
+#
+E%𝝓!
+CE%𝝓!
+CD 	. 
+(6) 
+In matrix form, we have 
+X
+cos(𝜑<)
+−sin(𝜑<)
+⋮
+⋮
+cos5𝜑F7
+−sin5𝜑F7
+Z X
+C𝑼$
+CD
+C𝑼$$
+CD
+Z =
+⎣
+⎢
+⎢
+⎢
+⎡−
+#
+E%𝝓'
+CE%𝝓'
+CD
+⋮
+−
+#
+E%𝝓(
+CE%𝝓(
+CD ⎦
+⎥
+⎥
+⎥
+⎤
+. 
+(7) 
+Let 𝐻 = X
+cos(𝜑<)
+−sin(𝜑<)
+⋮
+⋮
+cos5𝜑F7
+−sin5𝜑F7
+Z，X
+C𝑼$
+CD
+C𝑼$
+CD
+Z = b𝑼%D
+c
+𝑼%%D
+c d, then 
+b𝑼%D
+c
+𝑼%%D
+c d = (𝑯′𝐻),<𝐻′
+⎣
+⎢
+⎢
+⎢
+⎡−
+#
+E%𝝓'
+CE%𝝓'
+CD
+⋮
+−
+#
+E%𝝓(
+CE%𝝓(
+CD ⎦
+⎥
+⎥
+⎥
+⎤
+. 
+(8) 
+Thus, the second data consistency term (DC) of phase 
+gradient can be defined as 
+Loss:;? = f𝜕𝑼%
+𝜕𝑥 − 𝑼%D
+c f
+?
++ f𝜕𝑼%
+𝜕𝑦 − 𝑼%G
+c f
+?
++f𝜕𝑼%%
+𝜕𝑥 − 𝑼%%D
+c f
+?
++ f𝜕𝑼%%
+𝜕𝑦 − 𝑼%%G
+c f
+?
+.
+ (9) 
+4) Optimization Using Dual-DC 
+The displacement extraction problem in MRE including FFT 
+and unwrapping can be treated as an optimization problem (10) 
+by combining (5) and (9) as a loss function. Here, the adaptive 
+Momentum (ADAM) algorithm was used to solve this 
+optimization problem.  
+ 
+min
+𝑼 i∑
+f
+𝑰"
+𝑰# − 𝑒#*𝑼$⋅="#6𝑼$$⋅>"#-f
+?
+(,)
++
+𝜆 l
+m
+C𝑼$
+CD − 𝑼%D
+c m
+? + m
+C𝑼$
+CG − 𝑼%G
+c m
+?
++ m
+C𝑼$$
+CD − 𝑼%%D
+c m
+? + m
+C𝑼$$
+CG − 𝑼%%G
+c m
+?
+no, 
+(10) 
+where 𝜆 is a weighting parameter. 
+ 
+Algorithm 1 Dual-DC phase unwrapping 
+Input: Measured complex MR images 𝑰", Phase offset 𝜑" 
+for each image. 
+Output: Main displacement components 𝑼∗ = 𝑼′ + 𝑖 ⋅ 𝑼′′ 
+1. 
+Initialize the 𝑼∗  to be updated and set weighting 
+parameter 𝜆  and maximum number of iterations 
+𝑚𝑎𝑥𝐼𝑡𝑒𝑟. Set number of iterations 𝐿 = 0. 
+2. 
+Calculate cross phase differences 
+𝑰"
+𝑰#. (3) 
+3. 
+Calculate coefficients 𝐸() and 𝐹() using 𝜑". (4) 
+4. 
+Calculate phase gradient 𝑼%D
+c , 𝑼%G
+c , 𝑼%%D
+c , 𝑼%%G
+c  from MR 
+images 𝑰". (8) 
+5. 
+repeat 
+6. 
+      Update Loss:;<. (5)  
+7. 
+      Update the gradient of  𝑼∗, 
+C𝑼$
+CD ,
+C𝑼$
+CG ,
+C𝑼$$
+CD ,
+C𝑼$$
+CG . 
+8. 
+      Update Loss:;?. (9) 
+9. 
+      Update 𝑼∗ using ADAM algorithm. (10) 
+10.       𝐿 = 𝐿 + 1 
+11. until 𝐿 ≥ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟 
+12. return 𝑼∗ = 𝑼′ + 𝑖 ⋅ 𝑼′′ 
+B. TWENN: Traveling Wave Expansion-based Neural 
+Network Modulus Estimation 
+The TWE model was used to generate training data with 
+noise. It was also used to construct and train a complex value 
+covariance neural network to mine the mapping from wavefield 
+to wavenumber, and finally use multi-frequency multi-
+directional fusion to obtain the complex shear modulus. 
+1) Traveling Wave Expansion 
+For homogeneous, incompressible, isotropic, and viscoelastic 
+material, at spatial location 𝒓, the complex value displacement  
+is 𝑢(𝒓). It can be expressed as a superposition of 𝑀 traveling 
+waves [25]: 
+ 
+𝑢(𝒓) = Q
+𝑤H
+I
+HJ<
+	
+= Q
+𝑎H ⋅ 𝑒#⋅*K$6#⋅K$$-⋅𝒏)
+M ⋅𝒓	
+I
+HJ<
+ 
+(11) 
+where 𝑤H is the 𝑚th traveling waves, 𝑎H is the complex value 
+amplitude of the displacement, 𝑚 is the number of traveling 
+waves, 𝒏H
+{  is the unit vector of the propagation direction of the 
+𝑚th traveling wave, 𝑘∗ = 𝑘% + 𝑖 ⋅ 𝑘′′  is the local complex 
+wavenumber, 𝑘%  is the real wavenumber related to material 
+elasticity, 𝑘′′  is imaginary wavenumber related to material 
+viscosity. The complex wavenumber can be normalized by 
+vibration frequency 𝜔: 
+ 
+𝑘∗
+c = 𝑘∗
+𝜔 = 𝑘%
+𝜔 + 𝑖 ⋅ 𝑘′′
+𝜔 	
+= 𝑘%} + 𝑖 ⋅ 𝑘%%
+c. 
+(12) 
+Thus, the aim of the inversion is to find the operator ℱO: 𝑢 ⟶
+𝑘∗
+c.   
+2) Training Data Generation 
+To construct a neural network for realizing the mapping 
+operator ℱO, training data set was prepared using TWE model. 
+The training set was constructed by varying different numbers 
+of traveling waves 𝑀 , propagation direction 𝒏H
+{ , complex 
+amplitude 𝑎H and complex wavenumber 𝑘. In this study, an 
+isotropic patch of 3×3 (2D) or 3×3×3 (3D) was used for the 
+inversion of 𝑘 at each location. The relatively small patch used 
+is to increase the spatial resolution. To obtain ℱO with better 
+noise robustness, noise 𝜁(𝒓) was added to the training set. 
+Suppose ℂ(𝒓)  is the normalized, complex-valued Gaussian 
+noise, 𝑏 is the intensity of the noise,	𝜁(𝒓) = 𝑏 ⋅ ℂ(𝒓).The detail 
+of dataset generation rules is shown in Appendix. The training 
+data 𝑑(𝑟) can be written as: 
+ 
+𝑑(𝑟) = 𝑢(𝒓) + 𝜁(𝒓)	
+= Q
+𝑎H ⋅ 𝑒#⋅*K$6#⋅K$$-⋅𝒏)
+M ⋅𝒓	
+I
+HJ<
++ 𝑏 ⋅ ℂ(𝒓) 
+(13) 
+3) Covariance Complex Neural Network 
+Studies have shown complex value neural networks are more 
+suitable for solving complex-valued problems [26]. Therefore, 
+A complex-valued neural network was constructed to process 
+these complex-valued wavefields. If the weight matrix of the 
+linear layer is W = A + 𝑖B, and the complex input is h = a +
+𝑖b, the output is Wh = Aa − Bb + 𝑖(Ab + Ba). To preserve the 
+
+4 
+ 
+phase angle, and to keep the magnitude within a certain range, 
+a new activation function named modSigmoid is designed [27]: 
+ 
+modSigmoid(z) = Sigmoid(|z| + b)e#O*, 
+(14) 
+where Sigmoid(x) =
+<
+<6P+,.  
+Because the covariance can better expose the information 
+within the wave field [28], a covariance term 𝑫 =
+vec(𝑑)vec(𝑑)Q  is used as the model input. 𝑘%}  and 𝑘%%
+c  are 
+estimated separately by two parallel multi-layer full connection 
+networks (Fig. 1). The two separate networks are used so that  
+𝑘%} 	and	 𝑘%%
+c		 can	 be	 estimated	 with	 more	 flexible	 range	
+without	 combing	 the	 loss	 functions	 together	 with	 more	
+hyperparameters.	 In	 this	 study,	 the network was trained 
+using ADAM algorithm. ℱO(𝜔) was trained and obtained at 
+different frequencies 𝜔.	
+4) Mutli-frequency and Mutli-direction Fusion 
+For the wavefield 𝑼R),S-
+∗
+ obtained from MRE at different 
+frequencies 𝜔H(𝑚 = 1, … , 𝑀)  and in different directions 
+𝑑T(𝑛 = 1, … , 𝑁) , the corresponding normalized complex 
+wavenumber 𝑘∗
+c(𝜔H, 𝑑T) is obtained using the trained network 
+ℱO(𝜔H).  
+For complex shear modulus 𝐺∗ = 𝐺% + 𝑖𝐺′′ , using 
+corresponding principle [17], 
+ 
+𝐺∗ = 𝜌
+R.
+(K$,#⋅K$$).	 =
+X
+*K$
+Y,#⋅K$$
+Y -
+.	. 
+(15) 
+Thus, the storage modulus 𝐺% and loss modulus 𝐺%% can be 
+estimated from the average of  𝑘∗
+c(𝜔H, 𝑑T): 
+ 
+𝐺% + 𝑖𝐺%% =
+X
+Z∑
+0$
+1(3),𝑑-)
+),-
++%⋅∑
+0$$
+1 (3),𝑑-)
+),-
+78
+\
+.. 
+(16) 
+This method does not do any pre-filtering to filter the noise, 
+and the final modulus is filtered by a 3×3 median filter. 
+ 
+Algorithm 2 Modulus estimation with TWENN 
+Input: Wavefields of multi-frequency and multi-direction 
+𝑼R),S-
+∗
+, information of spatial resolution and vibration 
+frequencies 𝜔H(𝑚 = 1, … , 𝑀) 
+Output: Complex shear modulus 𝐺∗ = 𝐺% + 𝑖𝐺′′ 
+1. 
+Set the patch size, and intensity of the noise 𝑏. 
+2. 
+Generate training data based on spatial resolution and 
+vibration frequencies using the traveling wave 
+expansion model. (13) 
+3. 
+Construct covariance complex neural network ℱO . 
+(14,15) 
+4. 
+Training  ℱO(𝜔)	using ADAM. 
+5. 
+Calculate 𝑘∗
+c(𝜔T, 𝑑H) using ℱO. 
+6. 
+Estimate 𝐺∗ = 𝐺% + 𝑖𝐺′′ using multi-frequency multi-
+directional data fusion. (17) 
+7. 
+return 𝐺∗ = 𝐺% + 𝑖𝐺′′ 
+III. VALIDATION AND VERIFICATION 
+Wrapped phase images from simulation liver dataset and in 
+vivo brain MRE experiment were used for the validation of the 
+proposed phase unwrapping method with Dual-DC. Modulus 
+estimation with TWENN was validated using simulated 
+phantom, brain, and liver wavefields. The proposed pipeline 
+combining Dual-DC and TWENN was validated using 
+phantom, brain, or liver MRE images. A summary of the data 
+sets and methods used is shown in TABLE I. Results are also 
+compared with those from state-of-the-art methods. 
+A. Validation of Dual-DC phase unwrapping 
+1) Simulated Liver Dataset 
+ Artificially wrapped phase was generated using 2D 
+wavefield data for simulated liver images. The displacements 
+were normalized with a maximum phase of 4𝜋. Phases at 4 
+different temporal points within one cycle were generated. 
+Complex Gaussian noise with different intensities was added to 
+the complex image to evaluate the robustness of the algorithm. 
+2) In vivo 3D Brain Dataset 
+The in vivo brain MRE images were acquired using a 3T 
+scanner (uMR790, United Imaging Healthcare, Shanghai, 
+China) with TR/TE=4000/65ms, MEG=40mT/m, resolution=3 
+mm × 3 mm × 3 mm, and 8 phase offsets [29]. The study 
+protocol was reviewed and approved by the Institution Review 
+Board of Shanghai Jiao Tong University.  
+3) Algorithm Comparison  
+Performance of phase unwrapping method are compared with 
+that from SR algorithm [12] and LBE [13], [30] that are 
+commonly used in postprocessing MRE images [31]. 
+Fig. 1.  A flow chart of the pipeline of MRE image processing for displacement extraction, training data generation, structure of network and multi-
+data fusion. Complex displacement field was extracted from phase images using an optimization of an objective function with dual-DC.  Training 
+data was generated by TWE model for network training. The normalized complex wavenumber was estimated by TWENN. Complex shear 
+modulus was obtained by combining multi-frequency and multi-direction data. 
+ 
+ 
+
+Re(d)
+Re(w1)
+Re(W2)
+Re()
+,1
++
+Wrapped phase
+Im(d)
+Im(w1)
+Im(W2)
+Im()  Label
+Multi-data
+Data
+Training data generation
+fusion
+Re(U*)
+using TWE model
+k(w1,d1)
+k*(w1, d2)
+Re(D)
+K(w1, d3)
+Rel:
+Cross phase
+ differences
+Phase gradient
+k*(wm,di)
+*(wm,d2)
+i
+Im(·)
+Im(U*)
+Covariance
+*(wm,d3)
+Loss for
+Loss for
+G"
++
+FFT
+unwrapping
+Complex linear layer
+Complex
+k
+ModSigmoid activation
+Complex shear
+Neural Network
+Dual-DC phase unwrapping
+displacement
+TWENN modulus estimation
+modulus5 
+ 
+Simulation results are compared with the ground truth while 
+mean errors are calculated for validation. Interlayer continuity 
+of the unwrapped phase is compared for brain MRE images. 
+B. Validation of Modulus Estimation using TWENN 
+1) Test Data Set  
+The test data set contained 107 examples for model evaluation. 
+Wavefields with different signal-to-noise ratios (SNRs) and 
+complex wavenumbers were generated using the TWE model, 
+with a patch size of 3×3 and a resolution of 3 mm × 3 mm at 
+60Hz. SNRs were uniformly distributed from 12 dB to 38 dB, 
+real normalized wavenumbers were uniformly distributed from 
+0.35s/m to 1.35s/m, while imaginary normalized wavenumbers 
+were uniformly distributed between 0s/m and 0.28s/m. 
+Estimation of 𝑘%}  and 𝑘%%
+c  at different SNRs and complex 
+wavenumbers were compared with that from conventional DI 
+(fitting window=3×3) statistically. 
+2) Simulated Dataset 
+ The size of the simulated phantom using COMSOL had two 
+circular inclusions of 10 mm radius with complex shear moduli 
+of 4 + i0.48 and 6 + i0.84 kPa. The background had a complex 
+shear modulus of 2 + i0.2 kPa. Gaussian noise was added to the 
+phantom with a SNR of 28dB. Simulated liver and brain MRE 
+images with 3D multi-frequency and multi-direction wavefield 
+and known 𝐺" were also used for evaluation. Root Mean 
+Squared Error (RMSE) was used to evaluate estimating 𝐺" and 
+𝐺′′: 
+RMSE = ¤‖𝐺E]^
+%
+− 𝐺_`
+% ‖?
+?
+𝑁
+	
+(18) 
+RMSE of Estimated 𝐺"  and the fine structures were 
+compared. 
+C. Validation for the Proposed Pipeline 
+1) Comparison with Other Pipelines 
+Both phantom and human MRE experiment images were used 
+to evaluate the performance of the proposed pipeline for MRE 
+image processing.  
+The proposed pipeline contains a cascading of phase 
+unwrapping method using Dual-DC and TWENN for modulus 
+estimation. Comparisons were made with two state-of-the-art 
+multi-frequency MRE processing pipelines: LBE and k-MDEV, 
+PG and MDEV.  
+TABLE I 
+A SUMMARY OF DATA SETS AND METHODS USED FOR COMPARISON. 2D: 2D MODULUS INVERSION; 3D: 3D MODULUS INVERSION; BIOQIC: DATASETS OR 
+MRE PROCESSING PIPELINE WERE FROM HTTPS://BIOQIC-APPS.CHARITE.DE; COMSOL: DATA WAS SIMULATED USING COMSOL (COMSOL AB, 
+STOCKHOLM, SWEDEN); 1.5T/3T SCANNER: MRE IMAGES WERE ACQUIRED FROM A 1.5T/3T SCANNER. IMAGE RESOLUTION, VIBRATION FREQUENCY, AND 
+THE MOTION ENCODING DIRECTIONS (RO: READ OUT; PE: PHASE ENCODING; SS: SLICE SELECTION;) ARE PROVIDED IN THE DETAILS OF DATA. 
+ 
+
+Data
+Method
+Methods for
+Evaluation
+Application
+Data Set
+Details of Data
+Source
+Proposed
+Comparison
+Metrics
+Simulated
+2mm×2mm;
+BIOQIC
+SG
+Mean error
+Phase
+Liver
+42Hz: RO
+Dual-DC
+unwrapping
+3mm×3mm×3mm;
+3T
+Interlayer
+Brain MRE
+LBE
+50Hz; RO
+scanner
+continuity
+3mmx3mm;
+TWE
+Test data
+DI (2D)
+Mean error
+60Hz; SS
+model
+TWENN
+Simulated
+1.5mm×1.5mm;
+(2D)
+COMSOL
+phantom
+60,80,100Hz; SS
+k-MDEV
+Modulus
+1.5mm×1.5mm×1.5mm;
+Simulated
+estimation
+24, 28, 32, 36, 40, 44, 48, 52
+MDEV
+brain
+RMSE
+56, 60Hz; RO, PE, SS
+TWENN
+BIOQIC
+2mm×2mm×2mm
+(3D)
+(BIOQIC)
+Simulated liver
+30, 36, 42, 48Hz;
+RO, PE, SS
+1.5mmx1.5mm;
+Regional
+Phantom
+30, 40, 50, 60, 70, 80, 90.
+mean
+100Hz; SS
+BIOQIC
+LBE
+2mm×2mm;
+Match with
++
+Normal brain
+20, 25, 30, 35, 40, 45Hz;
+Dual-DC
+structural
+k-MDEV
+RO, PE, SS
++
+image
+Pipeline
+Normal liver
+TWENN
+PG
+performance
+Hepatic
+2.7mm×2.7mm;
+(2D)
+Regional
+1.5T
++
+siderosis &
+30, 40, 50, 60Hz;
+mean
+scanner
+MDEV
+Cirrhosis
+RO, PE, SS
+Liver tumor
+(BIOQIC)
+Dual-DC
+3mm×3mm×3mm;
+3T
+CNR
+Brain tumor
++TWENN
+30. 40. 50Hz: RO. PE. SS
+scanner
+(3D)6 
+ 
+2) Phantom MRE 
+A publicly available multi-frequency phantom dataset  
+containing four cylindrical inclusions was used for evaluation 
+[20]. The moduli estimated by each method were averaged in 
+the z-direction. Regional mean values and ground truth of 𝐺% 
+and 𝐺′′ were compared.  
+3) Brain MRE 
+ Brain MRE images of a patient with a meningioma were 
+acquired using a 3T scanner (uMR790, United Imaging 
+Healthcare, Shanghai, China) using an electromagnetic actuator 
+[29]. MRE was implemented using a single-shot spin-echo 
+echo-planar imaging sequence (TR/TE=4000/65ms) with a 
+motion-encoding gradient of 40 mT/m and 8 phase offsets. 
+Contrast-to-noise ratio (CNR) of 𝐺% with respect to the tumor 
+region was used for evaluation:  
+CNR =
+25𝐺aKb
+%
+ªªªªªª −	𝐺^cHde
+%
+ªªªªªªªªª7
+?
+𝜎aKb
+?
++ 𝜎^cHde
+?
+,
+(19) 
+where 𝐺aKb
+%
+ªªªªªª and 𝐺^cHde
+%
+ªªªªªªªªª are the mean value of shear modulus, 
+𝜎aKb	and 𝜎^cHde	 are the standard deviation of shear modulus, 
+for background tissue and tumor tissue, respectively. The 
+tumor region was delineated from the T1 structural image, and 
+the background neighborhood was delineated as the region of 
+one tumor radius outside of the tumor. A public data set of 
+normal brain MRE images was also used for evaluation. 
+4) Liver MRE  
+    Liver MRE images from a healthy volunteer, a patient with a 
+liver tumor, and a patient with hepatic siderosis & cirrhosis 
+were acquired using a 1.5T pneumatic MRE system [34] 
+(Magnetom Aera, Siemens, Erlangen, Germany, BIOQIC). 
+CNR of 𝐺% was used to evaluate the modulus estimation of liver 
+tumor data. For normal liver data and iron-deposited cirrhotic 
+data, the mean values of 𝐺%  of circled liver tissue were 
+compared. 
+D. Implementation Details for Dual-DC and TWENN 
+For all datasets unwrapped in this paper, the same 
+hyperparameters of Dual-DC were used (learning rate = 0.005, 
+𝜆 = 1000, the number of iterations = 4000). 
+TWENNs had 9-layer fully connected complex-value neural 
+networks, with the number of neurons in each layer being 
+60,50,40,30,20,15,10,6,1. The networks were trained using the 
+ADAM algorithm with a batch size of 500 and a step number 
+of 12000. The noise intensity 𝑏 of the training set for TWENN 
+for all in vivo liver and brain MRE images was 0.3, while for 
+other datasets 𝑏 was 0.001. If the wavefield was continuous 
+between imaging slices, a 3×3×3 patch was used. Otherwise, 
+3×3 patch was used. 
+The proposed methods were implemented with PyTorch 
+1.12.0 and CUDA 11.6 on an Ubuntu 20.04 LTS (64-bit) 
+operating system equipped with an AMD Ryzen 9 5950x 
+central processing unit (CPU) and NVIDIA RTX 3080Ti 
+graphics processing unit (GPU, 12 GB memory).  
+IV. RESULTS 
+A. Phase unwrapping test 
+1) Simulation test 
+Results of unwrapping simulation data showed the proposed 
+optimization-based algorithm using dual-DC could unwrap 
+phases with a complex Gaussian noise level up to 𝜎 = 0.4 
+without observing the offset of the background phase (Fig. 2a). 
+However, SG did not perform unwrap at some complex edges 
+without noise, and a large number of unwrapping failures were 
+observed with increased noise levels (Fig. 2a, b). Although LBE 
+had no obvious unwrapping failures, it introduced background 
+offset even without noise (Fig. 2b). In contrast, an optimization-
+based algorithm using Dual-DC had better performance at 
+different noise levels (Fig. 2c). 
+2) In vivo dataset test 
+It was observed that SG was prone to unwrapping failure. The 
+background offset introduced by the LBE could be seen in both 
+coronal and sagittal views leading to an obvious 3D wavefield 
+discontinuity. All these methods performed unwrapping at the 
+transverse layers. However, an optimization-based algorithm 
+using Dual-DC still ensured continuity at coronal and sagittal 
+views (Fig. 3). The absolute differences also showed that Dual-
+DC produced the smoothest phase. 
+ 
+Fig. 3.  (a) A comparison of extracted displacement field from brain MRE 
+images. (b) The corresponding absolute differences of displacement at 
+cross sections marked by the white dotted line. 
+Fig. 2.  (a) A comparison of the unwrapping performance of the three methods at different noise levels. (b) The differences between ground 
+truth and outputs of three methods. (c) Mean error at different noise levels. 
+
+Dual-DC
+Cor
+SG
+20
+LBE
+N
+0
+50
+100
+20
+10
+0
+0
+50
+100
+30
+Tra
+20
+10
+0
+0
+50
+100o Dual-DC
+error (rad)
+:SG
+ALBE
+0.8
+0.6
+Mean
+0.4
+0.2
+0
+0
+0.01 0.05
+0.1
+0.2
+0.4
+07 
+ 
+B. Modulus estimation test 
+1) Simulation data set 
+A comparison of inversion between DI and TWENN with 
+different complex wavenumbers and signal-to-noise ratios 
+(SNR) in 2D wavefield (patch size=3×3) showed TWENN 
+performed better than that of DI in estimating 𝑘%}  and 𝑘%%
+c. As 
+SNR decreased, the advantages of TWENN over DI increased 
+(Fig. 4a, b).  
+Results of the estimated shear moduli showed TWENN could 
+recover the inclusions better than other methods (Fig. 5). Values 
+of RMSE showed TWENN was the best at estimating complex 
+shear moduli of the inclusions in most cases (TABLE II). 
+ 
+Fig. 4.  Distribution of mean errors of estimating 𝑘!"  (left) and 𝑘!!
+# (right) 
+with respect to different SNR values and normalized complex 
+wavenumber.	
+ 
+Fig. 5.  Comparisons of estimated (a) 𝐺′ and (b) 𝐺′′ maps using different 
+methods based on phantom simulation. A noise of 28dB was added to 
+the simulated wave images.	
+2) Simulation test of brain and liver 
+Based on the simulated complex wavefield of the brain and 
+liver, it was observed that TWENN had the best estimation 
+accuracy with RMSEs of 3.34 and 2.06kPa (TABLE II). The 
+other two algorithms did not perform as well as TWENN 
+probably due to the strong reflection of waves or the use of 
+derivatives, which might produce artifacts for boundaries (Fig. 
+6).  
+ 
+Fig. 6.  A comparison of estimated 𝐺′ using different algorithms for 
+simulated brain and liver. Regions were magnified to illustrate the 
+anatomical features of 𝐺′. 
+C. Performance Evaluation  
+1) Phantom MRE 
+Using multi-frequency MRE images of phantom, estimated 
+𝐺% and 𝐺′′ values showed the proposed pipeline had the best 
+performance in recovering the inclusion structures (Fig. 7a). 
+Algorithms using the Laplace operator could not recover the 
+structures well, probably because phantom images contained 
+various noise and wave deflections. With longer wavelengths 
+and lower SNR in regions with relatively high modulus, these 
+algorithms were prone to underestimate the high modulus. The 
+proposed pipeline produced the closest modulus estimation to 
+the ground truth for both 𝐺% and 𝐺′′ at most cases (Fig. 7b).  
+ 
+Fig. 7.  (a) A comparison of estimated 𝐺′	 and	 𝐺′′  maps of gelatin 
+phantom using different processing methods. (b)  A comparison of mean 
+values of 𝐺!	and 𝐺′′  of each method. The longitudinal axis is displayed 
+in a logarithmic form. 
+TABLE II 
+COMPARISONS OF RMSE IN ESTIMATING THE COMPLEX SHEAR MODULI OF A SIMULATED PHANTOM, SIMULATED BRAIN AND LIVER. VALUES OF AND CNR 
+ARE ALSO COMPARED FOR IMAGING BRAIN AND LIVER TUMORS.  THE BEST RESULTS ARE SHOWN IN BOLD FONTS. 
+
+DI
+0.4
+(s/m)
+DI
+0.4
+Mean error (s/m)
+TWENN
+TWENN
+0.3
+error
+0.3
+0.2
+0.2
+Mean 
+0.1
+0.1
+0
+0
+40
+30
+40
+30
+20
+10
+0
+20
+10 0.5
+SNR(dB)
+k"(s/m)
+SNR(dB)
+k'(s/m)structur
+GT
+GT
+b)
+^ k-MDEV
+k-MDEV
+。 MDEV
+MDEV
+• Proposed
+ Proposed
+Pa)
+10
+103RMSEG' +RMSEG"i (kPa)
+RMSE (kPa)
+CNR
+Method
+Brain
+Liver
+Brain
+Liver
+BK
+Inc#1
+Inc#2
+simulation
+simulation
+tumor
+tumor
+k-MDEV
+0.81+0.48i
+0.98+0.41i
+2.26+1.71i
+3.65
+2.4
+826
+357
+MDEV
+0.43+0.59i
+0.89+0.54i
+3.52+2.28i
+3.94
+2.67
+16
+79
+TWENN
+0.60+0.10i
+0.83+0.13i
+1.12+0.19i
+3.34
+2.06
+3069
+39068 
+ 
+ 
+2) Tumor MRE 
+For both brain and liver tumors, the proposed pipeline had 
+better reconstruction of boundaries (Fig. 8a, b), with more 
+details matching that of T1 images and the highest values of 
+CNR (TABLE II). This showed the ability of the proposed 
+pipeline to reconstruct the structural details. 
+3) Normal brain MRE 
+For the normal human brain dataset, it was observed that 𝐺% 
+values estimated from MDEV were relatively low (<1kPa). 
+This was probably affected by noise. Both the k-MDEV and the 
+proposed pipeline could estimate shear modulus maps which 
+match the structure image. k-MDEV introduced significant 
+boundary artifacts which is consistent with the results in Fig. 6. 
+The brain midline showed to be reconstructed better with the 
+proposed pipeline than other algorithms as shown in the white 
+arrow of Slice 1 (Fig. 8c). The white arrow of Slice 2 (Fig. 8c) 
+showed k-MDEV estimated relatively low modulus (<0.3 kPa), 
+but the proposed method provided a proper estimate (~1kPa). 
+This dataset demonstrated the capability of the proposed 
+pipeline to reconstruct anatomical features.  
+4) Liver dataset test 
+  For liver MRE (Fig. 8e, f), the estimated 𝐺% values of the 
+normal liver from k-MDEV, MDEV, and the proposed pipeline 
+were 1.62, 1.31, and 1.97 kPa, respectively. These values were 
+all in line with the reported normal range [35]. However, for 
+cirrhotic cases with iron deposition, the estimated 𝐺% values 
+were 1.70 and 1.09 for k-MDEV and MDEV, respectively. 
+This was probably due to the iron deposition that produced low 
+SNR. The proposed pipeline provided an estimate of 3.13 kPa 
+that fell inside the reported range [35].  
+V. DISCUSSION 
+A pipeline for estimating complex moduli from MRE images 
+is proposed in this study. The processing pipeline contains two 
+major steps: displacement extraction and modulus estimation. 
+For displacement extraction, an optimization-based method 
+with Dual-DC terms is used for both calculating principal 
+components and phase unwrapping. For modulus estimation, a 
+complex neural network using covariance as input is proposed 
+based on the TWE model.  
+ Since discontinuities of unwrapped phase images can 
+introduce serious artifacts of the derived modulus distribution, 
+accurate phase unwrapping is an important prerequisite. The 
+search-like or discrete greedy update phase unwrapping 
+methods such as SG algorithm are hard to deal with complex or 
+noisy wavefields. Laplacian-based estimation (LBE) can 
+perform unwrapping well in noisy, but extra background offset 
+noise could be introduced. The dual-DC based continuous 
+optimization can solve the complex wrapped and noise 
+problems simultaneously without phase offset. The proposed 
+method uses phase gradient information for unwrapping, which 
+has better adaptability to complex scenarios than discrete 
+unwrapping. Compared to LBE, Dual-DC does not introduce 
+new background noise. Phase images were unwrapped 
+successfully for wavefield data from phantom, brain and liver, 
+using the same set of hyperparameters. This verifies the 
+generalizability of the proposed algorithm. In addition, the 
+Fig. 8. T1-weighted (T1w) images, magnitude images of MRE, 𝐺′ and 𝐺′′ maps estimated from different algorithms were compared for (a) brain 
+tumor, (b) liver tumor, and (c, d) two difference slices of normal brain images. Magnified views of the tumor region were also provided.  Magnitude 
+images of MRE, 𝐺′ and 𝐺′′ maps estimated from (e) a healthy volunteer and (f) a patient with hepatic siderosis & cirrhosis were compared. Regions 
+of interest were delineated with white line, and the corresponding values of mean±standard deviation of 𝐺′ were provided.	
+
+MDEV
+Proposed
+(c)
+MRE mag
+T1w
+k-MDEV
+k-MDEV
+MDEV
+Proposed
+k-MDEV
+MDEV
+Proposed
+(a)
+Slice 1
+0
+G'
+2.5kPa
+G"
+1.5kPa
+(d)
+MRE mag
+0
+7kPa
+Slice 2
+0
+G'
+4kPa
+0
+G"
+1.5kPa
+Brain tumor
+Normal brain
+0
+G"
+ 4kPa
+(b)
+T1 map
+k-MDEV
+MDEV
+Proposed
+k-MDEV
+MDEV
+Proposed
+k-MDEV
+MDEV
+Proposed
+Normal liver
+1.62±0.79
+1.31±0.39
+1.97±0.61
+(f)
+MRE mag
+6kPa
+0
+G'
+1.70±1.21
+1.09±0.33
+3.13±0.52
+Hepatic siderosis
+Liver tumor
+0
+G'4kPa
+0
+G"
+4kPa
+ 2kPa
+0
+& Cirrhosis9 
+ 
+computation time of MRE images of the whole brain (3 
+frequencies and 3 directions) was about 3 minutes.  
+For modulus inversion, where noise robustness, structural 
+resilience and computational complexity impose contradicting 
+constraints, TWENN manages to reach a tradeoff among these 
+three metrics. In terms of noise robustness, TWENN does not 
+use explicit differentiation computation, which also gives 
+benefit on the structural resolution to reducing potential edge 
+artifacts. Except for the LFE-type algorithm, most of the current 
+algorithms use differential computation, which can be sensitive 
+to noise. By adding noise to the training set, TWENN can 
+achieve both denoising and inversion simultaneously, 
+improving its performance at regions with low SNR. For 
+example, TWENN can estimate the modulus of cirrhotic tissue 
+at low signal-to-noise ratios induced by iron deposition (Fig. 8f). 
+Since iron deposition is an important reason for the failure of 
+the liver MRE [35], TWENN has the potential to improve the 
+clinical performance of liver MRE. 
+ Considering structural resilience, a small patch of 3×3 or 
+3×3×3 pixels is applied, so even if assuming local homogeneity, 
+TWENN can still distinguish details of the image objects. 
+Benefiting from the absence of differential calculations, no 
+significant artifacts are observed at the structural edges. 
+Compare with MDEV and k-MDEV which had differential 
+operations, TWENN has better performance in preserving 
+anatomical details on both simulated and in vivo liver and brain 
+datasets (Fig. 6, Fig. 8). 
+If the estimation kernel is chosen in a trial-and-error fashion, 
+it is difficult to ensure the noise robustness of small patches. 
+Therefore, neural networks are used here to establish the 
+nonlinear mapping between noisy wavefields and complex 
+wavenumbers. Even training without noise, TWENN still has 
+better noise robustness than conventional DI using a small 
+kernel (Fig. 4). The dimension of the kernel also affects the 
+inversion. Using dejittering method, it has been shown that 
+interslice phase inconsistencies introduced in signal acquisition 
+can be removed, providing robust 2D inversions, especially in 
+the abdomen. Here, with improved phase unwrapping, the 
+smooth and physically accurate phase can result in better 
+modulus estimation from 3D inversion.  
+In terms of computational complexity, FEM-like algorithms 
+require constant iterative updates for better wavefield fitting, 
+producing high computational costs. TWENN uses neural 
+networks to pre-learn wavefield structure information for 
+inversion. The time of training data generation and training 
+using a desktop PC did not exceed 3 minutes. The trained 
+network can be directly used to process MRE images with the 
+same vibration frequency and resolution. In this study, the 3D 
+multi-directional and multi-frequency inversion time of a whole 
+brain did not exceed 15 seconds, showing its potential to be 
+deployed clinically. 
+The modulus inversion of MRE is equivalent to estimating the 
+wavenumber for an array of wave points, which can be 
+considered as a variant of the Direction of Arrival (DOA) 
+problem in array signal processing. Strategies commonly used 
+in DOA solution problems [28], [36] can be drawn on, where 
+second-order moments can expose harmonic information well. 
+The possibility of estimating modulus from an array signal 
+processing perspective was validated in our previous work [37]. 
+From the perspective of generalizability, the training data set 
+in TWENN is produced using a wave equation. Theoretically, 
+any local wavefield can be represented using the TWE model, 
+including reflected standing waves. Therefore, the training set 
+can cover most of the wave propagation conditions, ensuring 
+the generalizability of TWENN. Existing neural network 
+training modulus estimation methods [23], [24] is usually 
+trained using FE simulation for a limited number of scenarios, 
+which could hamper their generalization performance. In 
+addition, most of the elastography processing pipelines used 
+various complex filters for either wave extraction or denoising 
+[20], [21], [22]. However, the proposed pipeline does not use 
+any filters other than a median filter with fixed parameters. 
+Therefore, complicated parameter tuning procedures are 
+prevented, improving its generalization performance. 
+ Although multi-frequency and multi-directional information 
+are used for better inversion performance [2], single-frequency 
+and single-direction inversion are also available for TWENN. 
+In addition to 𝐺%  and 𝐺′′  values, shear wave speed and 
+penetrating rate that are closely related with stiffness and 
+damping [2] can also be estimated. 
+Although TWENN can estimate a relatively smoother 
+viscosity distribution from the noisy wavefield, the validation 
+was carried out for simulation cases. In this study only limited 
+human imaging data were used for validation. Future work 
+includes applying the proposed pipeline to a larger cohort of 
+clinical data sets such as neurological and liver diseases. The 
+unwrapping method proposed in this paper is readily 
+transferable to other fields where unwrapping is required. Also, 
+the data-driven method proposed could be used in other 
+elastography cases, such as optical elastography and ultrasound 
+elastography where wave equations apply. Future work will 
+also explore the potential of using this noisy data-driven method 
+for mining inverse operators to solve other inverse problems 
+such as electrical resistance tomography. 
+APPENDIX  
+TRAINING DATA GENERATION  
+The training data set was generated using the following 
+parameter settings: 
+1) The number 𝑀 of traveling waves 1,2,3,4,5,6,7,8 was 
+set to be distributed at ratios of 6:4:3:2:1:1:1:1:1. 
+2) The complex amplitude |𝑎H|  was uniformly 
+distributed in [0,1] and angle(𝑎H)  was uniformly 
+distributed in [0,2𝜋]. 
+3) The unit vector of the wave propagation direction is 
+𝒏H
+{ = [𝑥, 𝑦, 𝑧].  
+a) 
+In 2D cases, 𝑥 = cos(𝜃) , 𝑦 = sin(𝜃) , 𝑧 = 0, 𝜃 
+was uniformly distributed in [0,2𝜋].  
+b) 
+In 3D cases,  𝑧  is uniformly distributed in 
+[−1,1], 𝜃 was uniformly distributed in [0,2𝜋], 
+𝑥 = √1 − 𝑧? cos(𝜃)，𝑦 = √1 − 𝑧? sin(𝜃). 
+4) 𝑘′  and 𝑘%%  were uniformly distributed within the 
+preset range. 
+ACKNOWLEDGMENT 
+We thank Prof. Ingolf Sack from Charite, Germany for 
+
+10 
+ 
+providing part of the testing data sets and helpful discussions.  
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diff --git a/y9E0T4oBgHgl3EQfcwB8/content/tmp_files/load_file.txt b/y9E0T4oBgHgl3EQfcwB8/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,1011 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf,len=1010
+page_content='1  MR Elastography with Optimization-Based  Phase Unwrapping and Traveling Wave  Expansion-based Neural Network (TWENN)  Shengyuan Ma, Runke Wang, Suhao Qiu, Ruokun Li, Qi Yue, Qingfang Sun, Liang Chen, Fuhua Yan, Guang-  Zhong Yang, Yuan Feng  Abstract— Magnetic Resonance Elastography (MRE) can  characterize biomechanical properties of soft tissue for  disease diagnosis and treatment planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However,  complicated wavefields acquired from MRE coupled with  noise pose challenges for accurate displacement extraction  and modulus estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Here we propose a pipeline for  processing  MRE  images  using  optimization-based  displacement extraction and Traveling Wave Expansion-  based Neural Network (TWENN) modulus estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Phase  unwrapping and displacement extraction were achieved by  optimization of an objective function with Dual Data  Consistency (Dual-DC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A complex-valued neural network  using displacement covariance as input has been  constructed for the estimation of complex wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A  model of traveling wave expansion is used to generate  training datasets with different levels of noise for the  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The complex shear modulus map is obtained by a  fusion of multifrequency and multidirectional data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Validation using images of brain and liver simulation  demonstrates the practical value of the proposed pipeline,  which can estimate the biomechanical properties with  minimum root-mean-square-errors compared with state-of-  the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Applications of the proposed method for  processing MRE images of phantom, brain, and liver show  clear anatomical features and that the pipeline is robust to  noise and has a good generalization capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Index  Terms—Magnetic  resonance  elastography,  Modulus estimation, Neural network, Traveling waves,  Phase unwrapping  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' INTRODUCTION  agnetic  Resonance  Elastography  (MRE)  can  noninvasively  measure  viscoelastic  mechanical  parameters of soft tissues [1], [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In recent years, studies  have investigated the potential of using MRE for the diagnosis  of liver cirrhosis [3], brain tumors [4], liver tumors [5], and  Parkinson’s diseases [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Post-processing of MRE images  includes two major steps: 1) extraction of wavefield from the  original wrapped phase;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' and 2) estimation of biomechanical  properties based on the measured wavefield [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, it is  This work was supported by grant 32271359 from the National Natural Science Foundation of China, grant 22ZR1429600 from the Natural  Science Foundation of Shanghai, and grant 2022YFB4702700 from the National Key R&D Program of China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (Corresponding author: Yuan Feng)  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Ma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Qiu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Yang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Feng are with the School of Biomedical Engineering, Shanghai Jiao Tong University, Shanghai,  200030, China (e-mail: fengyuan@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Li, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Yan, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Feng are with the Department of Radiology, Ruijin Hospital, School of Medicine, Shanghai Jiao Tong University, Shanghai,  China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Sun is with the Department of Neurosurgery, Ruijin Hospital, School of Medicine, Shanghai Jiao Tong University, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Yue and Liang Chen are with the Department of Neurosurgery, Huashan Hospital, Fudan University, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Yang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Feng are with the Institute of Medical Robotics, Shanghai Jiao Tong University, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  still a challenge to extract accurate displacement from wrapped  phase images with noise [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In vivo wavefields of soft tissues  with complicated anatomical structures usually contain  deflections and reflections with acquisition noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' These bring  challenges to the accurate estimation of the biomechanical  properties of soft tissues [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  As a first step of processing MRE images, the quality of the  wavefield extracted determines the overall performance of the  derived biomechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Since wave information is  encoded with motion encoding gradient in phase images, the  magnitude of the motion can introduce a wrapping effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Thus,  phase unwrapping is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Conventional approaches  involve unwrapping a single image frame before performing  Fast Fourier transform (FFT) in the time domain to extract the  principal component [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In addition, spatiotemporal  information can also be used for phase unwrapping [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Unwrapping algorithms such as Sorting by Reliability (SG)  algorithm  [12] and Dilate-Erode-Propagate (DE) [8] work in a  search-like or discrete greedy update manner, which can fail in  complicated  wrapped  scenarios  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Laplacian-based  Estimation (LBE) carries out phase unwrapping and FFT in the  frequency domain simultaneously [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' LBE is robust to noise  but is lossy can induce new background offset noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' This can  produce offsets to the estimated biomechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The  phase gradient method (PG) did not need direct phase  unwrapping, but was sensitive to noise and had to be used  together with specially designed modulus estimation algorithms  [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Therefore, an accurate wavefield extraction method that  can perform both phase unwrapping and extraction of principle  component, and is robust to noise is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  MRE is a shear wave based elastography, where the complex  shear modulus is to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Essentially, it is to find an  operator that can estimate the complex wavenumber with  acquired wave images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Many methods have been proposed so  far [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Typical algorithms include Direction Inversion (DI)  using Helmholtz equation [15] and local frequency estimation  (LFE) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, DI is sensitive to noise and easy to bring  edge artifacts due to the use of Laplace operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Iterative  algorithms with prior information such as Multifrequency  M  2  Elasticity Reconstruction using Structured Sparsity and  ADMM (MERSA) [17] and MRE Inversion by Compressive  Recovery (MICRo) [18] were proposed to use DI as an  estimation kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' But these methods had similar problems due  to the differentiation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' LFE is more resilient to noise  but is limited in recovering anatomical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' To provide  viscosity information, Enhanced Complex Local Frequency  (EC-LFE) was proposed but still had the structural resilience  problem [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  To overcome the limitations of the DI-based method,  Multifrequency Dual Elasto-Visco inversion (MDEV) was  proposed to improve noise robustness using multi-frequency  data averaging [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, differentiation operation could  still introduce structural edge artifacts, making it sensitive to  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' To improve the capability of distinguishing anatomical  features, k-MDEV applied directional filter banks to suppress  noise and new estimation kernels with a single-wave  assumption [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, the use of the Laplace operator may  over-enhance the object boundary and the assumption of single-  wave may be invalid when there are strong reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Elastography Software Pipeline (ESP)  [22] used a series of  filters for denoising and Gabor wavelets for inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' ESP  could recover refined anatomical details but required many  parameter-tuning steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Finite Element (FE) based Non-Linear Inversion (NLI)  algorithms can provide good inversion [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, the  computation cost was high and the boundary condition settings  could greatly influence the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Recently, data-driven  algorithms using Deep Learning (DL) were proposed using the  training set from FE simulation [23], [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' These methods also  relied on specific FE models for specific application scenarios,  hampering their generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Traveling Wave  Expansion (TWE) was first introduced to solve the inversion  problem by using large fitting windows [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, the  inverse of the ill-conditioned matrix limited the spatial  resolution that can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Therefore, an algorithm that is  noise-resilient, low in computation cost, good at distinguishing  anatomical features but with strong generalizability is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Paper Contribution  In this study, we proposed an optimization-based phase  unwrapping method with Dual Data Consistency (Dual-DC) to  achieve phase unwrapping and FFT simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A TWE-  based Neural Network (TWENN) algorithm was proposed for  the inversion of complex shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Simulation, phantom,  and human studies were carried out to demonstrate the accuracy  and robustness of the proposed pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Paper Organization  The rest of the paper is organized as follows: The proposed  pipeline including Dual-DC and TWENN is presented in  Section  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The  test  datasets,  evaluation  metrics,  implementation details of the proposed pipeline, and the  algorithms used for comparison are introduced in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The results are described in Section IV, and a detailed  discussion of the results is presented in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' METHODS  The proposed pipeline for processing MRE images includes  an optimization-based phase unwrapping method with Dual-  DC, and a Traveling Wave Expansion-based Neural Network  (TWENN) modulus estimation to estimate complex shear  modulus (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Optimization-based Phase Unwrapping with Dual  Data Consistency  The displacement extraction of phase images acquired from  MRE generally consists of two steps: phase unwrapping and  computation of the principal components using FFT [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Here,  an optimization-based method was proposed to accomplish  these two tasks simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  1) Problem Formulation  If 𝑼∗ = 𝑼′ + 𝑖 ⋅ 𝑼′′  is the complex displacement field  rotating at a frequency 𝜔, representing the oscillating motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  In the process of image acquisition, at a specific time point  𝑡"(𝑗 = 1, … , 𝐽) within the motion cycle where the phase offset  is 𝜑" = 𝜔𝑡" , the image phase recorded by motion encoding  gradient is 𝝓":  𝜙" = Re5𝑈∗ ⋅ e#$!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='7 = 𝑼% ⋅ cos5𝜑"7 − 𝑼%% ⋅ sin5𝜑"7, (1)  Therefore, the image 𝑰" acquired at 𝑡" is  𝐼" = |𝐴| ⋅ e#& ⋅ e#\'!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=',  (2)  where |𝑨| is the magnitude of image and 𝚽 is the background  phase of 𝑰".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The purpose of displacement extraction is to obtain  𝑼∗ from 𝑰"(𝑗 = 1 … 𝐽), which includes two tasks: computation  of principal component and phase unwrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Two loss  functions are proposed to solve them, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2) Loss Function for Principal Components  Cross differences among the acquired images are used to  remove the background phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' For images acquired at two  temporal  points  𝑡(  and  𝑡)  ,  (𝑝, 𝑞) ∈  {(𝑥, 𝑦)|𝑥 < 𝑦, 1 ≤ 𝑥, 𝑦 ≤ 𝐽}:  𝑰(  𝑰)  = 𝑒#*𝝓",𝝓#-  = 𝑒#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='𝑼$⋅1234*$"-,234*$#-56𝑼$$⋅1,478*$"-6478*$#-59  (3)  Define the constant coefficients 𝐸() and 𝐹(),  𝐸() = cos5𝜑(7 − cos5𝜑)7  𝐹() = −sin5𝜑(7 + sin5𝜑)7  (4)  The relationship between unknown 𝑼∗  and known 𝑰"  is  established so that the principal components are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Thus,  the first data consistency (DC) term of phase can be written as:  Loss:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='< = Q R  𝑰(  𝑰)  − 𝑒#*𝑼$⋅="#6𝑼$$⋅>"#-R  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (,)  (5)  3) Loss Function for Unwrapping  With Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (5) only, 𝑼∗ may still be wrapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The absolute  value of the wrapped phase gradient is much higher than that  after unwrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Using the chain rule, the gradient of phase  after unwrapping (true phase gradient) can be obtained directly  from the wrapped phase [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Thus, data consistency is achieved  between the true phase gradient and the gradient of 𝑼∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  If the background phase is ignored,  𝑰!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  B𝑰!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='B ≈ 𝑒#𝝓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' , using the  chain rule, the phase gradient is calculated from the wrapped  phase,  3  𝜕𝑒#𝝓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  𝜕𝑥  = 𝑖 ⋅ 𝑒#𝝓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' ⋅  𝜕 V𝑼% ⋅ cos5𝜑"7 − 𝑼%% ⋅ sin5𝜑"7W  𝜕𝑥  cos5𝜑"7  C𝑼$  CD − sin5𝜑"7  C𝑼$$  CD = −  #  E%𝝓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  CE%𝝓!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  CD  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content="  (6)  In matrix form, we have  X  cos(𝜑<)  −sin(𝜑<)  ⋮  ⋮  cos5𝜑F7  −sin5𝜑F7  Z X  C𝑼$  CD  C𝑼$$  CD  Z =  ⎣  ⎢  ⎢  ⎢  ⎡−  #  E%𝝓'  CE%𝝓'  CD  ⋮  −  #  E%𝝓(  CE%𝝓(  CD ⎦  ⎥  ⎥  ⎥  ⎤  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content="  (7)  Let 𝐻 = X  cos(𝜑<)  −sin(𝜑<)  ⋮  ⋮  cos5𝜑F7  −sin5𝜑F7  Z，X  C𝑼$  CD  C𝑼$  CD  Z = b𝑼%D  c  𝑼%%D  c d, then  b𝑼%D  c  𝑼%%D  c d = (𝑯′𝐻),<𝐻′  ⎣  ⎢  ⎢  ⎢  ⎡−  #  E%𝝓'  CE%𝝓'  CD  ⋮  −  #  E%𝝓(  CE%𝝓(  CD ⎦  ⎥  ⎥  ⎥  ⎤  ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (8)  Thus, the second data consistency term (DC) of phase  gradient can be defined as  Loss:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' = f𝜕𝑼%  𝜕𝑥 − 𝑼%D  c f  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  + f𝜕𝑼%  𝜕𝑦 − 𝑼%G  c f  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  +f𝜕𝑼%%  𝜕𝑥 − 𝑼%%D  c f  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  + f𝜕𝑼%%  𝜕𝑦 − 𝑼%%G  c f  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (9)  4) Optimization Using Dual-DC  The displacement extraction problem in MRE including FFT  and unwrapping can be treated as an optimization problem (10)  by combining (5) and (9) as a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Here, the adaptive  Momentum (ADAM) algorithm was used to solve this  optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  min  𝑼 i∑  f  𝑰"  𝑰# − 𝑒#*𝑼$⋅="#6𝑼$$⋅>"#-f  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (,)  +  𝜆 l  m  C𝑼$  CD − 𝑼%D  c m  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' + m  C𝑼$  CG − 𝑼%G  c m  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  + m  C𝑼$$  CD − 𝑼%%D  c m  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' + m  C𝑼$$  CG − 𝑼%%G  c m  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  no,  (10)  where 𝜆 is a weighting parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Algorithm 1 Dual-DC phase unwrapping  Input: Measured complex MR images 𝑰", Phase offset 𝜑"  for each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Output: Main displacement components 𝑼∗ = 𝑼′ + 𝑖 ⋅ 𝑼′′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Initialize the 𝑼∗  to be updated and set weighting  parameter 𝜆  and maximum number of iterations  𝑚𝑎𝑥𝐼𝑡𝑒𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Set number of iterations 𝐿 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Calculate cross phase differences  𝑰"  𝑰#.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (3)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Calculate coefficients 𝐸() and 𝐹() using 𝜑".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (4)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Calculate phase gradient 𝑼%D  c , 𝑼%G  c , 𝑼%%D  c , 𝑼%%G  c  from MR  images 𝑰".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (8)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  repeat  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Update Loss:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='<.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (5)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Update the gradient of  𝑼∗,  C𝑼$  CD ,  C𝑼$  CG ,  C𝑼$$  CD ,  C𝑼$$  CG .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Update Loss:;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (9)  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Update 𝑼∗ using ADAM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (10)  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝐿 = 𝐿 + 1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' until 𝐿 ≥ 𝑚𝑎𝑥𝐼𝑡𝑒𝑟  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' return 𝑼∗ = 𝑼′ + 𝑖 ⋅ 𝑼′′  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' TWENN: Traveling Wave Expansion-based Neural  Network Modulus Estimation  The TWE model was used to generate training data with  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' It was also used to construct and train a complex value  covariance neural network to mine the mapping from wavefield  to wavenumber, and finally use multi-frequency multi-  directional fusion to obtain the complex shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  1) Traveling Wave Expansion  For homogeneous, incompressible, isotropic, and viscoelastic  material, at spatial location 𝒓, the complex value displacement  is 𝑢(𝒓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' It can be expressed as a superposition of 𝑀 traveling  waves [25]:  𝑢(𝒓) = Q  𝑤H  I  HJ<  = Q  𝑎H ⋅ 𝑒#⋅*K$6#⋅K$$-⋅𝒏)  M ⋅𝒓  I  HJ<  (11)  where 𝑤H is the 𝑚th traveling waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝑎H is the complex value  amplitude of the displacement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝑚 is the number of traveling  waves,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝒏H  {  is the unit vector of the propagation direction of the  𝑚th traveling wave,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝑘∗ = 𝑘% + 𝑖 ⋅ 𝑘′′  is the local complex  wavenumber,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝑘%  is the real wavenumber related to material  elasticity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝑘′′  is imaginary wavenumber related to material  viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The complex wavenumber can be normalized by  vibration frequency 𝜔:  𝑘∗  c = 𝑘∗  𝜔 = 𝑘%  𝜔 + 𝑖 ⋅ 𝑘′′  𝜔  = 𝑘%} + 𝑖 ⋅ 𝑘%%  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (12)  Thus, the aim of the inversion is to find the operator ℱO: 𝑢 ⟶  𝑘∗  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2) Training Data Generation  To construct a neural network for realizing the mapping  operator ℱO, training data set was prepared using TWE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The training set was constructed by varying different numbers  of traveling waves 𝑀 , propagation direction 𝒏H  { , complex  amplitude 𝑎H and complex wavenumber 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In this study, an  isotropic patch of 3×3 (2D) or 3×3×3 (3D) was used for the  inversion of 𝑘 at each location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The relatively small patch used  is to increase the spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' To obtain ℱO with better  noise robustness, noise 𝜁(𝒓) was added to the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Suppose ℂ(𝒓)  is the normalized, complex-valued Gaussian  noise, 𝑏 is the intensity of the noise, 𝜁(𝒓) = 𝑏 ⋅ ℂ(𝒓).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='The detail  of dataset generation rules is shown in Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The training  data 𝑑(𝑟) can be written as:  𝑑(𝑟) = 𝑢(𝒓) + 𝜁(𝒓)  = Q  𝑎H ⋅ 𝑒#⋅*K$6#⋅K$$-⋅𝒏)  M ⋅𝒓  I  HJ<  + 𝑏 ⋅ ℂ(𝒓)  (13)  3) Covariance Complex Neural Network  Studies have shown complex value neural networks are more  suitable for solving complex-valued problems [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Therefore,  A complex-valued neural network was constructed to process  these complex-valued wavefields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' If the weight matrix of the  linear layer is W = A + 𝑖B, and the complex input is h = a +  𝑖b, the output is Wh = Aa − Bb + 𝑖(Ab + Ba).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' To preserve the  4  phase angle, and to keep the magnitude within a certain range,  a new activation function named modSigmoid is designed [27]:  modSigmoid(z) = Sigmoid(|z| + b)e#O*,  (14)  where Sigmoid(x) =  <  <6P+,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Because the covariance can better expose the information  within the wave field [28], a covariance term 𝑫 =  vec(𝑑)vec(𝑑)Q  is used as the model input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 𝑘%}  and 𝑘%%  c  are  estimated separately by two parallel multi-layer full connection  networks (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The two separate networks are used so that  𝑘%}  and  𝑘%%  c   can  be  estimated  with  more  flexible  range  without  combing  the  loss  functions  together  with  more  hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In  this  study,  the network was trained  using ADAM algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' ℱO(𝜔) was trained and obtained at  different frequencies 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  4) Mutli-frequency and Mutli-direction Fusion  For the wavefield 𝑼R),S-  ∗  obtained from MRE at different  frequencies 𝜔H(𝑚 = 1, … , 𝑀)  and in different directions  𝑑T(𝑛 = 1, … , 𝑁) , the corresponding normalized complex  wavenumber 𝑘∗  c(𝜔H, 𝑑T) is obtained using the trained network  ℱO(𝜔H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  For complex shear modulus 𝐺∗ = 𝐺% + 𝑖𝐺′′ , using  corresponding principle [17],  𝐺∗ = 𝜌  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (K$,#⋅K$$).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' =  X  K$  Y,#⋅K$$  Y -  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (15)  Thus, the storage modulus 𝐺% and loss modulus 𝐺%% can be  estimated from the average of  𝑘∗  c(𝜔H, 𝑑T):  𝐺% + 𝑖𝐺%% =  X  Z∑  0$  1(3),𝑑-)  ),-  +%⋅∑  0$$  1 (3),𝑑-)  ),-  78  \\  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='.  (16)  This method does not do any pre-filtering to filter the noise,  and the final modulus is filtered by a 3×3 median filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Algorithm 2 Modulus estimation with TWENN  Input: Wavefields of multi-frequency and multi-direction  𝑼R),S-  ∗  , information of spatial resolution and vibration  frequencies 𝜔H(𝑚 = 1, … , 𝑀)  Output: Complex shear modulus 𝐺∗ = 𝐺% + 𝑖𝐺′′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Set the patch size, and intensity of the noise 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Generate training data based on spatial resolution and  vibration frequencies using the traveling wave  expansion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (13)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Construct covariance complex neural network ℱO .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (14,15)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Training  ℱO(𝜔) using ADAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Calculate 𝑘∗  c(𝜔T, 𝑑H) using ℱO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Estimate 𝐺∗ = 𝐺% + 𝑖𝐺′′ using multi-frequency multi-  directional data fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (17)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  return 𝐺∗ = 𝐺% + 𝑖𝐺′′  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' VALIDATION AND VERIFICATION  Wrapped phase images from simulation liver dataset and in  vivo brain MRE experiment were used for the validation of the  proposed phase unwrapping method with Dual-DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Modulus  estimation with TWENN was validated using simulated  phantom, brain, and liver wavefields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The proposed pipeline  combining Dual-DC and TWENN was validated using  phantom, brain, or liver MRE images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A summary of the data  sets and methods used is shown in TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Results are also  compared with those from state-of-the-art methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Validation of Dual-DC phase unwrapping  1) Simulated Liver Dataset  Artificially wrapped phase was generated using 2D  wavefield data for simulated liver images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The displacements  were normalized with a maximum phase of 4𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Phases at 4  different temporal points within one cycle were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Complex Gaussian noise with different intensities was added to  the complex image to evaluate the robustness of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2) In vivo 3D Brain Dataset  The in vivo brain MRE images were acquired using a 3T  scanner (uMR790, United Imaging Healthcare, Shanghai,  China) with TR/TE=4000/65ms, MEG=40mT/m, resolution=3  mm × 3 mm × 3 mm, and 8 phase offsets [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The study  protocol was reviewed and approved by the Institution Review  Board of Shanghai Jiao Tong University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  3) Algorithm Comparison  Performance of phase unwrapping method are compared with  that from SR algorithm [12] and LBE [13], [30] that are  commonly used in postprocessing MRE images [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A flow chart of the pipeline of MRE image processing for displacement extraction, training data generation, structure of network and multi-  data fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Complex displacement field was extracted from phase images using an optimization of an objective function with dual-DC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Training  data was generated by TWE model for network training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The normalized complex wavenumber was estimated by TWENN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Complex shear  modulus was obtained by combining multi-frequency and multi-direction data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Re(d)  Re(w1)  Re(W2)  Re()  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='1  +  Wrapped phase  Im(d)  Im(w1)  Im(W2)  Im()  Label  Multi-data  Data  Training data generation  fusion  Re(U*)  using TWE model  k(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='d1)  k*(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' d2)  Re(D)  K(w1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' d3)  Rel:  Cross phase  differences  Phase gradient  k*(wm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='di)  (wm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='d2)  i  Im(·)  Im(U*)  Covariance  (wm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='d3)  Loss for  Loss for  G"  +  FFT  unwrapping  Complex linear layer  Complex  k  ModSigmoid activation  Complex shear  Neural Network  Dual-DC phase unwrapping  displacement  TWENN modulus estimation  modulus5  Simulation results are compared with the ground truth while  mean errors are calculated for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Interlayer continuity  of the unwrapped phase is compared for brain MRE images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Validation of Modulus Estimation using TWENN  1) Test Data Set  The test data set contained 107 examples for model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Wavefields with different signal-to-noise ratios (SNRs) and  complex wavenumbers were generated using the TWE model,  with a patch size of 3×3 and a resolution of 3 mm × 3 mm at  60Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' SNRs were uniformly distributed from 12 dB to 38 dB,  real normalized wavenumbers were uniformly distributed from  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='35s/m to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='35s/m, while imaginary normalized wavenumbers  were uniformly distributed between 0s/m and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='28s/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Estimation of 𝑘%}  and 𝑘%%  c  at different SNRs and complex  wavenumbers were compared with that from conventional DI  (fitting window=3×3) statistically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2) Simulated Dataset  The size of the simulated phantom using COMSOL had two  circular inclusions of 10 mm radius with complex shear moduli  of 4 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='48 and 6 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='84 kPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The background had a complex  shear modulus of 2 + i0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='2 kPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Gaussian noise was added to the  phantom with a SNR of 28dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Simulated liver and brain MRE  images with 3D multi-frequency and multi-direction wavefield  and known 𝐺" were also used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Root Mean  Squared Error (RMSE) was used to evaluate estimating 𝐺" and  𝐺′′:  RMSE = ¤‖𝐺E]^  %  − 𝐺_`  % ‖?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  𝑁  (18)  RMSE of Estimated 𝐺"  and the fine structures were  compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Validation for the Proposed Pipeline  1) Comparison with Other Pipelines  Both phantom and human MRE experiment images were used  to evaluate the performance of the proposed pipeline for MRE  image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The proposed pipeline contains a cascading of phase  unwrapping method using Dual-DC and TWENN for modulus  estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Comparisons were made with two state-of-the-art  multi-frequency MRE processing pipelines: LBE and k-MDEV,  PG and MDEV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  TABLE I  A SUMMARY OF DATA SETS AND METHODS USED FOR COMPARISON.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 2D: 2D MODULUS INVERSION;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 3D: 3D MODULUS INVERSION;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' BIOQIC: DATASETS OR  MRE PROCESSING PIPELINE WERE FROM HTTPS://BIOQIC-APPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='CHARITE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='DE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' COMSOL: DATA WAS SIMULATED USING COMSOL (COMSOL AB,  STOCKHOLM, SWEDEN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5T/3T SCANNER: MRE IMAGES WERE ACQUIRED FROM A 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5T/3T SCANNER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' IMAGE RESOLUTION, VIBRATION FREQUENCY, AND  THE MOTION ENCODING DIRECTIONS (RO: READ OUT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' PE: PHASE ENCODING;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' SS: SLICE SELECTION;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=') ARE PROVIDED IN THE DETAILS OF DATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Data  Method  Methods for  Evaluation  Application  Data Set  Details of Data  Source  Proposed  Comparison  Metrics  Simulated  2mm×2mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  BIOQIC  SG  Mean error  Phase  Liver  42Hz: RO  Dual-DC  unwrapping  3mm×3mm×3mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  3T  Interlayer  Brain MRE  LBE  50Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' RO  scanner  continuity  3mmx3mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  TWE  Test data  DI (2D)  Mean error  60Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' SS  model  TWENN  Simulated  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5mm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (2D)  COMSOL  phantom  60,80,100Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' SS  k-MDEV  Modulus  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5mm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5mm×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Simulated  estimation  24, 28, 32, 36, 40, 44, 48, 52  MDEV  brain  RMSE  56, 60Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' RO, PE, SS  TWENN  BIOQIC  2mm×2mm×2mm  (3D)  (BIOQIC)  Simulated liver  30, 36, 42, 48Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  RO, PE, SS  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5mmx1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Regional  Phantom  30, 40, 50, 60, 70, 80, 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  mean  100Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' SS  BIOQIC  LBE  2mm×2mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Match with  +  Normal brain  20, 25, 30, 35, 40, 45Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Dual-DC  structural  k-MDEV  RO, PE, SS  +  image  Pipeline  Normal liver  TWENN  PG  performance  Hepatic  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='7mm×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='7mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  (2D)  Regional  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5T  +  siderosis &  30, 40, 50, 60Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  mean  scanner  MDEV  Cirrhosis  RO, PE, SS  Liver tumor  (BIOQIC)  Dual-DC  3mm×3mm×3mm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  3T  CNR  Brain tumor  +TWENN  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 50Hz: RO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' PE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' SS  scanner  (3D)6  2) Phantom MRE  A publicly available multi-frequency phantom dataset  containing four cylindrical inclusions was used for evaluation  [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The moduli estimated by each method were averaged in  the z-direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Regional mean values and ground truth of 𝐺%  and 𝐺′′ were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  3) Brain MRE  Brain MRE images of a patient with a meningioma were  acquired using a 3T scanner (uMR790, United Imaging  Healthcare, Shanghai, China) using an electromagnetic actuator  [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' MRE was implemented using a single-shot spin-echo  echo-planar imaging sequence (TR/TE=4000/65ms) with a  motion-encoding gradient of 40 mT/m and 8 phase offsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Contrast-to-noise ratio (CNR) of 𝐺% with respect to the tumor  region was used for evaluation:  CNR =  25𝐺aKb  %  ªªªªªª − 𝐺^cHde  %  ªªªªªªªªª7  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  𝜎aKb  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  + 𝜎^cHde  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  ,  (19)  where 𝐺aKb  %  ªªªªªª and 𝐺^cHde  %  ªªªªªªªªª are the mean value of shear modulus,  𝜎aKb and 𝜎^cHde  are the standard deviation of shear modulus,  for background tissue and tumor tissue, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The  tumor region was delineated from the T1 structural image, and  the background neighborhood was delineated as the region of  one tumor radius outside of the tumor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A public data set of  normal brain MRE images was also used for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  4) Liver MRE  Liver MRE images from a healthy volunteer, a patient with a  liver tumor, and a patient with hepatic siderosis & cirrhosis  were acquired using a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5T pneumatic MRE system [34]  (Magnetom Aera, Siemens, Erlangen, Germany, BIOQIC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  CNR of 𝐺% was used to evaluate the modulus estimation of liver  tumor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' For normal liver data and iron-deposited cirrhotic  data, the mean values of 𝐺%  of circled liver tissue were  compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Implementation Details for Dual-DC and TWENN  For all datasets unwrapped in this paper, the same  hyperparameters of Dual-DC were used (learning rate = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='005,  𝜆 = 1000, the number of iterations = 4000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  TWENNs had 9-layer fully connected complex-value neural  networks, with the number of neurons in each layer being  60,50,40,30,20,15,10,6,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The networks were trained using the  ADAM algorithm with a batch size of 500 and a step number  of 12000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The noise intensity 𝑏 of the training set for TWENN  for all in vivo liver and brain MRE images was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='3, while for  other datasets 𝑏 was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' If the wavefield was continuous  between imaging slices, a 3×3×3 patch was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Otherwise,  3×3 patch was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The proposed methods were implemented with PyTorch  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='0 and CUDA 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='6 on an Ubuntu 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='04 LTS (64-bit)  operating system equipped with an AMD Ryzen 9 5950x  central processing unit (CPU) and NVIDIA RTX 3080Ti  graphics processing unit (GPU, 12 GB memory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Phase unwrapping test  1) Simulation test  Results of unwrapping simulation data showed the proposed  optimization-based algorithm using dual-DC could unwrap  phases with a complex Gaussian noise level up to 𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='4  without observing the offset of the background phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  However, SG did not perform unwrap at some complex edges  without noise, and a large number of unwrapping failures were  observed with increased noise levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 2a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Although LBE  had no obvious unwrapping failures, it introduced background  offset even without noise (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In contrast, an optimization-  based algorithm using Dual-DC had better performance at  different noise levels (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2) In vivo dataset test  It was observed that SG was prone to unwrapping failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The  background offset introduced by the LBE could be seen in both  coronal and sagittal views leading to an obvious 3D wavefield  discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' All these methods performed unwrapping at the  transverse layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, an optimization-based algorithm  using Dual-DC still ensured continuity at coronal and sagittal  views (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The absolute differences also showed that Dual-  DC produced the smoothest phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (a) A comparison of extracted displacement field from brain MRE  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (b) The corresponding absolute differences of displacement at  cross sections marked by the white dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (a) A comparison of the unwrapping performance of the three methods at different noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (b) The differences between ground  truth and outputs of three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (c) Mean error at different noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Dual-DC  Cor  SG  20  LBE  N  0  50  100  20  10  0  0  50  100  30  Tra  20  10  0  0  50  100o Dual-DC  error (rad)  :SG  ALBE  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='6  Mean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='2  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='4  07  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Modulus estimation test  1) Simulation data set  A comparison of inversion between DI and TWENN with  different complex wavenumbers and signal-to-noise ratios  (SNR) in 2D wavefield (patch size=3×3) showed TWENN  performed better than that of DI in estimating 𝑘%}  and 𝑘%%  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' As  SNR decreased, the advantages of TWENN over DI increased  (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 4a, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Results of the estimated shear moduli showed TWENN could  recover the inclusions better than other methods (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Values  of RMSE showed TWENN was the best at estimating complex  shear moduli of the inclusions in most cases (TABLE II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Distribution of mean errors of estimating 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='" (left) and 𝑘!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  # (right)  with respect to different SNR values and normalized complex  wavenumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Comparisons of estimated (a) 𝐺′ and (b) 𝐺′′ maps using different  methods based on phantom simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A noise of 28dB was added to  the simulated wave images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2) Simulation test of brain and liver  Based on the simulated complex wavefield of the brain and  liver, it was observed that TWENN had the best estimation  accuracy with RMSEs of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='34 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='06kPa (TABLE II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The  other two algorithms did not perform as well as TWENN  probably due to the strong reflection of waves or the use of  derivatives, which might produce artifacts for boundaries (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' A comparison of estimated 𝐺′ using different algorithms for  simulated brain and liver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Regions were magnified to illustrate the  anatomical features of 𝐺′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Performance Evaluation  1) Phantom MRE  Using multi-frequency MRE images of phantom, estimated  𝐺% and 𝐺′′ values showed the proposed pipeline had the best  performance in recovering the inclusion structures (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Algorithms using the Laplace operator could not recover the  structures well, probably because phantom images contained  various noise and wave deflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' With longer wavelengths  and lower SNR in regions with relatively high modulus, these  algorithms were prone to underestimate the high modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The  proposed pipeline produced the closest modulus estimation to  the ground truth for both 𝐺% and 𝐺′′ at most cases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 7b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (a) A comparison of estimated 𝐺′  and  𝐺′′  maps of gelatin  phantom using different processing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' (b)  A comparison of mean  values of 𝐺!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' and 𝐺′′  of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The longitudinal axis is displayed  in a logarithmic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  TABLE II  COMPARISONS OF RMSE IN ESTIMATING THE COMPLEX SHEAR MODULI OF A SIMULATED PHANTOM, SIMULATED BRAIN AND LIVER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' VALUES OF AND CNR  ARE ALSO COMPARED FOR IMAGING BRAIN AND LIVER TUMORS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' THE BEST RESULTS ARE SHOWN IN BOLD FONTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  DI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='4  (s/m)  DI  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='4  Mean error (s/m)  TWENN  TWENN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='3  error  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='2  Mean  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='1  0  0  40  30  40  30  20  10  0  20  10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5  SNR(dB)  k"(s/m)  SNR(dB)  k\'(s/m)structur  GT  GT  b)  ^ k-MDEV  k-MDEV  。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' MDEV  MDEV  Proposed  Proposed  Pa)  10  103RMSEG\' +RMSEG"i (kPa)  RMSE (kPa)  CNR  Method  Brain  Liver  Brain  Liver  BK  Inc#1  Inc#2  simulation  simulation  tumor  tumor  k-MDEV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='81+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='48i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='98+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='41i  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='26+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='71i  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='65  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='4  826  357  MDEV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='43+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='59i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='89+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='54i  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='52+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='28i  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='94  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='67  16  79  TWENN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='60+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='10i  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='83+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='13i  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='12+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='19i  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='34  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='06  3069  39068  2) Tumor MRE  For both brain and liver tumors, the proposed pipeline had  better reconstruction of boundaries (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 8a, b), with more  details matching that of T1 images and the highest values of  CNR (TABLE II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' This showed the ability of the proposed  pipeline to reconstruct the structural details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  3) Normal brain MRE  For the normal human brain dataset, it was observed that 𝐺%  values estimated from MDEV were relatively low (<1kPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  This was probably affected by noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Both the k-MDEV and the  proposed pipeline could estimate shear modulus maps which  match the structure image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' k-MDEV introduced significant  boundary artifacts which is consistent with the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The brain midline showed to be reconstructed better with the  proposed pipeline than other algorithms as shown in the white  arrow of Slice 1 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 8c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The white arrow of Slice 2 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 8c)  showed k-MDEV estimated relatively low modulus (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='3 kPa),  but the proposed method provided a proper estimate (~1kPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  This dataset demonstrated the capability of the proposed  pipeline to reconstruct anatomical features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  4) Liver dataset test  For liver MRE (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 8e, f), the estimated 𝐺% values of the  normal liver from k-MDEV, MDEV, and the proposed pipeline  were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='62, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='31, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='97 kPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' These values were  all in line with the reported normal range [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, for  cirrhotic cases with iron deposition, the estimated 𝐺% values  were 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='70 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='09 for k-MDEV and MDEV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  This was probably due to the iron deposition that produced low  SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The proposed pipeline provided an estimate of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='13 kPa  that fell inside the reported range [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' DISCUSSION  A pipeline for estimating complex moduli from MRE images  is proposed in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The processing pipeline contains two  major steps: displacement extraction and modulus estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  For displacement extraction, an optimization-based method  with Dual-DC terms is used for both calculating principal  components and phase unwrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' For modulus estimation, a  complex neural network using covariance as input is proposed  based on the TWE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Since discontinuities of unwrapped phase images can  introduce serious artifacts of the derived modulus distribution,  accurate phase unwrapping is an important prerequisite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The  search-like or discrete greedy update phase unwrapping  methods such as SG algorithm are hard to deal with complex or  noisy wavefields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Laplacian-based estimation (LBE) can  perform unwrapping well in noisy, but extra background offset  noise could be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The dual-DC based continuous  optimization can solve the complex wrapped and noise  problems simultaneously without phase offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The proposed  method uses phase gradient information for unwrapping, which  has better adaptability to complex scenarios than discrete  unwrapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Compared to LBE, Dual-DC does not introduce  new background noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Phase images were unwrapped  successfully for wavefield data from phantom, brain and liver,  using the same set of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' This verifies the  generalizability of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In addition, the  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' T1-weighted (T1w) images, magnitude images of MRE, 𝐺′ and 𝐺′′ maps estimated from different algorithms were compared for (a) brain  tumor, (b) liver tumor, and (c, d) two difference slices of normal brain images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Magnified views of the tumor region were also provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Magnitude  images of MRE, 𝐺′ and 𝐺′′ maps estimated from (e) a healthy volunteer and (f) a patient with hepatic siderosis & cirrhosis were compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Regions  of interest were delineated with white line, and the corresponding values of mean±standard deviation of 𝐺′ were provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content="  MDEV  Proposed  (c)  MRE mag  T1w  k-MDEV  k-MDEV  MDEV  Proposed  k-MDEV  MDEV  Proposed  (a)  Slice 1  0  G'  2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5kPa  G"  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5kPa  (d)  MRE mag  0  7kPa  Slice 2  0  G\'  4kPa  0  G"  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='5kPa  Brain tumor  Normal brain  0  G"  4kPa  (b)  T1 map  k-MDEV  MDEV  Proposed  k-MDEV  MDEV  Proposed  k-MDEV  MDEV  Proposed  Normal liver  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='79  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='31±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='39  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='97±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content="61  (f)  MRE mag  6kPa  0  G'  1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='70±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='21  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='09±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='33  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='13±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='52  Hepatic siderosis  Liver tumor  0  G\'4kPa  0  G"  4kPa  2kPa  0  & Cirrhosis9  computation time of MRE images of the whole brain (3  frequencies and 3 directions) was about 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  For modulus inversion, where noise robustness, structural  resilience and computational complexity impose contradicting  constraints, TWENN manages to reach a tradeoff among these  three metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In terms of noise robustness, TWENN does not  use explicit differentiation computation, which also gives  benefit on the structural resolution to reducing potential edge  artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Except for the LFE-type algorithm, most of the current  algorithms use differential computation, which can be sensitive  to noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' By adding noise to the training set, TWENN can  achieve both denoising and inversion simultaneously,  improving its performance at regions with low SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' For  example, TWENN can estimate the modulus of cirrhotic tissue  at low signal-to-noise ratios induced by iron deposition (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 8f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Since iron deposition is an important reason for the failure of  the liver MRE [35], TWENN has the potential to improve the  clinical performance of liver MRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Considering structural resilience, a small patch of 3×3 or  3×3×3 pixels is applied, so even if assuming local homogeneity,  TWENN can still distinguish details of the image objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Benefiting from the absence of differential calculations, no  significant artifacts are observed at the structural edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Compare with MDEV and k-MDEV which had differential  operations, TWENN has better performance in preserving  anatomical details on both simulated and in vivo liver and brain  datasets (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 6, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  If the estimation kernel is chosen in a trial-and-error fashion,  it is difficult to ensure the noise robustness of small patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Therefore, neural networks are used here to establish the  nonlinear mapping between noisy wavefields and complex  wavenumbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Even training without noise, TWENN still has  better noise robustness than conventional DI using a small  kernel (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The dimension of the kernel also affects the  inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Using dejittering method, it has been shown that  interslice phase inconsistencies introduced in signal acquisition  can be removed, providing robust 2D inversions, especially in  the abdomen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Here, with improved phase unwrapping, the  smooth and physically accurate phase can result in better  modulus estimation from 3D inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  In terms of computational complexity, FEM-like algorithms  require constant iterative updates for better wavefield fitting,  producing high computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' TWENN uses neural  networks to pre-learn wavefield structure information for  inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The time of training data generation and training  using a desktop PC did not exceed 3 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The trained  network can be directly used to process MRE images with the  same vibration frequency and resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In this study, the 3D  multi-directional and multi-frequency inversion time of a whole  brain did not exceed 15 seconds, showing its potential to be  deployed clinically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The modulus inversion of MRE is equivalent to estimating the  wavenumber for an array of wave points, which can be  considered as a variant of the Direction of Arrival (DOA)  problem in array signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Strategies commonly used  in DOA solution problems [28], [36] can be drawn on, where  second-order moments can expose harmonic information well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  The possibility of estimating modulus from an array signal  processing perspective was validated in our previous work [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  From the perspective of generalizability, the training data set  in TWENN is produced using a wave equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Theoretically,  any local wavefield can be represented using the TWE model,  including reflected standing waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Therefore, the training set  can cover most of the wave propagation conditions, ensuring  the generalizability of TWENN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Existing neural network  training modulus estimation methods [23], [24] is usually  trained using FE simulation for a limited number of scenarios,  which could hamper their generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In  addition, most of the elastography processing pipelines used  various complex filters for either wave extraction or denoising  [20], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' However, the proposed pipeline does not use  any filters other than a median filter with fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Therefore, complicated parameter tuning procedures are  prevented, improving its generalization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Although multi-frequency and multi-directional information  are used for better inversion performance [2], single-frequency  and single-direction inversion are also available for TWENN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  In addition to 𝐺%  and 𝐺′′  values, shear wave speed and  penetrating rate that are closely related with stiffness and  damping [2] can also be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  Although TWENN can estimate a relatively smoother  viscosity distribution from the noisy wavefield, the validation  was carried out for simulation cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' In this study only limited  human imaging data were used for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Future work  includes applying the proposed pipeline to a larger cohort of  clinical data sets such as neurological and liver diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' The  unwrapping method proposed in this paper is readily  transferable to other fields where unwrapping is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Also,  the data-driven method proposed could be used in other  elastography cases, such as optical elastography and ultrasound  elastography where wave equations apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Future work will  also explore the potential of using this noisy data-driven method  for mining inverse operators to solve other inverse problems  such as electrical resistance tomography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  APPENDIX  TRAINING DATA GENERATION  The training data set was generated using the following  parameter settings:  1) The number 𝑀 of traveling waves 1,2,3,4,5,6,7,8 was  set to be distributed at ratios of 6:4:3:2:1:1:1:1:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  2) The complex amplitude |𝑎H|  was uniformly  distributed in [0,1] and angle(𝑎H)  was uniformly  distributed in [0,2𝜋].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  3) The unit vector of the wave propagation direction is  𝒏H  { = [𝑥, 𝑦, 𝑧].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  a)  In 2D cases, 𝑥 = cos(𝜃) , 𝑦 = sin(𝜃) , 𝑧 = 0, 𝜃  was uniformly distributed in [0,2𝜋].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  b)  In 3D cases,  𝑧  is uniformly distributed in  [−1,1], 𝜃 was uniformly distributed in [0,2𝜋],  𝑥 = √1 − 𝑧?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' cos(𝜃)，𝑦 = √1 − 𝑧?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' sin(𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  4) 𝑘′  and 𝑘%%  were uniformly distributed within the  preset range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content='  ACKNOWLEDGMENT  We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
+page_content=' Ingolf Sack from Charite, Germany for  10  providing part of the testing data sets and helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
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+page_content=' Bengio, “Unitary Evolution Recurrent  Neural Networks,” 33rd Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/y9E0T4oBgHgl3EQfcwB8/content/2301.02367v1.pdf'}
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+An infinite family of w1+∞ invariant theories on the celestial
+sphere
+Shamik Banerjee
+National Institute of Science Education and Research (NISER),
+Bhubaneswar 752050, Odisha, India
+and
+Homi Bhabha National Institute,
+Anushakti Nagar, Mumbai, India-400085∗
+Harshal Kulkarni
+Department of Physical Sciences, IISER Kolkata,
+Mohanpur, West Bengal 741246, India†
+Partha Paul
+Centre for High Energy Physics, Indian Institute of Science,
+C.V. Raman Avenue, Bangalore 560012, India.‡
+Abstract
+In this note we determine the graviton-graviton OPE and the null states in any w1+∞ symmetric
+theory on the celestial sphere. Our analysis shows that there exists a discrete infinite family of
+such theories. The MHV-sector and the quantum self dual gravity are two members of this infinite
+family. However, the bulk description of other theories are not known to us.
+∗Electronic address: banerjeeshamik.phy@gmail.com
+†Electronic address: hck18ms056@iiserkol.ac.in
+‡Electronic address: pl.partha13@gmail.com
+1
+arXiv:2301.13225v1  [hep-th]  30 Jan 2023
+
+Contents
+I. Introduction
+2
+II. General structure of w-invariant OPE
+3
+A. What is theory A?
+5
+III. Null states in the MHV sector
+6
+A. N 0,0
+k
+6
+B. N 0,1
+k
+6
+IV. Organization of the OPE at every order
+7
+A. Action of sl2(R)V on the null states
+8
+1. N 0,0
+k
+8
+2. N 0,1
+k
+9
+V. w invariance
+10
+A. w invariance at O(z0¯z0)
+11
+B. w invariance at O(z0¯z1)
+12
+VI. Finiteness and an infinite family of theories
+12
+VII. w invariant OPE coefficients
+13
+VIII. Knizhnik-Zamolodchikov type null states
+13
+IX. Acknowledgements
+14
+References
+15
+I.
+INTRODUCTION
+Celestial holography [1] is an attempt to formulate a dual theory of quantum gravity
+in asymptotically flat space time. When the bulk is (3 + 1) dimensional, the dual theory
+is conjectured to be two dimensional and it lives on the celestial sphere. The correlation
+2
+
+functions of this theory are given by the Mellin transform of the bulk S-matrix elements
+[2–4] and an important question is how to compute them in the dual formulation.
+Now in the absence of a Lagrangian description one can resort to the Bootstrap technique
+and the job is sometimes simplified if the theory has a large number of symmetries. For
+example, one can solve the minimal models in 2-D CFT by using the representation theory of
+Virasoro and other current algebras even when the Lagrangian description is not known. In
+celestial holography the two dimensional dual theory also has various (infinite dimensional)
+current algebra symmetries [5–26] and it is expected that the bootstrap approach should
+simplify considerably in this case1.
+In this paper we consider dual theories which are invariant under the w1+∞ symmetry2[19–
+22, 25, 26]. We determine the graviton-graviton OPE and the null states in such theories. So
+far two examples of such theories were known. One of them is the MHV-sector [16, 17] and
+the other one is the quantum self dual gravity [22]. However, we find that there is a discrete
+infinite family of such theories and the MHV-sector and the quantum self dual gravity are
+two members of this family. Bulk description of this family of theories is not known to us
+and we leave this to future work.
+II.
+GENERAL STRUCTURE OF w-INVARIANT OPE
+The general structure of the w-invariant OPE between two positive helicity outgoing
+graviton primaries with weights ∆1 and ∆2 can be written as,
+G+
+∆1(z1, ¯z1)G+
+∆2(z2, ¯z2) = − ¯z12
+z12
+∞
+�
+n=0
+B (∆1 − 1 + n, ∆2 − 1) ¯zn
+12
+n!
+¯∂nG+
+∆1+∆2(z2, ¯z2)
++
+∞
+�
+p,q=0
+zp
+12¯zq
+12
+n
+�
+k=1
+˜Ck
+p,q(∆1, ∆2) ˜Op,q
+k (∆1, ∆2, z2, ¯z2)
+(2.1)
+where the first line contains the universal terms [19] having simple pole in z12. The second
+line contains the non-singular terms in z12 and ¯z12 and they are linear combinations of the w
+descendants of a positive helicity graviton primary, denoted by ˜Op,q
+i . The positive integer n
+may be finite or infinite. Our goal is to determine the operators ˜Op,q
+i
+and the OPE coefficients
+1 Please see [29] for a discussion of conformal bootstrap approach to celestial holography. Also recently
+celestial holography has been used to shed light on S-matrix bootstrap in [30].
+2 To be more precise the wedge subalgebra of w1+∞.
+3
+
+˜Ci
+p,q(∆1, ∆2) which can appear in a w-invariant theory. Now before we proceed let us briefly
+describe the symmetry algebra that follows from the universal singular terms of the OPE
+(2.1).
+So we start by defining the tower of conformally soft [31] gravitons [19]
+Hk(z, ¯z) = lim
+∆→k(∆ − k)G+
+∆(z, ¯z), k = 1, 0, −1, −2, ...
+(2.2)
+with weights
+� k+2
+2 , k−2
+2
+�
+. It follows from the structure of the singular terms in (2.1) that we
+can introduce the following truncated mode expansion
+Hk(z, ¯z) =
+2−k
+2
+�
+m= k−2
+2
+Hk
+m(z)
+¯zm+ k−2
+2
+(2.3)
+and the modes Hk
+m(z) are the conserved holomorphic currents. The currents Hk
+m(z) can be
+further mode expanded
+Hk
+m(z) =
+�
+α∈Z− k+2
+2
+Hk
+α,m
+zα+ k+2
+2
+(2.4)
+and one can show [19] that the modes Hk
+α,m satisfy the algebra3
+�
+Hk
+α,m, Hl
+β,n
+�
+= − [n(2 − k) − m(2 − l)]
+� 2−k
+2 − m + 2−l
+2 − n − 1
+�
+!
+� 2−k
+2 − m
+�
+!
+� 2−l
+2 − n
+�
+!
+� 2−k
+2 + m + 2−l
+2 + n − 1
+�
+!
+� 2−k
+2 + m
+�
+!
+� 2−l
+2 + n
+�
+!
+Hk+l
+α+β,m+n
+(2.5)
+This is called the Holographic Symmetry Algebra (HSA). Now if we make the following
+redefinition (or discrete light transformation)[20]
+wp
+α,m = 1
+2 (p − n − 1)! (p + n − 1)!H−2p+4
+α,m
+(2.6)
+then (2.5) turns into the w1+∞ algebra4
+�
+wp
+α,m, wq
+β,n
+�
+= [m(q − 1) − n(p − 1)] wp+q−2
+α+β,m+n
+(2.7)
+For our purpose it is more convenient to work with the HSA (2.5) rather than the w1+∞
+algebra. However, we continue to refer to the HSA as the w algebra.
+3 Here we are assuming that κ = √8πGN = 2.
+4 This is the wedge subalgebra of w1+∞.
+4
+
+Now suppose we have two different w invariant theories which we call A and B. A and B
+have identical symmetry algebras (2.5) because there are no central charges and moreover,
+their operator contents are also the same. Therefore if the OPE is completely determined by
+w covariance then there must exist a universal form of the OPE which holds in both theory
+A and theory B. But how is this possible if the graviton scattering amplitudes in theory A
+and theory B are different ? This is still possible if the following holds:
+Let us denote the universal OPE by OPEw and the OPEs obtained from the graviton
+scattering amplitudes of theory A and theory B by OPEA and OPEB, respectively. Note
+that OPEw holds in both theory A and theory B. Therefore it must be true that
+OPEw = OPEA + Linear combination of Null-States of theory A
+OPEw = OPEB + Linear combination of Null-States of theory B
+(2.8)
+This is a major simplification if we know the OPE and null states of either theory A or
+theory B.
+A.
+What is theory A?
+Our task will be simplified if we can find a theory A such that the null states of theory
+B can be written as a linear combination of the null states of theory A. This does not
+mean that the theory A and theory B have the same null states but, this implies that the
+null states of theory A form a basis in terms of which the null states of theory B can be
+expanded. Once we find such a theory A we can write, as a consequence of (2.8),
+OPEB = OPEA + Linear combination of Null-States of theory A
+(2.9)
+Now in our case we take the theory A to be the MHV-sector. This is the simplest theory
+in the sense that although this is w invariant, the graviton-graviton OPE can be written
+as [16] a linear combination of supertranslation and sl2 current algebra5 descendants only.
+Therefore MHV sector has the maximum number of null states, i.e, maximally constrained
+among all the w invariant theories. Now the OPE and the null states in the MHV-sector
+are known [16, 17] and we will use them to construct the general w invariant OPE.
+5 Supertranslation and sl2 current algebras are subalgebras of the w1+∞ algebra. They are generated by
+the soft gravitons H1(z, ¯z) and H0(z, ¯z).
+5
+
+III.
+NULL STATES IN THE MHV SECTOR
+According to our discussion in the previous section, we can write
+G+
+∆1(z1, ¯z1)G+
+∆2(z2, ¯z2)|B = G+
+∆1(z1, ¯z1)G+
+∆2(z2, ¯z2)|MHV
++
+∞
+�
+p,q=0
+zp
+12¯zq
+12
+n
+�
+k=1
+Ck
+p,q(∆1, ∆2)N p,q
+k (z2, ¯z2)
+(3.1)
+where N p,q
+k
+are the null states of the MHV sector. Since the MHV-sector OPE contains
+the universal singular part of OPEB, the second line of (3.1) contains only the non-singular
+terms.
+We now discuss the MHV null states at O(z0
+12¯z0
+12) and at O(z0
+12¯z1
+12). This can be done at
+any order however, for the sake of simplicity, we focus on these two orders only. Moreover, the
+O(z0
+12¯z1
+12) terms in the OPEB give rise to interesting constraints on the graviton scattering
+amplitudes [16] in theory B. So it will be interesting to determine them.
+A.
+N0,0
+k
+The MHV null states that can appear at O(z0
+12¯z0
+12) are given by [16, 17]6,
+Φk(∆) =
+�
+H1−k
+k−3
+2 , k+1
+2
+�
+−H1
+− 1
+2 ,− 1
+2
+�k
+− (−1)k
+k!
+Γ(∆ + k − 2)
+Γ(∆ − 2)
+H1
+− 3
+2 , 1
+2
+�
+G+
+∆−1
+(3.2)
+where k = 1, 2, 3, · · · , ∞. However, it is more convenient to work with the new basis defined
+by
+Ωk(∆) =
+k
+�
+n=1
+1
+(k − n)!
+Γ(∆ + k − 2)
+Γ(∆ + n − 2)Φn(∆)
+(3.3)
+The physical significance of this basis will become clear in the next section.
+B.
+N0,1
+k
+The MHV null states that can appear at O(z0
+12¯z1
+12) are given by [16, 17],
+Ψk(∆) =
+�
+H−k
+k−2
+2 , k
+2
+�
+−H1
+− 1
+2 ,− 1
+2
+�k+1
+− (−1)k
+k!
+Γ(∆ + k − 2)
+Γ(∆ − 2)
+H0
+−1,0
+�
+−H1
+− 1
+2 ,− 1
+2
+�
+− (−1)kk
+(k + 1)!
+Γ(∆ + k − 2)
+Γ(∆ − 3)
+H1
+− 3
+2 ,− 1
+2
+�
+G+
+∆−2
+(3.4)
+6 The null states are obtained by starting from the graviton-graviton OPE in the MHV sector and then
+making one of the gravitons conformally soft. This is discussed in detail in [16, 17].
+6
+
+where k = 1, 2, 3 · · · , ∞. Again we introduce the new basis
+Πk(∆) =
+k
+�
+n=1
+1
+(k − n)!
+Γ(∆ + k − 2)
+Γ(∆ + n − 2)Ψn(∆)
+(3.5)
+which is more convenient for our purpose.
+There is a second set of null states at this order which involve the descendant L−1G+
+∆.
+These are the null states of the Knizhnik-Zamolodchikov(KZ) type and their decoupling
+gives rise to differential equations for the scattering amplitudes [16, 27] which are the gener-
+alization of the standard KZ equation to Celestial holography. However, we do not need the
+KZ-type null states to write down the OPEB because the generator L−17 does not belong
+to the w algebra. We will give the explicit form of these null states at a later part of the
+paper.
+IV.
+ORGANIZATION OF THE OPE AT EVERY ORDER
+We have argued that the OPE in the theory B can be written as,
+G+
+∆1(z, ¯z)G+
+∆2(0, 0)|B = G+
+∆1(z1, ¯z1)G+
+∆2(0, 0)|MHV
++
+∞
+�
+p,q=0
+zp¯zq
+n
+�
+k=1
+Ck
+p,q(∆1, ∆2)N p,q
+k (0, 0)
+(4.1)
+where N p,q
+k
+are the MHV null states. Now, an infinite number of null states can potentially
+appear at every order of the OPEB. So we need some guiding principle which tells us which
+null states can appear at a given order and hopefully, we have only a finite number of them.
+The guiding principle is nothing but the w-covariance of the OPE. In fact, if we
+want to classify the states which can appear at a given order of the OPE, then we
+can also work with the sl2(R) subalgebra of the w algebra generated by the operators
+{H1
+−1/2,−1/2, H0
+0,0, H−1
+1/2,1/2},
+�
+H0
+0,0, H1
+−1/2,−1/2
+�
+= H1
+−1/2,−1/2
+�
+H0
+0,0, H−1
+1/2,1/2
+�
+= −H−1
+1/2,1/2
+�
+H1
+−1/2,−1/2, H−1
+1/2,1/2
+�
+= −H0
+0,0
+(4.2)
+7 ¯L−1 is a generator of the w algebra.
+7
+
+We denote this by sl2(R)V 8.
+Now using the relations
+H0
+0,0G+
+∆(z, ¯z) = 2¯hG+
+∆(z, ¯z) = (∆ − 2) G+
+∆(z, ¯z)
+H1
+−1/2,−1/2G+
+∆(z, ¯z) = −G+
+∆+1(z, ¯z)
+H−1
+1/2,1/2G+
+∆(z, ¯z) = −1
+2
+�
+(∆ − 2)(∆ − 3) + 4(∆ − 2)¯z ¯∂ + 3¯z2 ¯∂2�
+G+
+∆−1(z, ¯z)
+(4.3)
+and the w-covariance of the OPE, i.e, both sides of the OPE transform in the same way
+under w transformations, one can easily check that:
+The MHV null states that can appear at O(zp¯zq) of the OPE, transform in a repre-
+sentation of the sl2(R)V subalgebra. Moreover, this is the unique sl2 subalgebra with this
+property.
+We now verify this explicitly.
+A.
+Action of sl2(R)V on the null states
+1.
+N0,0
+k
+Let us consider the action of the sl2(R)V on the MHV null states Ωk(∆) that can appear
+at O(z0¯z0). It is given by,
+H1
+− 1
+2 ,− 1
+2Ωk(∆) = −Ωk(∆ + 1)
+(4.4)
+H−1
+1
+2 , 1
+2Ωk(∆) = 1
+2(k + 1)(k + 2)Ωk(∆ − 1) − 1
+2(∆ − 4)(∆ − 5)Ωk(∆ − 1)
+−1
+2(k + 1)(k + 2)Ωk+1(∆ − 1)
+(4.5)
+and H0
+0,0 = 2¯L0 is obviously diagonal on these states.
+So we can see that the null
+states Ωk(∆) indeed form a representation of sl2(R)V .
+However, this representation
+is reducible.
+For let us consider the subspace spanned by the (infinite) set of states
+{Ωn+1(∆), Ωn+2(∆), Ωn+3(∆), · · · } where n ≥ 0.
+It is easy to see from (4.4) and (4.5)
+that this set is closed under the action of the sl2(R)V . Therefore the representation carried
+by the states {Ωk(∆)}k=1,2,... is reducible. As a consequence we can set the states
+Ωk+1(∆) = 0, k ≥ n ≥ 0
+(4.6)
+8 Here V stands for vertical. Please see Fig.1 for an explanation.
+8
+
+without violating the sl2(R)V symmetry. After doing this we are left with n null states
+{Ω1(∆), ..., Ωn(∆)} which, as a consequence of (4.5) and (4.6), again form a representation
+of sl2(R)V . The crucial point is that the value of the integer n is not fixed by the sl2(R)V
+symmetry.
+Now for a given n, there is another set of ⌈ n
+2⌉9 nontrivial10 states {χ1
+n(∆), ..., χ⌈n/2⌉
+n
+(∆)}
+defined as
+χ1
+n(∆) =
+n
+�
+p=1
+Ωp(∆)
+χi
+n(∆) =
+n
+�
+p=i
+2i−2
+�
+q=i
+(p − q)Ωp(∆), i = 2, 3, ..., ⌈n
+2⌉
+(4.7)
+which transform in a representation of the sl2(R)V as a consequence of (4.6). We can also
+set these states to zero
+χi
+n(∆) = 0
+(4.8)
+without violating the sl2(R)V symmetry.
+2.
+N0,1
+k
+The action of the sl2(R)V on the MHV null states Πk(∆) which can appear at O(z0¯z1)
+is given by,
+H1
+− 1
+2 ,− 1
+2Πk(∆) = −Πk+1(∆ + 1)
+(4.9)
+and
+H−1
+1
+2 , 1
+2Πk(∆) = 1
+2k(k + 1)Πk(∆ − 1) − 1
+2(∆ − 4)(∆ − 5)Πk(∆ − 1)
+−1
+2(k − 1)(k + 2)Πk+1(∆ − 1)
+(4.10)
+Therefore
+we
+can
+see
+the
+states
+Πk(∆)
+form
+a
+representation
+of
+the
+sl2(R)V .
+This
+representation
+is
+reducible
+because
+the
+subspace
+spanned
+by
+the
+states
+{Πn+1(∆), Πn+2(∆), Πn+3(∆), · · · } form a representation of the sl2(R)V .
+We can get a
+smaller representation spanned by the states {Π1(∆), · · · , Πn(∆)} if we set
+Πk+1(∆) = 0, k ≥ n ≥ 0
+(4.11)
+9 ⌈ n
+2 ⌉ = Smallest integer ≥ n
+2 .
+10 There are of course the n states {Ω1(∆), ..., Ωn(∆)} which transform in a representation of sl2(R)V but,
+we cannot set them to zero because that will lead us again to the MHV sector.
+9
+
+There is no other nontrivial representation of sl2(R)V in the subspace spanned by the states
+{Π1(∆), ..., Πn(∆)}.
+V.
+W INVARIANCE
+So far we have discussed the invariance of the OPE under sl2(R)V .
+There is an-
+other sl2 subalgebra which is generated by the global (Lorentz) conformal transforma-
+tions {H0
+0,1, H0
+0,0, H0
+0,−1}. We call this sl2(R) because this acts only on the ¯z coordinate.
+Now the w symmetry is generated by the infinite number of soft currents {Hk
+p(z)} where
+k = 1, 0, −1, −2, ... is the dimension (∆) of the soft operator and k−2
+2
+≤ p ≤ − k−2
+2 . For a
+fixed k, the soft currents {Hk
+− k−2
+2 (z), ..., Hk
+k−2
+2 (z)} transform in a spin 2−k
+2
+representation of
+the sl2(R). Now let us consider the currents {H1
+1
+2, H0
+1, ..., Hk
+2−k
+2 , ...} with the lowest sl2(R)
+FIG. 1: The figure shows the soft currents. The rows and the columns are indexed by the sl2(R)
+weights and the dimension (∆ = k = 1, 0, −1, −2, ...) of the conformally soft graviton Hk(z, ¯z)
+which generates the currents sitting in a row, respectively. sl2(R) acts horizontally along a row
+and sl2(R)V acts vertically along a column. In this way they generate the whole symmetry algebra
+starting from the current on the top left corner.
+10
+
+sl2(R)
+1
+12
++1
+1
+2
+0
++1
++2
+8weights. These currents transform in an irreducible highest weight representation of the
+sl2(R)V . This can be seen from the following commutation relations:
+�
+H−1
+1
+2 , Hk
+2−k
+2
+�
+= −1
+2(k − 2)(k − 3)Hk−1
+2−(k−1)
+2
+�
+H0
+0, Hk
+2−k
+2
+�
+= (k − 2)Hk
+2−k
+2
+�
+H1
+− 1
+2, Hk
+2−k
+2
+�
+= −Hk+1
+2−(k+1)
+2
+(5.1)
+Therefore, starting from the current H1
+1
+2(z) we can generate any other w current by the
+combined action of the sl2(R) and sl2(R)V (Fig.1).
+Now the way it is reflected in the structure of the OPE is the following. Suppose we
+consider the OPE of two positive helicity gravitons. The OPE is w invariant if the singular
+terms have the following universal structure
+G+
+∆1(z, ¯z)G+
+∆2(0, 0) ⊃ − ¯z
+z
+∞
+�
+n=0
+B (∆1 − 1 + n, ∆2 − 1) ¯zn
+n!
+¯∂nG+
+∆1+∆2(0, 0)
+(5.2)
+Now, it has been shown [28] that the leading term in ¯z is uniquely determined by the sl2(R)V
+invariance11. Once we know the leading term, the subleading terms in ¯z of O( ¯zq
+z ), q ≥ 2 are
+determined by the sl2(R) invariance12. Therefore if we make sure that the sl2(R)V and the
+sl2(R) symmetries are not broken then we are bound to have w invariance.
+A.
+w invariance at O(z0¯z0)
+We have shown that the equations
+Ωk+1(∆) = 0, k ≥ n ≥ 0
+(5.3)
+are sl2(R)V invariant. Now one can easily check that
+H0
+0,1Ωk(∆) = 0
+(5.4)
+where H0
+0,1 is, upto normalization, ¯L1. So the MHV null states Ωk(∆) are sl2(R) primaries
+and equations (5.3) are sl2(R) invariant. This implies that (5.3) are also w invariant.
+11 It is assumed that the leading term is of O( ¯z
+z). This follows from the structure of the universal leading
+term in the collinear limit of two positive helicity gravitons.
+12 Here the assumption is that the Vir is not part of the symmetry algebra.This is further discussed in [15].
+11
+
+For the same reason the additional null state relations (4.8)
+χi
+n(∆) = 0, i = 1, 2, ..., ⌈n
+2⌉
+(5.5)
+at O(z0¯z0) are also w invariant.
+B.
+w invariance at O(z0¯z1)
+We have shown that the equations
+Πk+1(∆) = 0, k ≥ n′ ≥ 0
+(5.6)
+are sl2(R)V invariant. Now the action of H0
+0,1 on the MHV null states Πk(∆) is given by,
+H0
+0,1Πk+1(∆) = −(∆ + k − 2)Ωk+1(∆ − 1) + (k + 3)Ωk+2(∆ − 1)
+(5.7)
+So if we want to impose (5.6) without breaking the sl2(R) invariance then we also have to
+impose
+Ωk+1(∆) = 0, k ≥ n′ ≥ 0
+(5.8)
+VI.
+FINITENESS AND AN INFINITE FAMILY OF THEORIES
+Our analysis in the last section shows that w invariance requires us to set
+Ωk+1(∆) = 0, k ≥ n ≥ 0, at O(z0¯z0)
+(6.1)
+and
+Πk+1(∆) = 0, k ≥ n ≥ 0, at O(z0¯z1)
+(6.2)
+simultaneously. Therefore we can construct a w invariant OPE at O(z0¯z0) and O(z0¯z1) by
+keeping only the finite13 number of states {Ω1(∆), ..., Ωn(∆)} and {Π1(∆), · · · , Πn(∆)}. The
+crucial point here is that the value of the integer n, is not fixed by w invariance. So we get
+different w invariant OPEs for different choices of the integer n. For example, n = 0 gives
+the MHV-sector. Similarly, direct calculation of graviton-graviton OPE using the scattering
+amplitudes shows that n = 4 gives the OPE of the quantum self-dual gravity theory which
+is known to be w invariant [22]. The existence of this discrete infinite family of w invariant
+OPEs is the main outcome of our analysis.
+13 There are still an infinite number of states {Ωp(∆ + Z)}1≤p≤n required by the global time translation
+invariance.
+12
+
+VII.
+w INVARIANT OPE COEFFICIENTS
+In this section we write down the w invariant OPE for a given value of n, at O(z0¯z0) and
+O(z0¯z1), using the states {Ω1(∆), ..., Ωn(∆)} and {Π1(∆), · · · , Πn(∆)}. At O(z0¯z0) we have
+G+
+∆1(z, ¯z)G+
+∆2(0, 0)
+��
+O(z0¯z0) ⊃ G+
+∆1(z, ¯z)G+
+∆2(0, 0)
+��
+MHV at O(z0¯z0)
++
+n
+�
+p=1
+B(∆1 − 1 + p, ∆2 − 1) Ωp(∆1 + ∆2)
+(7.1)
+and at O(z0¯z1)
+G+
+∆1(z, ¯z)G+
+∆2(0, 0)
+��
+O(z0¯z1) ⊃ G+
+∆1(z, ¯z)G+
+∆2(0, 0)
+��
+MHV at O(z0¯z1)
++¯z
+n
+�
+p=1
+B(∆1 + p, ∆2 − 1) Πp(∆1 + ∆2 + 1)
+(7.2)
+The MHV sector OPE at O(z0¯z0) and O(z0¯z1) are given in [16].
+As we have already discussed the states {Ω1(∆), ..., Ωn(∆)} are not all independent be-
+cause of the existence of the additional null state relations (4.8)
+χi
+n(∆) = 0, i = 1, 2, ..., ⌈n
+2⌉
+(7.3)
+We can further simplify (7.1) by using these null state relations. However, we choose to leave
+it in this form because the OPE coefficients are particularly nice when written in terms of
+these linearly dependent states. Now different values of n give OPEs in different w invariant
+theories. We now discuss the KZ type null states of these theories.
+VIII.
+KNIZHNIK-ZAMOLODCHIKOV TYPE NULL STATES
+KZ type null states occur at O(z0¯z1) of the OPE. The simplest way to derive it is to
+demand the commutativity of the OPE, i.e,
+G+
+∆1(z1, ¯z1)G+
+∆2(z2, ¯z2) = G+
+∆2(z2, ¯z2)G+
+∆1(z1, ¯z1)
+(8.1)
+If we use the consistency condition (8.1) and the OPE given in (7.1) and (7.2) we get the
+following KZ type null state
+Ξn(∆) = ξ(∆) +
+n
+�
+k=1
+Πk(∆ + 1) = 0
+(8.2)
+13
+
+where ξ(∆) is the KZ type null state in the MHV sector given by [16]
+ξ(∆) = L−1 G+
+∆ + H0
+0,−1H1
+− 3
+2 , 1
+2G+
+∆−1 + H0
+−1,0G+
+∆ + (∆ − 1) H1
+− 3
+2 ,− 1
+2G+
+∆−1
+(8.3)
+We have used that χ1
+n(∆) is a null state in this theory to arrive at the form (8.2).
+As consistency checks we can compute the actions of sl2(R) and sl2(R)V generators on
+Ξn(∆). For example,
+H0
+0,1Ξn(∆) = (n + 2)Ωn+1(∆) − (∆ − 3)χ1
+n(∆) = 0
+(8.4)
+because Ωn+1(∆) and χ1
+n(∆) are both null states in this theory. Therefore Ξn(∆) is a sl2(R)
+primary.
+Similarly, we have
+H−1
+1
+2 , 1
+2Ξn(∆) = −1
+2(∆ − 2)(∆ − 3)Ξn(∆ − 1)
+−1
+2(n + 2)(n − 1)Πn+1(∆) − H−1
+1
+2 ,− 1
+2χ1
+n(∆) − H0
+0,−1
+�
+(∆ − 1)χ1
+n(∆ − 1) + χ2
+n(∆ − 1)
+� (8.5)
+But, since Πn+1(∆), χ1
+n(∆) and χ2
+n(∆) are null states in the theory, we get
+H−1
+1
+2 , 1
+2Ξn(∆) = −1
+2(∆ − 2)(∆ − 3)Ξn(∆ − 1)
+(8.6)
+Therefore Ξn(∆) transforms under a representation of the sl2(R)V and we can consistently
+set it to zero without violating the sl2(R)V symmetry. Therefore, (8.2) is indeed w invariant.
+Decoupling of null states gives rise to differential equations which the graviton scattering
+amplitudes in this theory have to satisfy. So in principle we know a lot about these theories
+however, solving these equations may not be easy in practice. We hope to return to this in
+near future.
+IX.
+ACKNOWLEDGEMENTS
+We would like to thank Andrew Strominger and Sabrina Pasterski for useful discus-
+sion.
+The work of SB is partially supported by the Swarnajayanti Fellowship (File No-
+SB/SJF/2021-22/14) of the Department of Science and Technology and SERB, India and
+by SERB grant MTR/2019/000937 (Soft-Theorems, S-matrix and Flat-Space Holography).
+The work of HK is partially supported by the KVPY fellowship of the Department of Science
+and Technology (DST), Government of India and the Summer Students’ Visiting Program
+14
+
+at the Institute of Physics, Bhubaneswar, India. The work of PP is supported by an IOE
+endowed postdoctoral position at IISc, Bengaluru, India.
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+
diff --git a/yNFQT4oBgHgl3EQfBDVt/content/tmp_files/load_file.txt b/yNFQT4oBgHgl3EQfBDVt/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf,len=646
+page_content='An infinite family of w1+∞ invariant theories on the celestial  sphere  Shamik Banerjee  National Institute of Science Education and Research (NISER),  Bhubaneswar 752050, Odisha, India  and  Homi Bhabha National Institute,  Anushakti Nagar, Mumbai, India-400085∗  Harshal Kulkarni  Department of Physical Sciences, IISER Kolkata,  Mohanpur, West Bengal 741246, India†  Partha Paul  Centre for High Energy Physics, Indian Institute of Science,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Raman Avenue, Bangalore 560012, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='‡  Abstract  In this note we determine the graviton-graviton OPE and the null states in any w1+∞ symmetric  theory on the celestial sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Our analysis shows that there exists a discrete infinite family of  such theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The MHV-sector and the quantum self dual gravity are two members of this infinite  family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' However, the bulk description of other theories are not known to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  ∗Electronic address: banerjeeshamik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='phy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='com  †Electronic address: hck18ms056@iiserkol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='in  ‡Electronic address: pl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='partha13@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='com  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='13225v1  [hep-th]  30 Jan 2023  Contents  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Introduction  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' General structure of w-invariant OPE  3  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' What is theory A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  5  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Null states in the MHV sector  6  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' N 0,0  k  6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' N 0,1  k  6  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Organization of the OPE at every order  7  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Action of sl2(R)V on the null states  8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' N 0,0  k  8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' N 0,1  k  9  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' w invariance  10  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' w invariance at O(z0¯z0)  11  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' w invariance at O(z0¯z1)  12  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Finiteness and an infinite family of theories  12  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' w invariant OPE coefficients  13  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Knizhnik-Zamolodchikov type null states  13  IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Acknowledgements  14  References  15  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  INTRODUCTION  Celestial holography [1] is an attempt to formulate a dual theory of quantum gravity  in asymptotically flat space time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' When the bulk is (3 + 1) dimensional, the dual theory  is conjectured to be two dimensional and it lives on the celestial sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The correlation  2  functions of this theory are given by the Mellin transform of the bulk S-matrix elements  [2–4] and an important question is how to compute them in the dual formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Now in the absence of a Lagrangian description one can resort to the Bootstrap technique  and the job is sometimes simplified if the theory has a large number of symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' For  example, one can solve the minimal models in 2-D CFT by using the representation theory of  Virasoro and other current algebras even when the Lagrangian description is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' In  celestial holography the two dimensional dual theory also has various (infinite dimensional)  current algebra symmetries [5–26] and it is expected that the bootstrap approach should  simplify considerably in this case1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  In this paper we consider dual theories which are invariant under the w1+∞ symmetry2[19–  22, 25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' We determine the graviton-graviton OPE and the null states in such theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' So  far two examples of such theories were known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' One of them is the MHV-sector [16, 17] and  the other one is the quantum self dual gravity [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' However, we find that there is a discrete  infinite family of such theories and the MHV-sector and the quantum self dual gravity are  two members of this family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Bulk description of this family of theories is not known to us  and we leave this to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  GENERAL STRUCTURE OF w-INVARIANT OPE  The general structure of the w-invariant OPE between two positive helicity outgoing  graviton primaries with weights ∆1 and ∆2 can be written as,  G+  ∆1(z1, ¯z1)G+  ∆2(z2, ¯z2) = − ¯z12  z12  ∞  �  n=0  B (∆1 − 1 + n, ∆2 − 1) ¯zn  12  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  ¯∂nG+  ∆1+∆2(z2, ¯z2)  +  ∞  �  p,q=0  zp  12¯zq  12  n  �  k=1  ˜Ck  p,q(∆1, ∆2) ˜Op,q  k (∆1, ∆2, z2, ¯z2)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1)  where the first line contains the universal terms [19] having simple pole in z12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The second  line contains the non-singular terms in z12 and ¯z12 and they are linear combinations of the w  descendants of a positive helicity graviton primary, denoted by ˜Op,q  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The positive integer n  may be finite or infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Our goal is to determine the operators ˜Op,q  i  and the OPE coefficients  1 Please see [29] for a discussion of conformal bootstrap approach to celestial holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Also recently  celestial holography has been used to shed light on S-matrix bootstrap in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  2 To be more precise the wedge subalgebra of w1+∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  3  ˜Ci  p,q(∆1, ∆2) which can appear in a w-invariant theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now before we proceed let us briefly  describe the symmetry algebra that follows from the universal singular terms of the OPE  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  So we start by defining the tower of conformally soft [31] gravitons [19]  Hk(z, ¯z) = lim  ∆→k(∆ − k)G+  ∆(z, ¯z), k = 1, 0, −1, −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2)  with weights  � k+2  2 , k−2  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' It follows from the structure of the singular terms in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1) that we  can introduce the following truncated mode expansion  Hk(z, ¯z) =  2−k  2  �  m= k−2  2  Hk  m(z)  ¯zm+ k−2  2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3)  and the modes Hk  m(z) are the conserved holomorphic currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The currents Hk  m(z) can be  further mode expanded  Hk  m(z) =  �  α∈Z− k+2  2  Hk  α,m  zα+ k+2  2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='4)  and one can show [19] that the modes Hk  α,m satisfy the algebra3  �  Hk  α,m, Hl  β,n  �  = − [n(2 − k) − m(2 − l)]  � 2−k  2 − m + 2−l  2 − n − 1  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  � 2−k  2 − m  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  � 2−l  2 − n  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  � 2−k  2 + m + 2−l  2 + n − 1  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  � 2−k  2 + m  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  � 2−l  2 + n  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Hk+l  α+β,m+n  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5)  This is called the Holographic Symmetry Algebra (HSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now if we make the following  redefinition (or discrete light transformation)[20]  wp  α,m = 1  2 (p − n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' (p + n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='H−2p+4  α,m  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='6)  then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5) turns into the w1+∞ algebra4  �  wp  α,m, wq  β,n  �  = [m(q − 1) − n(p − 1)] wp+q−2  α+β,m+n  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='7)  For our purpose it is more convenient to work with the HSA (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5) rather than the w1+∞  algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' However, we continue to refer to the HSA as the w algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  3 Here we are assuming that κ = √8πGN = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  4 This is the wedge subalgebra of w1+∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  4  Now suppose we have two different w invariant theories which we call A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' A and B  have identical symmetry algebras (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5) because there are no central charges and moreover,  their operator contents are also the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Therefore if the OPE is completely determined by  w covariance then there must exist a universal form of the OPE which holds in both theory  A and theory B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' But how is this possible if the graviton scattering amplitudes in theory A  and theory B are different ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This is still possible if the following holds:  Let us denote the universal OPE by OPEw and the OPEs obtained from the graviton  scattering amplitudes of theory A and theory B by OPEA and OPEB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Note  that OPEw holds in both theory A and theory B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Therefore it must be true that  OPEw = OPEA + Linear combination of Null-States of theory A  OPEw = OPEB + Linear combination of Null-States of theory B  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='8)  This is a major simplification if we know the OPE and null states of either theory A or  theory B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  What is theory A?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Our task will be simplified if we can find a theory A such that the null states of theory  B can be written as a linear combination of the null states of theory A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This does not  mean that the theory A and theory B have the same null states but, this implies that the  null states of theory A form a basis in terms of which the null states of theory B can be  expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Once we find such a theory A we can write, as a consequence of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='8),  OPEB = OPEA + Linear combination of Null-States of theory A  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='9)  Now in our case we take the theory A to be the MHV-sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This is the simplest theory  in the sense that although this is w invariant, the graviton-graviton OPE can be written  as [16] a linear combination of supertranslation and sl2 current algebra5 descendants only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Therefore MHV sector has the maximum number of null states, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='e, maximally constrained  among all the w invariant theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now the OPE and the null states in the MHV-sector  are known [16, 17] and we will use them to construct the general w invariant OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  5 Supertranslation and sl2 current algebras are subalgebras of the w1+∞ algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' They are generated by  the soft gravitons H1(z, ¯z) and H0(z, ¯z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  5  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  NULL STATES IN THE MHV SECTOR  According to our discussion in the previous section, we can write  G+  ∆1(z1, ¯z1)G+  ∆2(z2, ¯z2)|B = G+  ∆1(z1, ¯z1)G+  ∆2(z2, ¯z2)|MHV  +  ∞  �  p,q=0  zp  12¯zq  12  n  �  k=1  Ck  p,q(∆1, ∆2)N p,q  k (z2, ¯z2)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1)  where N p,q  k  are the null states of the MHV sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Since the MHV-sector OPE contains  the universal singular part of OPEB, the second line of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1) contains only the non-singular  terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  We now discuss the MHV null states at O(z0  12¯z0  12) and at O(z0  12¯z1  12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This can be done at  any order however, for the sake of simplicity, we focus on these two orders only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Moreover, the  O(z0  12¯z1  12) terms in the OPEB give rise to interesting constraints on the graviton scattering  amplitudes [16] in theory B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' So it will be interesting to determine them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  N0,0  k  The MHV null states that can appear at O(z0  12¯z0  12) are given by [16, 17]6,  Φk(∆) =  �  H1−k  k−3  2 , k+1  2  �  −H1  − 1  2 ,− 1  2  �k  − (−1)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Γ(∆ + k − 2)  Γ(∆ − 2)  H1  − 3  2 , 1  2  �  G+  ∆−1  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2)  where k = 1, 2, 3, · · · , ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' However, it is more convenient to work with the new basis defined  by  Ωk(∆) =  k  �  n=1  1  (k − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Γ(∆ + k − 2)  Γ(∆ + n − 2)Φn(∆)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3)  The physical significance of this basis will become clear in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  N0,1  k  The MHV null states that can appear at O(z0  12¯z1  12) are given by [16, 17],  Ψk(∆) =  �  H−k  k−2  2 , k  2  �  −H1  − 1  2 ,− 1  2  �k+1  − (−1)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Γ(∆ + k − 2)  Γ(∆ − 2)  H0  −1,0  �  −H1  − 1  2 ,− 1  2  �  − (−1)kk  (k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Γ(∆ + k − 2)  Γ(∆ − 3)  H1  − 3  2 ,− 1  2  �  G+  ∆−2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='4)  6 The null states are obtained by starting from the graviton-graviton OPE in the MHV sector and then  making one of the gravitons conformally soft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This is discussed in detail in [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  6  where k = 1, 2, 3 · · · , ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Again we introduce the new basis  Πk(∆) =  k  �  n=1  1  (k − n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Γ(∆ + k − 2)  Γ(∆ + n − 2)Ψn(∆)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5)  which is more convenient for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  There is a second set of null states at this order which involve the descendant L−1G+  ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  These are the null states of the Knizhnik-Zamolodchikov(KZ) type and their decoupling  gives rise to differential equations for the scattering amplitudes [16, 27] which are the gener-  alization of the standard KZ equation to Celestial holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' However, we do not need the  KZ-type null states to write down the OPEB because the generator L−17 does not belong  to the w algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' We will give the explicit form of these null states at a later part of the  paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  ORGANIZATION OF THE OPE AT EVERY ORDER  We have argued that the OPE in the theory B can be written as,  G+  ∆1(z, ¯z)G+  ∆2(0, 0)|B = G+  ∆1(z1, ¯z1)G+  ∆2(0, 0)|MHV  +  ∞  �  p,q=0  zp¯zq  n  �  k=1  Ck  p,q(∆1, ∆2)N p,q  k (0, 0)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1)  where N p,q  k  are the MHV null states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now, an infinite number of null states can potentially  appear at every order of the OPEB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' So we need some guiding principle which tells us which  null states can appear at a given order and hopefully, we have only a finite number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  The guiding principle is nothing but the w-covariance of the OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' In fact, if we  want to classify the states which can appear at a given order of the OPE, then we  can also work with the sl2(R) subalgebra of the w algebra generated by the operators  {H1  −1/2,−1/2, H0  0,0, H−1  1/2,1/2},  �  H0  0,0, H1  −1/2,−1/2  �  = H1  −1/2,−1/2  �  H0  0,0, H−1  1/2,1/2  �  = −H−1  1/2,1/2  �  H1  −1/2,−1/2, H−1  1/2,1/2  �  = −H0  0,0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2)  7 ¯L−1 is a generator of the w algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  7  We denote this by sl2(R)V 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Now using the relations  H0  0,0G+  ∆(z, ¯z) = 2¯hG+  ∆(z, ¯z) = (∆ − 2) G+  ∆(z, ¯z)  H1  −1/2,−1/2G+  ∆(z, ¯z) = −G+  ∆+1(z, ¯z)  H−1  1/2,1/2G+  ∆(z, ¯z) = −1  2  �  (∆ − 2)(∆ − 3) + 4(∆ − 2)¯z ¯∂ + 3¯z2 ¯∂2�  G+  ∆−1(z, ¯z)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3)  and the w-covariance of the OPE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='e, both sides of the OPE transform in the same way  under w transformations, one can easily check that:  The MHV null states that can appear at O(zp¯zq) of the OPE, transform in a repre-  sentation of the sl2(R)V subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Moreover, this is the unique sl2 subalgebra with this  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  We now verify this explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Action of sl2(R)V on the null states  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  N0,0  k  Let us consider the action of the sl2(R)V on the MHV null states Ωk(∆) that can appear  at O(z0¯z0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' It is given by,  H1  − 1  2 ,− 1  2Ωk(∆) = −Ωk(∆ + 1)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='4)  H−1  1  2 , 1  2Ωk(∆) = 1  2(k + 1)(k + 2)Ωk(∆ − 1) − 1  2(∆ − 4)(∆ − 5)Ωk(∆ − 1)  −1  2(k + 1)(k + 2)Ωk+1(∆ − 1)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5)  and H0  0,0 = 2¯L0 is obviously diagonal on these states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  So we can see that the null  states Ωk(∆) indeed form a representation of sl2(R)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  However, this representation  is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  For let us consider the subspace spanned by the (infinite) set of states  {Ωn+1(∆), Ωn+2(∆), Ωn+3(∆), · · · } where n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  It is easy to see from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5)  that this set is closed under the action of the sl2(R)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Therefore the representation carried  by the states {Ωk(∆)}k=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' is reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' As a consequence we can set the states  Ωk+1(∆) = 0, k ≥ n ≥ 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='6)  8 Here V stands for vertical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Please see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1 for an explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  8  without violating the sl2(R)V symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' After doing this we are left with n null states  {Ω1(∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Ωn(∆)} which, as a consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='6), again form a representation  of sl2(R)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The crucial point is that the value of the integer n is not fixed by the sl2(R)V  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Now for a given n, there is another set of ⌈ n  2⌉9 nontrivial10 states {χ1  n(∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', χ⌈n/2⌉  n  (∆)}  defined as  χ1  n(∆) =  n  �  p=1  Ωp(∆)  χi  n(∆) =  n  �  p=i  2i−2  �  q=i  (p − q)Ωp(∆), i = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', ⌈n  2⌉  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='7)  which transform in a representation of the sl2(R)V as a consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' We can also  set these states to zero  χi  n(∆) = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='8)  without violating the sl2(R)V symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  N0,1  k  The action of the sl2(R)V on the MHV null states Πk(∆) which can appear at O(z0¯z1)  is given by,  H1  − 1  2 ,− 1  2Πk(∆) = −Πk+1(∆ + 1)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='9)  and  H−1  1  2 , 1  2Πk(∆) = 1  2k(k + 1)Πk(∆ − 1) − 1  2(∆ − 4)(∆ − 5)Πk(∆ − 1)  −1  2(k − 1)(k + 2)Πk+1(∆ − 1)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='10)  Therefore  we  can  see  the  states  Πk(∆)  form  a  representation  of  the  sl2(R)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  This  representation  is  reducible  because  the  subspace  spanned  by  the  states  {Πn+1(∆), Πn+2(∆), Πn+3(∆), · · · } form a representation of the sl2(R)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  We can get a  smaller representation spanned by the states {Π1(∆), · · · , Πn(∆)} if we set  Πk+1(∆) = 0, k ≥ n ≥ 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='11)  9 ⌈ n  2 ⌉ = Smallest integer ≥ n  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  10 There are of course the n states {Ω1(∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Ωn(∆)} which transform in a representation of sl2(R)V but,  we cannot set them to zero because that will lead us again to the MHV sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  9  There is no other nontrivial representation of sl2(R)V in the subspace spanned by the states  {Π1(∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Πn(∆)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  W INVARIANCE  So far we have discussed the invariance of the OPE under sl2(R)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  There is an-  other sl2 subalgebra which is generated by the global (Lorentz) conformal transforma-  tions {H0  0,1, H0  0,0, H0  0,−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' We call this sl2(R) because this acts only on the ¯z coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Now the w symmetry is generated by the infinite number of soft currents {Hk  p(z)} where  k = 1, 0, −1, −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' is the dimension (∆) of the soft operator and k−2  2  ≤ p ≤ − k−2  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' For a  fixed k, the soft currents {Hk  − k−2  2 (z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Hk  k−2  2 (z)} transform in a spin 2−k  2  representation of  the sl2(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now let us consider the currents {H1  1  2, H0  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Hk  2−k  2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='} with the lowest sl2(R)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' 1: The figure shows the soft currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The rows and the columns are indexed by the sl2(R)  weights and the dimension (∆ = k = 1, 0, −1, −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=') of the conformally soft graviton Hk(z, ¯z)  which generates the currents sitting in a row, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' sl2(R) acts horizontally along a row  and sl2(R)V acts vertically along a column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' In this way they generate the whole symmetry algebra  starting from the current on the top left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  10  sl2(R)  1  12  +1  1  2  0  +1  +2  8weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' These currents transform in an irreducible highest weight representation of the  sl2(R)V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This can be seen from the following commutation relations:  �  H−1  1  2 , Hk  2−k  2  �  = −1  2(k − 2)(k − 3)Hk−1  2−(k−1)  2  �  H0  0, Hk  2−k  2  �  = (k − 2)Hk  2−k  2  �  H1  − 1  2, Hk  2−k  2  �  = −Hk+1  2−(k+1)  2  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1)  Therefore, starting from the current H1  1  2(z) we can generate any other w current by the  combined action of the sl2(R) and sl2(R)V (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Now the way it is reflected in the structure of the OPE is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Suppose we  consider the OPE of two positive helicity gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The OPE is w invariant if the singular  terms have the following universal structure  G+  ∆1(z, ¯z)G+  ∆2(0, 0) ⊃ − ¯z  z  ∞  �  n=0  B (∆1 − 1 + n, ∆2 − 1) ¯zn  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  ¯∂nG+  ∆1+∆2(0, 0)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2)  Now, it has been shown [28] that the leading term in ¯z is uniquely determined by the sl2(R)V  invariance11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Once we know the leading term, the subleading terms in ¯z of O( ¯zq  z ), q ≥ 2 are  determined by the sl2(R) invariance12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Therefore if we make sure that the sl2(R)V and the  sl2(R) symmetries are not broken then we are bound to have w invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  w invariance at O(z0¯z0)  We have shown that the equations  Ωk+1(∆) = 0, k ≥ n ≥ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3)  are sl2(R)V invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now one can easily check that  H0  0,1Ωk(∆) = 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='4)  where H0  0,1 is, upto normalization, ¯L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' So the MHV null states Ωk(∆) are sl2(R) primaries  and equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3) are sl2(R) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This implies that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3) are also w invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  11 It is assumed that the leading term is of O( ¯z  z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' This follows from the structure of the universal leading  term in the collinear limit of two positive helicity gravitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  12 Here the assumption is that the Vir is not part of the symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='This is further discussed in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  11  For the same reason the additional null state relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='8)  χi  n(∆) = 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', ⌈n  2⌉  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5)  at O(z0¯z0) are also w invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  w invariance at O(z0¯z1)  We have shown that the equations  Πk+1(∆) = 0, k ≥ n′ ≥ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='6)  are sl2(R)V invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now the action of H0  0,1 on the MHV null states Πk(∆) is given by,  H0  0,1Πk+1(∆) = −(∆ + k − 2)Ωk+1(∆ − 1) + (k + 3)Ωk+2(∆ − 1)  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='7)  So if we want to impose (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='6) without breaking the sl2(R) invariance then we also have to  impose  Ωk+1(∆) = 0, k ≥ n′ ≥ 0  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='8)  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  FINITENESS AND AN INFINITE FAMILY OF THEORIES  Our analysis in the last section shows that w invariance requires us to set  Ωk+1(∆) = 0, k ≥ n ≥ 0, at O(z0¯z0)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1)  and  Πk+1(∆) = 0, k ≥ n ≥ 0, at O(z0¯z1)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2)  simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Therefore we can construct a w invariant OPE at O(z0¯z0) and O(z0¯z1) by  keeping only the finite13 number of states {Ω1(∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Ωn(∆)} and {Π1(∆), · · · , Πn(∆)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The  crucial point here is that the value of the integer n, is not fixed by w invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' So we get  different w invariant OPEs for different choices of the integer n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' For example, n = 0 gives  the MHV-sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Similarly, direct calculation of graviton-graviton OPE using the scattering  amplitudes shows that n = 4 gives the OPE of the quantum self-dual gravity theory which  is known to be w invariant [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The existence of this discrete infinite family of w invariant  OPEs is the main outcome of our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  13 There are still an infinite number of states {Ωp(∆ + Z)}1≤p≤n required by the global time translation  invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  12  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  w INVARIANT OPE COEFFICIENTS  In this section we write down the w invariant OPE for a given value of n, at O(z0¯z0) and  O(z0¯z1), using the states {Ω1(∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Ωn(∆)} and {Π1(∆), · · · , Πn(∆)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' At O(z0¯z0) we have  G+  ∆1(z, ¯z)G+  ∆2(0, 0)  ��  O(z0¯z0) ⊃ G+  ∆1(z, ¯z)G+  ∆2(0, 0)  ��  MHV at O(z0¯z0)  +  n  �  p=1  B(∆1 − 1 + p, ∆2 − 1) Ωp(∆1 + ∆2)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1)  and at O(z0¯z1)  G+  ∆1(z, ¯z)G+  ∆2(0, 0)  ��  O(z0¯z1) ⊃ G+  ∆1(z, ¯z)G+  ∆2(0, 0)  ��  MHV at O(z0¯z1)  +¯z  n  �  p=1  B(∆1 + p, ∆2 − 1) Πp(∆1 + ∆2 + 1)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2)  The MHV sector OPE at O(z0¯z0) and O(z0¯z1) are given in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  As we have already discussed the states {Ω1(∆), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', Ωn(∆)} are not all independent be-  cause of the existence of the additional null state relations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='8)  χi  n(∆) = 0, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=', ⌈n  2⌉  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3)  We can further simplify (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1) by using these null state relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' However, we choose to leave  it in this form because the OPE coefficients are particularly nice when written in terms of  these linearly dependent states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Now different values of n give OPEs in different w invariant  theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' We now discuss the KZ type null states of these theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  KNIZHNIK-ZAMOLODCHIKOV TYPE NULL STATES  KZ type null states occur at O(z0¯z1) of the OPE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The simplest way to derive it is to  demand the commutativity of the OPE, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='e,  G+  ∆1(z1, ¯z1)G+  ∆2(z2, ¯z2) = G+  ∆2(z2, ¯z2)G+  ∆1(z1, ¯z1)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1)  If we use the consistency condition (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1) and the OPE given in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2) we get the  following KZ type null state  Ξn(∆) = ξ(∆) +  n  �  k=1  Πk(∆ + 1) = 0  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2)  13  where ξ(∆) is the KZ type null state in the MHV sector given by [16]  ξ(∆) = L−1 G+  ∆ + H0  0,−1H1  − 3  2 , 1  2G+  ∆−1 + H0  −1,0G+  ∆ + (∆ − 1) H1  − 3  2 ,− 1  2G+  ∆−1  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='3)  We have used that χ1  n(∆) is a null state in this theory to arrive at the form (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  As consistency checks we can compute the actions of sl2(R) and sl2(R)V generators on  Ξn(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' For example,  H0  0,1Ξn(∆) = (n + 2)Ωn+1(∆) − (∆ − 3)χ1  n(∆) = 0  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='4)  because Ωn+1(∆) and χ1  n(∆) are both null states in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Therefore Ξn(∆) is a sl2(R)  primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Similarly, we have  H−1  1  2 , 1  2Ξn(∆) = −1  2(∆ − 2)(∆ − 3)Ξn(∆ − 1)  −1  2(n + 2)(n − 1)Πn+1(∆) − H−1  1  2 ,− 1  2χ1  n(∆) − H0  0,−1  �  (∆ − 1)χ1  n(∆ − 1) + χ2  n(∆ − 1)  � (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='5)  But, since Πn+1(∆), χ1  n(∆) and χ2  n(∆) are null states in the theory, we get  H−1  1  2 , 1  2Ξn(∆) = −1  2(∆ − 2)(∆ − 3)Ξn(∆ − 1)  (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='6)  Therefore Ξn(∆) transforms under a representation of the sl2(R)V and we can consistently  set it to zero without violating the sl2(R)V symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Therefore, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='2) is indeed w invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Decoupling of null states gives rise to differential equations which the graviton scattering  amplitudes in this theory have to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' So in principle we know a lot about these theories  however, solving these equations may not be easy in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' We hope to return to this in  near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  We would like to thank Andrew Strominger and Sabrina Pasterski for useful discus-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  The work of SB is partially supported by the Swarnajayanti Fellowship (File No-  SB/SJF/2021-22/14) of the Department of Science and Technology and SERB, India and  by SERB grant MTR/2019/000937 (Soft-Theorems, S-matrix and Flat-Space Holography).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  The work of HK is partially supported by the KVPY fellowship of the Department of Science  and Technology (DST), Government of India and the Summer Students’ Visiting Program  14  at the Institute of Physics, Bhubaneswar, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' The work of PP is supported by an IOE  endowed postdoctoral position at IISc, Bengaluru, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Strominger, “Lectures on the Infrared Structure of Gravity and Gauge Theory,”  [arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='05448 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Raclariu, “Lectures on Celestial Holography,” [arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='02075 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Pasterski, “Lectures on celestial amplitudes,” Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' C 81, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='12, 1062 (2021)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1140/epjc/s10052-021-09846-7 [arXiv:2108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='04801 [hep-th]]  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Pasterski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Pate and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Raclariu, “Celestial Holography,” [arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='11392 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Pasterski,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Shao and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Strominger,  “Flat Space Amplitudes and Confor-  mal Symmetry of the Celestial Sphere,”  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' D 96,  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' 6,  065026 (2017)  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content=' Fotopoulos, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Stieberger, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content=' Zhu, “Celestial Yang-Mills am-  plitudes and D = 4 conformal blocks,” JHEP 09, 182 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1007/JHEP09(2022)182  [arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='08979 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Hu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Pasterski, “Celestial Recursion,” [arXiv:2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='11635 [hep-th]]  17  [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content=' Strominger and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Yuan, “Celestial Operator Products of  Gluons and Gravitons,” arXiv:1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='07424 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content=' Melton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content=' Strominger, “Conformal Block Expansion in  Celestial CFT,” [arXiv:2104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='13432 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  [30] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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+page_content=' Raman and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Sinha, “Celestial insights into the S-matrix bootstrap,” JHEP  08, 216 (2022) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1007/JHEP08(2022)216 [arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='07617 [hep-th]]  [31] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Donnay, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Puhm and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Strominger, “Conformally Soft Photons and Gravitons,” JHEP  1901, 184 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1007/JHEP01(2019)184 [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='05219 [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Pate, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Raclariu and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Strominger, “Conformally Soft Theorem in Gauge Theory,”  arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='10831 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Fan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Fotopoulos and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Taylor, “Soft Limits of Yang-Mills Amplitudes and Confor-  mal Correlators,” JHEP 1905, 121 (2019) doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='1007/JHEP05(2019)121 [arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='01676  [hep-th]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Nandan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Schreiber, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Volovich and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Zlotnikov, “Celestial Amplitudes: Conformal  Partial Waves and Soft Limits,” arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='10940 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Adamo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Mason and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Sharma, “Celestial amplitudes and conformal soft theorems,”  arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='09224 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Puhm, “Conformally Soft Theorem In Gravity,” arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='09799 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content=' Guevara, “Notes on Conformal Soft Theorems and Recursion Relations in Gravity,”  arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='07810 [hep-th].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
+page_content='  18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/yNFQT4oBgHgl3EQfBDVt/content/2301.13225v1.pdf'}
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